All rights reserved. This work may not bereproduced in whole or in part, by photocopy
or other means, without permission of the author.
APPROVAL
Name:
Degree:
Title of Project:
Supervisory Committee:
Alexander RozovMengxin (Simon) Gan
Master of Arts
Comparison of the Principal Component-BasedFixed Income Hedging Strategies
Dr. Daniel Smith
Senior SupervisorAssociate Professor, Faculty of Business Administration
Dr. Christopher Perignon
Second ReaderAssociate Professor, Faculty of Business Administration
Date Approved:
ii
SIMON FRASER UNIVERSITYLIBRARY
Declaration ofPartial Copyright LicenceThe author, whose copyright is declared on the title page of this work, has granted toSimon Fraser University the right to lend this thesis, project or extended essay to usersof the Simon Fraser University Library, and to make partial or single copies only forsuch users or in response to a request from the library of any other university, or othereducational institution, on its own behalf or for one of its users.
The author has further granted permission to Simon Fraser University to keep or makea digital copy for use in its circulating collection (currently available to the public at the"Institutional Repository" link of the SFU Library website <www.lib.sfu.ca> at:<http://ir.lib.sfu.ca/handle/1892/112>)and,withoutchangingthecontent,totranslate the thesis/project or extended essays, if technically possible, to any mediumor format for the purpose of preservation of the digital work.
The author has further agreed that permission for multiple copying of this work forscholarly purposes may be granted by either the author or the Dean of GraduateStudies.
It is understood that copying or publication of this work for financial gain shall not beallowed without the author's written permission.
Permission for public performance, or limited permission for private scholarly use, ofany multimedia materials forming part of this work, may have been granted by theauthor. This information may be found on the separately catalogued multimediamaterial and in the signed Partial Copyright Licence.
While licensing SFU to permit the above uses, the author retains copyright in thethesis, project or extended essays, including the right to change the work forsubsequent purposes, including editing and publishing the work in whole or in part,and licensing other parties, as the author may desire.
The original Partial Copyright Licence attesting to these terms, and signed by thisauthor, may be found in the original bound copy of this work, retained in the SimonFraser University Archive.
Simon Fraser University LibraryBurnabY,BC,Canada
Revised: Summer 2007
ABSTRACT
This study analyses and compares various Principal Component-based
methods of fixed income security immunization. We show that the methods are
effective in terms of reducing the volatility of the security returns; however, none
of the Principal Component-based techniques outperforms the mean-variance
optimization. We conclude that the Principal Component-based immunization is
more effective when including all the available underlying securities in the
portfolio instead of hedging with the limited number of selected underlying
securities. We show that the Principal Component-based methods are not
effective for hedging short-term securities. We conclude that application of the
Principal Component-based immunization techniques requires balancing the
contrary effects of the model error and the estimation error on the results.
Keywords: Immunization; Hedging; Principal components; Fixed incomesecurities; Systematic and idiosyncratic risks
iii
ACKNOWLEDGEMENTS
We would like to thank Dr. Daniel Smith for sharing his ideas, knowledge
and expertise with us.
We also like to thank Dr. Christopher Perignon for his valuable comments
on the project.
iv
TABLE OF CONTENTS
Approval ii
Abstract iii
Acknowledgements iv
Table of Contents v
List of Figures vi
List of Tables vii
1 Introduction 1
2 Principal Component-based Fixed Income Hedging Strategies 6
3 Data Description 14
4 Empirical Results : 16
5 Conclusion 21
Appendices 23
Reference List. 35
v
LIST OF FIGURES
Figure 1 Annualized weekly returns on the fixed income securities 25
Figure 2 Factor loadings of the fixed income securities returns 28
vi
LIST OF TABLES
Table 1 Descriptive statistics 23
Table 2 Eigenvectors and eigenvalues for the fixed income securitiesreturns 27
Table 3 Comparative effectiveness of different immunization methods bythe example of weekly returns on 3-year US Treasury Bonds 29
Table 4 Comparative effectiveness of different immunization methods bythe example of weekly returns on 5-year zero coupon fixedincome security 30
Table 5 Comparative effectiveness of different immunization methods bythe example of weekly returns on 1-month US Treasury Bills 31
Table 6 Comparative effectiveness of different immunization methods bythe example of weekly returns on 6-month zero coupon fixedincome security 32
Table 7 Comparative effectiveness of different immunization methods bythe example of weekly returns on 10-year US Treasury Bonds 33
Table 8 Comparative effectiveness of different immunization methods bythe example of weekly returns on 20-year zero coupon fixedincome security 34
vii
1 INTRODUCTION
Financial institutions consider liability hedging an integral part of their
investment strategies. The problem is important for banks and financial
companies, which seek to mitigate their exposure to market risk, to reduce
regulatory capital charges, and to lower their cost of capital. Liability
immunization is especially important for pension funds and insurance companies,
which have significant cash outflows outstanding. The problem also arises for
non-financial companies that actively use debt funding, operational lease
financing and/or futures/forward agreements.
One of the methods aimed at reducing the risks associated with liability
stream volatility is duration immunization, a popular technique of managing risks
described in most course books on fixed income security analysis such as
Fabozzi (2006) or Tuckman (2002). The method assumes that the only driving
force of the change in security returns is the parallel shift in interest rates. In
practice, this assumption turns out to be unrealistic because it ignores other
drivers of movements of the security returns such as changes in the interest rate
curve slope and curvature. Besides, assuming that returns of the securities with
different maturities are independent, the method does not take into account the
correlation between the returns of the securities with different maturities (phoa,
2000).
The Principal Component-based methods of immunization benefit from the
lack of the shortcomings that the duration-based approaches have. The Principal
Components represent a set of independent factors that explain significant part
of the variation of security returns. Once knowing those factors, it is possible to
hedge certain liability stream by choosing underlying fixed income securities to
offset the variations of the factors.
The Principal Component-based methods have been proved effective in
fixed income hedging. The three most important factors that cause changes in
fixed income security returns are associated with level, slope and curvature of
the interest rate curve (Falkenstein and Hanweck, 1997). Litterman and
Scheinkman (1991) and Knez, Litterman, and Scheinkman (1994) show that the
first three principal components explain a large proportion (up to 98%) of bond
return variations. Based on effects that those three components produce on the
security returns, Litterman and Scheinkman (1991) interpret them as level, slope
and curvature factors. Diebold, Ji and Li (2004) confirm the previous findings and
show that these three factors describe a great proportion of the systematic risks
of bond returns.
Bliss (1997) compares the duration-based and the Principal Component
based methods and shows that the latter outperform the duration-based
approaches. The paper presents evidences that the Principal Component-based
hedge error is at least twice as small as that of the duration-based immunization.
Having tested the Principal Component-based immunization on the same set of
data using different subperiods, the author concludes that the three common
2
factors effectively capture volatility of the interest rates in all periods, though the
bond loadings on the factors slightly change over time.
The findings of the previous paper were further expanded and enriched by
Perignon and Villa (2006). They explore time-varying covariance matrix for
calculating the Principal Components for US interest rates. The authors reveal
that time-varying covariances and, as a result, time-varying factor loadings of the
returns are more consistent with the economic reality than constant covariances,
which are rejected in the most of the experiments conducted.
The works mentioned above apply the Principal Component Analysis to
sovereign debt markets. Bertocchi, Giacometti and Zenios (2005) test
effectiveness of the Principal Component Analysis by the example of the U.S.
corporate debt market. The conclusions of the paper are encouraging: the
common factors explain 98% of variation in the corporate bond yields and
spreads. Those results remain stable when the method is applied to the bonds
that have different credit ratings or belong to different corporate sectors.
Alongside with the advantages of the Principal Component-based hedging
and positive results obtained, the literature describes problems and limitations
associated with the methods. Thus, Reisman and Zohar (2004) show that the
hedging based on the Principal Component Analysis implies existence of
arbitrage, that is, the self-financing riskless hedging portfolios might have non
zero returns. This is a contradiction to the market efficiency theory. The
phenomenon can be compared with the 'volatility smile' effect in derivatives
markets.
3
Perignon, Smith, and Villa (2005) compare the effectiveness of the
Principal Component-based methods in different countries and conclude that,
while for each country the three common factors explain most of the volatility of
the returns volatility, only one factor is common across different countries. This is
due to the fact that the Principal Component Analysis takes into account
correlation between the bonds with different maturities but not of different
countries.
McGuire and Schrjjvers (2003) study emerging bond markets of 15
developing countries and reveal that the common factors are able to capture only
one third of the volatility on the bond returns; whereas the other two third is
caused by idiosyncratic factors. Thus, the Principal Component-based
immunization is significantly less effective in emerging markets.
While the literature on the Principal Component-based immunization
mostly focuses on how to effectively hedge the systematic risks, little is known
about the influence that different approaches to selecting the underlying
securities make on the effectiveness of hedge. Besides, no research has been
done on how to minimize effectively both systematic and idiosyncratic risks of
fixed income securities.
The purpose of this study is to compare various mechanisms of selecting
the underlying fixed income securities to the hedging portfolio, to extend the
Principal Component-based technique to hedge against both systematic and
idiosyncratic risks and to compare the effectiveness of different Principal
Component-based strategies of immunization of fixed income securities.
4
Specifically, we use the Principal Component Analysis to extract common factors
explaining the variation in fixed income security returns. We elaborate on seven
different approaches to selecting the underlying securities. Then we construct
self-financing portfolios which minimize the factor exposures of the liability
stream associated with the chosen fixed income security to the variations of the
factors. In the empirical analysis, we apply all the techniques to two datasets,
U.S. Treasury security returns over the last 25 years and bootstrapped U.S. zero
coupon fixed income security returns for the period from 1988 to 2004. Finally,
we compare effectiveness of different Principal Component-based methods.
The rest of the project is organized as follows. Section 2 presents the
Principal Component-based methods of a liability stream immunization. Section 3
provides data description and statistics. Section 4 describes the results obtained
by applying seven different hedging strategies. Section 5 summarizes our
findings and observations.
5
2 PRINCIPAL COMPONENT-BASED FIXED INCOMEHEDGING STRATEGIES
Suppose we need to immunize a liability stream. For this purpose we
create a portfolio of fixed income securities that mimics the liability stream but is
not exposed (or is exposed to the less extent than the liability stream) to
systematic risks. Then, we hedge our liability stream using that portfolio. As a
result, the new liability stream is exposed only to idiosyncratic risks of individual
securities.
One of the solutions of the problem is to create the global minimum
variance portfolio that includes the short position in the liability stream and long
(and possibly short) positions in the underlying securities. The mean-variance
optimization technique can be used to obtain such a hedging portfolio.
In matrix notation the mean-variance problem can be presented as
follows. The objective function is:
min w'..[ w.WERN
The following constraints have to be imposed in order to construct a self-
financing hedging portfolio for bond j, which creates the liability stream we wish
to immunize:
ejw=-1,
i'w = O.
6
where ej is a vector containing unity in the row j and zeros in the other rows, i is
a vectors of unities.
The mean-variance problem can be solved using the Lagrangian:
L =w'I w - A] (e;w + 1)- A2 (i'w).
In matrix notation, the first order conditions for the extremal values of L are the
following:
This expression allows us to compute the vector of weights w:
The Principal Component-based methods of immunization consist in
extracting the common factors affecting the returns of all the securities,
calculating the security loadings on the factors and then constructing a self-
financing portfolio with zero loadings on the factors. Mathematically, the Principal
Component Analysis transforms the set of correlated variables to a set of
orthogonal variables (factors). Linear combinations of the factors replicate the
original correlated variables (Amenc and Marellini, 2002).
In matrix notation, the M-factor model for N fixed income security returns
R, at time t can be presented as follows:
7
where R, is a vector of size N containing the returns on the securities at time t, A
is a matrix of size [NxM] comprising security loadings on the factors, Ft is a vector
of size M containing the values of the common factors at time t, Et is a vector of
size N containing the residuals at time t, which are specific for each fixed income
security. We assume that common factors F are orthogonal, and the factors and
residuals have zero mean. In this model A represents systematic risks and E
introduces idiosyncratic risks associated with each security.
The Principal Component Analysis approach includes decomposition of
the covariance matrix of the bond returns X as follows:
X =AI\A',
where 1\ is the covariance matrix of the factors. Such decomposition always
exists because X is a positive-semidefinite matrix. Since the factors are
orthogonal, 1\ is a diagonal matrix, whose diagonal elements are the eigenvalues
of the covariance matrix X. Since we assume that COV(Ft,Et) =0, when using only
M first common factors for the analysis, the covariance of the security returns can
be presented as follows:
X =A1-MI\1-MA1-M'+ n,
where A 1-Mis the first M columns of the matrix A, 1\1-M is the covariance matrix of
the factors 1, ... , M, n =E(EE') is the covariance matrix of the residuals.
Willing to hedge security j against variations in the common factors, we
impose linear constraints on the portfolio weights w. For example, for hedging
8
against the variation in the first factor only the following constraints have to be
imposed:
A1'w = 0,
ejw=-1,
i'w = 0,
where A 1 is the first column of the matrix A.
The system of linear constraints described above has a unique solution
only if the number of securities in the portfolio is equal to the number of factors to
be hedged against plus two. For example, to hedge certain security against the
variations in one factor, we need to include two more securities in the portfolio. In
this case the portfolio weights can be computed as follows:
[A;j-l
rO1
w =;~ ~ 1 .
By analogy with the case of hedging against one factor, the weights of the
portfolio components that guarantee protection against the volatility in the first
and second factor or the first, second and third factors can be calculated as
follows:
W=e'.
1
r e'1
r
A; -I aA; a
-1
a
and W=
9
A; -I aA' a2
A' a3
-1
a
, respectively.
Obviously, the unique solutions of these systems of equations exist only if the
number of bonds in the portfolio is equal to four and five, respectively.
When dealing with a number of securities available for hedging, the
problem that arises is which particular two (three or four) securities have to be
chosen to hedge certain liability stream. There might be a number of approaches
to solve this problem. The simplest approach is to choose the securities
randomly. However, in such a case the properties of different fixed income
instruments (volatility, specific risks, etc.) are not taken into account.
Another approach is to construct a 'butterfly' portfolio, which assumes
hedging the mid-term liability with short-term and long-term securities. The idea
of this approach is to create the hedging portfolio that is balanced in terms of
sensitivity to interest rate changes. One can also wish to find the underlying
bonds with the minimal residual variances o; which implies the lowest specific
risks.
All of the Principal Component-based approaches presented above have
a common shortcoming. They use the limited number of the underlying securities
and thus do not exploit all the opportunities in the market for the better hedge.
The next group of the Principal Component-based methods implies
construction of the portfolio using all available securities in the market. In order to
get rid of an uncertainty about which set of weights to use (the systems of linear
equations above have the infinite number of decisions when the number of linear
constraints is lower than the number of the unknown variables), an additional
criterion has to be used. For example, the criterion can consist in reducing
10
idiosyncratic risks of the portfolio or balancing the weights of the portfolio
individual components.
Numerically, the second group of methods requires solving the
optimization problem with the following criterion function:
min w'Cw,WER N
where C is a matrix of size [NxN] that defines the criterion, N is the number of all
the available bonds in the market, and the following linear constraints:
Ak'w=O, k= 1, ..., M,
ejw=-1,
i'w= 0,
where M - the number of the common factors to hedge against.
The optimization problem introduced above can be solved using the
Lagrangian:
The first order conditions for the extremal values of L in matrix notation are the
from which the vector of weights w can be calculated as it was shown above for
the case of the mean-variance problem.
As we have already said, using a criterion function to create the hedging
portfolio allows us to get additional benefits from hedging. In order not only to get
rid of systematic risks but also to mitigate idiosyncratic risks of the securities, we
assign the covariance matrix of the residuals 0 to the matrix C. Assuming that
the residuals of different securities are independent, which is a plausible
assumption since each residual presents specific (independent) risk for the
correspondent fixed income instrument, we can simplify our task and, instead of
0, use a diagonal covariance matrix containing variances of the residuals as the
diagonal elements.
Remarkably, when using the full covariance matrix as a criterion function,
we obtain the optimization problem which is equivalent to the mean-variance
one. Indeed, since I" = A 1-MI\A1-M'+0 and W'Ak=O, k = 1, ... , M,the following is
true:
min w'I w =min (W'A AA' W+ W'.f.MtJ =min w'n W .WERN WERN I-M I-M WER N
Thus, the two problems should give us the same result.
We can also focus on the purposes other than specific risk reduction when
constructing the matrix C. One of them can be diminution of differences in the
weights of the portfolio components. This can be achieved by assigning the
identity matrix I to the matrix C. Then the criterion for the optimization will be
minimization of the variations of the portfolio weights.
12
Regarding the practical applications of the Principal Component-based
methods, we would like to mention an empirical issue associated with the
calculation of the covariance matrices L and fl. Jobson and Korkie (1980) show
that when the number of financial instruments in the market is large and
comparable with the number of available historical observations in the sample,
the covariance matrix L is computed with plenty of errors, which might generate
unreliable results. When computing the covariance matrix of the residuals fl,
computational errors becomes even larger because, assuming that the common
factors capture most of the variability of the returns, the residuals tend to zero.
13
3 DATA DESCRIPTION
We use two datasets of returns on fixed income securities. The first one
consists of weekly returns on U.S. Treasury zero coupon bills with maturities 3, 6
months and 1 year and U.S. Treasury coupon bonds with maturities 2,3,5,7,10
years for the period from 1982 to 2007 1. Totally, we obtain 1324 observations for
the 8 securities. The second dataset consists of weekly returns on the
bootstrapped zero coupon securities with maturities 0.5, 1, 1.5,2,2.5, ... , 10
years (totally 20 securities) for the period from 1988 to 20042. The second
dataset numbers 886 observations. The dataset statistics is presented in Table 1.
For both datasets, the standard deviation of the returns, which is a
measure of volatility, becomes larger with increasing maturity of the securities.
The same observation can also be done from the Figure 1. The distributions of
the returns are skewed. The largest asymmetry appears in the first dataset for
the returns of the short-term securities. The distributions are leptokurtic: the
kurtosises of the returns of the zero coupon securities are slightly higher than
those of normal distributed variables; whereas the distributions of the returns on
the U.S. Treasury securities are more peaked. In both cases, the short-term
1 The returns are calculated based on the yields of the US Treasury securities with constantmaturities obtained from the USA Federal Reserve Board website:http://www.federalreserve.gov/releases/h15/data.htm The following technique is used for thecalculation of the coupon bond returns: on each weekly interval [t, t+1] the bonds areconsidered issued and traded at par at time t and priced with 1-week accrued interest at timet+1; the return calculated as annualized growth rate of the price for a week.
2 The returns are computed based on the bootstrapped zero coupon securities obtained fromChristopher S. Jones' page in the University of South California website: http://wwwrcf.usc.edu/-christojlresearch_wp.htm. We thank Christopher S. Jones for publishing the data.
14
security returns are more leptokurtic than the mid- and long-term ones. For the
first dataset, there are minor first-order autocorrelations of the returns; whereas
the returns from the second dataset demonstrate no significant autocorrelations.
The Principal Component Analysis applied to the both datasets reveals
that variation in the first factor capture the majority of the return volatility (Table
2). For the first dataset, 96.37% of the return volatility is explained by variability of
the first factor; whereas changes in the second and third factors contribute
additional 2.16% and 0.58% to the returns volatility, respectively. For the second
dataset, variability of the first factor captures 97.86% of the return fluctuations;
the second and third factors explain 1.80% and 0.28%, respectively.
The loadings of the returns on the first factor are negative for both
datasets (Figure 2). Their absolute values increase with maturities of the
securities; thus, changes in the first factor slightly affect the short-term security
returns and have more significant influence on the long-term security returns.
There is no similar pattern in which variations in the second and third factors
affect the returns of the securities of the two datasets. Yet, as in the case of the
first factor, the second and the third factors have different influence on the short
mid- and long-term security returns. On the whole, the Principal Component
Analysis reveals that the returns of the short-term securities are less sensitive to
the variations of the factors; whereas the returns of the long-term securities are
more volatile and unstable.
15
4 EMPIRICAL RESULTS
For both datasets described above we exploit the Principal Component
based immunization techniques using:
the limited (minimal necessary) number of underlying securities
that are either:
randomly selected, or
have minimal residual volatilities, or
form 'butterfly' portfolio (whenever applicable);
the entire number of available underlying securities minimizing:
the portfolio residual volatility using the full covariance matrix
of the residuals;
the portfolio residual volatility using the diagonal covariance
matrix of the residuals;
the variance of the portfolio weights.
We apply the techniques listed above to immunize against weekly variations in
one, two and three factors. We compare the Principal Component-based
methods with the mean-variance approach to the liability stream immunization.
We apply all the methods to hedge mid-term as well as short-term and long-term
16
liability streams using rolling and expanding window techniques. The results
obtained are presented in Tables 3-8.
In all the cases presented, the mean-variance optimization gives the best
results in terms of minimization of the out-of-sample volatility of the returns. So
does the Principal Component-based method that minimizes portfolio residual
volatility using the full covariance matrix of the residuals. As it was discussed in
Chapter 1, the two approaches are theoretically equivalent. However, in practice,
the covariance matrix of the residuals is estimated with a lot of errors in all the
cases", This is the reason that the results obtained using the two methods are
different in some cases. Significant estimation error makes the results obtained
using the Principal Component-based method that minimizes portfolio residual
volatility using the full covariance matrix of the residuals unreliable and its
practical application inexpedient. Our further discussion does not include this
method.
Despite the fact that the first factor captures the significant proportion of
the return volatility (96.37% for the first dataset and 97.86% for the second
dataset) (Table 2), whereas the second factor explains only 2.16% and 1.80%,
respectively, and the third factor captures even less (0.58% and 0.28%,
respectively), hedging against two and three factors improves the results
dramatically in most of the cases compare to hedging only against the first factor.
3 For the first dataset, the estimated covariance matrix of the residuals turns out not to bepositive-semidefinite in 414, 516, and 418 out of 804 rolling windows and in 416,494, and 410out of 804 expanding windows when hedging against variations in one, two and three factors,respectively. For the second dataset, the correspondent numbers are 193, 263, and 179 out of366 rolling windows and 197,264, and 188 out of 366 expanding windows.
17
Those results are consistent with the literature. Thus, Litterman and Scheinkman
(1991) show that, despite relatively little contribution of the second and third
factors to the explanation of the variability of the bond yields, the results of
hedging against three factors outperform those when immunizing against one
factor by 28%. Further, Bliss (1997) investigates the problem and concludes that
a single-factor hedging model is effective only if the changes in yields of different
securities are perfectly correlated, which is not the case in the real world where a
certain part of non-parallel shifts in yields is not correlated with parallel shifts.
However, when choosing the number of the factors to immunize against,
one should have in mind two types of errors arising: the model error and the
estimation error. As it has been said above, taking into account the small number
of the factors and ignoring other ones potentially increases the model error. On
the other hand, adding more factors into account the estimation error increases
and might offset the positive effect of the immunization. In some cases
presented, the largest decrease in volatility is achieved when hedging against
one or two factors, and immunization against more factors only worsens the
results".
Although the Principal Component-based methods that imply using all the
available underlying securities are more effective than those based on the
selection of the minimal necessary number of underlying securities, this is not the
case when immunizing the short-term security in the case of the first set of data
(Table 5). Choosing the underlying securities with minimal residual volatilities is
4 This situation is more often for the U.S. Treasury security dataset (Tables 3, 5 and 7).
18
more effective than including in the portfolio the entire set of the securities and
calculating their weights by minimizing the portfolio residual volatility or the
variance of portfolio weights. Moreover, for both datasets, immunization of the
short-term securities does not significantly reduce but in most cases increases
volatility of the returns". These results are consistent with those one could
expect: short-term fixed-income securities have low volatility of the returns, and
choosing more volatile mid- and long-term underlying securities for hedging
brings no or little effect.
Immunization is found to be more efficient when the larger number of
securities and more variety of maturities are available for hedging. The second
dataset, which includes 20 different securities, allows us to achieve more
significant reduction in the out-of-sample volatilities than the first dataset",
Moreover, when the Principal Component techniques based on the selection of
the minimal necessary number of underlying securities are applied, the limited
available range of maturities for hedging, as in the case of the first dataset, might
lead to the increase of the out-of-sample volatilities? The same factor, along with
the estimation error mentioned above, causes the increase in the out-of-sample
5 See, for example, Tables 5 and 6. The rolling window mean volatilities of returns on 1-month USTreasury Bill and 6-month zero coupon security are 0.0134 and 0.0259, respectively. All theimmunization techniques, except for the mean variance approach and the PrincipalComponent-based method that uses the full covariance matrix of the residuals to minimizetheir volatility, generate higher volatilities.
6 However, in practice, including a large number of securities in the hedging portfolio increasesthe transaction cost of hedging.
7 See, for example, Tables 3 and 7, both rolling and expanding windows, the method which usesminimum residual volatility as a criterion for the securities selection.
19
volatilities when adding more factors to immunize against. In some cases, the
resulting growth in volatility becomes extremely larqe",
For both datasets, the Principal Component-based methods involving the
use of all the available underlying securities allow us to achieve not only more
significant decrease in the out-of-sample volatilities than the methods which
imply constructing the hedging portfolio using a limited set of securities but also
lower standard deviation of the out-of-sample volatilities and thus more stable
results. This is proved by the results obtained from both datasets using both
rolling and expanding window approaches. Besides, in terms of reduction of the
out-of-sample volatilities, minimizing the portfolio residual volatility using the
diagonal covariance matrix of the residuals is more efficient in most cases than
minimizing the variance of the portfolio weiqhts".
6 See Table 3, expanding windows, and Table 7, rolling windows, in both cases the method whichuses minimum residual volatility as a criterion for the securities selection when hedging againstthree factors.
9 Especially when hedging against two and three factors.
20
5 CONCLUSION
In this study, we analyze different Principal Component-based methods of
liability stream immunization. The methods are found to be effective in terms of
offsetting the changes in the hedged liability streams. They provide
comprehensible and straightforward approach to hedging. Allowing an analyst to
choose to what extent and against which factors he or she is willing to hedge, the
methods imply flexible approach to immunization. Besides, some of the Principal
Component-based methods allow an analyst to impose additional requirements
on hedging portfolio. However, none of the Principal Component-based methods
considered in this study succeeds to outperform the mean-variance approach.
We conclude that the Principal Component immunization techniques
based on including in the hedging portfolio all the underlying bonds available for
hedge are more effective in terms of reduction of the portfolio volatility than the
methods based on selecting the minimal number of underlying securities required
for hedge. The results of this paper also evidence that greater variety of
maturities of the underlying securities available for hedge leads to more
significant decrease in the portfolio volatility.
Since short-term security returns are less volatile than those of mid- and
long-term securities, it is impossible to achieve satisfactory results applying the
Principal Component-based methods to immunize short-term liability streams.
21
We conclude that the immunization techniques other than the Principal
Component-based have to be used for hedging in that case.
Finally, we conclude that one who decides to use the Principal
Component-based methods for liability stream immunization has to balance the
positive effect achieved from choosing larger number of different factors to hedge
against (and thus from reduction of the model error) and the negative effect of
the estimation error that increases with adding more factors to the model.
22
APPENDICES
Table 1 Descriptive statistics
Panel A: US Treasury securities
Annualized weekly returns on the U.S. Treasury securities of 1-month, 6-month, 1-year, 2-year, 3year, 5-year, 7-year and 10-year constant maturities (8 issues) are obtained for the period from1982/01/08 to 2007/05/25 by converting weekly yields into returns. The following technique isused for the calculation of the coupon bond returns (with maturities 2-year, 3-year, 5-year, 7-yearand 10-year): on each weekly interval [t, t+1] the bonds are considered issued and traded at parat time t and priced with 1-week accrued interest at time t+1; the return calculated as annualizedgrowth rate of the price for a week. Totally 1324 observations are obtained. Std.Dev. stands forstandard deviation, Rho (1), Rho (2), Rho (3) for autocorrelations at lags 1, 2 and 3 weeks,respectively.
1 year 0.0032 0.0675 2.2115 20.4161 0.2541 0.0574 0.0480
2 years 0.0002 0.1590 0.1564 8.2346 -0.3661 -0.1227 -0.0269
3 years 0.0005 0.2337 0.2218 4.8311 -0.3705 -0.1324 0.0054
5 years 0.0011 0.3549 0.2431 3.2634 -0.3537 -0.1680 0.0416
7 years 0.0017 0.4512 0.2373 2.4414 -0.3549 -0.1723 0.0550
10 years 0.0028 0.5586 0.2586 2.4043 -0.3548 -0.1756 0.0640
23
Panel B: Zero coupon fixed income securities
Annualized weekly returns on the bootstrapped zero coupon fixed income securities of maturitiesfrom 0.5 to 10 years (20 issues) are obtained for the period from 1988/01/04 to 2004/12/31 byconverting daily bootstrapping zero coupon yields into weekly returns. The following technique isused for the return calculation: first Friday prices of the securities are calculated; then the returnsare calculated as annualized growth rates of the price for each week. Totally 886 observationsare obtained. Std.Dev. stands for standard deviation, Rho (1), Rho (2), Rho (3) forautocorrelations at lags 1, 2 and 3 weeks, respectively.
-'00iii iii i ! I Ii; i ill • ~ ,______________ allal
Panel B: Zero coupon fixed income securities (selected maturities)
1.,..r Zero Coupon Bond 2-)18.' Zero Coupon Bond
0.25
4-)ltar uro Coupon Bond
~ Ii I ~ i ~ ~g I! ! ! ! ! ! !
I ~iii I ~
~ I g~ ~ I 8 i
I! ! s s ~
----- ------ ------
0.50
1.00
000
I::050
000
I :~: ,
3-year Zero Coupon Bond
--------------l
1.00
1'0
-01SI:: ~~~--~--~~~~ ~ ~ ~ ~
------------ --
-_ ..._------
5-ye., Zero Coupon Bond 8-y8ar Zero Coupon Bond
~~~-150 I I ! , I ! I I I ·-i -T I I ! gTI I
~ ! ! ! ! ! ! ! ! ! ! ! ~ ~ ~~
"--_.~--- -. . .---
7.,.., Zero Coupon Bond .,..r Zero Coupon Bond
0.50
-0.50
-1.50
iii~
~ ~ ~ ii~! ! s ! !
9188' Zero Coupon Bond
1.00
0.50
0.00
-0.50
-1.00
l _
- ,g 0
~~--
26
Table 2 Eigenvectors and eigenvalues for the fixed income securities returns
The two panels contain the factor loadings (eigenvectors) of the fixed income securities returnson the first three factors obtained using the Principal Component Analysis. The factors are shownin the order of decreasing influence on the return variability. For each eigenvector, thecorrespondent eigenvalues is presented. % Variance shows the share of the total variability in theoriginal dataset captured by the factors.
Figure 2 Factor loadings of the fixed income securities returns
The two panels plot the factor loadings of the fixed income securities returns on the first threefactors obtained using the Principal Component Analysis.
Panel A: US Treasury securities
0.4
0.2 ~
o --
-0.2
-0.4
-0.6
-0.8
--+-1 st factor-------- --- --- ---
_2nd factor
-.- 3d factor
-1 -l-----..~--~1 rronth 6 rronth 1 year 2 years 3 years 5 years 7 years 10 years
Table 3 Comparative effectiveness of different immunization methods by theexample of weekly returns on 3-year US Treasury Bonds.
Whenever appropriate, seven different methods of immunization against one, two and threecommon factors are implemented to hedge the 3-year U.S. Treasury security. To estimate andcompare the effectiveness of the hedging methods, both rolling window and expanding windowapproaches are used. For rolling window analysis, 10-year period, which includes 520observations, is chosen. Totally, 804 rolling windows are obtained by rolling forward by weekintervals. In expanding window analysis, the first window corresponds to 10-year period (520observations). Totally, 804 windows are obtained by expanding windows by week intervals.
For each rolling and expanding window, 19 self-financing hedging portfolios are constructed.Then, the out-of-sample volatilities of the portfolios are calculated. Each out-of-sample interval isconstructed by adding one observation to the sample. The table below represents the mean andstandard deviation of the out-of-sample volatilities (Mean Vol and SD Vol, respectively). PCAstands for the Principal Component Analysis.
Method Factors Rolling Windows Expandin~ WindowsMean Vol SO Vol Mean Vol SO Vol
Table 4 Comparative effectiveness of different immunization methods by the exampleof weekly returns on 5-year zero coupon fixed income security.
Whenever appropriate, seven different methods of immunization against one, two and threecommon factors are implemented to hedge the 5-year zero coupon fixed income security. Toestimate and compare the effectiveness of the hedging methods, both rolling window andexpanding window approaches are used. For rolling window analysis, 10-year period, whichincludes 520 observations, is chosen. Totally, 366 rolling windows are obtained by rolling forwardby week intervals. In expanding window analysis, the first window corresponds to 10-year period(520 observations). Totally, 366 windows are obtained by expanding windows by week intervals.
For each rolling and expanding window, 19 self-financing hedging portfolios are constructed.Then, the out-of-sample volatilities of the portfolios are calculated. Each out-of-sample interval isconstructed by adding one observation to the sample. The table below represents the mean andstandard deviation of the out-of-sample volatilities (Mean Vol and SD Vol, respectively). PCAstands for the Principal Component Analysis.
Table 5 Comparative effectiveness of different immunization methods by the exampleof weekly returns on 1-month US Treasury Bills.
Whenever appropriate, six different methods of immunization against one, two and three commonfactors are implemented to hedge the 1-month U.S. Treasury security. To estimate and comparethe effectiveness of the hedging methods, both rolling window and expanding windowapproaches are used. For rolling window analysis, 10-year period, which includes 520observations, is chosen. Totally, 804 rolling windows are obtained by rolling forward by weekintervals. In expanding window analysis, the first window corresponds to 10-year period (520observations). Totally, 804 windows are obtained by expanding windows by week intervals.
For each rolling and expanding window, 16 self-financing hedging portfolios are constructed.Then, the out-of-sample volatilities of the portfolios are calculated. Each out-of-sample interval isconstructed by adding one observation to the sample. The table below represents the mean andstandard deviation of the out-of-sample volatilities (Mean Vol and SD Vol, respectively). PCAstands for the Principal Component Analysis.
Method Factors Rolling Windows Expandin~ WindowsMa;llnVal SDVol Ma;lln Vnl SDVol
Table 6 Comparative effectiveness of different immunization methods by the exampleof weekly returns on 6-month zero coupon fixed income security.
Whenever appropriate, seven different methods of immunization against one, two and threecommon factors are implemented to hedge the 6-month zero coupon fixed income security. Toestimate and compare the effectiveness of the hedging methods, both rolling window andexpanding window approaches are used. For rolling window analysis, 10-year period, whichincludes 520 observations, is chosen. Totally, 366 rolling windows are obtained by rolling forwardby week intervals. In expanding window analysis, the first window corresponds to 10-year period(520 observations). Totally, 366 windows are obtained by expanding windows by week intervals.
For each rolling and expanding window, 16 self-financing hedging portfolios are constructed.Then, the out-of-sample volatilities of the portfolios are calculated. Each out-of-sample interval isconstructed by adding one observation to the sample. The table below represents the mean andstandard deviation of the out-of-sample volatilities (Mean Vol and SD Vol, respectively). PCAstands for the Principal Component Analysis.
Table 7 Comparative effectiveness of different immunization methods by the exampleof weekly returns on 10-year US Treasury Bonds.
Whenever appropriate, six different methods of immunization against one, two and three commonfactors are implemented to hedge the 10-year U.S. Treasury security. To estimate and comparethe effectiveness of the hedging methods, both rolling window and expanding windowapproaches are used. For rolling window analysis, 10-year period, which includes 520observations, is chosen. Totally, 804 rolling windows are obtained by rolling forward by weekintervals. In expanding window analysis, the first window corresponds to 10-year period (520observations). Totally, 804 windows are obtained by expanding windows by week intervals.
For each rolling and expanding window, 16 self-financing hedging portfolios are constructed.Then, the out-of-sample volatilities of the portfolios are calculated. Each out-of-sample interval isconstructed by adding one observation to the sample. The table below represents the mean andstandard deviation of the out-of-sample volatilities (Mean Vol and SO Vol, respectively). PCAstands for the Principal Component Analysis.
Method Factors Rolling Windows Expandin~ WindowsMean Vol SO Vol Mean Vol SO Vol
Table 8 Comparative effectiveness of different immunization methods by the exampleof weekly returns on 20-year zero coupon fixed income security.
Whenever appropriate, seven different methods of immunization against one, two and threecommon factors are implemented to hedge the 10-year zero coupon fixed income security. Toestimate and compare the effectiveness of the hedging methods, both rolling window andexpanding window approaches are used. For rolling window analysis, 10-year period, whichincludes 520 observations, is chosen. Totally, 366 rolling windows are obtained by rolling forwardby week intervals. In expanding window analysis, the first window corresponds to 1O-yearperiod(520 observations). Totally, 366 windows are obtained by expanding windows by week intervals.
For each rolling and expanding window, 16 self-financing hedging portfolios are constructed.Then, the out-of-sample volatilities of the portfolios are calculated. Each out-of-sample interval isconstructed by adding one observation to the sample. The table below represents the mean andstandard deviation of the out-of-sample volatilities (Mean Vol and SO Vol, respectively). PCAstands for the Principal Component Analysis.
Amenc, N., and Martellini, L., 2002. Portfolio Optimization and Hedge Fund StyleAllocation Decisions. Working Paper, No 02-4, University of SouthernCalifornia, Marshall School of Business.
Bertocchi, M., Gizcometti, R, and Zenios, S.A., 2005. Risk Factor Analysis andPortfolio Immunization in the Corporate Bond Market. European Journal ofOperational Research, 161,348-363.
Bliss, RR, 1997. Movements in the Term Structure of Interest Rates. EconomicReview, Federal Reserve Bank of Atlanta, Forth Quarter 1997, 16-33.
Diebold, F.X., Ji, L., and Li, C., 2004. A Three-Factor Yield Curve Model: NonAffine Structure, Systematic Risk Sources, and Generalized Duration.Working Paper, University of California.
Fabozzi, F. J., 2006. Bond Markets, Analysis and Strategies. New York: PrenticeHall.
Falkenstein, E., and Hanweck J., 1996. Minimizing Basis Risk from Non-ParallelShifts in the Yield Curve. Journal of Fixed Income, Vol 7 No 1.
Jobson, J. D., and Korkie, B. M., 1980. Estimation for Markowitz EfficientPortfolios. Journal of the American Statistical Association, Vol 75 No 372,544-554.
Knez, P.J., Litterman, R, and Scheinkman, J., 1994. Exploration into FactorsExplaining Money Market Returns. Journal of Finance, Vol 49 No 5, 18611882.
35
Litterman, R., and Scheinkman, J., 1991. Common Factors Affecting BondReturns. Journal of Fixed Income, 1, 54-61.
McGuire, P., and Schrijvers, M.A., 2003. Common Factors in Emerging MarketSpreads. Quarterly Review, Bank for International Settlements, December2003.
Perignon, C., Smith, DR., and Villa, C., 2007. Why Common Factors inInternational Bond Returns are Not So Common. Journal of InternationalMoney and Finance, Vol 26 No 2, 284-304.
Perignon, C., and Villa, C., 2006. Sources of Time Variation in the CovarianceMatrix of Interest Rates. Journal of Business, Vol 79 No 3, 1535-1549.
Phoa, W. 2000. Yield Curve Risk Factors: Domestic and Global Contexts. In Theprofessional's handbook of financial risk management, ed. LevBorodovsky and Marc Lore. Woburn, MA: Butterworth-Heinemann.
Reisman, H. and Zohar, G., 2004. Excess Yields in Bond Hedging. WorkingPaper, Israel Institute of Technology, Haifa.
Tuckman, 8.,2002. Fixed Income Securities: Tools for Today's Markets.Hoboken: John Wiley & Sons.
