Essays in Macro-Finance and International Finance
Yu Liu
Submitted in partial fulfillment of the
requirements for the degree
of Doctor of Philosophy
in the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2015
c©2015
Yu Liu
All Rights Reserved
ABSTRACT
Essays in Macro-Finance and International Finance
Yu Liu
This dissertation contains three essays on macro-finance and international finance.
In Chapter 1, Richard Clarida and I study the term structure of US interest rates using
observable macro factors as inputs to a Taylor-type rule that can account for the time path
of the short term interest rate. Using a standard essentially affine model, we build directly
on the pioneering work of Ang and Piazzesi, Rudebusch and Wu, and others but extend their
analysis to a framework in which all macro factors are observable. We focus on the period
since 1997 when US inflation expectations have been well anchored and inflation indexed
bonds – which provide useful information on expected inflation and expected future real in-
terest rates - have been issued by the US government. In contrast to many previous studies
that – of necessity - focus on earlier periods when low frequency movements in expected infla-
tion appeared to dominate, in our sample variation in expected inflation at longer horizons is
modest and the yield curve is importantly driven by the evolution of the ‘neutral’ real policy
rate as estimated by Laubach and Williams. Deviations of the policy rate from the Taylor
rule path are found to have a marked impact at the front end of the yield curve. In any
factor model yields are linear combinations of factors, and principal components, are linear
combinations of yields. In our model, we can solve explicitly for the mapping from macro
factors to traditional ‘level’ ‘slope’ and ‘curvature’ factors. Our model exhibits surprising
robustness in a post-crisis out-sample study. We also propose a novel, but simple regression
based approach to generate initial values - required to implement the non-linear GMM esti-
mation technique we use - for the affine model’s deep structural parameters.
In both Chapters 2 and 3, I study the portfolio problem associated with currency carry trade.
In Chapter 2 specifically, I analyze the carry trade threshold portfolios. I prove that under
general assumptions, the optimal mean-variance portfolio gives a higher weight to carry trade
having larger forward premium. I then proposes a more robust version of the mean-variance
optimal portfolio: the threshold portfolio, where I construct carry trade threshold portfolios
using thresholds that depend upon forward premium. And I show that empirically, up to
the optimal threshold value, higher-threshold portfolios outperform lower-threshold portfo-
lios. The financial performance then decreases, as the threshold goes higher. I model the
threshold effect in a random-walk model of exchange rates. The model predicts the optimal
threshold value and the relative gain of an optimal threshold portfolio. The model is cali-
brated, and the predictions are tested. I also discuss the threshold effect in a model which
features global risk factor. Following Jurek (2014) and using crash-hedged portfolios, I test
the crash risk explanation for outperformance of threshold portfolio, I show that the crash
risk premium can explain around 25 percent of the excess performance of higher threshold
portfolios.
In Chapter 3, I study the hedging problem associated with currency carry trade. I propose
theoretical frameworks and divide hedging instruments into three categories: insurance, tech-
nical rule, and the market neutral strategy. I then propose and empirically test four hedging
strategies: FX options strategy, VIX future strategy, “Stop-loss” rule and CTA strategy.
Based upon empirical evidence from 2000 to 2012, I find that CTA is the preferred hedging
strategy because it upgrades both return and volatility. The stop-loss strategy reduces risk.
Both the currency options strategy and the VIX future strategy offer good hedges against
tail risk, while also reducing volatility. Unfortunately they are costly to implement. I also
compare the VIX strategy to various currency option strategies, to determine if VIX is a
cheaper form of systematic insurance as compared to the currency options. With respect
to CTA, I study its risk-return aspect, I also provide a new methodology for replicating the
returns of the benchmark CTA index.
Table of Contents
List of Figures v
1 Macro Fundamentals and the Yield Curve: Re-Interpreting Factors in the
Essentially Affine Model 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Modeling Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Stochastic discount factor and the Taylor rule . . . . . . . . . . . . . . 5
1.2.2 Interest rate term structure . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Econometric methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Orthogonality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Exactly identified model . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.3 Over-identified model . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 Measurement for real neutral interest rate . . . . . . . . . . . . . . . . 11
1.4.2 Measurement for output gap . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.3 Measurement for the expected inflation . . . . . . . . . . . . . . . . . 13
1.4.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Results of exactly identified model . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.1 Macro factors dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.2 Market price of risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.3 Impulse response function . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.4 Variance decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 18
i
1.5.5 Robustness checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6 Extension: Forecasting of Bond Excess Return . . . . . . . . . . . . . . . . . 23
1.6.1 Forecasting using forward yields . . . . . . . . . . . . . . . . . . . . . 23
1.7 Extension: Adding Bonds of Longer Maturities . . . . . . . . . . . . . . . . . 25
1.7.1 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . 26
1.7.2 Impulse response function . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7.3 Variance decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.7.4 Robustness checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.8 Robustness check and discussions . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.8.1 Testing model on alternative measurements: I . . . . . . . . . . . . . . 33
1.8.2 Testing model on alternative measurements: II . . . . . . . . . . . . . 35
1.8.3 Model using efficient interest rate . . . . . . . . . . . . . . . . . . . . . 37
1.8.4 Taylor rule under zero lower bound regime . . . . . . . . . . . . . . . 39
1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2 Carry Trade Threshold Portfolios 42
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2 Carry Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 Currency investor’s portfolio problem . . . . . . . . . . . . . . . . . . . . . . . 48
2.3.1 Mean-variance problem . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.3 Backtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.4 Robustness check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4 Threshold portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4.1 Threshold portfolios based on forward premium . . . . . . . . . . . . . 55
2.4.2 Robustness check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.4.3 Comparison with a quantile based sorting rule . . . . . . . . . . . . . 57
2.5 Crash-hedged threshold portfolios . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5.2 Performance of crash-hedged threshold portfolios . . . . . . . . . . . . 59
2.6 Random-walk model of threshold effects . . . . . . . . . . . . . . . . . . . . . 60
ii
2.6.1 Random-walk model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.6.2 Possible variations of the random-walk models . . . . . . . . . . . . . 62
2.6.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.6.4 Model with crash-hedged portfolios . . . . . . . . . . . . . . . . . . . . 63
2.6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.6.6 Threshold portfolios when prices of risks are the same . . . . . . . . . 65
2.6.7 Extending models to include currencies with correlations . . . . . . . . 65
2.7 One-factor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.7.1 One-factor model with equal correlations of all currency pairs . . . . . 67
2.7.2 One-factor model with heterogeneous correlations of currency pairs . . 68
2.7.3 Model with increasing downside correlation . . . . . . . . . . . . . . . 70
2.7.4 Implication of random-walk model and one-factor model . . . . . . . . 71
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3 Currency Carry Trade Hedging 73
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2 Framework and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2.2 Hedging performance evaluation . . . . . . . . . . . . . . . . . . . . . 79
3.2.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3 Hedging using currency options or VIX future . . . . . . . . . . . . . . . . . . 81
3.3.1 Crash-risk hedging strategies . . . . . . . . . . . . . . . . . . . . . . . 82
3.3.2 Financial performances of hedged carry trade portfolios . . . . . . . . 84
3.3.3 Who would buy the insurance? . . . . . . . . . . . . . . . . . . . . . . 86
3.3.4 Hedging effects during crash periods . . . . . . . . . . . . . . . . . . . 87
3.4 Stop-loss Rules Applied to FX Carry Trade . . . . . . . . . . . . . . . . . . . 89
3.4.1 Stop-loss rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.4.2 Financial performances of hedging strategies . . . . . . . . . . . . . . 90
3.4.3 Who would buy the insurance? . . . . . . . . . . . . . . . . . . . . . . 90
3.4.4 Robust check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.5 CTA trend following as a hedging strategy . . . . . . . . . . . . . . . . . . . . 92
iii
3.5.1 Constructing CTA portfolio . . . . . . . . . . . . . . . . . . . . . . . . 93
3.5.2 CTA as a hedging strategy . . . . . . . . . . . . . . . . . . . . . . . . 94
3.6 Out-Sample Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.7 Two Issues Related to the Hedging Strategies . . . . . . . . . . . . . . . . . . 97
3.7.1 Compare the price of VIX futures to currency options . . . . . . . . . 98
3.7.2 CTA risk/return analysis . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Bibliography 111
A Appendix for Chapter 1 121
B Appendix for Chapter 2 144
C Appendix for Chapter 3 160
iv
List of Figures
1.1 Price Fitting for Exactly Identified Model . . . . . . . . . . . . . . . . . . . . 16
1.2 Contemporaneous Impulse Responses . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 Impulse Responses at Future Horizon . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 In-sample and Out-sample Pricing . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 PCA Factor Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.6 Contemporaneous Impulse Responses . . . . . . . . . . . . . . . . . . . . . . . 27
1.7 Impulse Responses at Future Horizon . . . . . . . . . . . . . . . . . . . . . . . 28
1.8 In-sample and Out-sample Pricing . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1 Cumulative G10 currency carry trade returns . . . . . . . . . . . . . . . . . . 83
3.2 Cumulative hedging strategies returns . . . . . . . . . . . . . . . . . . . . . . 83
3.3 CTA Total Return Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.4 Benchmark Index and Replication Portfolio . . . . . . . . . . . . . . . . . . . 107
3.5 Out-Sample Return Replication . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A.1 Macro-factors in Taylor Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A.2 Model Implied Yields and Observed Yields . . . . . . . . . . . . . . . . . . . 123
A.3 Impulse-responses Error Bands . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.4 Macro-factors in Taylor Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A.5 Contemporaneous Impulse Responses . . . . . . . . . . . . . . . . . . . . . . . 131
A.6 Impulse Responses at Future Horizon . . . . . . . . . . . . . . . . . . . . . . . 131
A.7 Macro-factors in Taylor Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.8 Contemporaneous Impulse Responses . . . . . . . . . . . . . . . . . . . . . . . 137
v
A.9 Impulse Responses at Future Horizon . . . . . . . . . . . . . . . . . . . . . . . 137
A.10 Macro-factors in Taylor Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
A.11 Contemporaneous Impulse Responses . . . . . . . . . . . . . . . . . . . . . . . 143
A.12 Impulse Responses at Future Horizon . . . . . . . . . . . . . . . . . . . . . . . 143
B.1 Cumulative G10 carry trade portfolio returns . . . . . . . . . . . . . . . . . . 152
B.2 G10/USD currencies’ volatilities . . . . . . . . . . . . . . . . . . . . . . . . . 152
B.3 Threshold portfolios of G10 currencies versus US dollar: Part 1 . . . . . . . . 153
B.4 Threshold portfolios of G10 currencies versus US dollar: Part 2 . . . . . . . . 153
B.5 Mean-variance Optimal Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . 154
B.6 10 Delta Put hedged threshold portfolios of G10 currencies versus US dollar:
Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
B.7 10 Delta Put hedged threshold portfolios of G10 currencies versus US dollar:
Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
B.8 Threshold portfolios of G10 currencies versus US dollar: Part 1 . . . . . . . . 156
B.9 Threshold portfolios of G10 currencies versus US dollar: Part 2 . . . . . . . . 156
B.10 Unhedged and hedged portfolios of TR0 and TR2.5 . . . . . . . . . . . . . . . 157
B.11 Empirical distribution of forward premiums and fitted exponential distribution 157
B.12 Threshold portfolios’ Sharpe ratios versus random-walk model implied values 158
B.13 Sharpe ratios of crash-hedged threshold portfolio with equal implied volatility 159
B.14 Sharpe ratios of crash-hedged threshold portfolio with variable implied volatility159
C.1 Cumulative return of portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . 164
vi
Acknowledgments
I am deeply indebted to my advisor, Professor Richard Clarida, for his guidance and encour-
agement throughout my years in the Ph.D program. I am very fortunate to have him as my
close mentor, for he has given me a guided tour through the fascinating world of international
finance and macro-finance. I greatly appreciate his generous contributions of time, ideas, and
envouragement, which have helped me to make my own humble contribution to economic
research. I thank Professors Serena Ng and Ricardo Reis for their helpful comments and
suggestions on my research papers. I also wish to thank Professors Jushan Bai and Robert
Hodrick for serving on my thesis committee.
I thank Professors Padma Desai and Edward Lincoln for the great experience I have had
while working with them. I thank Stephane Daul for helpful discussions. I also want to
thank all of the colleagues that I have been so fortnate to have in the Ph.D program: Feiran
Zhang, Shaowen Luo, Lina Lu, Charles Maurin, Zheli He. I am grateful to have had this great
community of smart and motivated people who accompanied me on my journey in pursuit of
economic wisdom. Moreover, they have offered great friendship and shared many important
moments with me.
Finally, this dissertation could never have been written without the support and love of my
wife, Zoey Yi Zhao, and my parents, Hongli Liu and Xiaoneng Jiang. I dedicate it to all of
them.
vii
To my wife, Zoey Yi Zhao, and my parents, Hongli Liu and Xiaoneng Jiang
viii
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 1
Chapter 1
Macro Fundamentals and the Yield
Curve: Re-Interpreting Factors in
the Essentially Affine Model
1.1 Introduction
The Taylor rule (Taylor [1993]) is widely regarded as framework useful for interpreting how
central banks set policy rates - see Clarida et al. [1998a], Clarida et al. [1998b] and Clarida
et al. [1999]. According to the Taylor rule, the central bank policy rate should reflect the real
neutral rate of the economy1, the deviation of the expected inflation vis-a-vis the inflation
target and the GDP gap. It is tempting to investigate the impact of Taylor rule factors on the
bond yield curves. In this chapter, we study the term structure of interest rates using observ-
able macro factors as inputs to a Taylor-type rule. This is an interesting approach, because
the Taylor rule stands right at the intersection of finance and macroeconomics. Its output,
the short-term interest rate, is the building block for longer-term interest rates through the
expectation hypothesis. Its inputs, various macroeconomic fundamentals, reflect the general
states of the economy and are closely tracked by central banks. By using Taylor rule factors
1Real neutral interest rate refers to the real short-term interest rate under which the output converges to
potential, where potential is the level of output consistent with stable inflation.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 2
in a macro-finance model, we set out to explain term structure variations through observable
and interpretable macro factors.
Here we are exploring an arbitrage-free term structure model, where the arbitrage opportu-
nities are eliminated by restrictions on the yields. One approach in finance uses latent factors
to price yields, using an affine stochastic discount factor, see Duffie and Kan [1996]; Dai and
Singleton [2000]. There is also a large amount of literature delving into term structures by
imposing no-arbitrage conditions on yield curves, and the latter are modeled, in some papers,
by using macroeconomic factors. Ang and Piazzesi [2003] is a pioneering paper that models
yield curves using macroeconomic factors in a reduced-form VAR setting. In our model,
all four factors are observable and inputs to a Taylor-type rule: real neutral interest rate,
expected inflation, output gap, and a Taylor rule deviation. We allow for the Taylor rule
deviation to be a priced factor subjected to the restrictions of the essentially affine model.
Previous studies have found that imposing a no-arbitrage condition improves the forecasting
performance of a VAR and that using macro factors improves the forecasting performance,
see Ang and Piazzesi [2003]. Hordahl et al. [2006] models evolution of inflation and output
gap within a New Keynesian framework. They confirm that imposing a no-arbitrage con-
dition, and including macro factors, improve the forecasting performance. Rudebusch and
Wu [2008] combines a term structure model with a New Keynesian model of macro factors.
It shows bi-directional interaction between macro factors and term structure variables. For
more contributions to the macro-finance literature, see for example Piazzesi [2005], De Graeve
et al. [2009], and Bikbov and Chernov [2010]. For testing a general class of affine model, see
Ghysels and Ng [1998].
Our estimation results based on US data suggest that macro variables affect the term struc-
ture in various ways. Yield curve exhibits strong contemporaneous response to macro shocks,
with expected inflation and real neutral rate having the most persistent effect. Real neutral
rate explains most of the price variation at long horizon for all bonds. At short horizon,
expected inflation explains most of the price variation for bonds at belly and long end, and
Taylor rule deviation mainly explains movements of the yield curve on short end. When it
comes to the impact of macro factors on the yield curve, our results are in line with previous
studies. Ang and Piazzesi [2003] finds that macro factors mainly account for movements
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 3
at the short and medium ends of yield curves. Ludvigson and Ng [2009] finds that “real”
and “inflation” factors have forecasting power for US bonds’ excess returns. Hordahl et al.
[2006] finds that monetary policy shocks have an impact mainly at the short end of yield
curve. Dewachter and Lyrio [2006] models long-run inflation simultaneously with term struc-
ture. It imposes martingale condition on latent macro factors and finds that for long end
of yield curve, inflation expectation is all-important. For short end, both real rate and in-
flation expectation count. Wu [2006] proposes a DSGE model in an attempt to characterize
the underlying macro factors. It finds that movement of the slope factor can be explained
by monetary policy shocks. And the level factor movement can be explained by technology
shocks. Duffee [2006] offers a model having only observable factors. He finds a positive re-
lation between short-term rate and expected inflation. Bekaert et al. [2010] models macro
factors in a New Keynesian setting. It models pricing kernel consistently with IS curve. The
model shows strong contemporaneous responses of the yield curve to macro shocks. Our
results also feature robustness during the post-crisis period in which the policy rate has been
constrained at the zero lower bound. We use data from 1997 to early 2008 to calibrate the
model parameters. We then use the data from late 2008 to 2012 to do our model testing. For
the out-sample period, our model exhibits good pricing accuracy. The out-sample fitting for
short end curve is, however, worse than medium and long ends. We discuss this issue and find
that this is due to the zero-lower bound regime in the out-sample period which particularly
affects particularly the short end.
After studying an exactly identified model, we extend our analysis to an over-identified model.
We propose a novel method for generating initial parameter values required for our non-linear
GMM estimation framework.
We also discuss the measurement issues of macro factors. In the main text we use Laubach
and Williams [2003]’s measurements of GDP gap and real neutral rate. We test the model
on two alternative measurement specificaions for robustness check.
In addition, we apply our model to an alternative monetary policy rule, referred to as W
rule by Curdia et al. [2014]. For the W rule the output gap doesn’t affect the monetary
policy reaction function. We find similar results regarding the real neutral rate, the expected
inflation, and the rule deviation.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 4
Among the papers studying the Taylor rule the most related work to ours is Smith and Taylor
[2009]. Their interest lies in studying the response of coefficients in the affine equation to the
change in coefficients in the Taylor rule. It also derives the formula linking the coefficients
of the Taylor rule to the coefficients of the implied affine equations for bond yields, but it
doesn’t calibrate the affine term structure model using bond yields data. The interest of the
present chapter is, however, to study the bond pricing using factors derived from Taylor rule.
We now briefly review some other papers that study the interactions between term structure
and macro factors. Among the papers imposing no-arbitrage conditions, some study the im-
plications for macro-economy of term structure variables within this macro-finance setting.
Ang et al. [2006] finds that the short rate has more predictive power than the term spreads
when it comes to GDP forecasting. Rudebusch et al. [2007] studies the effect of bond term
premium on macro variables, in both structural and reduced VAR settings. Another approach
estimates term-structure model without imposing any no-arbitrage conditions. Diebold et al.
[2006], for instance, uses both latent factors and observable macroeconomic factors. It finds
an effect of macro factors on future yield curve, but also an effect in the other direction. It
thus provides evidence of a bidirectional flow of information between the macro-economy and
the financial market.
The standard model can be extended in various ways. Monch [2008] explores the data-rich
environment by extracting factors from a large macro dataset. It improves forecasting per-
formance at the middle and long ends of yield curve. Farka and DaSilva [2011] estimates
the contemporaneous relation between monetary policy shocks and term structures, using
high frequency data around announcements. Following the financial crisis of 2008, some
papers started using non-traditional factors in the standard model to model term structure.
Dewachter et al. [2014] introduces liquidity factor and return-forecasting factors. It finds that
for cross-sectional yield curve fitting, the model has an advantage over most models using
traditional factors. It also finds that financial shocks have an impact on yield curve.
The rest of this chapter is organized as follows. Section 2 introduces a no-arbitrage macro-
finance model so as to study the interest rate term structure dynamics. Section 3 explains
the econometric method. It proposes an initial value generation method for GMM. Section
4 discusses the measurement issues of macro factors. Section 5 explores an exactly identified
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 5
model and analyzes the results. Section 6 studies the forecasting of bond excess returns.
Section 7 extends the model to over-identified case. Section 8 tests the model to alternative
measurements of macro factors and alternative monetary policy rule. It also discusses the
mode’s performance under zero-lower regime. Section 9 concludes the chapter.
1.2 Modeling Framework
1.2.1 Stochastic discount factor and the Taylor rule
In the essentially affine model, the short term interest rate is assumed to be an exact linear
function of a small number of ‘factors’. In many papers, the factors are unobserved and must
be inferred from bond yields themselves. We pursue a different course by selecting observable
macro factor as inputs to a Taylor rule. As a matter of accounting, we can always write
it = π∗ + rnrt + 1.5(πt − π∗) + 0.5gt + ut (1.1)
where π∗ is the target inflation rate, it is the nominal short term interest rate, rnrt is an
estimate of the ‘neutral’ real policy rate, πt is an estimate of expected inflation, gt is an
estimate of the output gap, and ut is the deviation of the interest rate from the Taylor rule
path. We assume π∗ equal to 2% for the following. Although in Taylor’s original formulation
the neutral real interest rate was assumed to be constant, the theory of optimal monetary
policy (Clarida et al. [1999]; Woodford [2003]) as well as empirical evidence (Laubach and
Williams [2003]) supports allowing for it to be time varying. With a Taylor rule formulation
and observable macro factors, the affine term structure model’s requirement that the short
rate being an exact linear combination of factors identifies ut – the Taylor rule deviation –
as a fourth factor.
In this chapter we test three measurement-specifications of the Taylor rule’s factors. For
the first specification(Specification I), we use Williams and Laubach’s estimation for the
real neutral rate and the GDP gap. We use Haubrich et al’s estimation for the inflation
expectation. For the second specification(Specification II), we keep the estimation of real
neutral rate and of expected inflation from Specification I. But we estimate the GDP gap
by using Okun’s law. For the third specification(Specification III), we estimate the real
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 6
neutral rate by the five-year forward rate five year in the future minus the inflation risk
premium, the expected inflation by the 5-year break-even inflation rate minus the inflation
risk premium, and the GDP-gap by Okun’s law. In the main test, we discuss results of the
first specification. In the section of the robustness check we test the second and the third
specifications.
Crucially the affine term structure model does not require that ut be white noise or that
it be orthogonal to the other factors. It only requires that a first order vector auto regression
in the four factors captures the forecastable variation in the factors. Thus, let the state
Xt = rnrt, gt, πt, ut, where Xt follows a Gaussian vector autoregression process,
Xt = µ+ ΦXt−1 + Σεt (1.2)
where µ is a constant vector, Φ a constant matrix, εt a Gaussian white noise N(0, I4), and
Σ a lower triangular matrix. Thus, as is common in VAR studies, we impose a Cholesky
identification with order rnrt, gt, πt, ut. Assuming the no-arbitrage condition holds, there
will exists a stochastic discount factor that will price bonds conditional on 1.1 and 1.2.
We follow Ang and Piazzesi [2003] (and many others since) and assume that the nominal
stochastic discount factor mt is of the essentially affine class and can be written as
mt+1 = exp
(−rt −
1
2λ∗Tt λt − λ∗Tt εt+1
)(1.3)
where λ∗t is the time-varying market price of risks in the economy, and εt is the innovation to
state variable Xt. Following the affine term structure model literature, we assume that the
market price of risk λ∗t is an affine function of state variable Xt,
λ∗t = λ0 + λ1Xt (1.4)
where λ0 is a vector of constants.
1.2.2 Interest rate term structure
We now model the dynamics for short rate and zero-coupon bond yields, after specifying the
SDF. Since the state variable has been deduced from the Taylor rule equation for short rate,
the affine short rate equation can be specified as
rt = δ0 + δT1 Xt (1.5)
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 7
Where by adjusting ut we have δ0 = 0 and δ1 = [1, 1.5, 0.5, 1]′.
Given the no-arbitrage assumption, we have the following relation linking price of the bond
with maturity of n periods at time t to that with maturity of n− 1 periods at time t+ 1
Pnt = E t[mt+1Pn−1t+1 ]
We can recursively solve for the yield of an n-period zero coupon bond, ynt = −log(Pnt ), as a
linear function of the state variable Xt:
ynt = an + bTnXt (1.6)
an and bn are given by an = −An/n and bn = −Bn/n, where An and Bn can be computed
recursively:
An+1 = An +BTn (µ− Σλ0) +
1
2BTnΣΣTBn − δT0 (1.7)
BTn+1 = BT
n (Φ− Σλ1)− δT1 (1.8)
with A1 = −δ0 and B1 = −δ1. From equations 1.7 and 1.8, the time-invariant component,
λ0, affects only the time-invariant component of yield an, while the time-varying parameter
affects the time-varying component of yield bn. Thus, λ0 affects only average bond yield,
and λ1 determines solely the time variation of bond yields. In the following section, we
concentrate on the latter.
1.3 Econometric methods
Since our macro factors are observable, we use GMM to estimate the model. We discuss two
cases: an exactly identified model and an over-identified model.
1.3.1 Orthogonality Conditions
We follow the literature and assume that, in the presence of of measurement error, Eq. 1.6 is
satisfied up to an additive, orthogonal disturbance. Thus we model each yield in the following
way:
ynt = an + bTnXt + ent (1.9)
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 8
with the following orthogonality condition:
E [Xtent ] = 0 (1.10)
Given all factors are observable, non-linear GMM which incorporates the model’s cross equa-
tion restrictions is our framework for estimation and testing. In the next subsections we
discuss, respectively, the exactly identified model and the over-identified model.
1.3.2 Exactly identified model
We use Cholesky identification to estimate 1.2, from which we get µ, Σ, and Φ. δ0 and
δ1 are also known. Hence, from equations 1.7 and 1.8, An and Bn are functions of λ0 and
λ1, denoted as An(λ0, λ1) and Bn(λ1). Let us suppose we observe bonds with N different
maturities. For each bond with a specific maturity we have T observations. We then have
the following orthogonality condition:
1
T
T∑t=1
[ynt +
An(λ0, λ1) +BTn (λ1)Xt
n
]= 0 (1.11)
1
T
T∑t=1
[ynt +
An(λ0, λ1) +BTn (λ1)Xt
n
]·Xt = 0 (1.12)
If we have K factors, then we have N(1+K) orthogonality conditions, and we have K(1+K)
unknowns. When K = N , the model is exactly identified.
1.3.3 Over-identified model
Using the framework outlined in subsection 1.3.2, we see that when K < N , the model is
over-identified. Indeed, the number of orthogonality conditions is N(1+K), while the number
of unknowns is K(1 +K). Thus the model is over-identified when K < N . Our motivation is
to study the time variation of bond yield, which is affected solely by λ1 as explained in 1.2.2.
Thus we conduct a two-stage estimation. In the first stage, we estimate λ1 by exploring
demeaned factors and demeaned yields. We then estimate λ0 by using the original factors
and yields.
We have the following moment condition for demeaned factors dXt and demeaned bond yields
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 9
dynt :
G(λ1, n) =1
T
T∑t=1
[dynt +
BTn (λ1)dXt
n
]· dXt = 0 (1.13)
Because the model is over-identified, in general we will not find any λ1 that satisfies 1.13 for
all yields to maturity. Let us use MaturitySet to denote the set of bond maturities under
study. In section 1.7 we will estimate a model for bonds with 12 different maturities, such
that MaturitySet = 3M, 6M, 1Y, 2Y, ..., 9Y, 10Y .
Given a set of initial values, we can use the quasi-Newton method to minimize the GMM
objective function:
λ1 = minλ∗1
∑n∈MaturitySet
GT (λ∗1, n)G(λ∗1, n) (1.14)
Since the objective function is highly nonlinear, the performance of a nonlinear optimization
algorithm can depend crucially on the initial values selected for the structural parameters.
One of the contributions made by the present chapter is to innovate a method for obtaining
initial values.
1.3.3.1 Initial value generation
From Eq. 1.8, we see that λ1 is uniquely determined from the following condition, given four
pairs of B whose maturities differ by one month:
BTn+1 = BT
n (Φ− Σλ1)− δT1
For example if we know Bn for n ∈ 1, 2, 4, 5, 6, 7, 9, 10, then BTn+1 = BT
n (Φ− Σλ1) − δT1gives us 16 equations for n ∈ 1, 4, 6, 9. We then can uniquely solve λ1. In order to have
an initial value for λ1, we need an estimate of the reduced form Bn parameters. We obtain
those from a reduced form regression
y(i),nt =
A(U)n
n+B
(U)Tn Xt
n+ νnt (1.15)
Consequently, we solve initial value λInitial1 with the following condition
B(U)Tn+1 = B(U)T
n
(Φ− ΣλInitial1
)− δInitial1 (1.16)
In practice, existing data sets do not include yields to maturity for all tenors between 1 month
and 120 months. As a result, we use an interpolation method to generate approximate yields
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 10
to maturity for tenors contiguous to the tenors we study in the exactly identified model. We
follow the method of Nelson-Siegel.
1.3.3.2 Nelson-Siegel Yield Curve Model
Nelson and Siegel [1987] models the forward rate as follows:
f(τ) =[β0 β1 β2
]1
e−τ/λ
(τ/λ)e−τ/λ
where λ is the only model parameter, and τ is the maturity. The spot rate has the following
structure:
r(τ) =[β0 β1 β2
]1
λ/τ(1− e−τ/λ)
λ/τ(1− e−τ/λ)− e−τ/λ
Note that Nelson and Siegel [1987] models the three latent factors of yield curve “level”,
“slope” and “curvature” by using three functions: 1, λ/τ(1−e−τ/λ) and λ/τ(1−e−τ/λ)−e−τ/λ.
1.3.3.3 Generating Hypothetical Yields via the Nelson-Siegel Model
In order to use the Nelson-Siegel Model to generate hypothetic yields, we first fit it to observed
yields in order to estimate λ. We then generate the entire yield curve by computing the
function:
r(τ) =[β0 β1 β2
]1
λ/τ(1− e−τ/λ)
λ/τ(1− e−τ/λ)− e−τ/λ
After we have generated the interpolated yields, we use the methods in subsection 1.3.3.1 to
compute the initial value of λ1.
1.4 Data
We first discuss the measurement issue respectively for the real neutral rate, the output gap,
and the expected inflation. We then detail out three sets of specifications.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 11
1.4.1 Measurement for real neutral interest rate
Real neutral rate can be motivated from the growth theory and related to the long-run
economic equilibrium. This is the method originally used by Taylor in Taylor [1993]. A key
implication of this approach is that real neutral rate is linked to the steady state growth rate
of output because real neutral rate changes over time in responding to technology change
and population growth. Another approach used in Hall [2000] is to characterize the time-
variation of real neutral rate by using the intercept of Taylor rule itself. This is an interesting
approach, especially by treating the real neutral rate as an unobservable variable in a Bayesian
approach.
Here following Archibald et al. [2001] and Laubach and Williams [2003] we focus on the
medium-run concept of real neutral interest rate, which abstracts from the short-run price
and output effects. Real neutral interest rate refers to the real short-term interest rate under
which the output converges to potential, where potential is the level of output consistent with
stable inflation (see Bomfim [1997]). The practical intuition of this assumption is: it may
take several quarters for the effect of the interest rate to take place on inflation. In contrast
to the short horizon, it is this longer horizon that is relevant for our discussion here. Clarida
et al. [1999] and Woodford [2003] have derived the time-varying natural interest rate for the
optimal monetary policy. We review below two methods to measure the real neutral rate.
The first method is based on the decomposition of observed inflation-indexed bond yields.
Note TIPSt the yield of an inflation-indexed bond at time t, rrt the real interest rate at time
t, and πrpt the risk premium of inflation. We then have the following relation
TIPSt = rrt + πrpt (1.17)
which says that the yield of inflation indexed bond is the sum of real rate and inflation risk
premium. We can further decompose the real interest rate into the real neutral interest rate
and a cyclical factor, which represents the monetary policy effect against inflationary pressure
on real rate. Let us denote as mt the monetary policy induced cyclical factor of real rate.
We thus have:
rrt = rnrt +mt (1.18)
TIPSt = rnrt +mt + πrpt (1.19)
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 12
Eq. 1.19 offers a way to estimate real neutral rate rnrt = TIPSt − (mt + πrpt ). Among the
three variables on the right-hand side, TIPSt is directly observable and πrpt can be estimated
in a joint nominal and real term-structure model. For mt we note that we can average out
the effects of mt by taking the average of Eq. 1.19. Note F TIPSnt the forward yield of a
one period inflation-indexed bond n periods in the future, rnrnt , mnt , and πrp,nt respectively
the expected real neutral rate, expected cyclical factor and expected inflation risk premium
for n periods in the future. We then have
F TIPSnt = rnrnt +mnt + πrp,nt
We estimate the rnrt by the following approximation:
rnrt =
∑119t=60 F TIPSnt
60− πrpt (1.20)
The second method we discuss is a Bayesian approach. Laubach and Williams [2003] jointly
estimates the output gap and the real neutral rate in a reduced-form equation. In their
model, the output gap is related to its own lags and the lags of real interest rate gap, which
is the difference between the real short-term rate and the real neutral rate. And the real neu-
tral rate is related to the potential output through the trend growth of the latter. Laubach
and Williams [2003] estimates the real neutral rate and output potential together by using
Kalman filter.
1.4.2 Measurement for output gap
We discuss here the output gap measures in the literature. One large class of detrending
method decomposes the log of output, yt into a trend component µt, and a cyclical component,
gaot.
yt = µt + gapt (1.21)
There are several methods to estimate the output trend µt and the output gap gapt. a).
Deterministic trends. This method approximates the output trend as a deterministic
function, with the linear trend and the quadratic trend being the simplest examples. b).
Unobserved component model. This kind of model assumes more complex dynamics
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 13
for the trend and the cyclical components. A classical example is HP filter proposed by
Hodrick and Prescott [1997]. For other examples, Watson [1986] models the trend process
to be random walk and cyclical process to be of AR(2). Harvey [1985] and Clark [1987]
both model trend process to be of stochastic trend and cyclical process to be of AR(2). c).
Unobserved component model with a Philips curve. This method adds a Phillips
curve to the univariate model described in b), see for example Kuttner [1994] and Gerlach
and Smets [1999].
The detrending methods have been shown to exhibit poor real time performance. Orphanides
and Van Norden [2002] shows that the reliability of detrending methods for real time output
gap estimation is poor mainly due to unreliability of the end-of-sample estimates of the trend
in output. We discuss next two alternative methods. The first approach is our preferred
approach, which is the joint estimation of output gap and real neutral rate in Laubach and
Williams [2003]. Their method was discussed previously in the subsection 1.4.1. The second
approach is the Okun’s law.
gapt = k(Ut − U∗) (1.22)
where Ut is the unemployment rate at time t, U∗ is the natural rate of unemployment, and
k measures the change in output per unit of change in the unemployment rate. This method
was originally proposed in Okun [1970], for more recent updates see Ball et al. [2013] and
Owyang and Sekhposyan [2012]. Ball et al. [2013] finds that Okun’s law did not change
substantially during the Great Recession. Owyang and Sekhposyan [2012] doesn’t finds
statistical significance of the slope change during the Great Recession for our specification.
Literature estimates k to be between 2 and 3 for US.
1.4.3 Measurement for the expected inflation
A common estimation for the expected inflation is the break-even inflation computed from
nominal and inflation-indexed bond yields. This method, however, doesn’t take into account
the inflation risk premium embedded in inflation-indexed bonds. If one wants to tackle this
issue, one has to model jointly the nominal and real interest rate term structure models.
Haubrich et al. [2012] and Christensen et al. [2010] are two examples. Both studies have
obtained close to zero inflation risk premium for recent years. Both papers also estimate the
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 14
expected inflation. We also note that Hilscher, Raviv, and Reis [2014] constructs risk-adjusted
distributions of inflation by using inflation option data.
1.4.4 Data
We use monthly US data from September 1997 to August 2012, which is a total of 180 months.
The 13 zero coupon bond yields of 1, 3, and 6 months as well as of 1, 2, 3, 4, 5, 6, 7, 8, 9 and
10 years all come from the US Treasury Department. For expected inflation we use results
of five-year expected inflation and five-year inflation risk premium in Haubrich et al. [2012],
which covers our whole sample period.
With respect to real neutral rate and GDP gap, our estimations in the main text come from
Laubach and Williams [2003]. As we discussed before the attraction of Laubach and Williams
[2003] is that it models real neutral rate and output gap jointly. As a robustness test we test
our model on two alternative sets of measurements. For the first alternative specification, we
keep the estimation of real neutral rate and of expected inflation from the main text. But
we estimate the GDP gap by using Okun’s law. For the second alternative specification, we
estimate the real neutral rate by the five-year forward rate five years in the future minus the
inflation risk premium, the expected inflation by the 5-year break-even inflation rate minus
the inflation risk premium, and the GDP gap by the Okun’s law.
Lastly the Taylor rule deviation is defined as in equation 1.1 with the short term rate it being
approximated by using the one month maturity bond yield.
1.5 Results of exactly identified model
Following Ang and Piazzesi [2003] and Rudebusch and Wu [2008] we first test the model with
bonds of maturity equal to 1, 3, 12, 36 or 60 months. Under our four factors specification,
the model is exactly identified.
1.5.1 Macro factors dynamics
We plot in Fig. A.1 in the appendix the contribution of each macro factor in Taylor rule
together with the Fed fund rate. Table 1.1 reports the estimates for state dynamics 1.2 and
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 15
the corresponding t-ratios. The diagonal of Φ determines the persistence of the macro factors.
We see that the diagonal terms are all higher than 0.8 and all highly statistically significant,
which suggest the strong persistence of macro variables.
Table 1.1: Macro Factors’ Dynamics
µ(×100) Φ Σ
r g π u r g π u
r -0.01 0.99 0.22
(0.00) (66.50)
g -3.39 0.01 0.95 0.02 0.21
(0.78) (0.61) (35.11) (1.40)
π -6.37 0.05 0.02 0.90 0.018 0.006 0.20
(1.33) (1.98) (0.58) (26.89) (1.15) (0.40)
u 52.02 -0.31 0.10 0.32 0.87 -0.27 -0.23 -0.16 0.30
(5.74) (6.09) (2.37) (5.56) (35.29) (11.91) (10.04) (7.24)
The table presents the estimates for the state dynamics following a VAR(1) process. The ab-
solute value of the t-ratio of each estimate is reported. The sample period is from September
1997 to August 2012.
1.5.2 Market price of risk
Using the method detailed above, we compute λ1. We report in Table 1.2 the result. In
Figure 1.1 we plot observed yields and model implied yields. The figure shows that our
model does a superior job of pricing the bonds.
1.5.3 Impulse response function
Here we study, respectively, the contemporaneous response of yields to shocks and responses
at longer horizons. The shock from each factor is assumed to be of one standard deviation
of the factor’s first difference. In Table 1.3 we report the assumed shock size, the unit is
percentage point.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 16
Figure 1.1: Price Fitting for Exactly Identified Model
1997 2000 2002 2005 2007 2010 2012 2015−1
0
1
2
3
4
5
6
7Observed and model implied Yields for 0.25 Years Bond
Model Implied Yield
Observed Yield
1997 2000 2002 2005 2007 2010 2012 2015−1
0
1
2
3
4
5
6
7Observed and model implied Yields for 0.5 Years Bond
Model Implied Yield
Observed Yield
1997 2000 2002 2005 2007 2010 2012 2015−1
0
1
2
3
4
5
6
7Observed and model implied Yields for 1 Years Bond
Model Implied Yield
Observed Yield
1997 2000 2002 2005 2007 2010 2012 2015−1
0
1
2
3
4
5
6
7Observed and model implied Yields for 2 Years Bond
Model Implied Yield
Observed Yield
Note: Since this model is exactly identified, the fit of the model simply indicates that our four factors capture
virtually all of the contemporaneous variation in the yields included. We test the over identified version of
the model in Section 7.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 17
Table 1.2: Market Price of Risk
r g π u
r 1.87 -1.65 -0.59 0.28
(0.81) (0.53) (0.71) (0.17)
g 2.65 1.78 0.24 1.61
(2.07) (1.18) (0.21) (1.76)
π 0.17 0.64 -0.33 0.14
(0.47) (1.33) (2.28) (0.56)
u -0.59 -0.15 0.42 -0.37
(2.01) (0.43) (3.13) (1.93)
The table presents the estimates for the market price of essentially affine model. The
absolute value of the t-ratio of each estimate is reported. The sample period is from
September 1997 to August 2012. The standard deviation is corrected following Newey
and McFadden(1994).
Table 1.3: Macro Shock Size
RealRate GDPGap Inflation TYRes
0.22 0.42 0.14 0.52
1.5.3.1 Response of contemporaneous yields
The pricing equation ynt = an+ bTnXt leads us that the effect of each factor on the yield curve
is given by bn, which is in turn determined by the affine asset pricing structure. bn thus
represents the initial response to shocks made by different factors. In Figure 1.2 we plot the
initial responses of bonds having different maturities. The shock from each factor is assumed
to be of one standard deviation of the factor’s first difference.
We look at first the effects of real neutral rate and expected inflation. The responses are
very persistent, and the responses of bonds having different maturities are in the same order.
Thus, shocks to the neutral rate and expected inflation affect mainly the level factor. When
it comes to the other two shocks, the maximum response of yield curve is for bonds having
one month or three months maturity. Thus these shocks are mean reverting.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 18
Figure 1.2: Contemporaneous Impulse Responses
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Bond maturities in Months
Annuliz
ed y
ield
in %
Initial response
RealRate
GDPGap
Inflation
TYRes
1.5.3.2 Response at longer horizon
Our term structure model gives us the yield curve responses to macro factor shocks at long
horizons. We plot in Figure 1.3 impulse responses to yield of maturity 1Y, 3Y and 5Y. We
find that for all three bonds, shock from real neutral rate is most significant at the longer
horizon. At one-month horizon, shock from Taylor residue is most significant for the 1Y
bond. For the 3Y and 5Y bonds the shock from expected inflation is most significant at the
one-month ahead horizon.
1.5.4 Variance decomposition
Based on our VAR dynamics we can decompose the forecasting variance of yields into con-
tributions from four macro shocks. In Table 1.4 we report the results of respectively 1Y,
3Y, and 5Y bonds. For the 1Y bond at one-month ahead horizon Taylor residue contributes
most of the variance. At longer horizons all factors contribute to the variance. For the 3Y
and the 5Y bonds we see results having similar properties. For one-month ahead forecasting,
expected inflation contributes most of the variance. At longer horizons, the real neutral rate
contributes most of the forecasting variance. We thus conclude that at short horizon Taylor
Rule residue chiefly explains the movements of the yield curve at short end. At long horizon
for long maturity bonds, it is real neutral rate that mainly explains the movements of the
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 19
Figure 1.3: Impulse Responses at Future Horizon
0 10 20 30 40 50 600
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Months
Annuliz
ed y
ield
in %
Initial response of 1Y Bond
RealRate
GDPGap
Inflation
TYRes
0 10 20 30 40 50 600
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Months
Annuliz
ed y
ield
in %
Initial response of 3Y Bond
RealRate
GDPGap
Inflation
TYRes
0 10 20 30 40 50 600
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Months
Annuliz
ed y
ield
in %
Initial response of 5Y Bond
RealRate
GDPGap
Inflation
TYRes
yield curves.
Our results regarding impulse response analysis and variance decomposition confirm those
gained by many previous studies. The model of Bekaert et al. [2010] also shows strong
contemporaneous responses of the yield curve to macro shocks. Hordahl et al. [2006] finds
that monetary policy shocks impact chiefly at the short end of the yield curve. Dewachter
and Lyrio [2006] finds that at the short end of the yield curve, both real rate and inflation
expectation count.
1.5.5 Robustness checks
In order to test the robustness of our model, we decompose the sample period into two parts.
We then calibrate the model by using in-sample data and test it by using out-sample data.
The in-sample data period is from September 1997 to June 2008, the out-sample data period
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 20
Table 1.4: Variance Decomposition
1Y Bond
RealRate GDPGap Inflation TYRes
1Month 0.02 0.03 0.47 0.48
1Y 0.01 0.23 0.60 0.16
5Y 0.30 0.28 0.35 0.07
3Y Bond
RealRate GDPGap Inflation TYRes
1Month 0.04 0.01 0.55 0.41
1Y 0.10 0.11 0.65 0.14
5Y 0.56 0.13 0.27 0.04
5Y Bond
RealRate GDPGap Inflation TYRes
1Month 0.02 0.00 0.93 0.04
1Y 0.16 0.02 0.79 0.02
5Y 0.65 0.03 0.31 0.01
from July 2008 to August 2012. We report in Table 1.5 the price of risk λ1 that we have
obtained by calibrating the in-sample data.
In Figure 1.4 we plot the observed yields and model implied yields. The in-sample period
is from August 1997 to June 2008, the out-sample period from July 2008 to August 2012. In
both cases we use the first ten months’ data in each sample to match the constant term an.
We then measure the accuracy of our model’s pricing by computing the root-mean-square
error for the sample excluding the first ten months.
The figure shows that our model tracks the time variation of bond yields in the out-
sample period. The out-sample performance of 3Y and 5Y bonds are especially good before
September 2011, the onset of Operation Twist. And the model works better for the 3Y
and 5Y bonds than the 3M and 1Y bond. This is confirmed by the root-mean-square error
statistics. In Table 1.6 and Table 1.7 we report the pricing statistics for the in-sample and
the out-sample periods.
We see by looking at Table 1.7 that the pricing error of 5Y bonds increases much less
than for 3M and 1Y bond. In the subsection 1.8.4, when we study an over-identified model,
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 21
Table 1.5: Market Price of Risk for In-sample Period
r g π u
r 0.07 -0.20 -3.01 0.11
(0.08) (0.43) (2.15) (0.47)
g -1.10 0.34 1.40 -0.10
(2.07) (1.23) (1.06) (0.60)
π 0.17 0.24 -0.22 0.04
(3.14) (8.20) (1.01) (2.75)
u -0.74 0.02 -0.31 -0.35
(2.20) (0.14) (1.10) (3.88)
The table presents the estimates for the market price of risk of essentially affine model.
The absolute value of the t-ratio of each estimate is reported. The sample period is from
September 1997 to June 2008. The standard deviation is corrected following Newey and
McFadden(1994).
Table 1.6: In-sample Pricing Error
Observed
Mean
RMSE Rlt Error(%)
1M 3.18 0.00 0.00
3M 3.39 0.21 6.14
1Y 3.62 0.20 5.49
3Y 4.05 0.19 4.75
5Y 4.39 0.16 3.54
The table reports the in-sample pricing error. The Observed Mean is the mean of
observed bond yield. RMSE is the root mean square error. Rlt Error is the RMSE
divided by the Observed Mean. The sample period is from September 1997 to June
2008.
we will discuss the reason of the model’s better performance at medium and long ends of the
curve.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 22
Figure 1.4: In-sample and Out-sample Pricing
1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014−1
0
1
2
3
4
5
6
7Observed and model implied Yields for 3 Months Bond
← Out−sample starts
Model
Observed
1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 20140
1
2
3
4
5
6
7Observed and model implied Yields for 12 Months Bond
← Out−sample starts
Model
Observed
1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 20140
1
2
3
4
5
6
7Observed and model implied Yields for 36 Months Bond
← Out−sample starts
Model
Observed
1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 20140
1
2
3
4
5
6
7Observed and model implied Yields for 60 Months Bond
← Out−sample starts
Model
Observed
Operation Twist
Table 1.7: Out-sample Pricing Error
Observed
Mean
RMSE Rlt Error(%)
1M 0.07 0.00 0.00
3M 0.10 0.23 228.95
1Y 0.27 0.79 296.09
3Y 0.91 0.81 89.11
5Y 1.67 0.69 41.37
The table reports the out-sample pricing error. The Observed Mean is the mean of
observed bond yield. RMSE is the root mean square error. Rlt Error is the RMSE
divided by the Observed Mean. The sample period is from July 2008 to August 2012.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 23
1.6 Extension: Forecasting of Bond Excess Return
1.6.1 Forecasting using forward yields
Following Cochrane and Piazzesi [2002], we forecast the excess return of bonds using forward
rates. We find - as did they - a single factor that predicts the excess return of bonds having
maturity between two and five years. We can interpret this factor by using our macro factors.
We note fnt , the forward rate at time t for a loan extended between period t + n − 1 and
t+ n. We also note rxnt+1, the excess return of bond with maturity n years between period t
and period t+ 1.
1.6.1.1 Unrestricted regression
To obtain the unrestricted specification we regress the excess returns on forward rates for
bonds of maturity between two and five years(2 ≤ n ≤ 5):
rxnt+1 = βn0 +5∑i=1
βni fit
Table 1.8 provides the results of that regression:
Table 1.8: Forecasting of Bond Excess Return
β0 β1 β2 β3 β4 β5 R2
2YBond Est -2.82 0.69 -1.01 -1.18 2.46 -0.17 0.20
2YBond Std 2.07 0.64 0.61 1.30 1.57 0.87
3YBond Est -5.36 1.32 -2.37 -2.04 4.96 -0.43 0.18
3YBond Std 3.53 1.26 1.46 2.42 3.40 1.76
4YBond Est -8.00 1.89 -2.80 -4.22 6.93 0.23 0.20
4YBond Std 4.51 1.77 2.25 3.16 4.86 2.61
5YBond Est -10.85 2.37 -2.69 -6.36 7.01 2.28 0.24
5YBond Std 5.14 2.21 2.99 3.80 6.03 3.43
The standard deviation in the table was obtained by doing the Hansen-
Hodrick correction.
It shows that the joint test for all coefficients equal to 0 has p-value below 0.05 for all
bonds except the 4Y bond.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 24
1.6.1.2 Restricted regression
To do a restricted regression we first regress the average excess returns on forward rates:
rxt+1 = γ0 +5∑i=1
γifit
where rxt+1 = 14
∑4i=1 rx
it+1. In Table 1.9 we report the results.
Table 1.9: Bond Excess Return Forecasting Factor
γ0 γ1 γ2 γ3 γ4 γ5 R2
Avg Est -6.75 1.57 -2.22 -3.45 5.34 0.48 0.21
Std 3.79 1.46 1.83 2.62 3.94 2.12
The standard deviation in the table was obtained by doing the
Hansen-Hodrick correction.
In the second stage we regress the individual bond return on the single factor: γ0 +∑5
i=1 γifit .
rxnt+1 = bn(γ0 +5∑i=1
γifit )
In Table 1.10 we report the estimation of bn. We see that b has t-statistic with p-value below
0.001 for all bonds.
Table 1.10: Forecasting of Bond Excess Return
b std R2
2Y Bond 0.43 0.12 0.16
3Y Bond 0.80 0.22 0.17
4Y Bond 1.20 0.30 0.20
5Y Bond 1.57 0.34 0.23
The standard deviation in the table was obtained
by doing the Hansen-Hodrick correction.
In Table 1.11 we represent the factor γ0 +∑5
i=1 γifit by using macro factors. This reveals the
loadings on the macro factors.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 25
Table 1.11: Forecasting Factor Representation using Macro Factors
NEUT GDPGAP EI TAYRES
0.39 -0.63 4.33 -0.09
1.7 Extension: Adding Bonds of Longer Maturities
Here we extend our analysis to bonds of a maturity of up to ten years. More specifically we
look at 13 bonds having maturities of 1 month, 3 months, 6 months and 1 year through 10
years. Our estimation method is similar to that employed in Section 1.3. We report λ1 in
Table 1.12.
Table 1.12: Market Price of Risk
r g π u
r -0.09 -0.30 -0.22 0.10
(0.62) (0.67) (0.71) (0.66)
g 0.04 0.66 0.10 0.34
(0.23) (2.25) (0.42) (2.63)
π 0.29 0.31 -0.42 0.07
(3.98) (1.01) (2.73) (1.54)
u -0.74 0.08 0.49 -0.34
(6.64) (0.34) (2.54) (3.89)
The table presents the estimates for the market price of essentially affine model. The
absolute value of the t-ratio of each estimate is reported. The sample period is from
September 1997 to August 2012. The standard deviation is corrected following Newey
and McFadden(1994).
For the λ1 in Table 1.12, we have∑
1≤n≤13 GT (λ∗1, n)G(λ∗1, n) = 0.013. The J-statistic is
2.282, with a p-value less than 0.01%. Thus we are unable to reject the 32 over identifying
restrictions implied by the theory. We next compare the factor loadings of the over-identified
model with that of the unrestricted regression. In Table 1.13 we report the result. We see
that the factor loadings of the over-identified model are close to that of the unrestricted
2We use the results of the first stage estimation when estimating λ1. Thus the J-statistics is an approxi-
mation.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 26
Figure 1.5: PCA Factor Loadings
1 2 3 4 5 6 7 8 9 10−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8PCA Factor Loadings
Factor 1
Factor 2
Factor 3
regression. In Fig. A.2 in the appendix we plot the observed yields and the model implied
yields for bonds having maturities up to 10 years.
1.7.1 Principal Component Analysis
We first conduct principal component analysis of the yield curves. We then represent the
factors extracted from that analysis by using macro factors. We plot in Figure 1.5 the loadings
on bonds of the first three factors obtained from principal component analysis.
We see that the first factor has almost equal loadings on all bonds. This is the level factor
in the empirical asset pricing literature. The second factor has smaller loadings on bonds
having higher maturities. This is the slope factor. The third factor has first decreasing then
increasing loadings and is thus the curvature factor.
Based on the pricing equation ynt = an + bTnXt we can represent the three factors as a
function of macro factors. In Table 1.14 we report the result.
1.7.2 Impulse response function
Here we study, respectively, the contemporaneous response of yields to shocks and responses
at longer horizons. The shock from each factor is assumed to be of one standard deviation
of the factor’s first difference.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 27
1.7.2.1 Response of contemporaneous yields
The pricing equation ynt = an+ bTnXt leads us that the effect of each factor on the yield curve
is given by bn, which is in turn determined by the affine asset pricing structure. bn thus
represents the initial response to shocks made by different factors. In Figure 1.6 we plot the
initial responses of bonds having different maturities. The shock from each factor is assumed
to be of one standard deviation of the factor’s first difference.
It shows that for all factors other than expected inflation, the response of the bond having
Figure 1.6: Contemporaneous Impulse Responses
0 20 40 60 80 100 120−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Bond maturities in Months
Annuliz
ed y
ield
in %
Initial response
RealRate
GDPGap
Inflation
TYRes
longer maturity is smaller than that having shorter maturity. The expected inflation has a
hump shape response function. Among all four factors the shocks to expected inflation and
real rate have the most significant response for a longer maturity bond.
1.7.2.2 Response at longer horizon
Our term structure model gives us the yield curve responses to macro factor shocks at long
horizons. We plot in Figure 1.7 impulse responses of bond yields of maturity 1Y, 5Y and
10Y. We find that for all three bonds, shock from real neutral rate is most significant at the
longer horizon. At the one-month horizon, shock from Taylor residue is most significant for
the 1Y bond. For the 5Y and 10Y bonds the shock from expected inflation is most significant
at a one-month ahead horizon.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 28
Figure 1.7: Impulse Responses at Future Horizon
0 10 20 30 40 50 600
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Months
Annuliz
ed y
ield
in %
Initial response of 1Y Bond
RealRate
GDPGap
Inflation
TYRes
0 10 20 30 40 50 600
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Months
Annuliz
ed y
ield
in %
Initial response of 5Y Bond
RealRate
GDPGap
Inflation
TYRes
0 10 20 30 40 50 60−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Months
Annuliz
ed y
ield
in %
Initial response of 10Y Bond
RealRate
GDPGap
Inflation
TYRes
In Figure A.3 in Appendix A we report the error bands of impulse responses.
1.7.3 Variance decomposition
Based on our VAR dynamics, we can decompose the forecasting variance of yields into con-
tributions from four macro shocks. In Table 1.15 we report the results of respectively 1Y,
5Y, and 10Y bonds, respectively.
For the 1Y bond at one-month ahead horizon, Taylor residue contributes most of the
variance. At longer horizons, all factors contribute to the variance. For the 5Y and the 10Y
bonds, we see results having similar properties. For one-month ahead forecasting, expected
inflation contributes most of the variance. At longer horizons, the real neutral rate contributes
most of the forecasting variance.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 29
Table 1.13: Model Implied Factor Loadings and Unrestricted Regression Factor Loadings
NEUT GDPGAP EI TAYRES
1M Unres Reg 1.00 1.00 1.00 1.00
1M Over-Iden Model 1.00 1.00 1.00 1.00
3M Unres Reg 0.94 1.04 1.19 0.99
3M Over-Iden Model 0.96 1.03 1.08 0.98
6M Unres Reg 0.84 1.06 1.31 0.94
6M Over-Iden Model 0.90 1.03 1.20 0.94
1Y Unres Reg 0.74 0.94 1.42 0.82
1Y Over-Iden Model 0.79 0.96 1.38 0.84
2Y Unres Reg 0.54 0.67 1.70 0.59
2Y Over-Iden Model 0.61 0.69 1.62 0.62
3Y Unres Reg 0.49 0.44 1.68 0.46
3Y Over-Iden Model 0.47 0.42 1.71 0.45
4Y Unres Reg 0.44 0.25 1.64 0.34
4Y Over-Iden Model 0.38 0.21 1.72 0.31
5Y Unres Reg 0.38 0.07 1.59 0.23
5Y Over-Iden Model 0.31 0.06 1.67 0.22
6Y Unres Reg 0.32 -0.02 1.54 0.18
6Y Over-Iden Model 0.26 -0.03 1.61 0.16
7Y Unres Reg 0.26 -0.11 1.49 0.13
7Y Over-Iden Model 0.22 -0.09 1.53 0.11
8Y Unres Reg 0.24 -0.16 1.41 0.09
8Y Over-Iden Model 0.20 -0.13 1.44 0.07
9Y Unres Reg 0.23 -0.21 1.33 0.05
9Y Over-Iden Model 0.19 -0.16 1.36 0.04
10Y Unres Reg 0.22 -0.25 1.26 0.01
10Y Over-Iden Model 0.19 -0.18 1.27 0.01
The table presents the factor loadings of the over-identified model with that of
the unrestricted regression. For example, 1M Unres Reg is the factor loading
from unrestricted regressions. 1M Over-Iden Model is the factor loading of the
over-identified model. The sample period is from September 1997 to August
2012.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 30
Table 1.14: PCA Factors Representation using Macro Factors
Factor 1 Factor 2 Factor 3
RealRate 1.14 0.60 0.17
GDPGap 0.54 1.16 0.25
Inflation 4.84 0.24 -0.35
TYRes 0.89 0.81 0.17
Table 1.15: Variance Decomposition
1Y Bond
RealRate GDPGap Inflation TYRes
1Month 0.00 0.01 0.31 0.67
1Y 0.01 0.23 0.56 0.21
5Y 0.32 0.28 0.32 0.08
5Y Bond
RealRate GDPGap Inflation TYRes
1Month 0.02 0.01 0.93 0.04
1Y 0.16 0.02 0.80 0.02
5Y 0.65 0.03 0.31 0.01
10Y Bond
RealRate GDPGap Inflation TYRes
1Month 0.06 0.01 0.92 0.00
1Y 0.31 0.01 0.68 0.00
5Y 0.75 0.00 0.24 0.00
In Table A.1 in Appendix A we report the error band of variance
decomposition.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 31
1.7.4 Robustness checks
In order to test the robustness of our model, we decompose the sample period into two parts.
The in-sample period is from August 1997 to June 2008 and the out-sample period from July
2008 to August 2012. We then calibrate the model by using in-sample data and test it by
using out-sample data. We report in Table 1.16 the price of risk λ1 that we have obtained
by calibrating the in-sample data.
Table 1.16: Market Price of Risk for In-sample Period
r g π u
r -1.25 0.56 1.56 0.00
(2.84) (0.95) (1.51) (0.01)
g -1.28 1.02 2.81 -0.07
(1.90) (1.95) (3.07) (0.50)
π 0.14 0.23 -0.40 0.03
(0.92) (1.18) (1.27) (4.08)
u -0.93 0.11 0.59 -0.35
(3.96) (0.38) (1.26) (3.97)
The table presents the estimates for the market price of risk of essentially affine model.
The absolute value of the t-ratio of each estimate is reported. The sample period is from
September 1997 to June 2008. The standard deviation is corrected following Newey and
McFadden(1994).
In Figure 1.8 we plot observed yields and model implied yields for 1Y, 5Y, 7Y and 10Y
bonds. In all cases, we use the first ten months data in each sample to match the constant
term an. We then measure the accuracy of our model’s pricing by computing the root-mean-
square error for the sample, excluding the first ten months.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 32
Figure 1.8: In-sample and Out-sample Pricing
1998 2000 2002 2004 2006 2008 2010 20120
1
2
3
4
5
6
7
Observed and model implied Yields for 24 Months Bond
← Out−sample starts
Model
Observed
1998 2000 2002 2004 2006 2008 2010 20120
1
2
3
4
5
6
7
Observed and model implied Yields for 60 Months Bond
← Out−sample starts
Model
Observed
Operation Twist
1998 2000 2002 2004 2006 2008 2010 20120
1
2
3
4
5
6
7
Observed and model implied Yields for 84 Months Bond
← Out−sample starts
Model
Observed
Operation Twist
1998 2000 2002 2004 2006 2008 2010 20120
1
2
3
4
5
6
7
Observed and model implied Yields for 120 Months Bond
← Out−sample starts
Model
Observed
Operation Twist
The figure shows that our model tracks the time variation of bond yields in the out-sample
period. The out-sample performances of medium and long term bonds are especially good
before September 2011, the onset of Operation Twist. And the model works better for the
medium and long end bonds than the short end bonds. This is confirmed by the root-mean-
square error statistics. In Table 1.17 and Table 1.18 we report the pricing statistics for the
in-sample and the out-sample periods. The Observed Mean is the mean of observed bond
yield. RMSE is the root mean square error. Rlt Error is the RMSE divided by the Observed
Mean.
We see by looking at Table 1.18 that all bonds are priced accurately for out-sample period
except those having maturities less than 2 years. In the subsection 1.8.4 we discuss the reason
of the big pricing error at out-sample period for short end bonds.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 33
Table 1.17: In-sample Pricing Error
Observed
Mean
RMSE Rlt Error(%)
3M 3.39 0.20 5.93
6M 3.52 0.25 7.02
1Y 3.62 0.22 6.10
2Y 3.88 0.21 5.44
3Y 4.05 0.18 4.48
4Y 4.22 0.15 3.66
5Y 4.39 0.15 3.45
6Y 4.48 0.15 3.43
7Y 4.58 0.16 3.48
8Y 4.67 0.17 3.54
9Y 4.77 0.18 3.72
10Y 4.86 0.21 4.38
The table reports the in-sample pricing error. The Observed Mean is the mean of
observed bond yield. RMSE is the root mean square error. Rlt Error is the RMSE
divided by the Observed Mean. The sample period is from September 1997 to June
2008.
1.8 Robustness check and discussions
1.8.1 Testing model on alternative measurements: I
As a robustness test, we test our model on two alternative sets of measurements for the Taylor
rule factors. We will compare the results to that of the main text. We denote the measurement
in the main text as Williams-Laubach’s data. In the first alternative specification, we estimate
the output gap by using Okun’s law, as in Eq. 1.22, where we have gt = 2.5 ∗ (4.75 − uet).
And we keep the Williams-Laubach’s estimation of real rate and Haudrich et al’s estimation
for the expected inflation. We do all the exercises as in the section 1.7. We report the results
in Appendix A.
We plot in Fig. A.4 the contributions of macro factors in Taylor rule along with the Fed fund
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 34
Table 1.18: Out-sample Pricing Error
Observed
Mean
RMSE Rlt Error(%)
3M 0.10 0.25 243.97
6M 0.17 0.60 361.75
1Y 0.27 0.86 320.58
2Y 0.58 0.87 149.80
3Y 0.91 0.86 94.62
4Y 1.29 0.77 60.21
5Y 1.67 0.70 42.17
6Y 2.00 0.61 30.70
7Y 2.34 0.54 23.21
8Y 2.55 0.51 19.92
9Y 2.76 0.48 17.38
10Y 2.96 0.46 15.57
The table reports the out-sample pricing error. The Observed Mean is the mean of
observed bond yield. RMSE is the root mean square error. Rlt Error is the RMSE
divided by the Observed Mean. The sample period is from July 2008 to August 2012.
rate. In Table A.2 we report the λ1. For the moment condition we have:∑1≤n≤13
GT (λ∗1, n)G(λ∗1, n) = 0.10
The J-statistic is 18.8, with a p-value smaller than 0.03. For this alternative set of measure-
ments we are thus unable to reject the 32 over identifying restrictions implied by the theory.
In Table A.4, we compare the factor loadings of the over-identified model with that of the
unrestricted regression. We see that the factor loadings of the over-identified model are close
to that of the unrestricted regression. In the following of this subsection we analyze the
results of impulse-response analysis, variance decomposition and out-sample test.
1.8.1.1 Impulse-response analysis
Fig. A.5 plots the contemporaneous response of yields to shocks. As in the case of Williams-
Laubach’s data, among all four factors the shocks to expected inflation and real rate have
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 35
the most significant response for longer maturity bond. And for all factors other than the
expected inflation, the response of bond having longer maturity is smaller than that having
shorter maturity. The expected inflation has a hump shape response function, which is sim-
ilar to Williams-Laubach’s data.
We plot in Fig. A.6 the responses at longer horizon. We find that for all three bonds, shock
from real neutral rate is most significant at the longer horizon. At the one-month horizon,
shock from Taylor residue is most significant for the 1Y bond. For the 5Y and 10Y bonds
the shock from expected inflation is the most significant at one month ahead horizon. These
properties are similar to that of Williams-Laubach’s data.
1.8.1.2 Variance decomposition
In Table A.5 we report the results of variance decomposition for 1Y, 5Y, and 10Y bonds.
The properties are similar to that of Williams-Laubach’s data.
For the 1Y bond at one-month ahead horizon, Taylor residue contributes most of the variance.
At longer horizons, all factors contribute to the variance. For the 5Y and the 10Y bonds
we see results having similar properties. For one-month ahead forecasting, expected inflation
contributes most of the variance. At longer horizons, the real neutral rate contributes most
of the forecasting variance.
1.8.1.3 Out-sample analysis
We report in Table A.3 the price of risk λ1 that we have obtained by calibrating the in-sample
data. In Table A.6 and Table A.7 we report the pricing statistics for the in-sample and the
out-sample periods. We see by looking at Table A.7 that all bonds are priced accurately for
out-sample period except those having maturities less than 2 years.
1.8.2 Testing model on alternative measurements: II
We now test our model on the second alternative set of measurements. In this specification,
we take the real neutral rate to be the five-year forward rate five year in the future minus the
inflation risk premium as in Eq. 1.20, and the expected inflation to be the 5-year break-even
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 36
inflation rate minus the inflation risk premium. And we model the output gap by using the
Okun’s law, as in Eq. 1.22, where we take k equal to 2.75. We do all the exercises as in the
section 1.7. And we report in Appendix A the results.
We plot in Fig. A.7 the contributions of macro factors in Taylor rule along with the Fed fund
rate. In Table A.8 we report the λ1. For the moment condition we have:
∑1≤n≤13
GT (λ∗1, n)G(λ∗1, n) = 0.02
The J-statistic is 2.9, with a p-value smaller than 0.01. We thus see that for this alternative
set of measurements we are again unable to reject the 32 over identifying restrictions implied
by the theory.
In Table A.10, we compare the factor loadings of the over-identified model with that of the
unrestricted regression. We see that the factor loadings of the over-identified model are close
to that of the unrestricted regression. In the following of this subsection we analyze the
results of impulse-response analysis, variance decomposition and out-sample test.
1.8.2.1 Impulse-response analysis
Fig. A.8 plots the contemporaneous response of yields to shocks. As in the case of Williams-
Laubach’s data, among all four factors the shocks to expected inflation and real rate have
the most significant response for longer maturity bond. And for all factors other than real
neutral rate, the response of bond having longer maturity is smaller than that having shorter
maturity.
We plot in Fig. A.9 the responses at longer horizon. We find that for 5Y and 10Y bonds,
shock from real neutral rate is most significant at the longer horizon. For 1Y bond it’s GDP
gap. At the one-month horizon, shock from Taylor residue is most significant for the 1Y
bond. For 5Y bond shock from expected inflation is the most significant at one-month hori-
zon. Most of the properties are similar to that of Williams-Laubach’s data.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 37
1.8.2.2 Variance decomposition
In Table A.11 we report the results of variance decomposition for 1Y, 5Y, and 10Y bonds.
The properties are similar to that of Williams-Laubach’s data.
For the 1Y bond at one-month ahead horizon, Taylor residue contributes most of the variance.
At longer horizons, all factors contribute to the variance. For the 5Y and the 10Y bonds
we see results having similar properties. For one-month ahead forecasting, real neutral rate
and expected inflation together contribute most of the variance. At longer horizons, the real
neutral rate contributes most of the forecasting variance.
1.8.2.3 Out-sample analysis
We report in Table A.9 the price of risk λ1 that we have obtained by calibrating the in-sample
data. In Table A.12 and Table A.13 we report the pricing statistics for the in-sample and the
out-sample periods. We see by looking at Table A.13 that all bonds are priced accurately for
out-sample period except those having maturities less than 2 years.
1.8.3 Model using efficient interest rate
Curdia et al. [2014] studies an alternative specification of monetary policy reaction function,
which the authors refer to as W rules, where the output gap doesn’t affect the reaction
function. The paper argues that W rules fit the US data better than Taylor rule. Here we
test our model on W rules, where we have three factors.
it = rnrt + 1.5(πt − 2%) + wt (1.23)
where rnrt and πt are the same as in Taylor rule, and wt is the W rule deviation. The
model setup and the econometric method are the same as in the case of the Taylor rule. In
Appendix A we report the results.
We plot in Fig. A.10 the contributions of macro factors in W rule along with the Fed fund
rate. In Table A.14 we report the λ1. For the moment condition we have:
∑1≤n≤13
GT (λ∗1, n)G(λ∗1, n) = 0.01
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 38
The J-statistic is 2.31, with a p-value smaller than 0.01. We thus see that we are unable to
reject the 27 over identifying restrictions implied by the theory.
In Table A.16,we compare the factor loadings of the over-identified model with that of the
unrestricted regression. We see that the factor loadings of the over-identified model are close
to that of the unrestricted regression. In the following of this subsection we analyze the
results of impulse-response analysis, variance decomposition and out-sample test.
1.8.3.1 Impulse-response analysis
Fig. A.11 plots the contemporaneous response of yields to shocks. Interestingly, as in the
case of Williams-Laubach’s data, among all three factors the shock to expected inflation has
the most significant response for longer maturity bond. And for real neutral rate and W rule
deviation, the response of bond having longer maturity is smaller than that having shorter
maturity. The expected inflation has a hump shape response function, which is the same as
for the four-factor model.
We plot in Fig. A.12 the responses at longer horizon. We find that for all three bonds, shock
from real neutral rate is most significant at the longer horizon. At the one-month horizon,
shock from W rule residue is most significant for the 1Y bond. For the 5Y and 10Y bonds
the shock from expected inflation is the most significant at short horizon. These properties
are similar to that of the four-factor model.
1.8.3.2 Variance decomposition
In Table A.17 we report the results of variance decomposition for 1Y, 5Y, and 10Y bonds.
The properties are similar to that of the four-factor model.
For the 1Y bond at one-month ahead horizon, Taylor residue contributes most of the variance.
At longer horizons, all factors contribute to the variance. For the 5Y and the 10Y bonds
we see results having similar properties. For one-month ahead forecasting, expected inflation
contributes most of the variance. At longer horizons, the real neutral rate contributes most
of the forecasting variance.
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 39
1.8.3.3 Out-sample analysis
We report in Table A.15 the price of risk λ1 that we have obtained by calibrating the in-
sample data. In Table A.18 and Table A.19 we report the pricing statistics for the in-sample
and the out-sample periods. We see by looking at Table A.19 that all bonds are priced
accurately for out-sample period except those having maturities less than 2 years.
1.8.3.4 Comparison with the Taylor rule factor model
From the previous analysis we conclude that the W rule factor exhibits similar properties
to the Taylor rule factor. For both models the over identifying restrictions implied by the
theory are not rejected with low p-value. They also feature similar out-sample test results.
We believe it is interesting to further investigate the empirical performance of W rule model
and compare the latter with the Taylor rule model.
1.8.4 Taylor rule under zero lower bound regime
In the subsections 1.5.5 and 1.7.4 we studied, respectively, the out-sample test for the exactly-
identified model and the over-identified model. There we saw that the out-sample perfor-
mance for bonds at short end is much worse than that of bonds at medium and long ends.
Note that our out-sample period extends from June 2008 to June 2012. And for most of the
out-sample period, from December 2008 to August 2012, Fed fixed the short term interest
rate at 0.25%. Thus we are looking at the problem of zero lower bound here. This poses a
problem to our Taylor rule factor model because under the zero lower bound regime the long-
term relation binding bond yields to macro factors may break down. A legitimate question
to raise is: does the underperformance of the out-sample test for both the exactly identified
and over-identified models come from the zero lower bound? We study this question here.
Our strategy is to calibrate the model using the whole sample data from September 1997 to
August 2012. We then look at the fitting performance of the model for two subsamples, with
the first one being from September 1997 to June 2008 and the second one being from July
2008 to August 2012. We already solved the model in the section 1.7. We report in Table 1.19
the fitting results for the second sample. The first column shows the mean of the bond yield
during the period. RMSE 1 is the root mean square error for the baseline model calibrated
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 40
using the whole sample data from 1997 to 2012. Rlt Error 1 is the RMSE 1 divided by the
Observed Mean. In Table 1.19 we also recall the results of the out-sample analysis of the
subsection 1.7.4. In Table 1.19, RMSE 2 shows the RMSE of the out-sample test, the same
RMSE as Table 1.18. Rlt Error 2 is the RMSE divided by the Observed Mean. In Table 1.19
the Out-Sample Decay is the ratio between Rlt Error 2 and Rlt Error 1.
By our definition we have the following relation:
RltErrorn2 = RltErrorn1 ×OutSampleDecayn (1.24)
Eq. 1.24 says that the out-sample test’s relative error can be factored into two components:
the relative error of the whole sample model on the same period and the out-sample decay.
Looking at Table 1.19 we see that the out-sample decay is generally decreasing as a function
of the bond’s maturity. We can thus see that the model’s robustness at long end is also better
than that at short end.
Table 1.19: Comparison of Whole-sample Fitting and Out-sample Fitting
Obs Mean RMSE 1 RMSE 2 Rlt Err 1(%) Rlt Err 2(%) OS Decay
3M 0.10 0.14 0.25 133.72 243.97 1.82
6M 0.17 0.14 0.60 84.76 361.75 4.27
1Y 0.27 0.14 0.86 53.19 320.58 6.03
2Y 0.58 0.24 0.87 41.62 149.80 3.60
3Y 0.91 0.26 0.86 28.16 94.62 3.36
4Y 1.29 0.24 0.77 18.32 60.21 3.29
5Y 1.67 0.23 0.70 13.98 42.17 3.02
6Y 2.00 0.24 0.61 12.08 30.70 2.54
7Y 2.34 0.27 0.54 11.62 23.21 2.00
8Y 2.55 0.28 0.51 11.04 19.92 1.80
9Y 2.76 0.30 0.48 10.78 17.38 1.61
10Y 2.96 0.31 0.46 10.63 15.57 1.46
Obs Mean is the mean of the observed yields. OS Decay is the out-sample decay.
We note that the whole sample model shows good fitting for the zero-lower bound period
on the medium and long ends of the curve. The good performance for the long-term bond
maybe reflects the market’s expectation that the Taylor rule will still characterize the short-
CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 41
term interest rate policy for the long term.
1.9 Conclusion
In this chapter we have interpreted the term structure of US interest rates by using observable
macro factors as inputs to a Taylor-type rule. Consistent with the theory, we allow the
observed deviation from the Taylor-type rule to be a priced factor. We have explained term
structure time variations in terms of observable and interpretable macro factors. We have thus
constructed an arbitrage-free term structure model, one in which the arbitrage opportunities
are eliminated by imposing restrictions on the yields.
Our estimation results of US data suggest that macro variables affect the term structure in
various ways. Yield curve exhibits a strong contemporaneous response to macro shocks, with
expected inflation and real neutral rate having the most persistent effect. The real neutral
rate explains most of the price variation at long horizon for all bonds. At short horizon,
expected inflation explains most of the price variation for bonds at belly and long end. At
short horizon, the Taylor rule deviation, explains most of the movements of the yield curve
at short end. Our results also suggest our estimated model is reasonably robust.
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 42
Chapter 2
Carry Trade Threshold Portfolios
2.1 Introduction
Carry trade strategies profit from violations of uncovered interest rate parity. They involve
borrowing low interest rate currency and investing it in high interest rate currency. Carry
trade is related to uncovered interest rate parity (UIP), which states that high interest rate
currency tends to depreciate relative to low interest rate currency such that the expectation
of the carry trade return is zero. A large body of literature rejects UIP, however; see, for ex-
ample, Fama [1984]. Starting with Meese and Rogoff [1983], one view has been that exchange
rates are roughly random walks. Some researchers believe that the random-walk model does
a good job of predicting exchange rates. The predictions of the out-of-sample mean squared
error of the random-walk model are about the same as the models that use fundamental data.
A recent update is Cheung et al. [2005]. More recently Engel et al. [2012] finds that the factor
model improves the forecast for the long horizon (8 and 12 quarters). Overall, the literature
suggests that the random-walk model works well for a short horizon (less than 8 quarters).
A more nuanced view of UIP is related to the forward premium anomaly, which states that
a currency is expected to appreciate when its interest rate is high, see Fama [1984], Hodrick
[1987], and Engel [1996]. Both the random-walk model literature and the forward premium
puzzle literature suggest that currency carry trade strategies are likely to be profitable.
Many papers have studied the carry trade. Empirical research has shown that excess returns
of carry trade portfolios are statistically significant for various sample periods and sets of
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 43
currencies. Different theories have been proposed to explain the carry trade excess returns.
According to Brunnermeier et al. [2008], one possible explanation for the excess return of
carry trade is compensation for crash risk. There has been an ongoing debate regarding
whether the classical asset pricing model can explain carry trade return, see Lustig and
Verdelhan [2007], Lustig et al. [2011] and Burnside et al. [2011a]. Daniel, Hodrick, and Lu
[2014] gives a recent summary on carry trade’s risks. This paper contributes to the literature
by studying the carry trade threshold portfolios. It constructs threshold portfolios by using
thresholds that depend upon a forward premium (the interest rate difference between the two
currencies). For example, a threshold portfolio of threshold value equal to 2% includes only
carry trade having a forward premium greater than 2%. We show that from 1990 to 2012
for G10 currencies, higher threshold portfolios outperformed lower ones up to the threshold
value of 2.5%. We explore the connections of the threshold effect to the currency investor’s
portfolio problem. We also study carry trade’s return predictability as a motivation for the
threshold portfolio. To explain the threshold effect, we model the latter in a random walk
currency model. The model predicts the optimal threshold value and the relative gain of the
optimal threshold portfolio. The model is calibrated and the predictions are tested. We also
show the non-existence of the threshold effect in a one-factor currency model.
Related to the recent research on the carry trade, we test the crash risk explanation (Brun-
nermeier et al. [2008]) for the carry trade threshold effect. In order to test the effect of crash
risk, we construct threshold portfolios that are hedged against that risk. Following Jurek
[2014], we construct crash-hedged threshold portfolios using out-of-money put options. We
then use the difference-in-difference method to test the crash risk effect. We first compute
the difference in performance of hedged threshold portfolios versus hedged benchmark equally
weighted portfolios, and the difference in performance of unhedged threshold portfolios ver-
sus unhedged benchmark equally weighted portfolios. We then compare the two differences
in performance. The result suggests that the crash risk premium can explain around 25
percent of the excess performance of the higher threshold portfolios, relative to benchmark
equally weighted portfolios. The remainder of the introduction section explains in detail our
approaches and main results.
Regarding the predictability of the carry trade return, Bakshi and Panayotov [2013] inves-
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 44
tigates time series predictability. Lustig et al. [2011], Ang and Chen [2010] and Menkhoff
et al. [2012] identify risk factors that affect the average carry trade return. We motivate
the study of threshold portfolio by looking at the predictability of the carry trade using a
forward premium. A simple regression shows that the loading of the carry trade return on
the forward premium is positive with significant t-statistics. Therefore, a currency investor
allocating more capital on carry trades having a larger forward premium can expect to have
a larger return.
In general, our regression shows a positive loading of carry trade returns on the forward
premiums. The next step is to evaluate the implications of this finding on the perspective of
the currency investor maximizing risk adjusted return. Our setup of the problem has been
conducted in the same spirit as that utilized by Burnside et al. [2008] and Berge et al. [2010].
Burnside et al. [2008] documents the gain achieved by diversifying the carry trade across
different currencies. Berge et al. [2010] discovered that forecasting models that use interest
rates help to improve the returns of currency portfolios. Consistent with the literature of
mean-variance analysis, see Levy and Markowitz [1979], Kroll et al. [1984] and Markowitz
[1991], we approximate the problem of maximizing the investor’s expected utility by solving
a mean-variance problem. When it comes to the literature related to currency portfolios,
Hochradl and Wagner [2010] studies optimal carry-trade portfolios using the Sharpe ratio as
its chief criterion. We study the optimal portfolio that maximizes the expected Sharpe ratio.
Our contribution is to prove that under certain second moment conditions, the investor’s
optimal portfolio always allocates more funds to trades having bigger forward premiums.
We empirically test the model’s implication that the carry trade with larger forward pre-
miums should have a larger weight in the portfolio. We form threshold portfolios, with the
threshold based on the forward premium, and compare their performance. Given the thresh-
old value, our threshold portfolio includes only those carry trades whose forward premiums
are higher than the threshold values. We find empirically that up to the threshold value of
2.5 percent, the threshold portfolio outperforms the benchmark portfolio with respect to the
Sharpe ratio. In addition, the portfolios with higher thresholds beat the portfolios with lower
thresholds. That result is confirmed through the robustness checks of resampling bootstrap
and White’s Reality Check. Recent empirical regime switch studies of the forward premium,
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 45
done by Jorda and Taylor [2012], Baillie and Chang [2011], and Christiansen et al. [2011],
also suggest that our threshold portfolios should have different performances.
Garratt and Lee [2010] evaluates the forecast performance of models based upon their eco-
nomic value in the decision making setting of an investor. Model A works better than Model
B, if a portfolio constructed with Model A has a better financial performance than the one
constructed with Model B. In that framework, a test of the threshold model proves that a
model that posits a positive loading of a carry trade return on forward premiums works better
than a model that posits neutral loading. Our threshold approach also allows us to formulate
another portfolio construction rule. Recent studies Lustig et al. [2011] and Menkhoff et al.
[2012] construct portfolios based on forward discounts. Their approaches are similar and yet
different. Menkhoff et al. [2012] constructs portfolios based on the relative order of forward
premium. In their five portfolio example, the first portfolio includes the quintile having the
highest forward premiums. We compare the performance of threshold models based on hard
threshold and relative order. We find that for the sample period 1990-2012, the hard thresh-
old rule led to better financial performance among the G10 currencies.
One recent explanation of carry trade excess return is the crash risk premium. Jurek [2014]
shows that carry trade returns don’t come primarily from the peso problem. This is done
by constructing crash-hedged carry trade portfolios. In the same spirit as Jurek, this paper
tests the crash risk explanation for the superior performance of the high forward premium
carry trade relative to that of the low forward premium. It constructs crash-hedged threshold
portfolios using out-of-money put options. The paper finds persistence of excess return of
hedged higher threshold portfolios, relative to hedged lower threshold portfolios as in the
unhedged case. It compares the relative difference in performance between higher threshold
portfolios and an equally weighted portfolio in hedged and unhedged cases. It shows that
the crash risk premium can explain approximately 25 percent of the excess performance of
higher threshold portfolios, relative to the benchmark equally weighted portfolio.
The paper also finds that the Sharpe ratio of threshold portfolio initially increases, as the
threshold value goes up until it reaches its optimal value. The Sharpe ratio then decreases.
The paper models this threshold effect in a random walk currency model, in keeping with
Meese and Rogoff [1983] and Engel and West [2004]. Using general assumptions on the
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 46
volatilities of the currencies and the distribution of the forward premiums, the model pre-
dicts the threshold value most likely to maximize the Sharpe ratio. The model also predicts
the relative gain in the Sharpe ratio of the threshold portfolio as compared to the equally
weighted portfolio. We calibrate the model to G10 currency trades, using data running from
January 1990 to June 2012. We then test the model’s predictions.
Finally the paper extends the random-walk model to a one-factor model where currency’s
innovation has loading on global factor, like in Lustig and Verdelhan [2007] and Jurek and
Xu [2013]. The one-factor model exhibits heterogeneous correlations between currency pairs
and increasing downside correlations of such pairs as in Ang and Bekaert [2002]. The pa-
per analytically solves the one-factor model, and compares the result to that gained via the
random-walk model. It finds that the optimal threshold value doesn’t exist if the currency
returns are correlated and if idiosyncratic currency risks are perfectly diversified across all
threshold portfolios. The nonexistence of optimal threshold value in the one-factor model
tells us that the quasi-independence of currencies movement is a factor that must be taken
into account if we wish to understand the threshold portfolio effect for currency carry trades.
The rest of the paper is organized as follows: The second section studies the prediction power
of the forward premium with respect to the carry trade returns. The third section solves the
currency investor’s mean-variance problem. The fourth section explains the threshold model
and subjects it to empirical tests. The fifth section presents the results of our examination
of the crash-hedged threshold portfolios. The sixth section solves the random-walk model,
with our solution of the one-factor model coming in the seventh section. The eighth section
concludes.
2.2 Carry Trade
This paper studies the portfolio problem of the carry trade investor. It shows a novel stylized
effect, related to the carry trade threshold portfolios. In this section, we introduce the basic
notions of carry trade.
Carry trade tries to profit from the empirical deviation of an exchange rate away from UIP
(uncovered interest rate parity). A carry trade comprises a short position in a low interest
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 47
rate currency, and an equivalent long position in a high interest rate currency. Thus, the log
return of a carry trade is
et+1 = it − i∗t − (st+1 − st) (2.1)
where it denotes the log interest rate of the high yield country; i∗t that of the low yield coun-
try; st is the log exchange rate.
Carry trade is related to uncovered interest rate parity (UIP), which states that high interest
rate currency tends to depreciate relative to low interest rate currency such that the expec-
tation of the carry trade return is zero. A large body of literature rejects UIP, however; see,
for example, Fama [1984]. Starting with Meese and Rogoff [1983], one view has been that
exchange rates are roughly random walks. Some individuals believe that the random-walk
model does a good job of predicting exchange rates. The predictions of the out-of-sample
mean squared error of the random-walk model are about the same as the models that use
fundamental data. A recent update is Cheung et al. [2005]. More recently Engel et al. [2012]
finds that the factor model improves the forecast for the long horizon (8 and 12 quarters).
Overall, the literature suggests that the random-walk model works well for a short horizon
(less than 8 quarters). A more nuanced view of UIP is related to the forward premium
anomaly, which states that a currency is expected to appreciate when its interest rate is high,
see Fama [1984], Hodrick [1987], and Engel [1996]. Both the random-walk model literature
and the forward premium puzzle literature suggest that currency carry trade strategies are
likely to be profitable.
We implement the carry trades of the G10 currencies1 versus the US dollar. The sample
period is from January 1990 to June 2012. The carry trades are rebalanced monthly. Table
B.1 details the performance statistics of the carry trades, and also shows the performance
statistics of the equally weighted carry trade portfolio2. In Fig. B.1, we plot the cumulative
return of the equally weighted portfolio. The equally weighted portfolio is our benchmark
portfolio when we study the threshold portfolios.
1These are the G10 currencies: US dollar, Euro, Japanese yen, Canadian dollar, Swiss franc, British pound,
Australian dollar, New Zealand dollar, Norwegian krone, and Swedish krona.
2The equally weighted carry trade portfolio includes nine carry trade of G10 currencies versus USD. It
gives equal allocation to the nine carry trades.
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 48
Suppose that a carry trade investor wants to improve the portfolio returns of his carry trades.
What variable should he look at when constructing the portfolio? An intuitive answer is the
forward premium (the interest rate difference), because both the random-walk model of ex-
change rates and the forward premium puzzle suggest that the carry trade return should be
positively related to the interest rate difference at trade initiation. We now test the prediction
power of the interest rate difference with respect to realized carry trade return. Retaining
the notation of Eq. 2.1, we run the following regression:
et+1 = β(it − i∗t ) + εt+1 (2.2)
Doing so brings us this estimation result:
β
Estimation 0.075
Std (0.045*)
N (2430)
* denotes p-value at 10% significance level.
The positive loading of the carry trade return on the yield difference tells us that a currency
investor allocating more capital to carry trades of higher forward premium can expect to see
a higher return. It is this link between forward premium and realized carry trade return
that motivates us to study the threshold portfolios in Section 2.4. Our result also provides
evidence that interest rate term premiums are related to foreign exchange risk premiums, in
keeping with Campbell and Clarida [1987], Clarida and Taylor [1997] and Clarida et al. [2003].
2.3 Currency investor’s portfolio problem
Our simple regression in the last section has revealed that forward premiums predict carry
trade returns. Thus, an investor whose objective is to maximize the expected return would
be wise to allocate more capital to carry trades having higher forward premiums. Does this
maxim also hold true for investors who are maximizing a risk adjusted return? We explore
this question within this section. We also explore the implications of the fact that the forward
premium predicts carry trade returns as they are seen from the perspective of the individual
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 49
currency investor. With regard to the portfolio optimization of currency investors, Burnside
et al. [2008] documents the gain that is achieved by diversifying the carry trade across different
currencies. Berge et al. [2010] discovers that forecasting models by using interest rates helps
to augment the returns of currency portfolios. Here we seek to maximize the expected utility
gained by the currency investor by restricting the investor’s portfolio to carry trades. The
investor, therefore, has to determine the allocation weights of carry trades in his portfolio.
2.3.1 Mean-variance problem
Following the literature on mean-variance analysis, see Levy and Markowitz [1979], Kroll
et al. [1984] and Markowitz [1991], we approximate the problem of maximizing the investor’s
expected utility by solving a mean-variance problem. Levy and Markowitz [1979] and Kroll
et al. [1984] showed that for various utility functions and empirical return distributions, the
expected utility maximizing portfolio can be well approximated by constructing an mean-
variance efficient portfolio3.
We first set up the standard mean-variance framework for the carry trade setting. Suppose
the investment universe of carry trades includes N carry trades with expected return RCTi
for carry trade i. The covariance of carry trade retun RCTi and RCTj is Σi,j. The investor’s
mean-variance problem is stated as following
Problem 1 Given the expected return of N carry trades RCT and the covariance of expected
returns Σ, the investor chooses the portfolio weights ω to solve the following problem:
max : ωTRCT
s.t.√ωTΣω = V
where V is the investor’s tolerance of return risk.
The next proposition shows that the solution to the mean-variance problem has a nice struc-
ture.
3In the original model of Levy and Markowitz [1979] short sale is not allowed. Here we relax this constraint
in our currency investment setting, because currency investment by its very nature involves taking a long
position on one currency and a short position on another.
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 50
Proposition 2 Regardless of the investors’ risk tolerances V , the investors’ positions on two
carry trades have the same ratio. The investors adjust their total exposure to match their risk
preference. The optimal portfolio maximizes the Sharpe ratio(risk adjusted return) ωTRCT√ωTΣω
.
Proof. Suppose ω∗ is the portfolio that solves Problem 1 for V = 1, then for general V the
solution of Problem 1 is V ω∗. Note that ω∗ also maximizes the Sharpe ratio ωTRCT√ωTΣω
.
Hochradl and Wagner [2010] also seeks to achieve the optimal carry trade portfolio using
the Sharpe ratio as its criterion. We now give the solution to the mean-variance optimal
portfolio.
Lemma 1 Assuming a universe of N carry trades, with expected return RCT = [RCTi ]T and
covariance matrix Σ of expected return, the portfolio that solves the problem 1 has weight
ω = λΣ−1RCT, where ω is such that√ωTΣω = V .
Proof. Our problem can be stated thus:
max :ωTRCT
√ωTΣω
, s.t.√ωTΣω = V (2.3)
Problem 2.3 is equivalent to this one:
max : ωTRCT, s.t.ωTΣω = V 2 (2.4)
The associated Lagrangian of Problem 2.4 is:
L = ωTRCT − λ(ωTΣω − V 2)
The first order conditions give us
ω = RCT ∗ (λΣ)−1
ωTΣω = V 2
Thus we have proved the theorem.
Lemma 1 gives the solution of the mean-variance optimal portfolio. Let us now take a step
back to think our original question: in the mean-variance optimal portfolio, do we place larger
weight on carry trade having larger forward premium? In the same spirit of the G10/USD
carry trades, let’s consider a setting where we implement carry trades of foreign currencies
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 51
versus domestic currency. Then the question can be stated as: in the mean-variance optimal
portfolio, do we have ωi > ωj for |rfi − rd| > |rfj − rd|? We study in the next subsection this
question.
2.3.2 Model
Lemma 1 gives the solution of the optimal mean-variance portfolio ω = λΣ−1RCT, do we
have ωi > ωj for |rfi − rd| > |rfj − rd|? In order to compare the weights of two carry trades,
we need to model the first and the second order moments of the expected returns. Before
going into more general discussion, we first state several simple cases. Let’s first suppose that
the carry trade of foreign currency i versus domestic currency has expected return |rfi − rd|,
this property is satisfied under the random walk model of foreign exchange rates. We also
assume that the currencies are not correlated between them and all currencies have equal
volatilities. Thus the covariance matrix of ω is the identity matrix multiplied by a constant,
σI. We thus have that the weight of carry trade k is ωk = λ|rfk−rd|
σ . It then follows ωi > ωj
if |rfi − rd| > |rfj − rd|.
Let’s consider another case where all the previous assumptions are maintained except that
the volatility of currency i is proportional to the forward premium |rfi − rd|: σi = k|rfi − rd|.
Then by simple calculation we have ωi = ωj = λk . Thus all carry trades have the equal
weights in the optimal mean-variance portfolio regardless of the forward premiums. This
simple example shows that in general we need restrictions to have the property ωi > ωj given
rfi > rfj > rd. We now construct our model.
Carry trade return has two parts: et = it − i∗t − (st+1 − st). In line with the literatures
on the random-walk model of exchange rates and the forward premium puzzle, we model
E t[et] = β · (it − i∗t ), where β = 1 for the random-walk model and β > 1 if we assume that
high yield currency tends to appreciate. Thus RCT = [β · |rfi − rd|]Ti , where |rfi − rd| is the
forward premium of the ith carry trade. For the second moment of carry trade returns, we
make the following assumptions: volatility of the carry trade return is the same across all
currencies; covariance of two carry trades’ returns is the same across all currency pairs. In
Fig. B.2 we plot the historical volatilities of G10/USD currencies. We can see that apart
from Canadian dollar other currencies have roughly the same volatility levels. Under these
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 52
assumptions, let J be the n by n matrix which has 1 for each of its elements; let I be the
identity matrix, the covariance matrix Σ can be written as aI + bJ. We also assume that
the carry trades are not perfectly correlated, and thus a > 0. For example, the covariance
matrix Σ of RCT for three carry trades has the following form:a+ b b b
b a+ b b
b b a+ b
Now we give a lemma, which will be used in the following theorem:
Lemma 2 If a 6= −nb, then the inverse of aI + bJ is cI + dJ, where c = 1a and d = −b
a(a+nb) .
Proof. We have that J2 = nJ. Thus (aI + bJ)(cI + dJ) = acI + (ad+ bc+ nbd)J. Thus for
c = 1a and d = −b
a(a+nb) we find that (aI + bJ)(cI + dJ) = I.
Now we can give our main result:
Proposition 3 In the optimal Sharpe ratio portfolio, the weight of a higher forward premium
carry trade is higher than that of a lower forward premium carry trade.
Proof. Note that ω = Σ−1RCT. We need to prove that ωj > ωk whenever RCTj > RCT
k.
From the previous lemma we know that Σ−1 = cI + dJ, where c > 0. Thus, ωj = cRCTj +
d∑N
h=1 RCTh, which in turn means that ωj > ωk if RCT
j > RCTk.
Our result implies that a portfolio including carry trades of higher forward premia will have
a higher expected Sharpe ratio than a portfolio containing trades of lower forward premia.
This motivates our study of threshold portfolios in the next section. Lustig et al. [2011] and
Menkhoff et al. [2012] construct portfolios based on forward discounts, and they document
similar results. Menkhoff et al. [2012] constructs portfolios based on relative order of forward
premia. In their five-portfolio example the first portfolio includes the quintile of highest
forward premium, the fifth the quintile of lowest forward premium. The first portfolio has a
higher Sharpe ratio than the fifth.
2.3.3 Backtesting
In this subsection we backtest the historical performance of mean-variance optimal portfolio.
We set the expected return as the forward premium. Because the return from interest rate
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 53
difference is risk-free, the volatility of carry trade return is due entirely to the FX position.
Thus we approximate the covariance of two carry trade returns by the historical covariance
of the two exchange rates. Lemma 1 however solves the Sharpe ratio optimization problem
without imposing any restrictions on the weights of carry trades. An investor may not want
to short carry trades in his portfolio. Thus, we consider a long only version of the portfolio
optimization problem, doing so by adding an additional positivity constraint. We define the
long only carry trade mean-variance optimal portfolio as follows:
Definition 1 Using the notation introduced in Lemma 1, the long only carry trade mean-
variance optimal portfolio has the allocation weights that solve the following problem:
max :ωTRCT
√ωTΣω
s.t.
P∑j=1
ωj = 1, ωi ≥ 0,∀i
The positivity constraint deprives the problem of any analytical formulation of its solution.
Numerically, it is a quadratic programming problem and can be solved quickly.
We report two tests with different window lengths when estimating the currency correlations.
In the first test, we use a one-year fixed window length. In the second, we use two-year window
length. The results are the rows of Carry 1Y and Carry 2Y seen in Table B.3. The cumulative
return indices are plotted in Fig. B.5. We see that both mean-variance portfolios have better
Sharpe ratio compared to the benchmark equally weighted portfolio. But the improvements
are not significant. After discussing the robustness of the result in the next subsection, we
will return to discuss the implications of the results.
2.3.4 Robustness check
The literature takes two different approaches to tackle the problem of data snooping bias.
The first approach focuses on data, and tries to avoid reusing the same data set. Testing
a model by using a different but comparable data set may do this. In our case, however,
no such alternative dataset is available. Thus we have used, instead two methods based on
bootstrap: (1) Resampling with bootstrap, see Efron [1979]; (2) White’s Reality Check, see
White [2000] and Politis and Romano [1994]. Here we test equal performance of the long
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 54
only mean-variance optimal portfolio against the benchmark portfolio. Our criterion is the
Sharpe ratio. We report the result of Carry 2Y below.
P-value of bootstrap resampling test
Carry 2Y
Benchmark 0.35
P-value of White’s Reality Check resampling test
Carry 2Y
Benchmark 0.40
In neither tests, mean-variance portfolio outperforms sigfinicantly equally weighted portfo-
lio. One concern regarding the mean-variance approach is the estimation error. DeMiguel
et al. [2009] studies this issue. They conclude that ”out of sample, the gain from optimal
diversification is more than offset by estimation error”. In their simple simulation example
of 10 assets, one needs an estimation of 3,000 months in order for the mean-variance portfolio
to outperform the naive equally weighted portfolio. We propose threshold portfolios in the
next section, by drawing on the insights of the mean-variance approach and also retaining
the robustness of the naive equally weighted portfolios.
2.4 Threshold portfolios
A forward premium threshold portfolio includes only carry trades having forward premiums
higher than the threshold value. We now have two motivations for studying threshold port-
folios. In Section 2.2, we saw that a currency investor allocating more capital to carry trades
that have larger forward premiums can expect to have a higher return. In Section 2.3, in a
mean-variance setting, an investor seeking to maximize the risk adjusted return also allocates
a larger weight to a carry trade having a larger forward premium. In addition, recent em-
pirical regime switch studies about the forward premium— Jorda and Taylor [2012], Baillie
and Chang [2011], and Christiansen et al. [2011]— suggest that threshold portfolios based on
forward premiums should have different performances depending on the threshold values.
An equally weighted threshold portfolio gives equal weights to the carry trades in the port-
folio. Compared to the mean-variance optimal portfolio, the equally weighted threshold
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 55
portfolio keeps the essential property that carry trades with a larger forward premium have
larger weights. In this case, those carry trades with forward premiums below the threshold
value have weights of zero. The carry trades with forward premiums beyond the threshold
value have positive weights. In contrast to the mean-variance optimal portfolio, the equally
weighted threshold portfolio gives equal weights to all the trades in the portfolio. The equal
diversification among the portfolio is simple, yet robust, as we will see. In the next subsection,
we test threshold portfolios based on the forward premium.
2.4.1 Threshold portfolios based on forward premium
Unlike the the naive portfolio, a portfolio having a forward premium threshold changes its
composition over time. In the same spirit of the G10/USD carry trades setting, we assume
implementing carry trades of foreign currencies versus domestic currency. For a threshold
value γ, the equally weighted threshold portfolio with threshold γ at time period t includes all
carry trades of foreign currency i versus domestic currency such that |rfi − rd| > γ at t. The
intuition driving a threshold portfolio is that the investor wants to collect the maximum yield
given the constraint of capital. He wants to hold more carry trades having larger forward
premiums.
We first test our threshold portfolios for the universe of G10 currencies versus the US dollar.
Table B.2 details the performance statistics of those threshold portfolios. Figure B.3 plots
the historical cumulative returns of the threshold portfolios. We test 13 thresholds, ranging
from 0 to 6 with a step size of 0.5.
The portfolio’s performance improves as the threshold goes up, to 2.5. At that point, the
performance begins to deteriorate. This isn’t surprising if we look at the average number of
trades in the portfolio. When the threshold becomes too large, there are hardly any trades
in the portfolio.
Based on the empirical test, we have identified two threshold effects: 1) Up to the opti-
mal threshold value, the threshold portfolio outperforms the benchmark portfolio (without
threshold) in terms of risk adjusted return; 2) Up to the optimal threshold value, the portfolio
with the higher threshold value outperforms that of the lower threshold value.
Garratt and Lee [2010] evaluates the forecast performance of models based on their eco-
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 56
nomic value within the decision making setting of an investor. In their framework, Model
A works better than Model B if the portfolio constructed with Model A has better financial
performance than the portfolio constructed with Model B. Working within that framework,
we tested the threshold model and proved that a model that posits positive loading of the
carry trade return on the forward premium works better than one that posits neutral loading.
The reason is that threshold models beat benchmark equally weighted portfolios in terms of
financial performance.
2.4.2 Robustness check
A common concern vis-a-vis the threshold model is the data snooping issue. In this subsection,
we check the robustness of our previous results. Similar to the robustness check done in
Section 2.3, we use two methods: resampling with a bootstrap (Efron [1979]) and White’s
Reality Check (White [2000]). We test the two threshold effects. For the first threshold
effect test, we test up to the optimal threshold value if a threshold portfolio has a better
Sharpe ratio than does the benchmark equally weighted portfolio. For the second threshold
effect test, we test up to the optimal threshold value if a threshold portfolio having a higher
threshold has a better Sharpe ratio than does a threshold portfolio having a lower threshold.
Our H0 hypothesis is that a threshold portfolio with a threshold value of γj has an equal or
lower Sharpe ratio than does the benchmark portfolio and the threshold portfolio having a
threshold value γk such that γj > γk. First, we report the results of our resampling bootstrap
test.
P-value of bootstrap resampling test
TR 0.5 TR 1 TR 1.5 TR 2 TR 2.5
benchmark 0.17 0.11 0.06∗ 0.03∗∗ 0.01∗∗∗
TR 0.5 0.31 0.13 0.08∗ 0.03∗∗
TR 1 0.15 0.12 0.03∗∗
TR 1.5 0.22 0.05∗∗
TR 2 0.11
(∗) stands for p-value below 0.1, (∗∗) for p-value below 0.05, (∗ ∗ ∗) for p-value below 0.01
The resampling bootstrap statistics support our contention that up to the optimal threshold
value, the threshold portfolio outperforms the benchmark equally weighted portfolio. The
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 57
p-values of testing threshold portfolios of threshold values 1.5%, 2%, and 2.5% having the
same performances as the benchmark portfolio are respectively: 0.06, 0.03, and 0.01. This
finding also implies that up to the optimal threshold value, a higher threshold portfolio beats
a lower one. The p-value of testing 2.5% threshold portfolio having equal performance as
0.5%, 1%, and 1.5% threshold portfolios are 0.03, 0.03, and 0.05.
We report next the result of White’s Reality Check. It shows that the interpretations of
p-values are similar to those gained by resampling bootstrap. Namely, up to the optimal
threshold value threshold portfolio outperforms the benchmark portfolio, and higher threshold
portfolios beat the lower ones.
P-value of White’s Reality Check
TR 0.5 TR 1 TR 1.5 TR 2 TR 2.5
benchmark 0.15 0.11 0.03∗∗ 0.03∗∗ 0∗∗∗
TR 0.5 0.31 0.11 0.06∗ 0.02∗∗
TR 1 0.16 0.10∗ 0.03∗∗
TR 1.5 0.22 0.04∗∗
TR 2 0.08∗
(∗) stands for p-value below 0.1, (∗∗) for p-value below 0.05, (∗ ∗ ∗) for p-value below 0.01
2.4.3 Comparison with a quantile based sorting rule
Recent studies — Lustig et al. [2011] and Menkhoff et al. [2012] — construct portfolios based
on forward discounts. Their approach is similar but different. Menkhoff et al. [2012] con-
structs portfolios based on relative order of forward premium. In their five portfolios example,
the first portfolio includes the quintile of highest forward premiums. We note that the objec-
tive of their portfolio construction is not to improve the financial performance but rather to
capture the risk factor of carry trade. It is interesting however, to compare the two methods
of portfolio construction. We name our sorting rule a “value based threshold rule”, while
the rule of Menkhoff et al. [2012] a “quantile based threshold rule”. We then compute the
performances of the constructed portfolios using the “quantile based threshold rule”. The
results are reported in Table B.4. Comparing Table B.2 to Table B.4, we see that for two
portfolios containing roughly the same number of trades, the one constructed on the basis of
a “value based threshold rule” outperforms the one constructed on the basis of a “quantile
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 58
based threshold rule”.
2.5 Crash-hedged threshold portfolios
One recent explanation of carry trade excess return relates to crash risk premium, see Brun-
nermeier et al. [2008], and Farhi and Gabaix [2008]. Following Brunnermeier et al. [2008],
we propose a crash-risk explanation for the threshold effects. We then test the crash-risk
explanation following Jurek [2014].
Brunnermeier et al. [2008] showed that FX crash risk increases with the interest rate differ-
ential and the past FX carry trade returns. The paper argues that carry trade is exposed
to crash risk, which limits arbitrage. In the same spirit, we propose a crash-risk explanation
for the threshold effects, which is that the carry trade of larger forward premium is more
exposed to the crash risk. Therefore, in normal times, the carry trade of a larger forward
premium earns a larger return as a compensation for its bigger loss at crash time. Therefore,
up to the optimal threshold value, a portfolio of higher threshold value outperforms that of
a lower threshold value during normal periods.
Jurek [2014] tests the crash risk explanation for carry trade excess return by constructing a
crash hedged carry trade portfolio. By adding an option overlay, the carry trade’s exposure
to exchange rate movements is eliminated beyond the strike price. If the carry trade’s ex-
cess return comes from the crash-risk exposure, then the crash-hedged carry trade portfolio
shouldn’t exhibit an excess return. By examining the return profile of crash-hedged the carry
trade portfolio, Jurek [2014] concludes that crash-risk can explain only part of the carry
trade excess return. In the same spirit as Jurek, we test the crash risk explanation for the
threshold effects by constructing crash-hedged threshold portfolios. We add an option overlay
to carry trades in the threshold portfolio such that the carry trade’s exposure to exchange
rate movements is eliminated beyond the strike price. The crash risk premium explanation
states that the excess return of higher threshold portfolios comes from the crash risk pre-
mium. If this is true, then crash-hedged higher threshold portfolios should perform just as
well as the crash-hedged portfolios at the lower threshold. We use the difference-in-difference
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 59
method to test the crash risk effect. First, we compute the difference in the performance of
hedged threshold portfolios versus hedged benchmark equally weighted portfolios, then the
difference in performance of unhedged threshold portfolios versus the unhedged benchmark
equally weighted portfolios. Finally, we compare the two differences in performance.
2.5.1 Data
We use price data on FX options covering G10/USD pairs from January 1999 to June 2012.
The FX option conventions are explained in Section 3 of Jurek [2014]. We use the most
out-of-money put options. In other words, we use the 10δ put options to hedge the crash
risk.
If we let Sf (T ) denote the price of one unit of domestic currency in terms of foreign currency
f , and let Pt denote the price of the out-money put option, then the pay-off of the option
overlaid portfolio is:
ef (t) = erft τ
Sf (t)
Sf (t+ τ)− erdt τ (1 +
erft τPtSf (t)
) + erft τ ·max(
Sf (t)
X−
Sf (t)
Sf (t+ τ), 0) (2.5)
2.5.2 Performance of crash-hedged threshold portfolios
This subsection reports the results of hedged threshold portfolios and compares them to those
of unhedged threshold portfolios. Table B.5 shows the results of hedged threshold portfolios
for the period between January 1999 and June 2012. Table B.6 shows the unhedged results
for the same period. Figures B.6 and B.7 plot the total return indices for hedged threshold
portfolios for the period January 1999 to June 2012. Figures B.8 and B.9 plot the total return
indices for unhedged threshold portfolios for the same period.
We next compare the performances of hedged versus unhedged portfolios. Table B.7 re-
ports summary statistics comparing the return, volatility, and Sharpe ratio of unhedged and
hedged portfolios for the period between January 1999 and June 2012. Hedged threshold
portfolios, in general, have lower returns and lower volatilities compared to unhedged ones,
with the change in return being more pronounced than that in volatility. This is shown by
the lower Sharpe ratio of hedged portfolios compared to unhedged ones. Figure B.10 plots
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 60
the unhedged and hedged portfolios of threshold values 0 and 2.5 taken together.
The next step is to see if the performance difference between unhedged threshold portfolios
and unhedged benchmark portfolios persists for hedged portfolios. We report our results in
Table B.8. Hedged Sharpe Diff shows the difference in the Sharpe ratio of the hedged thresh-
old portfolios versus the hedged equally weighted portfolios. Unhedged Sharpe Diff shows
the difference in the Sharpe ratio of the unhedged threshold portfolios versus the unhedged
equally weighted portfolios. Relative change is the relative percentage difference between
Hedged Sharpe Diff and Unhedged Sharpe Diff. We see that the difference in the Sharpe
ratio decreases by around 25% from unhedged to hedged portfolios.
We find here, like in the unhedged case, the persistence of an excess return of higher threshold
portfolios, relative to hedged lower threshold portfolios. This result suggests that the crash
risk premium doesn’t account for all of the outperformance of higher forward premium carry
trades relative to lower forward premium carry trades. The crash risk premium can explain
around 25% of the excess performance of higher threshold portfolios relative to benchmark
equally weighted portfolios.
2.6 Random-walk model of threshold effects
Section 2.4 finds two threshold effects: 1) Up to the optimal threshold value, the threshold
portfolio outperforms the benchmark portfolio (without threshold) in terms of the risk ad-
justed return. 2) Up to the optimal threshold value, the portfolio of the higher threshold
value outperforms that of the lower threshold value. In this section, we propose a model
which exhibits the two threshold effects. In our model, currencies have only idiosyncratic
shocks. Using a parametric probability distribution as an approximation of the distribution
of the forward premiums, the model predicts the best threshold value which is the one that
maximizes the Sharpe ratio. The model also predicts the relative gain in the Sharpe ratio
of the threshold portfolios as compared to the equally weighted portfolio. We calibrate the
model to G10 currency trades, using data ranging from January 1990 to June 2012.
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 61
2.6.1 Random-walk model
We denote as Si(t) the exchange rate of foreign currency i at time t. In our model, each
currency has only idiosyncratic risk and equal volatility. Thus,
dSi(t)
Si(t)= σ∗i dWi(t)
Meese and Rogoff [1983] notes that the random-walk model works well for exchange rates.
Engel and West [2004] finds that, under certain conditions, the exchange rate under the
rational expectation present value model, behaves just like random walk. We denote as ri(t)
the short term interest rate of the foreign currency i at time t, and rd(t) the short term interest
rate of the domestic country. We denote as carryi the carry trade of domestic currency and
foreign currency i. According to the random-walk model of currency rates, the expected
return of carryi from time t to time t + 1 is |rd(t) − ri(t)|. In order to model the threshold
effect, we assume that the forward premiums follow a parametric distribution. We also fit
the forward premium by an exponential distribution which has PDF λe−λx.
We next compute the average return and volatility of threshold portfolios. A threshold
portfolio of parameter y includes all those carry trades whose forward premia are greater
than y. We denote that portfolio as Ptfy. We also assume that Ptfy allocates equally to all
the carry trades in the portfolio. We can compute the expected return of Ptfy as
E [rt+1(Ptfy)] =
∫∞y e−λxxλ dx.
e−λy
Let σ20 denote the variance of return for the portfolio having a threshold zero. Now we prove
that the variance of return for the portfolio having threshold y is eλyσ20. Indeed, the mass
of currency having forward premiums higher than y is∫∞y λe−λxdx. = e−λy. The mass of
currency in the portfolio having threshold zero is 1. Both portfolios give equal allocation
to their carry trades. Given that each currency has the same volatility, the ratio of the
variances of the two portfolios should be the inverse of the ratio of the masses of currencies
in the portfolio. Thus we have
Var [rt+1(Ptfy)]
Var [rt+1(Ptf0)]=
1
e−λy= eλy
The volatility is
Vol [rt+1(Ptfy)] = σ0e12λy
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 62
And therefore the Sharpe ratio is
Sh[rt+1(Ptfy)] =E [rt+1(Ptfy)]
Vol [rt+1(Ptfy)]=
∫∞y e−λxxλdx.
σ0e− 1
2λy
Sh[rt+1(Ptfy)] is maximized at
y =1
λ(2.6)
For y = 1λ , the Sharpe ratio of Ptfy is 2
e1/2λσ0. Note that the Sharpe ratio of Ptf0 is 1
λσ0.
Thus,Sh[rt+1(Ptf1/λ)]
Sh[rt+1(Ptf0)]= 2e−1/2 = 1.21 (2.7)
Our model makes two key predictions for threshold portfolios:
1. Given λ, the optimal threshold that maximizes the Sharpe ratio is 1λ .
2. The threshold portfolio with threshold value y has Sharpe ratio with the value being the
Sharpe ratio of threshold zero portfolio multiplied byλ2
∫∞y e−λxxdx.
e−12λy
.
2.6.2 Possible variations of the random-walk models
The random-walk model makes two assumptions.
1. The model assumes the currency movement to be random walk. Other studies have found
that currency has momentum and that currency returns are dependent, see Asness et al.
[2013].
2. The currency has only idiosyncratic shocks in the model. Several studies have found that
currencies have loadings on global shock, see Lustig et al. [2011].
Both of those assumptions can be relaxed, as we shall see in Section 2.7 on the one-factor
model.
2.6.3 Calibration
We calibrate the model to G10 currency carry trade using data between January 1990 and
June 2012. In Section 2.4 we computed the financial performances of threshold portfolios,
with those statistics being reported in Table B.2.
We fit the forward premium distribution by exponential distribution. By the maximum
likelihood method, we obtain λ∗ = 0.45. We plot in Fig. B.11 the CDF of both the empirical
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 63
distribution and the fitted exponential distribution. The model predicts that the optimal
threshold portfolio will have a threshold value of 1λ∗ = 2.23. In Table B.2, we saw that
the optimal portfolio is that having threshold values of 2.5. Our model, therefore, makes a
reasonable prediction.
When it comes to another prediction, in Fig. B.12 we plot the Sharpe ratios of threshold
portfolios versus the Sharpe ratio implied by the model where λ = 0.45. We see that the
model works well up to a threshold value of 4%, over which a large deviation begins to appear.
This may be due to the relatively small size of the investment universe, which implies a sizable
variance in test results.
2.6.4 Model with crash-hedged portfolios
In this subsection we test the random-walk model on crash-hedged portfolios. Our motivation
is the same as in Section 2.5. Given the domestic and foreign interest rates rdt and rit, Si(t)
denotes the price in domestic currency of one unit of foreign currency i. This means that the
price of a put option, having respectively strike price X and volatility σt(X, i), is :
Pt(Si(t), X, τ, rdt , rit) = e−rdt τ [Fi(t, τ)N(d1)−XN(d2)]
where Fi(t, τ) = Si(t)e(rdt−rit)τ is the forward rate for currency with maturity τ . And
d1 =ln(Fi(t, τ))/X
σt(X, i)√τ
+1
2σt(X, i)
√τ
d2 = d1 − σt(X, i)√τ
The option-overlaid portfolio’s payoff was given in Eq. 2.5. Because of the non-linear pay-off
of the option, we can’t solve analytically the optimal threshold value. Therefore, we study
the threshold effects by doing numerical simulations. We assume that the investor wants to
hedge the downside currency risk beyond 10% loss. Thus, investors buy put options having
strike 0.9Fi(t, τ) so as to hedge their currency crash risk. We simulate two cases of volatility
specifications. In the first case, the implied volatility is constant for all currency pairs. In
the second case, the implied volatility grows as the forward premiums do.
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 64
2.6.4.1 Equal implied volatilities for currency options
We assume that the implied volatilities for currency options are the same. Based on empirical
evidence as to the option premium, we assume a volatility premium of implied volatility
compared to historical volatility. We then do a numerical simulation in order to test the
threshold effect. We simulate 1000 carry trades occurring in 500 periods. In each period, the
funding currency interest rate has a uniform distribution on [0, 2%]. The forward premium
has an exponential distribution of mean 2%. We test three cases of currency volatility: 10%,
15% and 20%. We test two cases of implied volatility premium: 5% and 15%. If, for example,
the currency volatility is 10% and the implied volatility premium is 15%, then the implied
volatility is 11.5%. We plot the simulation results in Fig. B.13 and see that the optimal
threshold values fall between 2% and 3%. The Sharpe ratio rises as the threshold value
rises from 0 to between 2% and 3%, after which it declines. What we find is similar to the
result gained in Section 2.5: when currency movement follows random walk, the crash-hedged
threshold portfolios have profiles similar to the non-hedged portfolios.
2.6.4.2 Variable implied volatilities for currency options
We now do another simulation, with all the other assumptions, aside from those relating to
implied volatilities, being the same as those discussed in the previous subsection. We assume
that the option-implied volatility rises along with the forward premium.
σ∗i = (1 + premium) ∗ σ + 0.2 ∗ (1− e−fpi/fp)σ
where σ∗i is the implied volatility for currency i, σ is the historic volatility, and fp is the
mean of the forward premiums. We plot the simulation result in Fig. B.14 and see that the
optimal threshold value in the range of 2% to 3%. The Sharpe ratio rises as the threshold
value rises, from 0 to between 2% and 3%, after which the Sharpe ratio declines. Thus, the
monotonicity property is similar to the case of equal implied volatility that we looked at in
the subsection 2.6.4.1.
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 65
2.6.5 Conclusion
Both simulation cases feature two threshold effects: 1) Up to the optimal threshold value,
the threshold portfolio outperforms the benchmark portfolio (without threshold) in terms of
risk adjusted return. 2) Up to the optimal threshold value, the portfolio of higher threshold
value outperforms that of the lower threshold value. Thus, the essential implications of the
crash-hedged models are the same as the unhedged model.
2.6.6 Threshold portfolios when prices of risks are the same
In the previous model, prices of risks are different across different currency markets. This
means that the price of risk λi for currency i is
λi =abs(ri − rd)
σ
Now we prove that if the price of risk is the same across different currency markets, then the
best portfolio’s allocation depends on the volatility of currencies. Therefore the threshold
effect doesn’t exist for the forward premium but does exist for the volatilities.
Proposition 4 In a setting of N currencies of equal volatility, and if abs(ri−rd)σ is the same
for all currencies i, then the portfolio that optimizes the Sharpe ratio gives currency i weight
µ/σi, where µ is a renormalization constant.
Proof. The variance covariance matrix Σ is diagonal that has σ2 on the main diagonal. We
know from mean-variance portfolio theory, that the optimal allocation weight ωi is equal to
ν abs(ri−rd)σ2 , where ν is a renormalization constant. We also know that abs(ri−rd)
σ is the same
for all currencies i. Hence ωi =µ/σi, where µ is a renormalization constant.
Thus in a setting where prices of risk are the same for all currencies, we are well advised to
investigate threshold portfolios with thresholds depending on currency volatilities. Pursuing
that direction of research ever further would be a reasonable extension of the present project.
2.6.7 Extending models to include currencies with correlations
We now extend our baseline model to the case where currencies are correlated. We assume
that all two currency pairs have the same correlation ρ, and that the volatility of each currency
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 66
is σ. This means that the variance-covariance matrix of currency returns is
Σ = σ2
1 ρ ... ρ ρ
ρ 1 ... ρ ρ
ρ ρ ... ρ ρ
ρ ρ ... 1 ρ
ρ ρ ... ρ 1
All the other assumptions are the same as in Section 2. Our model suggests that
si(t) =Si(t+ 1)− Si(t)
Si(t)= σi(t)εi(t), 1 ≤ i ≤ N
where:
cov(εi(t), εj(t)) =
1 if i = j
ρ if i 6= j
We assume that the distribution of forward premiums follows the exponential distribution
with parameter λ. When computing the expected return of threshold portfolios, the correla-
tion structure doesn’t change the computation; thus as in the subsection 2.6.1 we have
E [rt+1(Ptfy)] =
∫∞y e−λxxλ dx.
e−λy
In the next section, we provide an example in which all currencies have equal correlations.
We will solve the model analytically, just as we did in subsection 2.6.1.
2.7 One-factor model
Here, following Lustig et al. [2011] and Jurek and Xu [2013], we extend our random-walk
model into a one-factor model, where each currency has loadings on the global factor. We
have:
si(t) = wiεg(t) +√
1− w2i εi(t) (2.8)
εg(t) and εi(t) are independent normal variables with zero means and the same standard
deviation: σ, εg(t) ∼ N(0, σ2), εi(t) ∼ N(0, σ2), where εg(t) is the global risk factor and εi(t)
is the idiosyncratic risk factor of currency i.
Our main interest is to study if the one-factor model features the threshold effects, which is
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 67
there exists a optimal threshold portfolio in terms of risk adjusted return. And if the optimal
threshold value exists, if the model feature two other effects: 1) Up to the optimal threshold
value, threshold portfolio outperforms benchmark portfolio(without threshold) in terms of
risk adjusted return. 2) Up to the optimal threshold value, portfolio of higher threshold
value outperforms that of lower threshold value.
We first compute the variance and covariance of currencies:
Var (si(t)) = w2i Var (εg(t)) + (1− w2)Var (εi(t)) = σ2
Cov (si(t), sj(t)) = wiwjσ2
If each currency has the same loading on the global risk factor, wi = w, then all currency
pairs will have the same correlations: Corr (si(t), sj(t)) = w2. On the other hand, given ρ,
we can construct a one-factor model in which wi = w =√ρ, such that Corr (si(t), sj(t)) = ρ.
Thus we see that the model posited on an equal correlation of all currency pairs is a special
case of the one-factor model. In the ensuing subsections we study the one factor model while
imposing various restrictions on wi. First, however, we study the case of equal correlations
of all currency pairs.
2.7.1 One-factor model with equal correlations of all currency pairs
Our assumptions with respect to forward premium are the same as those made in Section
2.6. Thus for a threshold portfolio having threshold value y we have
E [rt+1(Ptfy)] =
∫∞y e−λxxλ dx.
e−λy
Next we consider the variance of rt+1(Ptfy). Supposing that the portfolio includes N carry
trades with currency movement sij , sij (t) = wεg(t) +√
1− w2εij (t). Thus we have
Var [rt+1(Ptfy)] = w2Var εg(t) +
∑Nj=1(1− w2)Var εij (t)
N2
Var [rt+1(Ptfy)] = (w2 +1− w2
N)σ2
Now we assume that N is sufficiently large relative to 1−ω2
ω2 , such that we could omit the
second term in the first order approximation.
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 68
Assumption 1 We assume that N 1−ω2
ω2 for all threshold portfolios, such that the variance
of portfolio return is dominated by the covariance of different trades.
Under this assumption we have
Var [rt+1(Ptfy)] ' w2σ2 (2.9)
In this case the Sharpe ratio of threshold portfolio is
Sh[rt+1(Ptfy)] 'y + 1/λ
wσ(2.10)
Thus under Assumption 1 in the one-factor model, idiosyncratic currency risks are perfectly
diversified within each threshold portfolio. All threshold portfolios therefore have the same
risk, which is the global systematic risk. Hence the Sharpe ratio having a higher threshold
value is always higher than that having a lower threshold value. Different from the random
walk model, in the case of one-factor model there is no optimal threshold portfolio in terms
of risk adjusted return.
In practice Assumption 1 is false, due to the limited number of carry trades. Higher threshold
portfolios, which include a smaller number of carry trades, are more prone to idiosyncratic
risk. In fact the idiosyncratic risk grows so quickly that higher threshold portfolios’ perfor-
mances drop dramatically, just as we found in Section 2.4.
2.7.2 One-factor model with heterogeneous correlations of currency pairs
Some research, for example Brunnermeier et al. [2008], suggest that the currency pairs of
higher forward premiums tend to comove. In this subsection we model this property in our
one-factor framework. We assume that there are in total N currency pairs, and without loss
of generality we assume that fpi < fpj for i < j. In our model the loading of currency pair
on the global risk factor depends upon its forward premium:
si(t) =i
Nεg(t) +
√1− (
i
N)2εi(t) (2.11)
Thus the currency pair of higher forward premiums has a higher loading on the global risk
factor, with Cov (si(t), sj(t)) = ijN2 . The correlation between currency pairs of higher forward
premiums is thus greater than that of lower forward premiums. We next compute the return
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 69
volatility of threshold portfolios.
Assume that the portfolio includes N carry trades, with loadings wij (1 ≤ j ≤ N) on the
global risk factor. Thus
Var [rt+1(Ptfy)] =(∑N
j=1wij )2Var εg(t)
N2+
∑Nj=1(1− w2
ij)Var εij (t)
N2
Var [rt+1(Ptfy)] = (wij )2σ2 +
(1− w2ij
)σ2
N
where wij and w2ij
are the means of wij and w2ij
, respectively. We now make the following
assumption, similar to Assumption 1
Assumption 2 We assume that N 1−w2
ij
(wij )2for all threshold portfolios, such that the vari-
ance of portfolio return is dominated by the covariance of different trades.
Under Assumption 2, we have
Var [rt+1(Ptfy)] ' (wij )2σ2
std [rt+1(Ptfy)] ' wijσ
We now compute the return volatility of the threshold portfolio. The portfolio with threshold
value y includes (1 − e−λy)N carry trades having biggest forward premiums , and thus the
average loading of portfolio on global factor is wyij = 1− e−λy
2 . From which we have
std [rt+1(Ptfy)] ' (1− e−λy
2)σ
Sh[rt+1(Ptfy)] 'y + 1/λ
(1− e−λy
2 )σ
It is easy to verify that Sh[rt+1(Ptfy)] is an increasing function for y greater than 1/λ; thus,
the result gained for heterogeneous correlations is similar to that for equal correlations. Under
Assumption 2, idiosyncratic currency risks are perfectly diversified within each threshold
portfolio. When the investor increases the threshold value, the increase in expected return
outdoes the increase in volatility, and therefore the investor always wants to invest in higher
threshold portfolios. In the real world of a small universe of carry trades, Assumption 2 is
also wrong, just as we pointed out about Assumption 1.
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 70
2.7.3 Model with increasing downside correlation
It is well known that correlation between assets increases during period of high market volatil-
ity; this phenomenon has been discussed for example in both Ang and Bekaert [2002] and
Bouchaud and Potters [2001]. Now let us show that our one-factor model features precisely
such an increasing downside correlation. We use the following conditional correlation to
measure the downside correlation:
ρ(si(t), sj(t))|εg(t)|>ν =Cov (si(t), sj(t))|εg(t)|>ν√
Var (si(t))|εg(t)|>νVar (sj(t))|εg(t)|>ν
ρ(si(t), sj(t))εg(t)<−ν =Cov (si(t), sj(t))εg(t)<−ν√
Var (si(t))εg(t)<−νVar (sj(t))εg(t)<−ν
where ρ(si(t), sj(t))|εg(t)|>ν is the correlation between si(t) and sj(t), conditional on the global
factor having high volatility. ρ(si(t), sj(t))εg(t)<−ν is conditional on the downside return of
the global factor. We have the following relationship.
Proposition 5 ρ(si(t), sj(t))|εg(t)|>ν and ρ(si(t), sj(t))εg(t)<−ν are increasing functions of ν.
Proof. We prove for ρ(si(t), sj(t))|εg(t)|>ν , and the proof for ρ(si(t), sj(t))εg(t)<−ν is similar.
We have
ρ(si(t), sj(t))|εg(t)|>ν =
wiwjVar (εg(t))|εg(t)|>ν√w2i Var (εg(t))|εg(t)|>ν + (1− w2
i )Var (εi(t))√w2jVar (εg(t))|εg(t)|>ν + (1− w2
j )Var (εj(t))
Note x = Var (εg(t))|εg(t)|>ν , we have
ρ(si(t), sj(t))|εg(t)|>ν =wiwjx
((w2i x+ (1− w2
i )σ2)(w2
jx+ (1− w2j )σ
2))1/2
Thus,
dρ(si(t), sj(t))|εg(t)|>ν
dx=
wiwj(w2i x+ (1− w2
i )σ2)(w2
jx+ (1− w2j )σ
2)− 12wiwjx(2w2
iw2jx+ (w2
i + w2j )σ
2)
((w2i x+ (1− w2
i )σ2)(w2
jx+ (1− w2j )σ
2))3/2
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 71
It is easy to verify thatdρ(si(t),sj(t))|εg(t)|>ν
dx > 0, and thus we are done.
The last proposition shows that the one-factor model features increasing downside correla-
tions.
2.7.4 Implication of random-walk model and one-factor model
The random walk model in the section 2.6 features two threshold effects: 1) Up to the optimal
threshold value, threshold portfolio outperforms benchmark portfolio(without threshold) in
terms of risk adjusted return. 2) Up to the optimal threshold value, portfolio of higher
threshold value outperforms that of lower threshold value. We see that these two threshold
effects disappear in the one-factor model, where investor always wants to invest in higher
threshold portfolio. The reason is in a universe of many carry trades, idiosyncratic risks
are perfectly diversified in threshold portfolios. Thus volatility of threshold portfolio return
is determined by the correlation between currencies. As the threshold goes up, the effect
of increase in average return dominates that of increase in volatility. The non-existence of
optimal threshold value in one-factor model shows that quasi-independence of currencies is
important to understand the threshold effects for currency carry trades.
2.8 Conclusion
This paper studies threshold portfolios. It constructs carry trade threshold portfolios using
thresholds depending on the forward premium. It shows that from 1990 to 2012 portfo-
lios with a higher threshold up to the optimal threshold value outperform lower threshold
portfolios. The robustness of the results is tested. Following Jurek [2014] it tests if the out-
performance of higher threshold portfolios is compensation for crash risk. It constructs the
hedged threshold portfolio using currency options and computes the returns of crash-hedged
threshold portfolios. It finds that crash risk could explain around 25% of the excess perfor-
mance of higher threshold portfolios relative to a benchmark equally weighted portfolio.
Empirically, the paper finds that the Sharpe ratio of threshold portfolio initially increases
as the threshold value goes up until the optimal threshold value. The Sharpe ratio then
decreases. The paper models the effects of threshold portfolios in a random walk currency
CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 72
model. Under general assumptions about the volatilities of the currency and the distribution
of forward premiums, the model predicts the best threshold value that maximizes Sharpe ra-
tio. The model also predicts the relative gain in the Sharpe ratio of the threshold portfolios
compared to the equally weighted portfolio. It calibrates the model to G10 currency trades
using data from January 1990 to June 2012. It finds that both key predictions are verified.
The paper also explores a one-factor model which exhibits rich properties of currencies: het-
erogeneous correlations and increasing downside correlations. It finds that in a one-factor
model, the optimal threshold value doesn’t exist if the currency returns are correlated and id-
iosyncratic currency risks are perfectly diversified in threshold portfolios. The non-existence
of an optimal threshold value in the one-factor model shows that quasi-independence of cur-
rencies is important to understand the threshold effects for currency carry trades.
Related to threshold effects, the paper studies the portfolio problem of a currency investor. It
approximates the problem of maximizing the investor’s expected utility by a mean-variance
problem. It solves the investor’s optimal portfolio given the expected returns and correlations
of currencies. It proves that under certain second moment conditions, the investor’s optimal
portfolio always allocates more funds to trades having larger forward premiums.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 73
Chapter 3
Currency Carry Trade Hedging
3.1 Introduction
Portfolio hedging1 is an important issue in investment. The available literature has exten-
sively studied hedging options using underlyings2 and hedging spots with futures3, but the
currency hedging problem4 has not been widely studied. Thus, this paper fills that gap in
the literature in seeking to solve the hedging problem for currency investors. It proposes the-
oretical frameworks for currency investment hedging, while then proposing and empirically
testing various hedging strategies.
This paper studies the hedging problems of a popular currency investment strategy - currency
carry trade, which profits from the violations of uncovered interest rate parity. Carry trade
has been pointed out as a popular currency investment strategy: for example Galati et al.
[2007] and Clarida et al. [2009]. Also see Daniel, Hodrick, and Lu [2014] for a summary of
the carry trade’s risks and drawdowns. Carry trade portfolios are prone to large drawdowns
in liquidity crisis periods and times of high currency volatility; see Brunnermeier et al. [2008]
1The term portfolio hedging refers to techniques used by investors to reduce their overall risk exposure
within an investment portfolio. In other words, hedging uses one investment to minimize the negative impact
of adverse price swings in another. The mathematical framework is provided in Section 3.2.1.
2For example, the option sellers hedge exposures to underlying assets by buying or selling the latter.
3For example, airline companies buy oil futures so as to eliminate risks related to future oil prices.
4By currency hedging we mean specifically hedging as part of a deliberate currency investment strategies.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 74
and Clarida et al. [2009]. In Section 2, we study the hedging problem in the context of the
general utility maximization framework of a currency investor. The solution provided by
the model reveals that investors employ hedging instruments for two reasons: either to 1)
boost the portfolio’s expected return or 2) reduce the portfolio’s variance/risk. We refer to
the position taken for reason 1) as the directional position, and for reason 2) as the variance
reduction position. The model’s solution shows that the investor takes the directional posi-
tion if the hedging instrument has a positive expected return. The model also shows that
the investor takes the variance reduction position only if the covariance between the hedging
instrument’s return and the carry trade’s return is negative. We then divide hedging instru-
ments into three categories, according to their expected return rh and their covariance with
the carry trade’s return rc: Cov [rc, rh]. 1) Insurance, which in general reduces the return
and also the risk to the portfolio, rh < 0,Cov [rc, rh] < 0; 2) Technical rule, which similar to
insurance reduces the portfolio’s risk but doesn’t worsen its return, rh ≈ 05, Cov [rc, rh] < 0;
and 3) Market neutral strategy, which boosts the return of the portfolio but doesn’t affect
its risk, rh > 0,Cov [rc, rh] ≈ 0. After setting up the theoretical framework we then propose
various hedging strategies. We also discuss some additional questions related to the hedging
strategies.
We consider four types of hedging strategies: FX options6 strategy, VIX future strategy7,
stop-loss strategy8 and CTA Trend Following strategy9. The first two are examples of insur-
ace, Stop-loss is a technical rule and CTA is a market neutral strategy. We show that based
5For the following we mean the technical rule doesn’t worsen the carry trade portfolio’s return when we
say the technical rule has rh ≈ 0.
6Three kinds of currency options are actively traded: Delta 10, Delta 25, and Delta 50, they are quoted by
the Delta of the option computed in the Garman–Kohlhagen model.
7VIX index tracks market estimate of expected volatility. VIX is calculated by using the real-time prices
of S&P 500 options.
8Stop-loss rules are rules that reduce a portfolio’s position after it has reached a cumulative loss.
9CTA (Commodity trading advisor) is an asset manager trading chiefly in futures. The predominant tactic
of CTAs is the trend-following strategy. CTA tries to profit from the market trends by observing the current
direction and deciding whether to buy or sell based on the momentum. Henceforth we will refer to CTA
trend-following strategy as simply the CTA strategy.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 75
on empirical evidence from 2000 to 2012, CTA is the preferred hedging strategy because
it increases significantly the average portfolio return and decreases dramatically portfolio
volatility and drawdown. Stop-loss strategy reduces portfolio’s risk while decreasing signifi-
cantly minimum drawdown and minimum monthly return. Currency option and VIX future
strategies offer good hedge against tail risk. They also reduce portfolio return volatility.
However they are costly to implement and significantly worsen average return and Sharpe
ratio. Delta 10 is the least expensive, while Delta 25 and VIX are almost equally costly.
Delta 50 is the most expensive.
Our results shed light on the crash risk explanation for the carry trade. According to the
crash risk theory, carry trade portfolio hedged against crash risk shouldn’t have significant
return. In Chapter 2 we saw that a option hedged carry trade portfolio shows significant
return for the period from 1990 to 2012. In the current paper we show that carry trade
portfolio hedged with stop-loss rule also shows significant return. And the stop-loss hedged
portfolio has positive skewness, in contrast to unhedged carry trade portfolio. Our results
thus indicate that crash risk explains only part of the carry trade excess return.
This paper discusses the intuitions and rationales of employing each hedging strategy. Cur-
rency option is the most straightforward hedging candidate because it eliminates the downside
currency risk beyond the option strike price. Previous studies, see Jurek [2014] and Burnside
et al. [2011a], have constructed hedged carry trade portfolios using out-of-money put options.
Actively traded currency options include the Delta 10, Delta 25 and Delta 50 options. We
construct three hedging portfolios using these three options, and find that all three currency
option hedging strategies reduce the volatility of a carry trade portfolio. The hedgings are
costly, however, for they also reduce the average returns. Overall, hedgings worsen the Sharpe
ratio, but they do offer good hedging against tail events because all of them significantly re-
duce the minimum monthly return while also slightly reducing the minimum drawdown.
The second approach we study is the VIX strategy, whereby we take long position in VIX
future as hedging. Clarida et al. [2009] points out that carry trades suffer losses when un-
dertaken within a high volatility regime. Brunnermeier et al. [2008] finds that rising global
volatility, proxied by VIX, predicts an increasing chance of crash. Menkhoff et al. [2012] finds
that the depreciation of high interest rate currency is positively correlated with the increase of
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 76
global currency volatility. Their findings suggest that carry trade crash risk could be hedged
by buying global volatility insurance. Thus we propose to hedge carry trade portfolio by
hedging volatility risk. Lacking as we are a traded FX volatility index, we use VIX futures as
a rolling hedging vehicle. We find that VIX hedging exhibits properties similar to those dis-
played by currency option hedgings; namely, it reduces both the return and the volatility of
a carry trade portfolio. Overall, it worsens the Sharpe ratio, but offers good hedging against
tail events like currency options because it reduces significantly the minimum monthly return,
and much less significantly, the minimum drawdown.
The third hedging strategy we consider is the stop-loss rule. Kaminski and Lo [2013] studies
the stop-loss rules, which are rules that reduce a portfolio’s position after reaching a cumu-
lative loss. Kaminski and Lo [2013] define as the ”stopping premium” the marginal impact
of stop-loss rules on a portfolio’s expected return. They show that the stopping premium
can be positive if portfolio returns are characterized by momentum. They also show that
the stop-loss rule reduces portfolio variance if the stop-loss rule involves switching to cash
when the stop-loss threshold is reached. We make the contribution by applying stop-loss
rule to carry trade portfolios. Following our rule, we reduce position in a currency when
the cumulative loss during a month reaches the threshold. We find that the stop-loss rule
improves the average return of carry trade portfolio while reducing the volatility. It is known
that foreign exchange rates exhibit momentum; see Asness et al. [2013]. Burnside et al.
[2011b] documents that carry trade strategies exhibit momentum. Thus our results can be
explained by the model of Kaminski and Lo [2013]. We also find that the stop-loss rule offers
good hedging against tail event. It reduces significantly both the absolute size of minimum
monthly return and the minimal drawdown.
The fourth and last hedging strategy we consider is CTA. As we pointed out previously,
carry trade is prone to volatility risk, and volatility tends to spike after significant market
movement. Brunnermeier et al. [2008] argues that the excess return of carry trade portfolio
is compensation for crash risk, which may correlate with losses in other markets. We thus
suggest to hedge carry trade portfolio with a market neutral strategy. Previous research on
CTA strategies has pointed out the following two stylized facts:
1) CTA payoff is a nonlinear function of market payoff; it’s not correlated with market return.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 77
2) The Beta of CTA strategy is small. The Beta is positive when the market goes up; it is
negative when the market goes down.
These two characteristics make CTA an ideal hedging strategy. Empirically we find that
CTA offers the best hedging by all criteria (detailed below) among the four hedging strate-
gies. We construct an equally volatility weighted portfolio comprising carry trade portfolio
and CTA portfolio. Compared to unhedged carry trade portfolio, it increases the average
monthly return by 32% and reduces monthly return volatility by 31%. Thus, it increases the
Sharpe ratio by 93% and reduces the absolute size of minimum monthly return by 63%, and
the minimal drawdown by 67%.
Besides studying hedging problems the paper also studies two issues related to different hedg-
ing instruments. Related to CTA the paper studies the risk-return aspect of CTA strategy.
In addition we compare VIX strategy to various currency option strategies. The objective
is to determine if VIX is a cheaper form of systematic insurance as compared to currency
options.
The rest of the paper is organized as following: The second section presents the theoretical
framework and the data. The third section studies the currency option and VIX hedging
strategies. The fourth section studies the stop-loss rule. The fifth section presents results
of CTA hedging strategy. The sixth section conducts out-sample test for hedging strategies.
The seventh section explores two issues related to the hedging strategies. The eighth section
summarizes results and concludes.
3.2 Framework and Data
3.2.1 Framework
The purpose of this section is to set up the framework for the complications involved with
carry trade hedging. The literature exploring futures hedging is extensive; see Lien and Tse
[2002] for a literature review, and Cotter and Hanly [2006] for an empirical summary. Within
the setting of futures hedging, the hedger has exposure in the spot market. Thus, carry
trade hedging and futures hedging are similar in that agents want to eliminate risks in both
situations. Nonetheless, there is an essential difference distinguishing one from the other.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 78
Futures generally are regarded as having zero expected return, and generally that is not
true for carry trade hedging instruments. As a result, the minimum variance hedging widely
studied by the analysis of the futures hedging problem, under the assumption that futures
having zero expected return, can be extended to the carry trade setting. To be more specific,
let us suppose that the hedger has utility function U , where U is an increasing function with
negative second-order derivative. We denote as rc the return of carry trade, and as rh the
return of the hedging instrument. Let E denote the expectation with respect to rc and rh.
The optimal hedging ratio αh is determined via this optimization problem:
maximizeα
E [U(rc + αrh)]
subject to 0 ≤ α(3.1)
The first order condition of 3.1 is:
E [U ′(rc + αrh)rh] = 0 (3.2)
which is
Cov [U ′(rc + αrh), rh] + E [rc + αrh]E [rh] = 0 (3.3)
Assume that rc = β0 + β1rh + ε, where rh and ε are independent. Then under the common
assumption E [rh] = 0, where the hedging instrument has zero expected return, the solution
of 3.1 satisfies αh = −Cov [rc,rh]V ar[rh] , which also minimizes the variance within a hedged portfolio.
When E [rh] 6= 0, we follow Lien and Tse [2002] in their way of solving problem 3.1 for mean-
variance utility function. In other words assume that U(r) = E [r] − A2 Var [r] with A
2 the
Arrow-Pratt risk aversion coefficient. The solution of Problem 3.1 then satisfies
αh =E [rh]
A− Cov [rc, rh]
V ar[rh](3.4)
We can interpret αh as being the sum of directional position E [rh]A and minimum variance
position −Cov [rc,rh]V ar[rh] .
Now we look at the possible hedging instruments of carry trade by the light of Eq. 3.4. In
general, we only use a hedging instrument when Cov [rc, rht] ≈ 0 or Cov [rc, rht] < 0.
Insurance : Insurance strategy has return rhi such that E [rhi] < 0, Cov [rc, rhi] < 0. The
examples we study are currency options and VIX futures.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 79
Technical rule : Technical rule strategy has return rht such that E [rht] ≈ 0, Cov [rc, rht] <
0. The example we study is the stop-loss rule.
Market neutral strategy : Market neutral strategy has return rhn such that E [rhn] > 0,
Cov [rc, rht] ≈ 0. The example we study is the CTA.
The above information may become clearer when it is viewed thus:
Cov [rc, rhi] < 0 Cov [rc, rhi] ≈ 0
E [rhi] < 0 Insurance Utility non-improving
E [rhi] ≈ 0 Technical Rule Utility non-improving
E [rhi] > 0 Market Neutral Strategy
In the ensuing sections, we study in detail the performance of these strategies, while also
providing their theoretical motivations. Note, however, that the intuitions of all of them are
well explained by Eq. 3.4. The carry trade investor employs a hedging strategy only when
doing so reduces the variance or enhances the expected return. In the latter case, the hedging
strategy generally should be market neutral, in order to qualify as a hedging strategy. Both
the Insurance and the Technical Rule strategies reduce the variance of the carry portfolio.
The CTA strategy is of positive return, and uncorrelated with the carry strategy. In addition,
CTA is market neutral. After we have introduced the various hedging strategies, we are ready
to discuss the criteria for the evaluation of hedging performance.
3.2.2 Hedging performance evaluation
Markowitz [1952] laid the foundation of modern portfolio analysis in the framework of return
and risk. While there is consensus that the return can be analyzed by expected return,
there is no universally accepted risk measure. The most popular performance metric for
futures hedging is the variance; see, for example, Cotter and Hanly [2006]. Variance is an
intuitive measure employed when the expected return of a hedging strategy is zero, but the
hedging instruments we propose do not all have zero expected returns; insurance strategies
have negative expected returns, whereas the CTA strategy has a positive average return. We
thus propose two categories of performance metrics.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 80
Financial performance indicators : Sharpe ratio; minimum drawdown; minimum monthly
return; moments of return distributions.
Utility effect for trader : We compute the utility gain for a trader who has CRRA utility.
In order to quantify the hedging effect, we compute the minimal amount of risk-aversion
needed in order for the trader to break even. That is the risk-aversion coefficient of the
trader who is indifferent between employing the hedging strategy or not.
Financial performance indicators in the first category are commonly used for characteriz-
ing portfolio performance. Sharpe ratio was proposed in Sharpe [1966], as a measure of
risk-adjusted return. Minimum drawdown and minimum monthly return are both popular
downside risk measures studied, for example, in Young [1998]. Variance is the original risk
measure proposed by Markowitz [1952].
The risk-aversion indicator (Pratt [1964]) in the second category is used specifically for char-
acterizing hedging strategies. Our intuition is that if a trader of low risk-aversion is willing
to take a hedging strategy, then a trader of high risk-aversion will also be. We will test
empirically this proposition.
We use the above two categories of metrics to compare all hedging strategies. We also com-
pare the hedging performances arrived at via currency options and VIX futures. As a preview
of our result, we find that currency options and VIX futures are all expensive as gauged by
the above two performance metrics, but have a strong tail-risk hedging effect. That is to
say, they offer a good hedge during the worst performance months of carry trades. We thus
employ two special metrics to compare currency options to VIX futures. We compute the
utility gain for a trader who is employing currency options or VIX futures and who has CRRA
utility during crisis periods. We also compute the utility cost to the trader during non-crisis
periods. We then look at the following metrics
Insurance Quality : The ratio of utility gain in crisis periods over utility cost in non-crisis
periods.
Tail hedging effect : Returns on currency options and VIX futures strategy during crisis
periods.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 81
3.2.3 Data
We use the following data: FX price data covering G10/USD pairs from January 1990 to
June 2012; price data on FX options covering G10/USD pairs from January 1999 to June
2012; price data on VIX futures from March 2004 to June 2012; price data on 74 liquid
futures from June 2000 to January 2014; Newedge CTA index from June 2000 to January
2014. The Newedge CTA index data comes from Newedge, while the other price data comes
from Datastream. The FX option conventions are explained in Section 3 of Jurek [2014]. We
use on-the-run VIX futures.
3.3 Hedging using currency options or VIX future
In this section we first report the summary statistics of the unhedged G10 carry trade port-
folio. We then implement the currency option hedging strategy. Previous studies, see Jurek
[2014] and Burnside et al. [2011a], have constructed hedged carry trade portfolios using out-
of-money put options. Actively traded currency options include the Delta 10, Delta 25 and
Delta 50 options. We construct three hedging portfolios using these three options, and find
that all three currency option hedging strategies reduce the volatility of a carry trade port-
folio. The hedgings are costly, however, for they also reduce the average returns. Overall,
hedgings worsen the Sharpe ratio, but they do offer good hedging against tail events, because
all of them significantly reduce the minimum monthly return, while also slightly reducing the
minimum drawdown.
The second approach we study is the VIX strategy, whereby we take long positions in VIX
future as hedging. Clarida et al. [2009] points out that carry trades suffer losses when un-
dertaken within a high-volatility regime. Brunnermeier et al. [2008] finds that rising global
volatility, proxied by VIX, predicts an increasing chance of crash. Menkhoff et al. [2012]
finds that the depreciation of high-interest-rate currency is positively correlated with the
increase of global currency volatility. Their findings suggest that carry trade crash risk could
be hedged by buying global volatility insurance. Thus, we propose to hedge carry trade
portfolios by hedging volatility risk. Lacking as we are a traded FX volatility index, we
use VIX futures as a rolling hedging vehicle. We find that VIX hedging exhibits properties
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 82
similar to those displayed by currency option hedgings; namely, it reduces both the return
and the volatility of a carry trade portfolio. Overall, it worsens the Sharpe ratio, but offers
good hedging against tail events, like currency options, because it reduces significantly the
minimum monthly return, and much less significantly, the minimum drawdown.
For an interesting question related to currency options we compare in the subsection 3.7.1
VIX strategy to various currency option strategies. Our objective there is to determine if
VIX is a cheaper form of systematic insurance as compared to currency options. We report
next the summary statistics of the equally weighted G10 currency carry trade strategy, as
well as of the option-holding and VIX-holding strategies.
3.3.1 Crash-risk hedging strategies
Carry trades are well known for “going up by stairs and going down by elevators”. Fig. 3.1
illustrates the total return indices for G10 currency carry trades over the period ranging from
January 1990 to June 2012. Table 3.1 reports the summary statistics for the carry trade
portfolio.
Table 3.1: Carry Trade Portfolio Performance Statistics
Mean Std Skewness Kurtosis Minimum Maximum Sharpe N
3.40 5.97 -0.81 5.36 -6.61 5.18 0.57 270.00
Table 1 reports summary statistics for currency carry trades implemented in G10 currencies. Summary
statistics are reported over Jan.1990-Jun. 2012 (N = 270 months). Means, volatilities and Sharpe ratios
(SR) are annualized; Min and Max report the smallest and largest observed monthly return.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 83
Figure 3.1: Cumulative G10 currency carry trade returns
1990 1992 1995 1997 2000 2002 2005 2007 2010 20121
1.5
2
2.5
Figure 3.1 plots total return indices for currency carry trades implemented in G10 currencies. The time period
is from January 1990 to June 2012.
In both places we see that carry trade portfolios tend to achieve positive average return
and a relatively big drawdown. We show in Fig. 3.2 the cumulative returns of currency
option instruments and VIX instrument.
Figure 3.2: Cumulative hedging strategies returns
2004 2005 2006 2007 2008 2009 2010 2011 2012 20130.7
0.75
0.8
0.85
0.9
0.95
1
1.05
50 Delta
25 Delta
10 Delta
VIX
Figure 3.2 plots total return indices for currency option portfolio implemented in G10 currencies. It also plots
total return index for investing in the on-the-run VIX future. The time period is from March 2004 to June
2012. We normalize the VIX strategy so that it has the same volatility as the Delta 25 strategy.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 84
Note that the crash-hedging strategies have average negative returns, reflecting the hedg-
ing cost. In Table 3.2 we show the correlations between the carry trade portfolio and the
various hedging portfolios.
Table 3.2: Correlations between the carry trade portfolio and various hedging instruments
Carry 50 Delta 25 Delta 10 Delta VIX
Carry 1.00 -0.73 -0.62 -0.54 -0.60
50 Delta 1.00 0.95 0.84 0.49
25 Delta 1.00 0.94 0.40
10 Delta 1.00 0.29
VIX 1.00
Table 3.2 reports correlations between carry trade portfolio and different hedging portfolios for the period
from March 2004 to June 2012.
The table confirms the hedging effects of both the currency option and the VIX portfo-
lios. The correlations of carry trade portfolio with the three currency option portfolios are
respectively, -0.73, -0.62, and -0.54. The correlation of the carry trade portfolio with the
VIX portfolio is -0.6. Table 3.2 also reveals high correlations among the currency options
portfolios, with the average correlation being 0.91. Last but not least we see that the VIX
portfolio is positively correlated with all of the currency option portfolios.
3.3.2 Financial performances of hedged carry trade portfolios
Here we compute the average return, volatility, Sharpe ratio, drawdown, and minimum
monthly return of hedged carry trade portfolios versus unhedged portfolios. In Table 3.3
we report the results of a carry trade portfolio hedged with position in the option contract
or the VIX future. The position in hedging instrument is such that its volatility is equal to
10% of that of the carry trade portfolio.
In order to compare currency option strategies and VIX strategy with stop-loss strategy and
CTA strategy later, we report in Table C.1 in appendix the results for the period from June
2000 to June 2012. Because VIX future data is missing from June 2000 to February 2004,
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 85
Table 3.3: Carry Trade Hedged with Options or VIX Futures
Sharpe*10 Rlt ∆ Mean Rlt ∆ Std Rlt ∆ DD Rlt ∆ Min Rlt ∆
Carry 0.64 0.04 1.95 -22.48 -6.55
50 Delta 0.44 -30.67 0.03 -30.65 1.95 0.03 -22.87 1.76 -6.50 -0.79
25 Delta 0.52 -19.11 0.03 -19.20 1.95 -0.12 -22.70 0.98 -6.48 -1.18
10 Delta 0.56 -12.13 0.03 -12.29 1.95 -0.18 -22.53 0.25 -6.48 -1.10
VIX 0.54 -15.13 0.03 -15.85 1.94 -0.85 -22.43 -0.22 -6.42 -1.97
Table 3.3 reports summary statistics for unhedged and hedged carry trade portfolios. The construction rule
is detailed in the text. Hedged portfolio includes position in hedging strategy which has 10% of return
volatility of that of unhedged carry trade portfolio. The time period is from March 2004 to June 2012.
Unhedged denotes the unhedged carry trade portfolio. Sharpe is the Sharpe ratio. Mean is the average
monthly return. Std is the volatility of monthly return. DD is the minimum drawdown. Min is the minimum
monthly return. Rlt ∆ denotes the change of the indicator compared to the unhedged carry portfolio.
for that period we replicate VIX strategy using currrency options.
We see that adding an option overlay decreases not only the portfolio’s average return but
also its Sharpe ratio. The volatility of the hedged portfolio is lower than that of the un-
hedged portfolio. The hedged portfolio also has a bigger minimum monthly return than the
unhedged portfolio. Both the Delta 10 option portfolio and the VIX portfolio have smaller
drawdowns than the unhedged portfolio.
Now we propose another measure to evaluate the impact of hedging overlay: the derivative of
the financial performance indicators. Note for example f(ruh) the statistics of unhedged port-
folio, f(ruh + trh) that of a hedged portfolio which includes 1 position in unhedged portfolio
and t position in hedged strategy. We define the derivative f′
of f as following:
f′(ruh)h = lim
t→0
f(ruh + trh)− f(ruh)
t
We report in Table C.2 in the appendix the derivative of financial performance indicators.
We compute f′(ruh)h by numerical methods. We see that all option hedging strategies and
VIX strategy reduce the maximal monthly loss. VIX and Delta 10 improve the minimum
drawdowns.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 86
Next we study the expensiveness of hedging instruments in the framework of investor’s utility
maximization problem.
3.3.3 Who would buy the insurance?
We assume that the trader has CRRA utility function, and he maximizes expected utility
period by period. St is the trader’s wealth at time period t; ωt is the trader’s portfolio at the
beginning of period t; pt is the price of securities. Trader’s problem at period t is:
maxωt+1E t[Uγ(St+1)]
s.t. : ωt · pt = ωt+1 · pt, St+1 = ωt+1 · pt+1
where Uγ(St+1) =S1−γT+1
1− γ
We showed previously that buying currency option or VIX future is on average costly. Adding
currency option overlay or VIX overlay reduces the average return and the Sharpe ratio of
the portfolio. In order to measure the trade-off between cost and payout of hedging overlay,
we compute the break-even risk-aversion coefficient for which trader is indifferent between
taking an additional position in hedging portfolio or not. Empirically, given the return of
unhedged portfolio ruh(t) and that of hedging instrument rh(t), the break-even risk-aversion
coefficient for a trader who takes k position of hedged portfolio is γh such that:∑Tt=1 Uγ(ruh(t))
T=
∑Tt=1 Uγ(ruh(t) + krh(t))
T
Later we will also calculate break-even risk-aversion coefficient for other hedging strategies.
We show in Table 3.4 the results computed for currency option hedging strategies and VIX
future strategy.
Note γh the break-even risk-aversion coefficient for hedging strategy h. Then intuitively
any trader with risk-aversion coefficient γ > γh will also prefer to have the hedging overlay.
That’s what we have verified empirically by running simulations.
Remark 1 Consider two hedging strategies h1 and h2 with break-even risk-aversion coeffi-
cients γh1 and γh2 respectively. We say that h1 is more expensive for CRRA utility trader
than h2 if and only if γh1 > γh2.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 87
Table 3.4: Break-even Risk-aversion Coefficients
k 50 Delta 25 Delta 10 Delta VIX
0.01 26 21 18 14
0.03 26 21 18 14
0.05 27 21 19 14
0.07 27 21 19 14
0.09 27 21 19 14
Table 3.4 computes empirically the break-even risk-aversion coefficient for which trader is indifferent between
taking an additional position in hedging portfolio or not. The time period is from March 2004 to June 2012.
Surprisingly VIX is the best hedging strategy for CRRA utility trader. For example trader
with risk-aversion coefficient of 15 does not gain by buying any currency option but gains by
buying VIX future. Recall that the hedged portfolio overlaid with Delta 10 option delivers
highest Sharpe ratio. We thus find that the cheapest hedging strategy in terms of return
reduction is not the cheapest when we consider the utility of trader. We analyze more in
detail the hedging effects for crash periods next.
3.3.4 Hedging effects during crash periods
In this subsection, we first identify the periods during which carry trade portfolios suffered
huge losses. We then analyze the hedging effects of different hedging strategies during these
periods. In Table C.3, in the appendix, we report the ten months of biggest losses for carry
trade portfolios. For each month, we also report the return of carry portfolios and different
hedging portfolios. We report respectively average portfolio returns for those ten months, as
well as for the six months of biggest carry portfolio losses. On the last row, we report the
number of months on which the hedging portfolio has a negative return, and thus doesn’t
pay off.
Carry portfolios suffered losses greater than 3% during six particular months over the last
several years. Among them, three are associated with the onset of the last financial crisis
(August, September and October of 2008), with losses of -4.46%, -3.17% and -6.55%. Two
ample wreckages take place with the European Debt Crises (September 2011, May 2012),
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 88
with losses between 4.3% and 4.5%. The last one is May 2010, with a -5.5% depletion, which
corresponds to the Flash Crash. We now analyze the effects of different hedging strategies;
in particular, we are interested in whether the VIX strategy offers good hedges.
1. Average hedging returns.
All hedging portfolios have, on average, positive returns. The average return of VIX is 0.88%,
between that of Delta 10 at 0.54% and Delta 25 at 1.17%. The Delta 50 strategy, which is
the most expensive among all hedging strategies, also sees the highest payoffs when carry
portfolios suffer considerable losses. In studying average hedging returns, we see that VIX
offers agreeable hedges against currency crash risk.
2. Number of positive hedging returns.
For the aforementioned ten months, all hedging strategies had negative payoffs for certain
periods. Delta 10 had four months of negative returns, while VIX had three months, which
is the same as Delta 25; Delta 50 had only 1 month. By this criterion, VIX again qualifies
as a favorable hedger.
Among the ten months, we study the five months for which Delta 10 strategies showed the
highest returns. We identify these five months as crash periods. They are also the five months
for which carry portfolios suffered the most extensive losses. The five months represent three
events: the onsets of the Financial Crisis, European Debt Crisis and Flash Crash. Specifically,
we study hedging performances during these three periods. In Table 5, we report the gains of
hedging strategies and losses of carry portfolios during the three periods. For the Financial
Crisis period, we include the three-month period from August to October 2008. In European
Debt Crisis, we include September 2011 and May 2012. Flash Crash corresponds to May
2010. We report in the next table hedging portfolio performances during the three major
crisis periods:
We see that all hedging portfolios have, on average, positive returns. The average return
of VIX is 1.42%, between that of Delta 10 at 0.97% and Delta 25 at 1.96%. The Delta 50
strategy, which is the most expensive among all hedging strategies, also has the highest payoff
when carry portfolios suffer from heavy losses. Upon studying average hedging returns, we
see that VIX offers commendable hedges against currency crash risk.
We conclude that currency option strategies and the VIX strategy all offer hedging against
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 89
Table 3.5: Hedging Effects During Tail Events
Carry 50 Delta 25 Delta 10 Delta VIX
Financial Crisis -14.18 7.94 6.01 3.49 5.05
European Debt Crisis -8.83 5.29 4.07 1.68 1.82
Flash Crash -5.54 2.47 1.71 0.66 1.62
Month Mean -4.76 2.62 1.96 0.97 1.42
Financial Crisis period includes three months from August to October 2008. European Debt Crisis includes
September 2011 and May 2012. Flash Crash corresponds to May 2010.
crash risk.
3.4 Stop-loss Rules Applied to FX Carry Trade
The third hedging strategy we consider is the stop-loss rule. Kaminski and Lo [2013] studies
the stop-loss rules, which are rules that reduce a portfolio’s position after reaching a cumu-
lative loss. Kaminski and Lo [2013] define as the ”stopping premium” the marginal impact
of stop-loss rules on a portfolio’s expected return. They show that the stopping premium
can be positive if portfolio returns are characterized by momentum. They also show that
the stop-loss rule reduces portfolio variance if the stop-loss rule involves switching to cash
when the stop-loss threshold is reached. We make the contribution by applying stop-loss
rules to carry trade portfolios. Following our rule, we reduce position in a currency when the
cumulative loss during a month reaches the threshold. We find that the stop-loss rule im-
proves the average return of carry trade portfolios while reducing the volatility. It is known
that foreign exchange rates exhibit momentum; see Asness et al. [2013]. Burnside et al.
[2011b] documents that carry trade strategies exhibit momentum. Thus, our results can be
explained by the model of Kaminski and Lo [2013]. We also find that the stop-loss rule offers
good hedging against tail events. It reduces significantly both the absolute size of minimum
monthly returns and the minimal drawdown.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 90
3.4.1 Stop-loss rule
Our baseline portfolio comprises carry trades of G10 currency versus USD. It is equally
weighted for nine trades. The portfolio is rebalanced monthly. We form stop-loss portfolios
by adding a stop-loss rule overlay. We apply a stop-loss threshold for each carry trade
position. For each currency pair, we exit the current carry trade position if the month to day
loss is bigger than the stop-loss threshold. We test 10 stop-loss thresholds from 1% to 10%
with step 1%.
3.4.2 Financial performances of hedging strategies
Table 3.6 shows the test result for the period from January 1999 to June 2012. Unhedged is the
equally weighted portfolio comprising G10/USD paris. TR 1 denotes the equally weighted
portfolio overlaid with stop-loss threshold 1%. Similarly for TR 2, etc. Delta 10 is the
hedged portfolio hedged using out-of-money Delta 10 put option, as in Jurek [2014]. Figure
C.1 in appendix plots the cumulative returns for Unhedged, Delta 10 and TR3 portfolio. We
compare stop-loss portfolios respectively to unhedged equally weighted portfolio and hedged
equally weighted portfolio. For simplicity we use the stop-loss portfolio with threshold 3%.
Compared to unhedged equally weighted portfolio, stop-loss portfolio has higher return, lower
volatility, more positive skewness and higher Sharpe ratio. The reduction in loss is achieved
without giving away return. This suggests momentum property of carry trade returns, as
documented in Burnside et al. [2011b].
Compared to 10δ hedged portfolio, the TR3 portfolio has almost the same mean return,
smaller volatility, roughly the same skewness and higher Sharpe ratio.
3.4.3 Who would buy the insurance?
We find that stop-loss overlays of certain threshold values decrease the average monthly
return. Similar to Section 3.5 we consider the utility gain for a trader who has CRRA utility
function. We compute the break-even risk-aversion coefficient for which trader is indifferent
between taking an additional position in hedging portfolio or not. For the setup of the model
please refer to Section 3.3. The minimal risk-aversion coefficient we test is 2. We show in
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 91
Table 3.6: Carry Trade Hedged using Stop-loss Rules
Mean Std Skewness Kurtosis Min Max Sharpe
TR 1 2.02 4.87 1.57 7.16 -1.77 7.11 0.41
TR 2 2.72 5.57 0.90 4.66 -3.11 6.83 0.49
TR 3 3.38 5.87 0.43 4.00 -4.14 6.69 0.58
TR 4 3.38 6.01 0.22 4.08 -4.36 6.48 0.56
TR 5 3.43 6.32 -0.03 4.21 -5.58 6.17 0.54
TR 6 3.15 6.45 -0.23 4.67 -6.07 6.41 0.49
Unhedged 3.23 6.64 -0.63 6.51 -8.73 6.47 0.49
10 Delta 3.17 7.17 0.28 5.01 -5.92 7.94 0.44
Table 3.6 reports summary statistics for carry trade portfolios with stop-loss overlay. TR 1 is the stop-loss
portfolio with 1% as stop-loss threshold, similar for TR 2, TR 3, etc. Unhedged is the unhedged equally
weighted G10 carry trade portfolio. Delta 10 is the hedged portfolio hedged using Delta 10 put option. The
time period is from January 1999 to June 2012.
Table 3.7 the results.
Table 3.7: Break-even Risk-aversion Coefficients
TR 1 TR 2 TR 3 TR 4 TR 5 TR 6
0.1 2 2 2 2 2 2
0.3 2 2 2 2 2 2
0.5 2 2 2 2 2 2
0.7 2 2 2 2 2 2
0.9 2 2 2 2 2 2
Table 3.7 computes empirically the break-even risk-aversion coefficient for which trader is indifferent between
including a specified position of stop-loss strategy in the portfolio or not. The time period is from January
1999 to June 2012.
The results show that all traders of risk-aversion coefficients higher than 2 improve their
utility by taking positions in the stop-loss strategy. This is in contrast to currency option
or VIX strategy. For the latter case only traders of high risk-aversion coefficients improve
utility by taking hedging positions. We thus show that stop-loss strategy is a more efficient
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 92
hedging strategy.
3.4.4 Robust check
Our test shows that stop-loss portfolio with threshold 3%(TR3) has higher Sharpe ratio and
more postive skewnwess than unhedged portfolio. In this subsection we test if the results
are robust. We use resampling bootstrap, the p-value that TR3 has lower or equal Sharpe
ratio than unhedged portfolio is 0.1. The p-value that TR3 has more negative skewness than
unhedged portfolio is 0. We thus have strong belief that stop-loss rule reduces losses and
improves Sharpe ratio.
We do the same test for TR3 versus hedged Delta 10 portfolio. The p-value that TR3 has
lower or equal Sharpe ratio than hedged Delta 10 portfolio is 0.37. The p-value that TR3 has
more negative skewness than hedged Delta 10 portfolio is 0.39. There is thus no significance
for the comparison of financial performance between stop-loss strategy and Delta 10 hedging
strategy.
3.5 CTA trend following as a hedging strategy
The fourth and last hedging strategy we consider is CTA. As we pointed out previously
carry trade is prone to volatility risk, and volatility tends to spike after significant market
movement. Brunnermeier et al. [2008] argues that the excess return of carry trade portfolio
is compensation for crash risk, which may correlate with losses in other markets. We thus
suggest to hedge carry trade portfolio with a market neutral strategy. Previous research on
CTA strategies has pointed out the following two stylized facts:
1) CTA payoff is a nonlinear function of market payoff; it’s not correlated with market return.
2) The Beta of CTA strategy is small. The Beta is positive when the market goes up; it is
negative when the market goes down.
These two characteristics make CTA an ideal hedging strategy. Empirically, we find that CTA
offers the best hedging by all criteria (detailed below) among the four hedging strategies. We
construct an equally volatility-weighted portfolio comprising carry trade portfolios and CTA
portfolios. Compared to unhedged carry trade portfolios, it increases the average monthly
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 93
return by 32% and reduces monthly return volatility by 31%. Thus, it increases the Sharpe
ratio by 93%, and reduces the absolute size of minimum monthly return by 63%, and the
minimal drawdown by 67%.
In the first part of this section, we detail the portfolio construction rule for CTA. In the
second part, we study CTA as a carry trade hedging instrument. For an interesting question
related to CTA, in the subsection 3.7.2 we study the risk/return analysis for CTA strategy.
3.5.1 Constructing CTA portfolio
We construct a CTA trend following strategy which has the following features:
1) Long/Short portfolio: Portfolio takes long or short positions in different assets.
2) Risk management in two levels: a) In the portfolio level: portfolio volatility is controlled.
b) In individual asset level: individual asset volatility is also controlled.
Our investment universe comprises 74 liquid commodity futures. All major asset classes are
included: equity, bond, FX and commodity. We note Pi(t) the price of future i at time t.
We have 32 equity index futures, 11 bond futures, 25 commodity futures and 10 currency
forwards. Equity index futures include benchmark indices of major developed countries.
Bond futures include those of US, UK, Germany, Japan and Australia. Commodities futures
include the most liquid commodity futures. Currency forwards include all G10 currencies
versus the US dollar.
We specify now the portfolio construction rule. The portfolio construction depends on two
hyper parameters: target volatility of portfolio σ∗ and lookback period τ .
1. The lookback period τ is fixed to be 6 months, τ = 6Month. The signal given by the
trend of the lookback period is Si(t) = sign(Pi(t)−Pi(t− τ)), the sign of portfolio’s position
of asset i is determined by Si(t).
2. The absolute size of position of asset i is determined by two criteria: a) The overall
volatility of portfolio is equal to the targeted volatility. b) The volatility of individual asset
is equal across different assets. Note σi(t) the volatility estimated at time t for asset i, Σt the
estimated covariance matrix of all futures at time t. We thus have that the position ωi(t) of
asset i at time t satisfies the following condition:
ωi(t) = k ∗ sign(Pi(t)− Pi(t− τ))/σi(t) (3.5)
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 94
Where k is a constant such that: √ω(t)TΣtω(t)T = σ∗ (3.6)
From 3.6 we can solve k and deduce the position of individual assets.
3. The portfolio is rebalanced each week. We set transaction cost to be at 0.05% for all assets,
which is a conservative estimation of transaction cost, as our assets are all liquid futures.
3.5.1.1 Simulation of constructed CTA portfolio
We report in the table below the summary statistics of constructed CTA portfolio from
June 2000 to January 2014. We plot in Figure 3.3 the total return index of simulated CTA
portfolio.
Table 3.8: CTA Performance Statistics
Mean Volatility Skewness Kurtosis Minimum Maximum Sharpe
CTA 7.96 14.94 0.30 2.80 -9.04 12.57 0.53
Table 3.8 reports summary statistics for simulated CTA portfolio. The time period is from June 2000 to
January 2014.
3.5.2 CTA as a hedging strategy
We study in this subsection the hedging performance of CTA. We compute the Sharpe ratio,
drawdown and minimum monthly return of hedging strategies versus unhedged strategies.
We report in the following table financial performance indicators of a portfolio which has
90% position in unhedged strategy and 10% position in CTA strategy.
We see that compared to unhedged carry trade portfolio, CTA hedged portfolio has higher
average return, lower return volatility, higher Sharpe ratio, lower drawdown and lower mini-
mum monthly return.
3.5.2.1 Who would buy the insurance?
Similar to Section 3.3 we consider the utility gain for a trader who has CRRA utility function.
We compute the break-even risk-aversion coefficient for which trader is indifferent between
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 95
Table 3.9: Carry Trade Hedged using CTA
Sharpe Rlt ∆ Mean Rlt ∆ Std Rlt ∆ DD Rlt ∆ Min Rlt ∆
Unhedged 0.413 0.224 1.877 -22.763 -5.548
CTA hedged 0.499 20.773 0.246 9.678 1.705 -9.187 -17.453 -23.330 -5.011 -9.681
Table 3.9 reports summary statistics for unhedged and hedged carry trade portfolios. The construction rule
is detailed in the text. Hedged portfolio has 90% position in unhedged strategy and 10% position in CTA
strategy. The time period is from June 2000 to June 2012. Sharpe is the Sharpe ratio. Mean is the average
monthly return. Std is the volatility of monthly return. DD is the minimum drawdown. Min is the minimum
monthly return.
taking an additional position in hedging portfolio or not. For the setup of the model please
refer to Section 3.3. The minimal risk-aversion coefficient we test is 2. We show in Table
3.10 the results. The results show that all traders of risk-aversion coefficients higher than
Table 3.10: Break-even Risk-aversion Coefficients
CTA
0.1 2
0.3 2
0.5 2
0.7 2
0.9 2
Table 3.10 computes empirically the break-even risk-aversion coefficient for which trader is indifferent
between including a specified position of CTA strategy in the portfolio or not. The time period is from June
2000 to January 2014.
2 improve their utility by taking positions in the CTA strategy. This is the same result as
for the stop-loss strategies. Recall that for currency option or VIX strategy, only traders of
high risk-aversion coefficients improve utility by taking hedging positions. We thus show that
CTA strategy is a more efficient hedging strategy than currency options or VIX.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 96
3.6 Out-Sample Analysis
In this section we do an out-sample analysis for all hedging strategies. We divide the whole
sample period into two sub-periods. The in-sample period is from May 2000 to December
2007. The out-sample period is from January 2008 to June 2012. We first calibrate the
optimal hedging ratio using in-sample data. We then test the hedging portfolio using the
out-sample data.
We use Sharpe ratio as the objective function to compute optimal hedging ratio for in-sample
data. We report in Table 3.11 the optimal hedging ratios for the in-sample period.
Table 3.11: Optimal Hedging Ratio
Delta 50 Delta 25 Delta 10 VIX SL 1 SL 2 SL 3 SL 4 SL 5 CTA
OHR 0.00 0.00 0.00 0.00 0.20 0.25 1.00 0.00 0.00 0.12
This table reports the optimal hedging ratios for various hedging strategies.
We report next in Table 3.12 financial performance of hedged portfolio for the in-sample
period.
Table 3.12: In-sample Hedging Performance
Mean Volatility Skew Kurtosis Min Max Sharpe
Unhedged 4.84 4.41 -0.17 2.96 -2.77 3.87 1.10
SL1 4.63 4.18 0.07 3.08 -2.22 4.07 1.11
Gain -4.32 5.15 19.95 5.21 0.88
SL2 4.75 4.30 0.01 2.97 -2.24 4.10 1.10
Gain -1.84 2.48 18.95 5.97 0.66
SL3 5.07 4.44 0.03 3.18 -2.47 4.48 1.14
Gain 4.71 -0.71 120.11 7.38 10.61 15.59 3.97
CTA 7.55 5.80 -0.18 2.41 -3.25 3.84 1.30
Gain 55.83 -31.48 -8.91 -18.55 -17.58 -0.94 18.52
The table presents the in-sample financial performances of hedged portfolios. Gain denotes the improvement
in percentage relative to unhedged portfolio. The in-sample period is from May 2000 to December 2007.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 97
Based on in-sample Sharpe ratio, stop-loss hedging with threshold 3% is chosen among
all stop-loss strategies. We report in Table 3.13 the financial performance of hedged portfolio
for out-sample period.
Table 3.13: Out-sample Hedging Performance
Mean Std Skewness Kurtosis Min Max Sharpe
Unhedged -0.69 9.72 -0.35 3.93 -8.73 6.47 -0.07
SL3 -0.42 8.00 0.76 3.27 -4.14 6.69 -0.05
Gain 39.90 17.71 52.54 3.37 26.96
CTA 0.66 9.68 -0.08 3.11 -6.57 7.49 0.07
Gain 195.79 0.38 24.80 15.69 196.15
The table presents the out-sample financial performances of hedged portfolios. Gain denotes the
improvement in percentage relative to unhedged portfolio. The out-sample period is from January 2008 to
June 2012.
We see that the hedging strategies have robust performance for out-sample period.
3.7 Two Issues Related to the Hedging Strategies
Here we study two issues related to the hedging instruments. In the first part of this section,
we compare the VIX strategy to various currency option strategies. The objective is to de-
termine if VIX is a cheaper form of systematic insurance as compared to currency options.
Caballero and Doyle [2012] studies this question and argues that compared to VIX, currency
options provide a cheap form of systematic insurance. We tackle the issue by comparing
the effects of different hedging strategies on the carry trade portfolio. First we compare the
financial performance of VIX hedging with that of currency options hedging. Secondly, we
look at the utility gain achieved by traders employing different hedging strategies. Lastly, we
study the tail hedging effects.
Overall, our findings, with respect to both financial performance and tail risk hedging quality,
suggest that Delta 10 is the least expensive in terms of hedging, followed by Delta 25 and
VIX, while Delta 50 is the most expensive. This result helps to shed light on the mispricing
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 98
issue with vis-a-vis FX options, and makes a contribution to the literature testing crash risk
explanations of carry trade excess returns.
In the second part of this section, we study the risk-return analysis of CTA strategy. We
provide a new methodology for replicating the returns to a benchmark CTA index. We find
that the risks and returns of CTA benchmark index can be matched with simple trend follow-
ing strategy. The trend following strategy provides transparency over its risk management
and trend-picking period. Fung and Hsieh [2001] provides evidence that the non-linear risk
exposure of CTA strategy resembles that of lookback straddle options. However, this paper
doesn’t provide evidence that the CTA funds implement momentum strategy through look-
back straddle options. We construct a CTA index by specifying technical trading rules, and
we find that our CTA index replicates with statistical significance the return of the bench-
mark CTA index. The replication delivers high R2. The constructed CTA index also does a
superior job of matching the drawdown patterns of the benchmark CTA index. Some previ-
ous papers construct passive CTA index by specifying technical trading rules; see Schneeweis
et al. [2001]. Our innovation is to replicate the benchmark CTA index by portfolio generation
using explicit rules.
3.7.1 Compare the price of VIX futures to currency options
After studying the hedging effects of currency options and VIX, we compare now the VIX
strategy to various currency option strategies. The objective is to determine if VIX is a
cheaper systematic insurance compared to currency options. Caballero and Doyle [2012]
studies this question; they argue that, compared to VIX, currency options provide a cheap
form of systematic insurance. We tackle the issue by comparing the hedging effects of different
hedging strategies on the carry trade portfolio. In doing this, we extensively draw on previous
results. First we compare the financial performance statistics in 3.3.2. We look at the hedging
effects on minimum drawdown, minimum monthly return, return volatility, average return
and Sharpe ratio, and we find that currency options and VIX offer the same return smoothing
and tail hedging effect. While Delta 10 is the least expensive, Delta 50 is the most expensive,
with Delta 25 and VIX in between. Secondly, we study in the framework of the investor’s
utility maximization problem, as in 3.3.3. The result shows that currency options are more
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 99
expensive than VIX for a currency trader.
Thirdly, we study the tail hedging effects. We compare the hedging returns for the ten months
during which carry trade suffered the most substantial losses, as in 3.3.4. We also propose
a measure called hedging quality. For a CRRA trader, we compute the average utility gain
during a crisis period, thanks to hedging pay-off. We also compute the average utility cost in
non-crisis periods due to constructing the hedging portfolio. We define the ratio of hedging
utility gain and hedging utility cost as hedging quality. We find that Delta 10 has the highest
hedging quality, followed by Delta 25, VIX and Delta 50. We also conduct a robustness test.
3.7.1.1 Financial performance comparison
We have the relevant results in Table C.1 in the appendix. We find that currency options
and VIX reduce, by roughly the same amount, minimum monthly return and monthly return
volatility, ranging respectively between 37% and 40%, and 13% and 15%. They are, however,
all costly to buy, while more out-of-money options are less costly on average. The Delta 10
strategy decreases the average return by 30%. Both Delta 25 and VIX decrease by 37%, while
Delta 50 decreases by 50%. More out-of-money options perform better in terms of Sharpe
ratio. The Delta 10 strategy decreases the Sharpe ratio by 19%. Delta 25 and VIX decrease
between 26% and 27%, and Delta 50 decreases by 41%. As the options are costly to buy,
the effect on minimum drawdown is less significant. The Delta 10 strategy reduces minimum
drawdown by 9%, VIX by 8%, Delta 25 by 4% and Delta 50 by 2%. In general, financial
performance statistics show that currency options and VIX offer the same return smoothing
and tail hedging effect. While Delta 10 is the least expensive, followed by Delta 25 and VIX,
Delta 50 is the most expensive.
3.7.1.2 Utility improvement for traders
We study in the framework of the investor’s utility maximization problem, as in 3.3.3.We
have the results in Table 3.4. We find that VIX delivers the lowest break-even risk-aversion
coefficient, followed by Delta 10, Delta 25 and Delta 50. It suggests that currency options
are more expensive than VIX for the CRRA utility trader.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 100
3.7.1.3 Tail hedging effects
We compare now the tail hedging effects between different option hedging strategies and the
VIX future strategy. We recall the result in 3.3.4, where we compared the hedging returns
for the ten months during which carry trade suffered the largest losses. We found that Delta
50 gives the highest hedging return, followed by Delta 25, VIX and Delta 10.
We now propose a new measure called tail-hedging quality. For a CRRA trader, we compute
the average utility gain during a crisis period, thanks to hedging pay-off. We also compute the
average utility cost in non-crisis periods, due to constructing the hedging portfolio. We define
the ratio of hedging utility gain and hedging utility cost as hedging quality. We find that
Delta 10 has the highest hedging quality, followed by Delta 25, VIX and Delta 50. This result
is robust for traders with different risk-aversion coefficients. Using a resampling bootstrap,
we find it significant that the more out-of-money currency option has higher hedging quality,
and it is not significant that currency option strategies have higher hedging quality than VIX
strategy. We describe in the following the definition of tail hedging quality, the empirical
result and the robustness test.
Definition
We study the hedging gain and cost from the view of currency trader. We model the trader
to have CRRA utility function.
In a crisis period T + 1, assume the carry portfolio suffers loss of −rc, without hedging the
trader’s utility is thus U(ST (1− rc)). With hedging strategy hi the trader’s utility would be
U(ST (1− rc + rhi )), where rhi is the return of hedging strategy i. The utility gain of hedging
strategy i is thus :
GT+1i (rc, r
hi ) =
U(ST (1− rc + rhi ))− U(ST (1− rc))abs(U(ST (1− rc)))
= 1− (1− rc + rhi
1− rc)1−γ (3.7)
Denote by ΩT+1C the set of events corresponding to crisis at period T + 1. Given the joint
distribution FC(rc, rhi ) of rc, rhi conditional on ΩT+1
C , 3.7 computes the expected utility
gain of hedging strategy i conditional on ΩT+1C , which is:
ET [GT+1i |ΩT+1
C ] =
∫GT+1i (rc, r
hi ) dF (rc, r
hi ) =
∫1− (
1− rc + rhi1− rc
)1−γ dFC(rc, rhi ) (3.8)
We now compute hedging gains given the observation on portfolio returns during crisis period.
Suppose we observe M crisis periods, with carry trade portfolio loss rtc, hedging portfolio
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 101
return rh,ti for crisis period t, with 1 ≤ t ≤ M . Given the risk-aversion coefficient γ, the
average hedging gain of strategy i for M crisis periods is
Gi(γ) =
∑Mt=1 1− (
1−rtc+rh,ti
1−rtc)1−γ
M(3.9)
Similarly we compute the hedging cost. In a non-crisis period T+1, assume the carry portfolio
gains return of rc, hedging strategy i suffers loss of rhi . The utility cost of hedging strategy i
is thus :
CT+1i (rc, r
hi ) =
U(ST (1− rc))− U(ST (1− rc + rhi ))
abs(U(ST (1− rc)))= (
1 + rc − rhi1 + rc
)1−γ − 1 (3.10)
Denote by ΩT+1NC the set of events corresponding to non-crisis at period T +1. Given the joint
distribution FNC(rc, rhi ) of rc, rhi conditional on ΩT+1
NC . The following equation computes
the expected utility cost of hedging strategy i conditional on ΩT+1NC :
ET [CT+1i |ΩT+1
NC ] =
∫CT+1i (rc, r
hi ) dFNC(rc, r
hi ) =
∫(1 + rc − rhi
1 + rc)1−γ − 1 dFNC(rc, r
hi )
(3.11)
Again suppose we observe K non-crisis periods, with carry trade portfolio return rtc, hedging
portfolio loss rh,ti for non-crisis period t, with 1 ≤ t ≤ K. Given the risk-aversion coefficient
γ the average hedging cost of strategy i for M crisis periods is thus:
Ci(γ) =
∑Kt=1(
1+rtc−rh,ti
1+rtc)1−γ − 1
M(3.12)
We use the ratio of Gi(γ) and Ci(γ) to measure the hedging quality Qi of hedging strategy i.
Qi =Gi(γ)
Ci(γ)(3.13)
Result
Jackwerth [2000] and Bliss and Panigirtzoglou [2004] estimate the CRRA risk aversion coef-
ficient using equity option data. They find γ to be between 2 and 10. We report in Table
C.4 and Table C.5, in the appendix, the average utility gain for crisis periods and average
utility loss for non-crisis periods of different hedging strategies. We take into consideration
six months listed in Table C.3, where carry trade suffered the largest losses as a crisis period.
Another 86 months are considered as a non-crisis period. In computation, we assume that the
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 102
hedged portfolio takes one position in unhedged portfolios and one position in currency op-
tions or VIX future portfolios. The VIX future portfolio is normalized such that the volatility
of the VIX portfolio is the same as that of the Delta 25 currency option strategy.
We see that utility gain is an increasing function of risk aversion coefficient. Traders who
are more risk-averse value more the hedging gains. Across the different hedging strategies,
Delta 50 has the highest utility gain, followed by Delta 25, VIX and Delta 10, which has the
least utility gain. Utility cost is also an increasing function of risk aversion coefficient. Better
insurance thus demands higher prices. Across the different hedging strategies, Delta 50 has
the highest utility cost, followed by Delta 25, VIX and Delta 10, which has the lowest utility
cost. We report in Table 3.14 the hedging quality Qi, computed from 3.13.
Table 3.14: Hedging Quality
50 Delta 25 Delta 10 Delta VIX
γ = 2 4.80 6.59 8.30 6.18
γ = 3 4.68 6.48 8.21 6.05
γ = 4 4.57 6.36 8.12 5.93
γ = 5 4.46 6.25 8.04 5.81
γ = 6 4.36 6.14 7.96 5.70
γ = 7 4.26 6.03 7.87 5.59
γ = 8 4.16 5.93 7.79 5.48
γ = 9 4.06 5.83 7.71 5.37
γ = 10 3.97 5.72 7.63 5.27
Mean 4.37 6.15 7.96 5.71
Table 3.14 computes empirically the hedging quality for trader who takes one position in unhedged portfolio
and one position in hedging strategy. The VIX future portfolio is normalized such that the volatility of VIX
portfolio is the same as that of Delta 50 currency option strategy. The time period is from March 2004 to
June 2012.
We see that Delta 10 has the best hedging quality, followed by Delta 25 and VIX. Delta
50 has the lowest quality, which is 55% of that of Delta 10.
As a particularly interesting result, we find from Table 3.14 that a large variation of hedging
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 103
quality exists across currency options. The hedging quality increases as the options become
more out-of-money. Delta 50’s hedging quality is 55% of that of Delta 10. Delta 25’s is 79%
of that of Delta 10. VIX’s hedging quality is slightly below Delta 25. From the table, we
conclude that VIX’s hedging quality is better than Delta 50, worse than Delta 10, and almost
the same as Delta 25.
Robustness Test
In the last section, we conclude that hedging quality increases as options become more out-
of-money, and VIX has roughly the same hedging quality as the Delta 25 option. We now
test the robustness of our conclusion relative to varying risk aversion coefficients and different
samples of crisis periods.
From Table 3.14 we see that the relative order of hedging quality is robust to varying
risk-aversion coefficients. To test the robustness of the result relative to selection of crisis
period, we test for a larger sample of crisis period, including all ten months of Table C.3. We
compute the ratio of hedging qualities of all hedging strategies relative to that of Delta 10.
We then compare the ratios to the results of Table 3.14. We report in Table C.6 in appendix
the results. For example, the 1.06 for Delta 50 and γ = 3 is computed from Ratio10Ratio6
, where
Ratio10 = Q50DeltaQ10Delta
is the ratio of hedging qualities considering 10 months of Table 3.14 as
crisis period, and Ratio6 = Q50DeltaQ10Delta
is the ratio of hedging qualities taking 6 months of Table
3.14 where carry trade suffers biggest losses as crisis period. We see that the results are
robust, because the hedging quality ratio doesn’t change after using a larger sample of crisis
period.
Lastly we use bootstrap resampling to test robustness. We test hypothesis that the hedging
quality of one strategy is better than the othee one. We resample 1000 times. We report in
Table 3.15 the p-values. For example, the p-value of the hypothesis QDelta50 >= QDelta10 is
0.02.
From the above table we see that increasing hedging quality as option becomes more out-
of-money is robust. There is no significance for the comparison between VIX strategy and
currency option strategies.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 104
Table 3.15: Bootstrap Resampling
VIX 25 Delta 10 Delta
50 Delta 0.29 0.01** 0.02**
VIX 0.45 0.33
25 Delta 0.10*
Table 3.15 reports the p-value that one strategy has better hedging quality than other one from bootstrap
resampling.
3.7.1.4 Discussions and Implications
Overall, the financial performance and tail risk hedging quality suggest that Delta 10 is the
least expensive in terms of hedging, followed by Delta 25 and VIX, while Delta 50 is the most
expensive. Our result helps to shed light on the mispricing issue of FX options, which is
important for the literature testing crash risk explanation of carry trade excess returns. We
approach the problem by comparing the expense of currency options to VIX futures, which
is a benchmark insurance against systematic risk. Following Brunnermeier et al. [2008], one
possible explanation for the excess return of carry trade portfolios is compensation for crash
risk. In order to test the effect of crash risk, Jurek [2014] constructs crash hedged carry trade
portfolios using out-of-money put options. By comparing returns of hedged and unhedged
carry trade portfolios, it concludes that crash risk could explain only 25% of carry trade excess
returns. One assumption of Jurek’s approach is that currency options are correctly priced. If
currency options were mispriced and implied crash risks were lower than the accurate crash
risk, then hedged carry trade portfolios could earn higher returns. Jurek [2014] discusses
this issue by comparing the option implied variance and skewness with realized variance and
skewness. It finds variance premium and skewness premium for currency options, which
suggests that options are not mispriced. Our computation shows that, overall, VIX hedging
is as expensive as Delta 25 hedging, and more expensive than Delta 10 hedging, while VIX
hedging is less expensive than Delta 50 hedging. Our evidence shows that the more in-the-
money Delta 25 and Delta 50 are not underpriced compared to VIX futures.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 105
3.7.2 CTA risk/return analysis
We constructed a CTA portfolio with explicit rules in Section 3.5. We now show that the
benchmark CTA return index can be replicated by our simulated CTA portfolio. We use the
Newedge CTA Index as the benchmark index, which tracks the largest 20 (by asset under
management) CTAs. This is representative of the managed future funds. Funds selected for
Newedge CTA Index meet the following two criteria: 1) Funds are open to new investment.
2) Funds report returns on a daily basis.
We set the target volatility of the simulated portfolio such that the volatility of the portfolio
for the whole sample is the same as the volatility of the CTA benchmark index. We show in
Table 3.16 the summary statistics of the simulated CTA portfolio and Newedge CTA Index.
We plot in Figure 3.3 the total return indices of our simulated portfolio and the Newedge
CTA Index.
Table 3.16: CTA Performance Statistics
Mean Volatility Skewness Kurtosis Min Max Sharpe
Simulated 7.96 14.94 0.30 2.80 -9.04 12.57 0.53
Newedge 8.84 14.94 0.18 3.62 -13.45 14.61 0.59
Table 3.16 reports summary statistics for the simulated CTA portfolio and Newedge CTA index. The time
period is from June 2000 to January 2014.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 106
Figure 3.3: CTA Total Return Indices
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 20150.5
1
1.5
2
2.5
3
3.5Total return indices
Simulated Portfolio
NewEdge
Figure 3.3 plots the total return indices of the simulated CTA portfolio and Newedge CTA index for the period
from June 2000 to January 2014.
From the graph above we see that the simulated portfolio and the benchmark index
comove. We study next the question of replicating CTA benchmark index using the simulated
CTA portfolio. We first run the following regression:
rrtb = βrrts + εt (3.14)
Where in the equation above rrtb is the return of the benchmark index, rrts is the return of
the simulated portfolio. We have the following result: We see β = 0.7 is significant at 0.01%
Table 3.17: Regression Analysis
value std
α · 100 0.06 0.05
β 0.67 0.02
R2 0.51
level. R-square is 0.52.
We construct a replication portfolio which has return rtrep = βrrts+(1− β)rrs at period t. We
plot next the total return indices of the benchmark portfolio and the replication portfolio.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 107
Figure 3.4: Benchmark Index and Replication Portfolio
2000 2002 2005 2007 2010 2012 20150.5
1
1.5
2
2.5
3
3.5Total return indices
NewEdge
Replication
Figure 3.4 plots the total return indices of Newedge CTA Index and the replication portfolio.
We see from the figure above that the replication portfolio replicates the return of the
benchmark portfolio. In addition the replication portfolio has lower volatility than the bench-
mark portfolio.
We conduct next out-sample analysis.
3.7.2.1 Out-sample Analysis
In this subsection we do the out-sample analysis for CTA benchmark return index replication.
We first look at the out-sample behavior of the return’s replication.
Replication of return
We divide the whole sample to the in-sample and the out-sample periods. The in-sample
period is from Juanary 2000 to July 2009. The out-sample period is from August 20009 to
January 2014. We estimate the β by using Eq. 3.16 for respectively the in-sample period(βIS)
and the out-sample period((βOS)). We then test the H0 hypothesis that βIS 6= βOS . We
report in Table 3.18 the results.
We see from Table 3.18 that the H0 hypothesis βIS 6= βOS is rejected with p-value 0.51. We
next construct the out-sample replication portfolio rtrep,OS = ˆβISrrts,OS + (1− ˆβIS)rrs,OS by
using the βIS estimated using the in-sample data. We plot in Fig. 3.5 the total return indices
of the replication portfolio and Newedge index for the out-sample period.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 108
Table 3.18: Replication Residue
β std R2 t-Stat(H0) p-value(H0)
In Sample 0.69 0.03 0.51
Out Sample 0.66 0.04 0.55 -0.66 0.51
Std is the standard deviation of β. p-value is for the hypothesis that βIS 6= βOS .
Figure 3.5: Out-Sample Return Replication
2009 2010 2011 2012 2013 2014 20150.9
0.95
1
1.05
1.1
1.15
1.2
1.25Out Sample Total return indices
NewEdge
Replication
Figure 3.5 plots the total return indices of Newedge CTA Index and the replication portfolio.
We see in Fig. 3.5 that the out-sample replication portfolio tracks well the benchmark
total return index. We next do similar out-sample analysis for the drawdown time series.
Replication of drawdown
We define the drawdown seris of a return time series as the following:
ddt = min(0, rt) (3.15)
We denote the drawdown series of the benchmark index return as ddtb, the drawdown series
of the simulated portfolio return as ddts. We look at the replication behavior for drawdown
series, where we run the following regression respectively for the in-sample data and the
out-sample data.
ddtb = βddts + εt (3.16)
We then test the H0 hypothesis that βIS 6= βOS . We report in Table 3.19 the results.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 109
Table 3.19: Out-Sample Drawdown Replication
β std R2 t-Stat(H0) p-value(H0)
In Sample 0.59 0.06 0.30
Out Sample 0.62 0.07 0.42 0.44 0.66
Std is the standard deviation of β. p-value is for the hypothesis that βIS 6= βOS .
We see from Table 3.19 that the H0 hypothesis βIS 6= βOS is rejected with p-value 0.66.
The out-sample analysis shows that the replication is robust. We conduct next factor analysis
of CTA risk/return.
3.7.2.2 Explanation using traditional factors
We test the explanation of CTA benchmark index using traditional factors. In Table 3.20 we
report the result for Fama-French’s three factors model and Fung-Hsieh’s five factors model.
We also report the results of combining the above model with our rule based portfolio. We see
that traditional factors models don’t explaint the CTA benchmark index return. The loadings
on the rule based portfolio are significant in models combining traditional factos and rule-
based factor. In the table Mkt-Rf, SMB and HML are Fama-French’s three factors(Fama and
French [1993]). PTFSBD, PTFSFX, PTFSCOM, PTFSIR and PTFSSTK are Fung-Hsieh’s
five factors.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 110
Table 3.20: Explanation Using Traditional Factors and Simulated CTA Index
1 2 3 4 5 6 7
Intercept 0.83 0.70 1.03 0.25 0.22 0.17 0.30
0.34 0.34 0.36 0.26 0.26 0.27 0.28
Mkt-RF -0.14 -0.15 0.04 0.02
0.07 0.08 0.06 0.06
SMB 0.10 0.13
0.13 0.10
HML 0.19 0.00
0.11 0.08
PTFSBD 0.03 0.04
0.04 0.03
PTFSFX -0.01 -0.03
0.03 0.02
PTFSCOM 0.06 0.04
0.04 0.03
PTFSIR -0.00 -0.01
0.02 0.02
PTFSSTK 0.05 -0.02
0.04 0.03
RuleBased 0.65 0.66 0.67 0.66
0.06 0.06 0.06 0.06
R2 0.02 0.04 0.05 0.44 0.44 0.45 0.46
The table reports coefficients from monthly return regressions under several risk models over the period from
Junuary 2000 to June 2014. The dependent variable is the return of Newedge CTA index. Specification 1
corresponds to a CAPM-style model with a single factor computed as the S&P500. Specification 2
corredponds to the Fama-French model. Specification 3 corresponds to Fung-Hsieh’s 5-factor model.
Specification 4 corresponds to the model of rule based CTA. Specification 5 corresponds to Specification 1
plus the rule based CTA model. Specification 6 corresponds to Specification 2 plus the rule based CTA
model. Specification 7 corresponds to Specification 3 plus the rule based CTA model.
CHAPTER 3. CURRENCY CARRY TRADE HEDGING 111
3.8 Conclusion
In this chapter, I have studied the hedging problem associated with currency carry trade. I
have proposed theoretical frameworks and divided hedging instruments into three categories
according to the expected return of the hedging instrument and its correlation with the carry
trade: insurance, technical rule, and the market neutral strategy.
I then proposed and empirically tested four hedging strategies: FX options strategy, VIX
future strategy, “Stop-loss” rule and CTA strategy. Based upon empirical evidence from
2000 to 2012, I have found that CTA is the preferred hedging strategy because it upgrades
both return and volatility. The stop-loss strategy reduces risk. Both the currency options
strategy and the VIX future strategy offer good hedges against tail risk, while also reducing
volatility. However they worsen significantly the portfolio’s average return.
BIBLIOGRAPHY 112
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APPENDIX A. APPENDIX FOR CHAPTER 1 121
Appendix A
Appendix for Chapter 1
Table A.1: Variance Decomposition Error Bands
1Y Bond
rr ErrBand g ErrBand π ErrBand u ErrBand
1Month 0.00 [0.00,0.04] 0.01 [0.00,0.07] 0.31 [0.21,0.41] 0.67 [0.54,0.77]
1Y 0.01 [0.00,0.14] 0.23 [0.04,0.45] 0.56 [0.32,0.71] 0.21 [0.13,0.34]
5Y 0.32 [0.02,0.66] 0.28 [0.02,0.59] 0.32 [0.13,0.64] 0.08 [0.04,0.23]
5Y Bond
rr ErrBand g ErrBand π ErrBand u ErrBand
1Month 0.02 [0.00,0.07] 0.01 [0.00,0.04] 0.93 [0.87,0.95] 0.04 [0.03,0.06]
1Y 0.16 [0.03,0.41] 0.02 [0.00,0.18] 0.80 [0.51,0.91] 0.02 [0.01,0.04]
5Y 0.65 [0.07,0.79] 0.03 [0.00,0.26] 0.31 [0.17,0.78] 0.01 [0.00,0.03]
10Y Bond
rr ErrBand g ErrBand π ErrBand u ErrBand
1Month 0.06 [0.02,0.13] 0.01 [0.00,0.06] 0.92 [0.84,0.97] 0.00 [0.00,0.00]
1Y 0.31 [0.11,0.53] 0.01 [0.00,0.10] 0.68 [0.44,0.86] 0.00 [0.00,0.00]
5Y 0.75 [0.17,0.85] 0.00 [0.00,0.13] 0.24 [0.14,0.74] 0.00 [0.00,0.00]
This table shows the error band for variance decomposition. ErrBand is the 90% error band computed
by using bootstrap. rr represents the real neutral rate, g represents the GDP gap, π represents the expected
inflation, and u represents the Taylor rule residue.
APPENDIX A. APPENDIX FOR CHAPTER 1 122
Figure A.1: Macro-factors in Taylor Rule
APPENDIX A. APPENDIX FOR CHAPTER 1 123
We plot below observed yields and model implied yields obtained in the section 1.7 for
bonds with maturity up to 10 years.
Figure A.2: Model Implied Yields and Observed Yields
1997 2000 2002 2005 2007 2010 2012 20150
1
2
3
4
5
6
7Observed and model implied Yields for 0.25 Years Bond
Model Implied Yield
Observed Yield
1997 2000 2002 2005 2007 2010 2012 20150
1
2
3
4
5
6
7Observed and model implied Yields for 0.5 Years Bond
Model Implied Yield
Observed Yield
1997 2000 2002 2005 2007 2010 2012 2015−1
0
1
2
3
4
5
6
7Observed and model implied Yields for 1 Years Bond
Model Implied Yield
Observed Yield
1997 2000 2002 2005 2007 2010 2012 20150
1
2
3
4
5
6
7Observed and model implied Yields for 2 Years Bond
Model Implied Yield
Observed Yield
1997 2000 2002 2005 2007 2010 2012 20150
1
2
3
4
5
6
7Observed and model implied Yields for 3 Years Bond
Model Implied Yield
Observed Yield
1997 2000 2002 2005 2007 2010 2012 20150
1
2
3
4
5
6
7Observed and model implied Yields for 4 Years Bond
Model Implied Yield
Observed Yield
APPENDIX A. APPENDIX FOR CHAPTER 1 124
1997 2000 2002 2005 2007 2010 2012 20150
1
2
3
4
5
6
7Observed and model implied Yields for 5 Years Bond
Model Implied Yield
Observed Yield
1997 2000 2002 2005 2007 2010 2012 20150
1
2
3
4
5
6
7Observed and model implied Yields for 6 Years Bond
Model Implied Yield
Observed Yield
1997 2000 2002 2005 2007 2010 2012 20150
1
2
3
4
5
6
7Observed and model implied Yields for 7 Years Bond
Model Implied Yield
Observed Yield
1997 2000 2002 2005 2007 2010 2012 20151
2
3
4
5
6
7Observed and model implied Yields for 8 Years Bond
Model Implied Yield
Observed Yield
1997 2000 2002 2005 2007 2010 2012 20151
2
3
4
5
6
7Observed and model implied Yields for 9 Years Bond
Model Implied Yield
Observed Yield
1997 2000 2002 2005 2007 2010 2012 20151
2
3
4
5
6
7Observed and model implied Yields for 10 Years Bond
Model Implied Yield
Observed Yield
APPENDIX A. APPENDIX FOR CHAPTER 1 125
Figure A.3: Impulse-responses Error Bands
This figure plots the 90% error band for impulse-response.
APPENDIX A. APPENDIX FOR CHAPTER 1 126
Appendix 2
Table A.2: Market Price of Risk
r g π u
r 0.92 -0.52 -0.16 0.14
(1.82) (1.75) (0.61) (1.22)
g 0.54 0.09 -0.35 0.32
(0.62) (0.20) (1.02) (1.95)
π -0.02 0.23 -0.48 -0.00
(0.11) (2.70) (2.46) (0.08)
u -0.45 -0.35 0.40 -0.33
(2.32) (2.76) (1.47) (4.07)
The table presents the estimates for the price of risk of the essentially affine model. The absolute value
of the t-ratio of each estimate is reported. The sample period is from September 1997 to August 2012. The
standard deviation is corrected following Newey and McFadden(1994).
Table A.3: Market Price of Risk for In-sample Period
r g π u
r 1.31 -0.65 -1.40 0.32
(2.00) (1.44) (1.25) (1.48)
g -0.29 -0.20 0.77 -0.04
(0.27) (0.62) (0.90) (0.16)
π 0.03 0.54 -0.71 -0.01
(0.12) (3.10) (2.03) (0.29)
u -0.44 -0.18 -0.59 -0.39
(1.64) (0.60) (1.05) (2.23)
The table presents the estimates for the market price of risk of essentially affine model. The absolute
value of the t-ratio of each estimate is reported. The sample period is from September 1997 to June 2008.
The standard deviation is corrected following Newey and McFadden(1994).
APPENDIX A. APPENDIX FOR CHAPTER 1 127
Table A.4: Model Implied Factor Loadings and Unrestricted Regression Factor Loadings
NEUT GDPGAP EI TAYRES
1M Unres Reg 1.00 1.00 1.00 1.00
1M Over-Iden Model 1.00 1.00 1.00 1.00
3M Unres Reg 0.87 1.01 1.18 0.96
3M Over-Iden Model 0.89 1.01 1.07 0.96
6M Unres Reg 0.68 1.00 1.28 0.89
6M Over-Iden Model 0.77 0.99 1.18 0.90
1Y Unres Reg 0.53 0.88 1.38 0.75
1Y Over-Iden Model 0.63 0.91 1.35 0.79
2Y Unres Reg 0.41 0.63 1.68 0.55
2Y Over-Iden Model 0.50 0.67 1.58 0.59
3Y Unres Reg 0.39 0.47 1.68 0.42
3Y Over-Iden Model 0.45 0.46 1.69 0.44
4Y Unres Reg 0.36 0.33 1.65 0.31
4Y Over-Iden Model 0.42 0.30 1.71 0.33
5Y Unres Reg 0.33 0.19 1.62 0.21
5Y Over-Iden Model 0.39 0.18 1.68 0.25
6Y Unres Reg 0.33 0.12 1.58 0.18
6Y Over-Iden Model 0.36 0.10 1.61 0.18
7Y Unres Reg 0.32 0.05 1.54 0.15
7Y Over-Iden Model 0.33 0.04 1.52 0.13
8Y Unres Reg 0.30 0.01 1.47 0.10
8Y Over-Iden Model 0.30 0.01 1.43 0.09
9Y Unres Reg 0.27 -0.04 1.39 0.05
9Y Over-Iden Model 0.27 -0.02 1.34 0.05
10Y Unres Reg 0.24 -0.08 1.31 0.01
10Y Over-Iden Model 0.24 -0.04 1.24 0.02
The table presents the factor loadings of the over-identified model with that of the unrestricted regression.
For example, 1M Unres Reg is the factor loading from unrestricted regressions. 1M Over-Iden Model is the
factor loading of the over-identified model. The sample period is from September 1997 to August 2012.
APPENDIX A. APPENDIX FOR CHAPTER 1 128
Table A.5: Variance Decomposition
1Y Bond
RealRate GDPGap Inflation TYRes
1Month 0.02 0.07 0.28 0.64
1Y 0.02 0.13 0.56 0.29
5Y 0.52 0.12 0.25 0.10
5Y Bond
RealRate GDPGap Inflation TYRes
1Month 0.03 0.01 0.90 0.06
1Y 0.17 0.02 0.78 0.04
5Y 0.69 0.02 0.28 0.01
10Y Bond
RealRate GDPGap Inflation TYRes
1Month 0.09 0.00 0.91 0.00
1Y 0.34 0.00 0.65 0.00
5Y 0.77 0.00 0.23 0.00
Table A.6: In-sample Pricing Error
Observed Mean RMSE Rlt Error(%)
1M 3.18 0.00 0.00
3M 3.39 0.21 6.23
6M 3.52 0.28 7.98
1Y 3.62 0.26 7.20
2Y 3.88 0.24 6.25
3Y 4.05 0.20 4.93
4Y 4.22 0.17 3.94
5Y 4.39 0.16 3.65
6Y 4.48 0.17 3.74
7Y 4.58 0.18 3.89
8Y 4.67 0.18 3.92
9Y 4.77 0.18 3.87
10Y 4.86 0.19 3.99
The table reports the in-sample pricing error. The Observed Mean is the mean of observed bond yield.
APPENDIX A. APPENDIX FOR CHAPTER 1 129
RMSE is the root mean square error. Rlt Error is the RMSE divided by the Observed Mean. The sample
period is from September 1997 to June 2008.
Table A.7: Out-sample Pricing Error
Observed Mean RMSE Rlt Error(%)
1M 0.07 0.00 0.00
3M 0.10 0.13 125.32
6M 0.17 0.20 121.54
1Y 0.27 0.32 120.20
2Y 0.58 0.37 64.91
3Y 0.91 0.46 50.32
4Y 1.29 0.47 36.93
5Y 1.67 0.51 30.54
6Y 2.00 0.51 25.61
7Y 2.34 0.53 22.46
8Y 2.55 0.56 22.09
9Y 2.76 0.63 22.84
10Y 2.96 0.75 25.23
The table reports the out-sample pricing error. The Observed Mean is the mean of observed bond yield.
RMSE is the root mean square error. Rlt Error is the RMSE divided by the Observed Mean. The sample
period is from July 2008 to August 2012.
APPENDIX A. APPENDIX FOR CHAPTER 1 130
Figure A.4: Macro-factors in Taylor Rule
APPENDIX A. APPENDIX FOR CHAPTER 1 131
Figure A.5: Contemporaneous Impulse Responses
0 20 40 60 80 100 120−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Bond maturities in Months
Annuliz
ed y
ield
in %
Initial response
RealRate
GDPGap
Inflation
TYRes
Figure A.6: Impulse Responses at Future Horizon
0 10 20 30 40 50 600
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Months
Annuliz
ed y
ield
in %
Initial response of 1Y Bond
RealRate
GDPGap
Inflation
TYRes
0 10 20 30 40 50 600
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Months
Annuliz
ed y
ield
in %
Initial response of 5Y Bond
RealRate
GDPGap
Inflation
TYRes
0 10 20 30 40 50 60−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Months
Annuliz
ed y
ield
in %
Initial response of 10Y Bond
RealRate
GDPGap
Inflation
TYRes
APPENDIX A. APPENDIX FOR CHAPTER 1 132
Appendix 3
Table A.8: Market Price of Risk
r g π u
r 0.30 -0.43 -0.20 0.01
(1.39) (1.83) (1.18) (0.16)
g -0.09 -0.01 0.11 0.01
(0.27) (0.03) (0.31) (0.04)
π -0.65 0.90 0.15 0.03
(1.58) (2.00) (0.43) (0.20)
u 0.14 -0.36 -0.05 -0.07
(0.85) (2.20) (0.28) (0.61)
The table presents the estimates for the price of risk of the essentially affine model. The absolute value
of the t-ratio of each estimate is reported. The sample period is from September 1997 to August 2012. The
standard deviation is corrected following Newey and McFadden(1994).
Table A.9: Market Price of Risk for In-sample Period
r g π u
r -0.33 -1.03 -0.80 0.11
(0.97) (2.25) (1.83) (0.45)
g -0.43 0.05 -0.12 0.18
(1.42) (0.07) (0.24) (2.04)
π 0.28 1.74 1.15 -0.11
(0.47) (2.31) (1.55) (0.28)
u -0.32 -0.33 -0.51 -0.33
(1.54) (1.02) (2.02) (1.71)
The table presents the estimates for the market price of risk of essentially affine model. The absolute
value of the t-ratio of each estimate is reported. The sample period is from September 1997 to June 2008.
The standard deviation is corrected following Newey and McFadden(1994).
APPENDIX A. APPENDIX FOR CHAPTER 1 133
Table A.10: Model Implied Factor Loadings and Unrestricted Regression Factor Loadings
NEUT GDPGAP EI TAYRES
1M Unres Reg 1.00 1.00 1.00 1.00
1M Over-Iden Model 1.00 1.00 1.00 1.00
3M Unres Reg 1.07 1.03 1.05 1.02
3M Over-Iden Model 1.00 1.01 1.02 0.98
6M Unres Reg 1.05 1.03 1.06 0.99
6M Over-Iden Model 1.01 1.01 1.03 0.95
1Y Unres Reg 1.05 0.96 1.00 0.90
1Y Over-Iden Model 1.04 0.97 1.02 0.89
2Y Unres Reg 1.17 0.83 0.97 0.77
2Y Over-Iden Model 1.11 0.84 0.97 0.77
3Y Unres Reg 1.18 0.70 0.88 0.65
3Y Over-Iden Model 1.16 0.71 0.91 0.65
4Y Unres Reg 1.18 0.58 0.80 0.53
4Y Over-Iden Model 1.18 0.59 0.84 0.55
5Y Unres Reg 1.17 0.47 0.72 0.43
5Y Over-Iden Model 1.18 0.48 0.77 0.46
6Y Unres Reg 1.15 0.40 0.69 0.39
6Y Over-Iden Model 1.16 0.39 0.70 0.39
7Y Unres Reg 1.14 0.32 0.66 0.35
7Y Over-Iden Model 1.12 0.32 0.63 0.32
8Y Unres Reg 1.09 0.28 0.60 0.29
8Y Over-Iden Model 1.08 0.26 0.57 0.26
9Y Unres Reg 1.04 0.23 0.54 0.23
9Y Over-Iden Model 1.02 0.22 0.51 0.21
10Y Unres Reg 0.99 0.18 0.49 0.17
10Y Over-Iden Model 0.96 0.20 0.45 0.17
The table presents the factor loadings of the over-identified model with that of the unrestricted regression.
For example, 1M Unres Reg is the factor loading from unrestricted regressions. 1M Over-Iden Model is the
factor loading of the over-identified model. The sample period is from September 1997 to August 2012.
APPENDIX A. APPENDIX FOR CHAPTER 1 134
Table A.11: Variance Decomposition
1Y Bond
RealRate GDPGap Inflation TYRes
1Month 0.00 0.03 0.14 0.83
1Y 0.01 0.05 0.27 0.67
5Y 0.02 0.20 0.30 0.48
5Y Bond
RealRate GDPGap Inflation TYRes
1Month 0.38 0.01 0.29 0.31
1Y 0.44 0.02 0.27 0.26
5Y 0.55 0.06 0.20 0.18
10Y Bond
RealRate GDPGap Inflation TYRes
1Month 0.68 0.01 0.25 0.05
1Y 0.76 0.01 0.17 0.05
5Y 0.86 0.02 0.09 0.03
Table A.12: In-sample Pricing Error
Observed Mean RMSE Rlt Error(%)
1M 3.18 0.00 0.00
3M 3.39 0.20 5.81
6M 3.52 0.26 7.41
1Y 3.62 0.32 8.84
2Y 3.88 0.50 12.89
3Y 4.05 0.50 12.29
4Y 4.22 0.48 11.36
5Y 4.39 0.46 10.49
6Y 4.48 0.46 10.31
7Y 4.58 0.46 10.01
8Y 4.67 0.45 9.57
9Y 4.77 0.43 9.01
10Y 4.86 0.42 8.57
The table reports the in-sample pricing error. The Observed Mean is the mean of observed bond yield.
APPENDIX A. APPENDIX FOR CHAPTER 1 135
RMSE is the root mean square error. Rlt Error is the RMSE divided by the Observed Mean. The sample
period is from September 1997 to June 2008.
Table A.13: Out-sample Pricing Error
Observed Mean RMSE Rlt Error(%)
1M 0.07 0.00 0.00
3M 0.10 0.11 104.25
6M 0.17 0.23 140.68
1Y 0.27 0.41 154.14
2Y 0.58 0.54 93.46
3Y 0.91 0.65 71.72
4Y 1.29 0.65 50.59
5Y 1.67 0.63 37.66
6Y 2.00 0.56 28.10
7Y 2.34 0.49 21.04
8Y 2.55 0.45 17.62
9Y 2.76 0.40 14.55
10Y 2.96 0.35 11.83
The table reports the out-sample pricing error. The Observed Mean is the mean of observed bond yield.
RMSE is the root mean square error. Rlt Error is the RMSE divided by the Observed Mean. The sample
period is from July 2008 to August 2012.
APPENDIX A. APPENDIX FOR CHAPTER 1 136
Figure A.7: Macro-factors in Taylor Rule
APPENDIX A. APPENDIX FOR CHAPTER 1 137
Figure A.8: Contemporaneous Impulse Responses
0 20 40 60 80 100 1200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Bond maturities in Months
Annuliz
ed y
ield
in %
Initial response
RealRate
GDPGap
Inflation
TYRes
Figure A.9: Impulse Responses at Future Horizon
0 10 20 30 40 50 60−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Months
Annuliz
ed y
ield
in %
Initial response of 1Y Bond
RealRate
GDPGap
Inflation
TYRes
0 10 20 30 40 50 600
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Months
Annuliz
ed y
ield
in %
Initial response of 5Y Bond
RealRate
GDPGap
Inflation
TYRes
0 10 20 30 40 50 600
0.05
0.1
0.15
0.2
0.25
Months
Annuliz
ed y
ield
in %
Initial response of 10Y Bond
RealRate
GDPGap
Inflation
TYRes
APPENDIX A. APPENDIX FOR CHAPTER 1 138
Appendix 4
Table A.14: Market Price of Risk
r π u
r 0.10 -6.43 -2.15
(0.07) (4.18) (1.47)
π 0.30 0.52 0.38
(1.38) (2.02) (1.67)
u -0.34 -0.49 -0.47
(1.61) (2.88) (3.06)
The table presents the estimates for the price of risk of the essentially affine model. The absolute value
of the t-ratio of each estimate is reported. The sample period is from September 1997 to August 2012. The
standard deviation is corrected following Newey and McFadden(1994).
Table A.15: Market Price of Risk for In-sample Period
r π u
r 1.81 -3.85 -0.02
(4.73) (3.36) (0.08)
π -0.13 0.04 0.04
(1.47) (0.26) (1.50)
u -0.20 -0.65 -0.31
(0.81) (2.28) (3.24)
The table presents the estimates for the market price of risk of essentially affine model. The absolute
value of the t-ratio of each estimate is reported. The sample period is from September 1997 to June 2008.
The standard deviation is corrected following Newey and McFadden(1994).
APPENDIX A. APPENDIX FOR CHAPTER 1 139
Table A.16: Model Implied Factor Loadings and Unrestricted Regression Factor Loadings
r π u
1M Unres Reg 1.00 1.00 1.00
1M Over-Iden Model 1.00 1.00 1.00
3M Unres Reg 0.97 1.18 1.00
3M Over-Iden Model 0.96 1.17 1.00
6M Unres Reg 0.91 1.29 0.97
6M Over-Iden Model 0.91 1.31 0.96
1Y Unres Reg 0.81 1.40 0.84
1Y Over-Iden Model 0.80 1.45 0.84
2Y Unres Reg 0.58 1.69 0.61
2Y Over-Iden Model 0.62 1.60 0.63
3Y Unres Reg 0.48 1.68 0.45
3Y Over-Iden Model 0.48 1.66 0.45
4Y Unres Reg 0.38 1.65 0.32
4Y Over-Iden Model 0.36 1.67 0.31
5Y Unres Reg 0.28 1.62 0.20
5Y Over-Iden Model 0.27 1.65 0.21
6Y Unres Reg 0.20 1.58 0.14
6Y Over-Iden Model 0.20 1.60 0.13
7Y Unres Reg 0.12 1.54 0.08
7Y Over-Iden Model 0.15 1.53 0.07
8Y Unres Reg 0.10 1.46 0.03
8Y Over-Iden Model 0.11 1.46 0.02
9Y Unres Reg 0.09 1.38 -0.01
9Y Over-Iden Model 0.08 1.38 -0.01
10Y Unres Reg 0.07 1.31 -0.05
10Y Over-Iden Model 0.05 1.30 -0.03
The table presents the factor loadings of the over-identified model with that of the unrestricted regression.
For example, 1M Unres Reg is the factor loading from unrestricted regressions. 1M Over-Iden Model is the
factor loading of the over-identified model. The sample period is from September 1997 to August 2012.
APPENDIX A. APPENDIX FOR CHAPTER 1 140
Table A.17: Variance Decomposition
1Y Bond
r π u
1Month 0.00 0.29 0.71
1Y 0.02 0.62 0.36
5Y 0.53 0.35 0.13
5Y Bond
r π u
1Month 0.02 0.93 0.05
1Y 0.20 0.77 0.03
5Y 0.71 0.28 0.01
10Y Bond
r π u
1Month 0.04 0.96 0.00
1Y 0.29 0.71 0.00
5Y 0.75 0.25 0.00
Table A.18: In-sample Pricing Error
Observed Mean Error Std Rlt Error(%)
1M 3.18 0.00 0.00
3M 3.39 0.24 7.17
6M 3.52 0.31 8.93
1Y 3.62 0.28 7.75
2Y 3.88 0.26 6.61
3Y 4.05 0.20 4.95
4Y 4.22 0.16 3.78
5Y 4.39 0.15 3.46
6Y 4.48 0.16 3.48
7Y 4.58 0.17 3.78
8Y 4.67 0.19 4.03
9Y 4.77 0.20 4.20
10Y 4.86 0.22 4.46
The table reports the in-sample pricing error. The Observed Mean is the mean of observed bond yield.
APPENDIX A. APPENDIX FOR CHAPTER 1 141
RMSE is the root mean square error. Rlt Error is the RMSE divided by the Observed Mean. The sample
period is from September 1997 to June 2008.
Table A.19: Out-sample Pricing Error
Observed Mean Error Std Rlt Error(%)
1M 0.07 0.00 0.00
3M 0.10 0.19 183.21
6M 0.17 0.44 268.27
1Y 0.27 0.63 235.58
2Y 0.58 0.64 110.25
3Y 0.91 0.67 73.95
4Y 1.29 0.63 49.17
5Y 1.67 0.61 36.33
6Y 2.00 0.56 27.86
7Y 2.34 0.53 22.46
8Y 2.55 0.52 20.43
9Y 2.76 0.52 18.84
10Y 2.96 0.52 17.59
The table reports the out-sample pricing error. The Observed Mean is the mean of observed bond yield.
RMSE is the root mean square error. Rlt Error is the RMSE divided by the Observed Mean. The sample
period is from July 2008 to August 2012.
APPENDIX A. APPENDIX FOR CHAPTER 1 142
Figure A.10: Macro-factors in Taylor Rule
APPENDIX A. APPENDIX FOR CHAPTER 1 143
Figure A.11: Contemporaneous Impulse Responses
0 20 40 60 80 100 120−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Bond maturities in Months
Annu
lized y
ield
in
%
Initial response
RealRate
Inflation
WRes
Figure A.12: Impulse Responses at Future Horizon
0 10 20 30 40 50 600
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Months
An
nu
lize
d y
ield
in
%
Initial response of 1Y Bond
RealRate
Inflation
WRes
0 10 20 30 40 50 600
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Months
An
nu
lize
d y
ield
in
%
Initial response of 5Y Bond
RealRate
Inflation
WRes
0 10 20 30 40 50 60−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Months
An
nu
lize
d y
ield
in
%
Initial response of 10Y Bond
RealRate
Inflation
WRes
APPENDIX B. APPENDIX FOR CHAPTER 2 144
Appendix B
Appendix for Chapter 2
Subsection 2.6.1
Proof for the maximization of Sharpe ratio for threshold portfolios:
E [rt+1(Ptfy)] =
∫∞y e−λxxλ dx.
e−λy= (y + 1/λ)
Thus :
Sh[rt+1(Ptfy)] = (y + 1/λ)e−12λy/σ
The derivative of Sh[rt+1(Ptfy)] is 12(1 − λy)e−
12λy/σ. Sh[rt+1(Ptfy)] is thus maximized at
y = 1λ .
APPENDIX B. APPENDIX FOR CHAPTER 2 145
Table B.1: Carry Trade Performance Statistics
Table B.1 reports summary statistics for currency carry trades implemented in G10
currencies. Summary statistics are reported over Jan.1990-Jun. 2012 (N = 270 months).
SR represents the Sharpe ratio. Means, volatilities and Sharpe ratios (SR) are annualized;
Min and Max report the smallest and largest observed monthly return.
AUD CAD CHF EUR GBP JPY NOK NZD SEK EQW
Mean 5.98 2.66 -0.61 3.59 0.64 1.22 2.59 4.72 4.65 3.40
Std 11.62 7.91 11.48 10.36 9.59 11.45 11.03 11.61 11.78 5.97
Skewness -0.54 0.74 -0.22 -0.31 -0.61 -0.43 -0.37 -0.22 -0.52 -0.81
Kurtosis 5.49 9.85 3.91 3.99 5.19 4.95 3.98 5.97 4.34 5.36
Minimum -16.12 -7.56 -13.04 -10.03 -11.88 -14.76 -11.91 -12.93 -14.44 -6.61
Maximum 9.59 14.84 11.69 9.74 8.86 11.41 7.99 14.25 9.56 5.18
SR 0.51 0.34 -0.05 0.35 0.07 0.11 0.24 0.41 0.39 0.57
N 270.00 270.00 270.00 270.00 270.00 270.00 270.00 270.00 270.00 270.00
APPENDIX B. APPENDIX FOR CHAPTER 2 146
Table B.2: Carry Trade Threshold Portfolios Performance Statistics
Table B.2 reports summary statistics for threshold portfolios of G10 currencies versus US
dollar. NrTr is the average number of trades in the portfolio.
Mean Std Skew Kurt Min Max Sharpe NrTr N
TR 0 3.40 5.97 -0.81 5.36 -6.61 5.18 0.57 9.00 270.00
TR 0.5 4.70 7.67 -0.60 6.78 -8.74 8.54 0.61 7.40 270.00
TR 1 5.27 8.42 -0.48 6.60 -10.41 9.42 0.63 6.10 270.00
TR 1.5 6.09 8.89 -0.66 6.71 -11.95 9.28 0.68 5.10 270.00
TR 2 6.81 9.49 -0.60 6.70 -12.73 11.37 0.72 4.20 270.00
TR 2.5 7.99 10.03 -0.42 5.59 -11.52 11.89 0.80 3.30 270.00
TR 3 7.32 10.39 -0.52 4.75 -10.79 9.59 0.70 2.50 270.00
TR 3.5 6.66 11.22 -0.52 5.00 -11.30 11.23 0.59 2.00 270.00
TR 4 7.31 10.77 -0.18 4.43 -9.57 11.41 0.68 1.60 270.00
TR 4.5 5.27 10.41 -0.58 6.71 -14.76 11.41 0.51 1.20 270.00
TR 5 1.04 8.03 -1.11 9.40 -13.25 7.44 0.13 0.70 270.00
TR 5.5 -0.14 5.93 -0.77 11.20 -8.73 7.53 -0.02 0.50 270.00
TR 6 0.67 5.16 -0.43 16.48 -8.26 7.83 0.13 0.40 270.00
Table B.3: Carry Trade Mean-variance Optimal Portfolios Performance Statistics
Table B.3 reports summary statistics for the long only mean-variance optimal portfolio, and
that of the equally weighted portfolio. Carry 1Y is the mean-variance portfolio using one
year window when estimating the currency correlations, Carry 2Y the one using two years.
The time period is from February 1992 to June 2012.
Mean Std Skewness Minimum Maximum Sharpe Ratio
EqWeighted 2.92 5.89 -0.76 -6.61 5.18 0.50
Carry 1Y 2.60 4.81 -0.57 -6.26 4.14 0.54
Carry 2Y 2.76 4.96 -0.69 -5.62 4.00 0.56
APPENDIX B. APPENDIX FOR CHAPTER 2 147
Table B.4: Carry Trade Threshold Portfolios Performance Statistics: Quantile Based Rule
Table B.4 reports summary statistics for threshold portfolios of G10 currencies versus USD.
Portfolios are constructed using ”quantile based threshold rule”. Portfolio1 contains the
carry trade having biggest forward premium. Portfolio2 contains 2 carry trades having
biggest forward premiums, etc.
Mean Std Skew Kurt Min Max Sharpe NrTr N
Portolio 1 5.80 12.46 -0.57 4.85 -14.23 9.79 0.47 1.00 270.00
Portolio 2 5.41 11.23 -0.38 4.96 -12.13 11.64 0.48 2.00 270.00
Portolio 3 5.59 10.14 -0.55 5.66 -13.35 9.28 0.55 3.00 270.00
Portolio 4 4.45 9.37 -0.60 5.96 -12.27 8.38 0.48 4.00 270.00
Portolio 5 4.28 8.74 -0.62 6.28 -11.87 8.48 0.49 5.00 270.00
Portolio 6 4.41 8.28 -0.68 6.76 -11.48 8.12 0.53 6.00 270.00
Portolio 7 4.32 7.62 -0.50 5.84 -7.94 8.17 0.57 7.00 270.00
Portolio 8 3.58 6.68 -0.63 5.67 -7.10 6.71 0.54 8.00 270.00
Portolio 9 3.42 5.98 -0.81 5.32 -6.61 5.18 0.57 9.00 270.00
APPENDIX B. APPENDIX FOR CHAPTER 2 148
Table B.5: Carry Trade Crash-Hedged Threshold Portfolios Performance Statistics
Table B.5 reports summary statistics for 10 Delta Put hedged threshold portfolios of G10
currencies versus US dollar for the period from January 1999 to June 2012. NrTr is the
average number of trades in the portfolio.
Mean Std Skew Kurt Min Max Sharpe NrTr N
TR 0 3.17 7.17 0.28 5.01 -5.92 7.94 0.44 9.00 161.00
TR 0.5 4.55 8.68 0.37 6.19 -7.16 10.41 0.52 6.92 161.00
TR 1 5.17 9.31 0.51 6.31 -7.68 11.41 0.56 5.57 161.00
TR 1.5 6.03 9.93 0.52 6.11 -7.93 11.41 0.61 4.63 161.00
TR 2 6.90 10.29 0.22 5.85 -9.25 11.41 0.67 3.73 161.00
TR 2.5 7.64 10.74 0.44 6.46 -9.25 13.25 0.71 2.65 161.00
TR 3 6.30 10.50 0.07 5.50 -9.25 10.81 0.60 1.89 161.00
TR 3.5 4.63 10.27 -0.27 5.83 -10.36 10.19 0.45 1.43 161.00
TR 4 4.87 10.21 -0.61 7.42 -12.49 10.19 0.48 1.05 161.00
TR 4.5 2.38 8.46 -1.45 13.08 -13.01 7.93 0.28 0.63 161.00
TR 5 -1.45 5.16 -5.12 42.94 -13.01 5.15 -0.28 0.25 161.00
TR 5.5 -0.50 2.87 -2.18 32.22 -5.81 5.15 -0.18 0.14 161.00
TR 6 -0.16 1.64 -6.80 60.86 -4.25 1.53 -0.10 0.09 161.00
APPENDIX B. APPENDIX FOR CHAPTER 2 149
Table B.6: Unhedged Carry Trade Threshold Portfolios Performance Statistics
Table B.6 reports summary statistics for unhedged threshold portfolios of G10 currencies
versus US dollar for the period from January 1999 to June 2012.
Mean Std Skew Kurt Min Max Sharpe N
TR 0 2.97 6.10 -0.57 5.14 -6.61 5.18 0.49 162.00
TR 0.5 5.03 8.61 -0.47 6.36 -8.74 8.54 0.58 162.00
TR 1 6.05 9.59 -0.40 6.00 -10.41 9.42 0.63 162.00
TR 1.5 7.27 10.19 -0.60 6.03 -11.95 9.28 0.71 162.00
TR 2 8.31 10.82 -0.57 6.19 -12.73 11.37 0.77 162.00
TR 2.5 9.24 11.14 -0.39 5.45 -11.52 11.89 0.83 162.00
TR 3 8.66 11.03 -0.43 4.86 -10.79 9.59 0.78 162.00
TR 3.5 7.29 11.48 -0.54 4.91 -11.30 9.59 0.63 162.00
TR 4 7.91 10.41 -0.11 4.63 -9.57 9.59 0.76 162.00
TR 4.5 4.88 9.04 -0.04 5.83 -8.92 9.59 0.54 162.00
TR 5 0.15 5.71 -0.32 9.58 -7.04 6.24 0.03 162.00
TR 5.5 -0.40 4.37 -0.70 14.92 -6.32 5.72 -0.09 162.00
TR 6 0.43 2.65 2.48 29.27 -3.23 5.72 0.16 162.00
APPENDIX B. APPENDIX FOR CHAPTER 2 150
Table B.7: Comparison of Hedged and Unhedged Carry Trade Threshold Portfolios
Table B.7 reports summary statistics for comparison of return, volatility and Sharpe ratio
between unhedged and hedged portfolios for the period from January 1999 to June 2012..
M H denotes the average return of hedged portfolio. M U denotes the average return of
unhedged portfolio. M Dif denotes the relative difference in percentage between M H and M
U. Vol H denotes the return volatility of hedged portfolio. Sh H denotes the Sharpe ratio of
hedged portfolio. Similarly Vol U, Vol Dif, Sh U, and Sh Dif are defined.
M H M U M Dif Vol H Vol U Vol Dif Sh H Sh U Sh Dif
TR 0 3.17 2.97 6.59 7.17 6.10 17.49 0.44 0.49 -9.28
TR 0.5 4.55 5.03 -9.63 8.68 8.61 0.85 0.52 0.58 -10.39
TR 1 5.17 6.05 -14.64 9.31 9.59 -2.97 0.56 0.63 -12.02
TR 1.5 6.03 7.27 -17.04 9.93 10.19 -2.51 0.61 0.71 -14.90
TR 2 6.90 8.31 -16.98 10.29 10.82 -4.89 0.67 0.77 -12.71
TR 2.5 7.64 9.24 -17.32 10.74 11.14 -3.65 0.71 0.83 -14.19
TR 3 6.30 8.66 -27.27 10.50 11.03 -4.89 0.60 0.78 -23.53
TR 3.5 4.63 7.29 -36.46 10.27 11.48 -10.56 0.45 0.63 -28.95
TR 4 4.87 7.91 -38.45 10.21 10.41 -1.98 0.48 0.76 -37.21
TR 4.5 2.38 4.88 -51.25 8.46 9.04 -6.47 0.28 0.54 -47.88
TR 5 -1.45 0.15 -1050.61 5.16 5.71 -9.62 -0.28 0.03 -1151.84
TR 5.5 -0.50 -0.40 26.04 2.87 4.37 -34.35 -0.18 -0.09 91.99
TR 6 -0.16 0.43 -137.83 1.64 2.65 -37.94 -0.10 0.16 -160.96
APPENDIX B. APPENDIX FOR CHAPTER 2 151
Table B.8: Comparison of Hedged and Unhedged Carry Trade Threshold Portfolios
Table B.8 reports summary statistics for relative strength of threshold portfolios versus
benchmark equally weighted portfolio in terms of Sharpe ratio for the period from January
1999 to June 2012. Hedged Sharpe Diff shows the difference of Sharpe ratio of hedged
threshold portfolios versus hedged equally weighted portfolio. Unhedged Sharpe Diff shows
the difference of Sharpe ratio of unhedged threshold portfolios versus unhedged equally
weighted portfolio. Relative change is the relative difference between Unhedged Sharpe Diff
and Hedged Sharpe Diff in percentage.
Hedged Sharpe Diff Unhedged Sharpe Diff Relative Change
TR 0.5 0.08 0.10 -15.97
TR 1 0.11 0.14 -21.33
TR 1.5 0.17 0.23 -27.01
TR 2 0.23 0.28 -18.68
TR 2.5 0.27 0.34 -21.19
APPENDIX B. APPENDIX FOR CHAPTER 2 152
Figure B.1: Cumulative G10 carry trade portfolio returns
1987 1990 1992 1995 1997 2000 2002 2005 2007 2010 20120.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Figure B.1 illustrates the total return indices for G10 currency carry trade portfolio over the period from
January 1990 to June 2012.
Figure B.2: G10/USD currencies’ volatilities
1987 1990 1992 1995 1997 2000 2002 2005 2007 2010 20120
0.005
0.01
0.015
0.02
0.025
0.03
AUD
CAD
CHF
EUR
GBP
JPY
NOK
NZD
SEK
Figure B.2 illustrates the G10/USD currencies’ volatilities from February 1990 to June 2012. The volatility
index is calculated by exponential weighted moving average formula: EMV 2t = α ∗ EMV 2
t−1 + (1 − α) ∗ r2t ,
where rt is the currency return, EMVt is the exponential weighted moving average volatility, α = 0.98.
APPENDIX B. APPENDIX FOR CHAPTER 2 153
Figure B.3: Threshold portfolios of G10 currencies versus US dollar: Part 1
1987 1990 1992 1995 1997 2000 2002 2005 2007 2010 20120
1
2
3
4
5
6Threshold portfolios of G10 currencies versus US dollar
TR 0
TR 0.5
TR 1
TR 1.5
TR 2
TR 2.5
Figure B.4: Threshold portfolios of G10 currencies versus US dollar: Part 2
1987 1990 1992 1995 1997 2000 2002 2005 2007 2010 20120
1
2
3
4
5
6Threshold portfolios of G10 currencies versus US dollar
TR 2.5
TR 3
TR 3.5
TR 4
TR 4.5
TR 5
TR 5.5
TR 6
Figure B.3 and Figure B.4 illustrate the total return indices for threshold portfolios of G10 currencies versus
US dollar over the period from January 1990 to June 2012.
APPENDIX B. APPENDIX FOR CHAPTER 2 154
Figure B.5: Mean-variance Optimal Portfolio
1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 20140.8
1
1.2
1.4
1.6
1.8
2
2.2Mean−variance optimal portfolio
EqWeighted
Carry 1Y
Carry SY
Figure B.5 illustrates the total return indices of two long only mean-variance optimal portfolio, and that of
the equally weighted portfolio. Carry 1Y is the mean-variance portfolio using one year window for estimating
the currency correlations, Carry 2Y the one using two years. The time period is from January 1990 to June
2012.
APPENDIX B. APPENDIX FOR CHAPTER 2 155
Figure B.6: 10 Delta Put hedged threshold portfolios of G10 currencies versus US dollar:
Part 1
1997 2000 2002 2005 2007 2010 20120.5
1
1.5
2
2.5
310Delta Put Hedged Threshold portfolios of G10 currencies versus US dollar
TR 0
TR 0.5
TR 1
TR 1.5
TR 2
TR 2.5
Figure B.7: 10 Delta Put hedged threshold portfolios of G10 currencies versus US dollar:
Part 2
1997 2000 2002 2005 2007 2010 20120.5
1
1.5
2
2.5
310Delta Put Hedged Threshold portfolios of G10 currencies versus US dollar
TR 2.5
TR 3
TR 3.5
TR 4
TR 4.5
TR 5
TR 5.5
TR 6
Figure B.6 and Figure B.7 illustrate the total return indices for 10 Delta Put hedged threshold portfolios of
G10 currencies versus US dollar over the period from January 1999 to June 2012.
APPENDIX B. APPENDIX FOR CHAPTER 2 156
Figure B.8: Threshold portfolios of G10 currencies versus US dollar: Part 1
1997 2000 2002 2005 2007 2010 20121
1.5
2
2.5
3
3.5Threshold portfolios of G10 currencies versus US dollar
TR 0
TR 0.5
TR 1
TR 1.5
TR 2
TR 2.5
Figure B.9: Threshold portfolios of G10 currencies versus US dollar: Part 2
1997 2000 2002 2005 2007 2010 20120.5
1
1.5
2
2.5
3
3.5Threshold portfolios of G10 currencies versus US dollar
TR 2.5
TR 3
TR 3.5
TR 4
TR 4.5
TR 5
TR 5.5
TR 6
Figure B.8 and Figure B.9 illustrate the total return indices for threshold portfolios of G10 currencies versus
US dollar over the period from January 1999 to June 2012.
APPENDIX B. APPENDIX FOR CHAPTER 2 157
Figure B.10: Unhedged and hedged portfolios of TR0 and TR2.5
1997 2000 2002 2005 2007 2010 20120.5
1
1.5
2
2.5
3
3.5Unhedged and hedged portfolios of TR0 and TR2.5
Hedged TR 0
Hedged TR 2.5
Unhedged TR 0
Unhedged TR 2.5
Figure B.10 illustrate the total return indices for unhedged and hedged threshold portfolios of threshold 0 and
2.5% over the period from January 1999 to June 2012.
Figure B.11: Empirical distribution of forward premiums and fitted exponential distribution
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
F(x
)
Empirical CDF
Exp(2.23)
Forward Premium
Figure B.11 plots the CDF of the empirical distribution of the forward premiums and the fitted exponential
distribution having mean 2.23.
APPENDIX B. APPENDIX FOR CHAPTER 2 158
Figure B.12: Threshold portfolios’ Sharpe ratios versus random-walk model implied values
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.8
0.9
1
1.1
1.2
1.3
1.4
1.5Threshold portfolios Sharpe ratios versus model implied values
Threshold value
Sh
arp
e r
atio
Model implied
Empirical test
Figure B.12 plots the Sharpe ratio of G10 currency carry trade threshold portfolios versus the Sharpe ratios
implied by the random-walk model with λ = 0.45. The Sharpe ratios of zero threshold portfolios are normalized
to 1.
APPENDIX B. APPENDIX FOR CHAPTER 2 159
Figure B.13: Sharpe ratios of crash-hedged threshold portfolio with equal implied volatility
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050.9
1
1.1
1.2
1.3
1.4
1.5
1.6
Threshold Value
Port
folio
Sharp
e R
atio
Crash Hedged Threshold Portfolio
V10 Pre5
V10 Pre15
V15 Pre5
V15 Pre15
V20 Pre5
V20 Pre15
Figure B.14: Sharpe ratios of crash-hedged threshold portfolio with variable implied volatility
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.051
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Threshold Value
Port
folio
Sharp
e R
atio
Crash Hedged Threshold Portfolio, Variable Volatility
V10 Pre5
V10 Pre15
V15 Pre5
V15 Pre15
V20 Pre5
V20 Pre15
Figure B.13 and B.14 plot the Sharpe ratio of simulated threshold portfolios. We simulate 1000 carry trades
for 500 periods. The funding currency interest rate has uniform distribution of [0, 2%]. The forward premium
has exponential distribution with mean 2%. The currency volatility is 20%. Figure B.13 plots the case for
equal implied option volatility of 24%. Figure B.14 plots variable implied volatility as explained in Section
6.4.2. Vol 10 Pre5 means currency volatility of 10% and implied volatility premium of 5% against historical
volatility. Similarly for others.
APPENDIX C. APPENDIX FOR CHAPTER 3 160
Appendix C
Appendix for Chapter 3
Table C.1: Carry Trade Hedged using Options or VIX Futures
Sharpe Rlt ∆ Mean Rlt ∆ Std Rlt ∆ DD Rlt ∆ Min Rlt ∆
Carry 0.418 0.227 1.883 -23.836 -8.731
Delta 10 0.337 -19.402 0.159 -30.120 1.632 -13.298 -21.729 -8.843 -5.526 -36.714
Delta 25 0.309 -26.177 0.144 -36.613 1.616 -14.136 -22.773 -4.462 -5.416 -37.967
VIX 0.306 -26.790 0.142 -37.376 1.610 -14.460 -21.986 -7.762 -5.263 -39.725
Delta 50 0.249 -40.526 0.114 -49.673 1.593 -15.380 -23.434 -1.690 -5.525 -36.720
Table C.1 reports summary statistics for carry trade portfolio hedged with currency options and VIX future.
The time period is from June 2000 to June 2012. VIX future data is missing from June 2000 to February
2004. For that period VIX future return is replicated from currency options. Carry represents the unhedged
carry trade portfolio.
APPENDIX C. APPENDIX FOR CHAPTER 3 161
Table C.2: Carry Trade Hedged using Options or VIX Futures: Change in Performance
Der Sh Der Mean Der Std Der DD ∆dd/∆mean Der Min ∆Min/∆mean
Delta 50 -0.0149 -0.0087 -0.0185 -0.0957 -11.0339 0.1028 11.8509
Delta 25 -0.0103 -0.0061 -0.0165 -0.0296 -4.8939 0.1137 18.7798
Delta 10 -0.0069 -0.0041 -0.0148 0.0748 18.1641 0.1027 24.9451
VIX -0.0097 -0.0057 -0.0165 0.0491 8.5626 0.1290 22.5209
Table C.2 reports estimation for the derivatives of financial performance indicators. Der Sh is the derivative
of Sharpe Ratio. Der Mean is the derivative of average monthly return. Der Std is the derivative of monthly
return volatility. Der DD is the derivative of minimum drawdown. Der Min is the derivative of minimum
monthly return.
Table C.3: Hedging Effects of Options and VIX Futures for Crisis Periods
Carry 50 Delta 25 Delta 10 Delta VIX
Oct-2008 -6.55 4.34 3.12 1.41 3.38
May-2010 -5.54 2.47 1.71 0.66 1.62
Sep-2011 -4.48 2.88 2.15 0.63 0.89
Aug-2008 -4.46 3.44 3.08 2.30 -0.39
May-2012 -4.35 2.41 1.92 1.05 0.94
Sep-2008 -3.17 0.16 -0.19 -0.21 2.06
Aug-2010 -2.61 0.26 -0.37 -0.21 -0.28
Jan-2009 -2.43 0.51 0.35 -0.23 0.77
Apr-2006 -2.24 1.51 1.15 0.51 -0.29
Nov-2008 -2.16 -0.39 -1.17 -0.51 0.14
Mean -3.80 1.76 1.17 0.54 0.88
Mean of 6 Months -4.76 2.62 1.96 0.97 1.42
Number of Negative Return 1 3 4 3
APPENDIX C. APPENDIX FOR CHAPTER 3 162
Table C.4: Average utility gain in crisis period
50 Delta 25 Delta 10 Delta VIX
γ = 2 2.67 2.02 1.01 1.46
γ = 3 5.25 3.98 2.00 2.88
γ = 4 7.74 5.89 2.97 4.26
γ = 5 10.16 7.75 3.93 5.62
γ = 6 12.49 9.56 4.87 6.94
γ = 7 14.75 11.32 5.80 8.22
γ = 8 16.93 13.04 6.71 9.48
γ = 9 19.04 14.71 7.61 10.71
γ = 10 21.09 16.34 8.49 11.91
Mean 12.24 9.40 4.82 6.83
Table C.4 computes empirically the average utility gain for trader who takes one position in unhedged
portfolio and one position in hedging strategy. The VIX future portfolio is normalized such that the
volatility of VIX portfolio is the same as that of Delta 25 currency option strategy. We take six months
listed in Table 5 as crisis period. The time period is from March 2004 to June 2012.
APPENDIX C. APPENDIX FOR CHAPTER 3 163
Table C.5: Average utility cost in non-crisis period
50 Delta 25 Delta 10 Delta VIX
γ = 2 0.56 0.31 0.12 0.24
γ = 3 1.12 0.61 0.24 0.48
γ = 4 1.69 0.93 0.37 0.72
γ = 5 2.27 1.24 0.49 0.97
γ = 6 2.86 1.56 0.61 1.22
γ = 7 3.46 1.88 0.74 1.47
γ = 8 4.07 2.20 0.86 1.73
γ = 9 4.69 2.53 0.99 1.99
γ = 10 5.31 2.85 1.11 2.26
Mean 2.89 1.57 0.61 1.23
Table C.5 computes empirically the average utility loss for trader who takes one position in unhedged
portfolio and one position in hedging strategy. The VIX future portfolio is normalized such that the
volatility of VIX portfolio is the same as that of Delta 25 currency option strategy. The time period is from
March 2004 to June 2012.
Table C.6: Ratios of relative hedging qualities
50 Delta 25 Delta 10 Delta VIX
γ = 2 1.05 0.98 0.95 1.00
γ = 3 1.06 0.97 0.94 1.00
γ = 4 1.06 0.97 0.94 1.00
γ = 5 1.06 0.97 0.94 1.00
γ = 6 1.06 0.97 0.94 1.00
γ = 7 1.06 0.97 0.94 1.00
γ = 8 1.06 0.97 0.94 1.00
γ = 9 1.06 0.97 0.94 1.00
γ = 10 1.06 0.97 0.94 1.00
Table C.6 reports the ratio of hedging quality taking ten months as crisis period and that taking six periods
as crisis period.
APPENDIX C. APPENDIX FOR CHAPTER 3 164
Figure C.1: Cumulative return of portfolios
1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 20130.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8Stop Loss vs Unhedged vs Hedged portfolio
TR3
Unhedged B
Hedged B
This figure plots the total return indices of unhedged equally weighted portfolio(Unhedged B), hedged equally
weighted portfolio(Hedged B), and stop-loss portfolio with threshold 3%(TR3) for the period from January
1999 to June 2012.