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Swiss Finance InstituteResearch Paper Series N10 31Alternative Models For HedgingYield Curve Risk: An EmpiricalComparisonNicolaCARCANOUniversit della Svizzera Italiana and Bank VontobelHakim DALL'OUniversit della Svizzera Italiana and Swiss Finance Institute
7/27/2019 Alternative Models for Hedging Yield Curve Risk an Empirical Comparison
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ALTERNATIVE MODELS FOR HEDGING YIELD CURVE RISK: AN EMPIRICALCOMPARISON 1
Nicola Carcano
urich.Universit della Svizzera Italiana, Lugano and Bank Vontobel, Z
[email protected] . Via Sole 14, CH 6977 Ruvigliana.
Hakim DallO
Universit della Svizzera Italiana, Lugano and Swiss Finance Institute.
[email protected]. Telephone:
Via Buffi 13, CH 6900 Lugano.
0041 058 666 4497 Fax: 0041 058 666 4734
ABSTRACTWe develop alternative models for hedging yield curve risk and test them by
hedging US Treasury bond portfolios through note/bond futures. We show that
traditional implementations of models based on principal component analysis,
duration vectors and key rate duration lead to high exposure to model errors and to
sizable transaction costs, thus lowering the hedging quality. Also, this quality varies
from one test case to the other, so that a clear ranking of the models is not possible.
We show that accounting for the variance of modeling errors substantially reduces
both hedging errors and transaction costs for all considered models. Also, this
allows to clearly rank these models: erroradjusted principal component analysis
ystematically and significantly outperforms alternative models.s
eywords: Yield curve risk, interest rate risk, immunization, hedging.K
EL codes: G11; E43J
1 We are grateful to Robert R. Bliss for having allowed us to use his yield curve estimates and to Ray Jireh
and Daniel Grombacher from the CME for having provided us with the relevant data underlying the bondfuture contracts. All errors or omissions should only be charged to the authors.
1
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1. IntroductionWe define yield curve risk as the risk that the value of a financial asset might change
due to shifts in one or more points of the relevant yield curve. As such, it represents
one of the most widely spread financial risks impacting a very diversified range of
entities: not only financial institutions, like banks (both central and private),
insurance companies, portfolio managers, and hedge funds, but also pension funds,
real estate as well as many other industrial companies. Generalizing, we may say
that each institution having to match future streams of assets and liabilities is
exposed up to a certain extent to yield curve risk.
The simplest way to cope with yield curve risk is to match positive with
negative cashflows as much as possible. This approach of cash-flowmatchingis not
only theoretically straightforward, but also very effective in minimizing yield curve
risk. Unfortunately, the dates and the amounts of future cashflows are often subject
to constraints in practice, so that implementing an accurate cashflow matching
might not be possible.
When cashflow matching is not possible, socalled immunization techniques
are employed to manage yield curve risks. These techniques have the goal of making
the sensitivity of assets and liabilities to yield curve changes as much as possible
similar to each other. The key idea behind these techniques is that if assets and
liabilities react in a similar way to a change in the yield curve, the overall balance
sheet will not be largely affected by this change.
Originally, academicians and practitioners focused on the concept ofduration
firstly introduced by Macaulay (1938) for implementing immunization
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techniques. Duration represents the first derivative of the priceyield relationship of
a bond and was shown to lead to adequate immunization for parallel yield curve
shifts1. Accordingly, we can claim that the first models relying on duration were
targeting generic interest rate risk and not really yield curve risk, since the different
points of the yield curve were not allowed to move independently from each other.
The first step to move from generic interest rate risk to yield curve risk was
made with the introduction of the concept of convexity (see, for example, Klotz
(1985)). Convexity is related to the second derivative of the priceyield relationship
of a bond. However, the impact of interest rate changes taking place over a few days
or weeks is normally wellapproximated by duration. Accordingly, the importance of
convexity is commonly not related to its added value in the description of the price
yield relationship. As highlighted by Bierwag et al. (1987) and recently confirmed by
Hodges and Parekh (2006), this importance is due to the fact that immunization
strategies relying on duration and convexitymatching are consistent with plausible
twofac otor processes describing n nparallel yield curve shifts.
Later research took the argument supporting duration and convexity
matching even further: socalled M-square and M-vectormodels were introduced by
Fong and Fabozzi (1985), Chambers et al. (1988), and Nawalka and Chambers
(1997). Similarly as for convexity, most of these models relied on the observation
that furtherorder approximations of the priceyield relationship lead to
immunization strategies which are consistent with multifactor processes accurately
describing actual yield curve shifts. We will identify this class of models as duration
vector (DV) models. An accurate review of them is given in Nawalka and et. (2003),
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who also introduce a generalization of the DV approach identified as generalized
duratio vn ector(GDV).
A parallel development of immunization models relied on a statistical
description of the factors underlying yield curve shifts. This description was based
on a technique known as principal component analysis (PCA). PCA identifies
orthogonal factors explaining the largest possible proportion of the variance of
interest rate changes. Litterman and Scheinkman (1988) showed that a PCA relying
on 3 components allows to capture the three most important characteristics
displayed by yield curve shapes: level, slope, and curvature. Accordingly,
immunization models matching the sensitivity of assets and liabilities to these three
components should lead to highquality hedging.
A third class of widely used immunization models relies on the concept ofkey
rate duration (KRD) introduced by Ho (1992). These models explain yield curve
shifts based on a certain number of points along the curve the key rates and on
linear approximations based on time to maturity for the remaining rates.
Yield curve hedging techniques used in practice very often rely on one of the
three abovementioned classes of models. However, we are not aware of a conclusive
evidence on the relative performance of these three approaches2. Moreover, several
studies performing empirical tests of these hedging models reported puzzling
results. Particularly, models capable to better capture the dynamics of the yield
curve were not always shown to lead to better hedging. This was the case of the
volatility and covarianceadjusted models tested by Carcano and Foresi (1997) and
of the 2component PCA tested by Falkenstein and Hanweck (1997) which was
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found to lead to better immunization than the corresponding 3component PCA. As
a result, the key question about the best model to use in order to minimize yield
curve risk has not found a conclusive answer, yet.
Carcano (2009) tested a model of PCAhedging which controls the exposure
to model errors. He found that by introducing this adjustment 3component PCA
does lead to better hedging than 2component PCA, as theory would suggest. On this
basis, he claimed that random changes in the exposure to model errors might be
responsible for the lack of conclusiveness displayed by previous empirical tests of
alterna h ntive edgi g models.
The goal of this paper is to provide relevant empirical evidence for
identifying the best model to minimize yield curve risk. Our expectation was that the
exposure to model errors plays a crucial role in determining the performance of the
alternative models. Once an adjustment for this error is introduced, the quality of
the tested models mainly depends on how well the underlying factor model catches
the actual dynamics of the yield curve. Accordingly, we extended all three
mainstream immunization approaches in order to account for model errors and
compared them among themselves and with their traditional implementations
which ignore model errors.
We relied on previous evidence that three factors are sufficient to explain the
vast majority of the yield curve dynamics and tested only threefactor models.
Accordingly, we expected the quality of the resulting erroradjusted hedging
strategies to be comparable. The PCA model is constructed in a way to explain the
largest possible part of the variance of yield curve shifts based on three orthogonal
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factors. Accordingly, we suspected that once we account for model error exposure
this model would slightly outperform the alternative models.
We tested the alternative strategies on a portfolio of US Treasury bonds and
notes hedged by four US Treasury note and bond future contracts. Our expectations
have been confirmed by the results of these tests. Even though it is not possible to
clearly rank the models based on their traditional implementations, the approach
relying on PCA consistently outperforms the other approaches when the error
adjustment is introduced. Additionally, this adjustment is shown to add very
significant value to all three tested approaches.
The remainder of the paper is organized as follows: section 2 presents the
hedging models we are going to test and their theoretical justification. Section 3
describes our dataset and testing approach. Section 4 reports our results, both on
the full sample as well as on three subsamples, while Section 5 concludes and
indicates some possible directions for future research.
2. TheHedgingMethodology2.1 The sensitivity of bond and future prices
We consider the problem of immunizing a riskfree bond portfolio which at time t
has a value Vt by identifying the optimal underlying value y to be invested in each
of the four US Tnote/Tbond futures (the 2year, the 5year, the 10year, and the
30year contracts). We group the cash flows of the bond portfolio and of the
cheapesttodeliver bonds underlying the futures in n time buckets. Following the
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most common approach to this immunization problem as in Martellini and Priaulet
(2001), we impose the socalled self-financingconstraint:
ttty VH=
4
,y=1
The latter constraint implies that the market value of the portfolio to be hedged
must be equal to the market value of the hedging portfolio when the latter consists
of bonds or to the market value of the underlying bonds when the latter consists of
derivative contracts (like in our case). In practice, the amounts to be invested in the
hedging portfolio are often constrained, even though the form of these constraints
can differ from the last equation. Accordingly, we felt that including a constraint
would
(1.)
make our empirical tests more realistic.
We intend to analyze the quality of alternative hedging models on a very
short hedging horizon, ideally tending to zero. For practical reasons, we set this
hedging horizon to one month. This choice was motivated by the fact that many
institutional investors and portfolio managers do have a time horizon of 1 to 3
months, when they set up their hedging strategies. After this period, they mostly
reconsider the whole hedging problem and determine a new strategy.
The market risk for the portfolio to be hedged comes from unexpected shifts
in the corresponding continuously compounded zerocoupon riskfree rates R(t,Dk),
where Dk indicates the duration and maturity of the corresponding time bucket. We
assume, for simplicity, that all rates are martingales: that is, E[dR(t,Dk)]=0 for every
kand t.
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Approximating the dynamics of the term structure through a limited number
of factors results in a difference between the modeled and actual dynamics of
interest rates, the modelerror. For a generic 3factor model of the term structure of
interest rates, we can describe the dynamics of the zerocoupon riskfree rate
R(t,Dk) of maturity Dk as:
(2.)( ) ( kl
tlkk DtFcDtdR ,,3
+ )l 1=
where Flt represents the change in the l-th factor between time t and t+1, clk
represents the sensitivity of the zerocoupon rate of maturity Dk to this change, and
represents the model error.
As reported in several papers, like Hodges and Parekh (2006), the impact of
monthly rate changes on the price of a zerocoupon bond can be wellapproximated
by its duration. Accordingly, we will follow this simplifying approach.
Estimating the sensitivity of future prices to changes in zero rates is
significantly more complex. This is due to the fact that as illustrated by Fleming
and Whaley (1994) future contracts embed 4 types of options. The first option is a
quality option that permits the short position to deliver the cheapest bond (the so
called CheapesttoDeliver or CTD) to the long position. The other three options are
defined time options3. So far, both academicians and practitioners working on
immunization have focused on the first option, considering the possible impact of
the time options on optimal hedging strategies to be negligible. We will follow the
same approach and restrict our next considerations to the quality option.
8
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VFG-3V72TB1-D&_user=835417&_coverDate=10%2F31%2F1998&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_rerunOrigin=scholar.google&_acct=C000045119&_version=1&_urlVersion=0&_userid=835417&md5=4dfb41ef574025ffa4e1aba78e16d965#bb24http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VFG-3V72TB1-D&_user=835417&_coverDate=10%2F31%2F1998&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_rerunOrigin=scholar.google&_acct=C000045119&_version=1&_urlVersion=0&_userid=835417&md5=4dfb41ef574025ffa4e1aba78e16d965#bb24http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VFG-3V72TB1-D&_user=835417&_coverDate=10%2F31%2F1998&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_rerunOrigin=scholar.google&_acct=C000045119&_version=1&_urlVersion=0&_userid=835417&md5=4dfb41ef574025ffa4e1aba78e16d965#bb24http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VFG-3V72TB1-D&_user=835417&_coverDate=10%2F31%2F1998&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_rerunOrigin=scholar.google&_acct=C000045119&_version=1&_urlVersion=0&_userid=835417&md5=4dfb41ef574025ffa4e1aba78e16d965#bb247/27/2019 Alternative Models for Hedging Yield Curve Risk an Empirical Comparison
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In the past, researchers implementing hedging strategies through note and
bond futures attempted to simplify the problem. One approach has been to calculate
the sensitivity of the futures through standard regression analysis (examples of this
approach can be found in Kuberek and Norman (1983) and more recently in
CollinDufresne et. al (2001) and in Elton et al. (2001)). Such an approach implicitly
assumes that the sensitivity of the future price including the value of the quality
option is constant over time. Intending to test highquality hedging strategies on
US Treasuries and knowing that the sensitivity of the future price varies
significantly with the underlying CTD bond, we decided not to follow this approach.
A second simplifying approach made the embedded quality option less complex
than it really is. For example, Grieves and Marcus (2005) assumed that this option
can be represented as a switching option between only two bonds. Unfortunately,
latest research has shown that this simplification is too crude to accurately describe
future price sensitivity (see, for example, Henrard (2006) and Grieves et al. (2010)).
As a result, numerical procedures based on arbitragefree term structure
models are currently recommended when an accurate evaluation of the quality
option is needed. However, we intend to minimize yield curve risk based on three
factor PCA, DV, and KRD models. Accordingly, we would need to rely on multifactor
term structure models capable of producing stable and robust estimates of the
sensitivity of future prices to these factors. The development of a model with these
characteristics goes beyond the scope of this paper and is left to further efforts. Our
goal here is to compare traditional and erroradjusted versions of the three hedging
models in the simplest possible way, so to make our results as general as possible.
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Our wayout of this dilemma relies on the observation made among others
by Rendleman (2004) and Grieves et al. (2010). They show that the value of the
delivery option has a very low impact on hedging strategies based on the next
expiring future contract, when yields are nottooclose to the notional coupon of the
future contract. "Too close" is normally defined as an absolute distance not greater
than 0.5% 1%. We will show later on that for the vast majority of our sample
yields are not too close to the notional coupon. Accordingly, we will estimate the
sensitivity of the future price ignoring the delivery option and perform a subsample
analysis to show if and how our results vary when yields are too close to the
notional coupon.
Within this framework, the quoted future price FP can be represented by the
following expression:
( )( )
+=skCTD eCF 1
where CFindicates the Conversion Factor, cfCTD,k indicates the cashflow paid by the
cheapesttodeliver bond at time k, and AICTD,s represents the accrued interests of
the cheapesttodeliver bond on the expiration date s of the future contract. The only
cashflows of this bond which are relevant for the valuation of the future contract
are the
(3.)
= n
sCTD
DDtR
DDtR
kCTD
t AIecf
FP sskk
,
,
,
,1
ones maturing after the expiration date s.
Approximating the effect of rate changes on the price of a zero bond by its
duration, the percentage sensitivity of the future price to these changes can be
expressed as:
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( )
( )
( ) tkCTDkDDtRkCTD
CTDtk eCFFPDtRkk,
for all rates maturing after the future contract (i.e.: k> s) and
DDtR
kt
t
DcfeDFP
FP
ss
,,,
,,
(4.)
( )
( )
( ) tsCTDs
n
skskCTDts eCFFPDtRkk
11, +=+=
for the zero rate with maturity equal to the expiration of the future contract, where
CTD,k, represents the percentage of the CTD future price related to the CTD cash
flow with maturity kand is defined based on the last two equations. The sensitivity
of the future price to changes in zero rates maturing before the future contract is
zero, which implies: = 0 for all k< s.
tkCTDs
n
DDtR
kCTDDDtR
st
t
DDcfeDFP
FP
ss
,,,,,
,,
=
(5.)
CTD,k,t
2.2 The development of the hedging equations
Given the approach to estimate the sensitivity of bond and future prices to zero rate
changes described in the previous chapter, we approximate the total unexpected
return provided by the combination of the two portfolios Vand Has follows:
( ) ( ) ,,4
,,,, +n
tkkk
n
tkykktyt ADDtdRDDtdR 1 11= ==y kk
(6.)
whereAi indicates the present value of the bond portfolio cashflows included in the
i-th time bucket. As in Carcano (2009), we assume that the error terms of two zero
rates of different maturity are independent from each other. Additionally, we
assume that the error term of the zero rate of maturity Dkis independent from the
fitted values of all considered zero rates, including the zero rate of maturity=
3
1l
l
tl Fc
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Dk4. On this basis and relying on the definition ofdR(t,Dk)given in (2.), the expected
squared value of the unexpected return can be approximated by:
[ ]
( )
= = = ===
+
+
n
tkyktytkkk
n
k
n
v l h
h
thv
l
tlkty
tvyvtytvvy
tkyktytkktt
DADDt
FcFcEDADDADE
24
,,,,
2
1 1
3
1
3
1
4
1,,,,
4
1,,,,
2
,
(7.)
= = k y1 1
If we construct the Lagrangian function as:
( ) ( ) ,1
,
2
= +
t
M
tytttt HEL 1 =y
and we set its first derivatives equal to zero, we obtain the selffinancing constraint
(8.)
(1.) and the following 4 equations for each futurejincluded in the hedging portfolio:
( ) tn
k
tktkyty
n
tkjkk
h
thv
l
tlktkvtvj ADDtFcFcEDD =
+
=1
,
4
,,,,,
223 3
,, ,2yv l h == = = 11 1 1
The proof of the second order condition of the minimization can be obtained
analogously as in Carcano (2009). For each erroradjusted hedging strategy, the
optimal weights y to be invested in each future have been calculated based on the
last set of equations. For the PCA model, the last equation can be simplified relying
on the independency among the principal components as follows:
(9.)
( )[ ] ( ) tn
k
tktkyty
n
tkjkk
l
ttlvlkkvtvj ADDtFEccDD =
+
=1,
4
,,,,,
223
2
,, ,2yv l == = 11 1
It can be easily seen that the traditional version of these models represents a
special case of their erroradjusted version when the volatility of the model errors
is set equal to zero. We will now briefly recall the hedging equations for the
(10.)
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traditional version of the analyzed models and the corresponding processes of zero
rate changes. The common idea behind the traditional hedging equations is that the
sensitivity of the portfolio to be hedged to the three risk factors must be exactly
replicated by the sensitivity of the hedging portfolio.
In the case of the PCA model, the factors included in equation (2.) are the
three principal components. In addition to the selffinancing constraint, the hedging
equations for the traditional version of this model are:
== =
== =
=
=
=
n
kktkkk
n
tkyty
n
k
kktkkk
n
k y
tkyty
n
k
kktkkk
n
k y
tkyty
DcADc
DcADc
DcADc
3,3
4
,,,
1
2,2
1
4
1
,,,
1
1,1
1
4
1
,,,
== = kk y 11 1
The factors, factor sensitivities, and error terms have been directly obtained by the
applica
(11.)
tion of the PCA methodology.
For the DV model, we refer to Chambers et al. (1988). The process underlying
this model can be considered as a special case of the generic process (2.), where the
three sensitivity parameters clk have been set equal to respectively 1, Dk,and Dk2.
In addition to the selffinancing constraint, the traditional version of the DV model
leads to the following system of hedging equations:
== =
== =
== =
=
=
=
n
k
ktkk
n
k y
tkyty
n
k
ktkk
n
k y
tkyty
n
k
ktkk
n
k y
tkyty
DAD
DAD
DAD
1
3
,
3
1
4
1
,,,
1
2
,
2
1
4
1
,,,
1
,
1
4
1
,,,
(12.)
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The linear 3factor process of zero rate changes which is consistent with this model
and minimizes the model error is represented by:
( ) ( )kktkttk DtDFDFFDtdR ,, +++ (13.)
where the factors and the error terms have been estimated applying the ordinary
least square technique to the changes in all considered zero rates between time t
and t+1. It is visible that if we multiply each term of the last equation by DK in order
to estimate the sensitivity of the bond price to the zero rate change, we obtain the
overall
2321
sensitivity to the factor changes on which the equations in (12.) are based.
A review of the DV methodology is given in Nawalkha et al. [2003] who
propose and test a generalization of it. They found out that for short immunization
horizons like the one we are going to assume a GDV model leading to lower
exponents for Dk than in (13.) leads to better immunization. They suggest that the
reason for this result might be that lower exponents are consistent with mean
reverting processes leading to higher volatility for shortterm rates than for long
term rates (a characteristic consistently displayed by yield curve shifts).
Particularly, they suggest a model which results in setting the three
sensitivity parameters of expression (2.) equal to respectively Dk-0.75, Dk-0.5,and Dk-
0.25. This leads to the following system ofhedging equations:
== =
== =
== =
=
=
=
n
k
ktkk
n
k y
tkyty
n
k
ktkk
n
k y
tkyty
n
k ktkk
n
k y tkyty
DAD
DAD
DAD
1
75.0
,
75.0
1
4
1
,,,
1
5.0
,
5.0
1
4
1
,,,
1
25.0
,
25.0
1
4
1 ,,,
(14.)
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Following the same reasoning described above, the linear 3factor process of
zero rate changes which is consistent with the last set of equations and minimizes
the model error is represented by:
( ) ( k )ttt
k DtFFF
DtdR ,,321
+++ (15.)kkk DDD
25.05.075.0
where the factors and the error terms have been estimated like in the DV model.
It should be highlighted that Nawalkha et. [2003] tested a version of DV and
GDV models including a minimization of the squared values of the weights y. This
was motivated by an excess of the hedging instruments over the hedging
constraints. This does not apply to our case, since we have four hedging constraints
(e.g.: for the GDV model, the three constraints reported under (14.) and the self
financing constraint) and four hedging instruments (the four bond/note future
contracts existing at the beginning of our dataset).
For the KRD model, we refer to Ho (1992). The resulting process of zero rate
changes can be described as:
( ) ( )kl
t
KRD
lkk DtFcDtdR ,,3
+l 1=
where in this case the factor Fl represents the lth key zero rate change and clk
represents the sensitivity of the zerocoupon rate of maturity Dk to this change
which h
(16.)
as been defined following Nawalkha et al. (2005).
In addition to the selffinancing constraint, the resulting system of hedging
equations for the tradition KRD model is:
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== =
== =
=
=
=
n
k
KRD
ktkk
KRD
k
n
tkyty
n
k
k
KRD
ktkk
KRD
k
n
k y
tkyty
n
k
k
KRD
ktkk
KRD
k
n
k y
tkyty
DcADc
DcADc
DcADc
,3,,3
4
,,,
1
,2,,2
1
4
1
,,,
1
,1,,1
1
4
1
,,,
== = kk y 11 1
Also in this case, the error terms have been estimated applying the ordinary least
square technique to the changes in all considered zero rates between time tand t+1,
where the 2year, 12year, and 22year zero rates have been used as key rates and
ave been assumed to be also exposed to model errors.
(17.)
h
3. The ata etandthe estingapproachWe tested the alternative hedging strategies on 144 monthly periods from
December 1996 to December 2008. The portfolio to be hedged is formed by 8 US
Treasury bonds and notes. We defined 8 time buckets with maturity equal to
respectively 2, 4, 6, 8, 10, 16, 20, and 26 years. In order to select the securities
included in the portfolio to be hedged, we impose three conditions: the bonds or
notes must have a publicly held face value outstanding of at least 5 billion US$, the
first coupon must already have been paid and the maturity date must be as close as
possibl
d s t
e to the one of the corresponding time bucket5.
The hedging portfolio was formed by the four US Tbond and Tnote future
contracts with denomination of respectively 2, 5, 10, and 30 years. We always
referred to the next expiring future contract.
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For each contract and each month, we identified the cheapesttodeliver
bond following the net basis method. As pointed out by Choundhry (2006), there is
no consensus about the best way to identify the CTD. The two most common
methods rely either on the net basis or on the implied repo rate (IRR). In academia,
the second method is the most widely used, while practitioners often argue that the
net basis approach should be used since as pointed out by Chance (1989) it
measures the actual profit and loss for a cashandcarry trade. The cheapestto
deliver bonds have been identified relying on the monthly baskets of deliverable
bonds and conversion factors (CF) kindly provided to us by the Chicago Mercantile
Exchange (CME).
We extracted all information related to US Treasury bonds and notes (both
for the securities included in the portfolio to be hedged as well as for the cheapest
todeliver bonds of the future contracts) from the CRSP database. This included both
mid prices and reference data. The closing price of the future contracts has been
provided by Datastream. From both databases, we only downloaded endofmonth
ata.d
In order to estimate the sensitivity of each financial instrument to the three
selected factors, we calculated the present value of each individual cashflow. For
the future contracts, this calculation was based on the cheapesttodeliver bonds.
The discount rate we used for this calculation relied on the Unsmoothed FamaBliss
zerocoupon rates. The methodology followed for the estimation of these rates has
been described in Bliss [1997]. We used the same set of zero rates between May
1975 and December 1991 to estimate the parameters of all tested hedging models.
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We test our hedging strategies by varying the weights invested in the 8
bonds of the portfolio to be hedged. The first 3 portfolios are identified as bullet
portfolios, because the vast majority of the bond positions matures in the same
period. For the shortbullet, this period is within 5 years; for the mediumbullet, it is
between 8 and 16 years, and for the longbulletit is over 20 years. The other three
portfolios replicate typical bond portfolio structures: ladders (evenly distributed
bond maturity), barbells (most bonds mature either in the short term or in the long
term), and butterflies (long positions in bonds maturing either in the short term or
in the long term and short positions in bonds maturing in the medium term). The set
of equations (9.) is solved at the end of each month for each hedging strategy and
each of the 6 bond portfolios; the hedging portfolio for the following month is based
on the resulting weights y for each future contract.
In order to asses the quality of a certain immunization strategy, we analyze
the Standard Error of Immunization (SEI), that is, the average absolute value of the
hedgingerror. The hedging error is the difference between the unexpected return of
the bond portfolio to be hedged and the unexpected return of the futures portfolio.
Lower SEI indicates higher quality of the immunization strategy. The unexpected
return of the bond portfolio is based on the excess return provided by the CRSP
database for the individual bonds (that is, the return in excess of what would have
been computed if the promised yield as of the end of last month had remained
constant throughout the hedging period). For the future contracts, the unexpected
return has been calculated in two different ways depending if the contract expired
during the hedging period or not. In the case of noexpiration, the unexpected return
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has been simply calculated as the percentage change in the quoted future price. In
the case of a contract expiration, the unexpected return has been calculated
assuming an opening of the future position at the end of the previous month and a
delivery of the cheapesttodeliver bond at the end of the expiration month. The
cheapesttodeliver bond has been identified as the bond with the highest delivery
volume based on the actual delivery statistics provided by the CME.
Given the dependency of different hedging strategies on the same case and
time, we estimate statistical significance following an approach of matched pairs
experiment. In other words, we calculated the difference between the absolute value
of the hedging errors generated by two strategies on the same case and holding
period. Our inference refers to the mean value of this difference. As a benchmark
model, we use the erroradjusted PCA, which was expected to be the best performer.
For each hedging problem, we also estimate the square root of the average
sum of the squared weights y expressed as percentages of the bond portfolio value.
This estimate is a useful proxy of the level of transaction costs implied by each
hedging strategy. In fact, these costs are normally proportional to the sum of the
absolute value of all long and short future positions. This statistic can also give a
broad idea of the exposure to model errors implied by a certain strategy.
Finally, we analyzed our dataset in order to assess when market yields
should be considered too close to the notional coupon of the future contracts. As
explained in section 2.1, we intend to base our subsample analysis on this
assessment. This will allow us to isolate the observations for which the impact of the
delivery option is likely to be tangible from the rest of the sample.
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We followed Grieves et al. (2010) in defining tooclose as an absolute distance
not greater than 0.5%. The next figure highlights the period during which market
yields were within this distance from the notional coupon. It was the period starting
from March 2000 (when the notional coupon was lowered from 8% to 6%) and
ending in June 2004 (when the market yield of the 30year bond briefly touched the
lower limit of our range). Accordingly, we will use these two dates to limit our sub
samples.
Figure 1
Assessing the distance between the yields of the 2year, 5year, 10year and 30year
reasury bonds and the future notional coupon (Source: Datastream)T
0
1
2
3
4
5
6
7
8
02.
12.
1996
02.
06.
1997
02.
12.
1997
02.
06.
1998
02.
12.
1998
02.
06.
1999
02.
12.
1999
02.
06.
2000
02.
12.
2000
02.
06.
2001
02.
12.
2001
02.
06.
2002
02.
12.
2002
02.
06.
2003
02.
12.
2003
02.
06.
2004
02.
12.
2004
02.
06.
2005
02.
12.
2005
02.
06.
2006
02.
12.
2006
02.
06.
2007
02.
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2007
02.
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02.
12.
2008
USBD30Y
USBD10Y
USBDS5Y
USBDS2Y
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4. TheresultsAfter estimating the parameters of the tested models between May 1975 and
December 1991, we analyzed the size of the model errors on the same sample. As
expected, all models explain a high proportion of the variance of interest rate
changes, but this proportion is slightly higher for the PCA model (circa 95%) than
for the DV model (circa 93%) and the KRD model (circa 92%). The main reason for
the worst performance of the latter models is their inability to correctly account for
the term structure of volatility (i.e.: the higher volatility of shortterm rates). The
GDV model shares this strength of the PCA model and leads to similar model errors.
The results of the strategies based on the PCA, DV, and KRD models are
reported in Exhibits 1 to 3. For the sake of brevity, we have not reported the results
provided by the GDV model which led to significantly worse hedging than the
simpler DV model. This outcome is not consistent with the abovementioned findings
of Nawalkha et al. [2003]. We believe that the reason for this inconsistency is the
relatively high sensitivity of the futures to changes in the zero rate of maturity s (the
expiration date of the contract), which affects the full costofcarry6. This leads to a
high exposure to model errors which overwhelms the relatively good quality of the
underlying process of interest rate changes.
Exhibit 1 shows a comparison of the results of the three methods in their
traditional forms. As expected, the relative performance largely depends on the
exposure to model errors: the model leading to lower squared weights statistics
generally leads to lower SEI. Since these statistics can randomly vary from one
hedging problem to the other, also the relative performance presents a high degree
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of randomness. On average, the hedging quality of the three models is comparable:
the PCA model happens to be the model leading to the lowest average exposure to
model errors and also to the lowest average SEI. The overall quality of the three
models is not outstanding: on average, the hedging error represents circa 20% of
the unexpected return volatility we intended to hedge.
Exhibit 1
Testing the most common hedging techniques in their traditional form. (December
1996 December 2008, 144 monthly observations)
CaseDescription
Portfolioto
be
hedged TraditionalPCA TraditionalKRD TraditionalDV
Standarddeviationof
unexpectedreturn SEI(1)
(2)
Squared
weights(3)
SEI(2)
Squared
weights SEI(2)
Squared
weights
ShortBullet 1.44% 0.27% *** 1.99 0.20% *** 1.18 0.22% *** 1.38
MediumBullet 1.84% 0.31% *** 2.22 0.24% *** 1.48 0.26% *** 1.68
LongBullet 2.18% 0.41% *** 3.23 0.46% *** 4.61 0.49% *** 4.85
Ladder 1.82% 0.33% *** 2.60 0.33% *** 2.92 0.35% *** 3.16
Barbell 1.81% 0.36% *** 3.02 0.43% *** 4.41 0.45% *** 4.58
Butterfly 1.81% 0.41% *** 3.64 0.57% *** 6.57 0.60% *** 6.62
Average 1.81% 0.35% 2.78 0.37% 3.53 0.39% 3.71
Note: (1) SEI (Standard Error of Immunization) represents the average absolute value of
the hedging error; the hedging error is the difference between the unexpected return of
the bond portfolio to be hedged and the unexpected return of the future portfolio. (2)
Statistical significance is related to the average difference between the absolute value of
the hedging errors for the tested strategy and the erroradjusted PCA: * indicates 10%
significance, ** indicates 5% significance, and *** indicates 1% significance. (3) It
indicates the square root of the average sum of the squared weights y expressed as
percentages of the value of the bond portfolio.
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Our second step is to analyze the performance of the three methods in their
corresponding erroradjusted versions. In Exhibit 2, we compare these results. They
support our initial hypothesis that controlling the exposure to the model errors
significantly improves the hedging quality. In particular, the error adjustment leads
to an average reduction in the SEI for the PCA model of 46% (from 0.35% to 0.19%),
whereas this reduction equals 38% for the KRD and 49% for the DV models. This
reduction is statistically significant for each model and each of the 6 tested bond
portfolios. The reduction in the squared weights statistics obtained for the error
adjusted models is also very substantial, thus highlighting a second important
advantage of this adjustment: the cut in transaction costs. If the costs of settingup
the hedging strategy are indeed proportional to our squared weights statistics, then
the reduction in these costs would be around 80% for each of the three reported
models.
A very relevant observation we can make on Exhibit 2 is that once we
control the exposure to the model errors the superior quality of the process of
interest rate changes underlying the PCA model seems to emerge consistently: on
each of the 6 tested bond portfolios, the PCA model outperforms both alternative
models and in several cases this outperformance is statistically significant. This
strongly differs from the randomness in the relative performance of the traditional
models displayed by Exhibit 1. The overall quality of the three models is now quite
good, especially if one considers that the hedging is based on different financial
instruments (i.e.: futures) than the ones included in the portfolio to be hedged (i.e.:
bonds) and their corresponding prices come from two different markets and data
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providers: the hedging error normally ranges between 10% and 12% of the
unexpected return volatility we intended to hedge.
Exhibit 2
Testing the most common hedging techniques in their erroradjusted form.
(December 1996 December 2008, 144 monthly observations)
CaseDescription
ErrorAdjustedPCA ErrorAdjustedKRD ErrorAdjustedDV
SEI(1)
(2)
Squared
weights(3)
SEI(2)
Squared
weights SEI(2)
Squared
weights
ShortBullet 0.16% 0.51 0.16% 0.54 0.17% ** 0.54
MediumBullet 0.19% 0.60 0.19% 0.62 0.20% * 0.59
Long
Bullet
0.22%
0.72
0.27% *** 0.91
0.22%
0.73
Ladder 0.19% 0.58 0.21% *** 0.67 0.19% 0.59
Barbell 0.19% 0.62 0.25% *** 0.78 0.20% * 0.60
Butterfly 0.20% 0.76 0.31% *** 0.98 0.24% *** 0.62
Average 0.19% 0.63 0.23% 0.75 0.20% 0.61
Note: (1) SEI (Standard Error of Immunization) represents the average absolute value of the
hedging error. The hedging error is the difference between the unexpected return of the
bond portfolio to be hedged and the unexpected return of the future portfolio. (2) Statistical
significance is related to the average difference between the absolute value of the hedging
errors for the tested strategy and the erroradjusted PCA: * indicates 10% significance,
** indicates 5% significance, and *** indicates 1% significance. (3) It indicates the
square root of the average sum of the squared weights y expressed as percentages of the
value of the bond portfolio.
Finally, we report in Exhibit 3 the hedging quality statistics we would have
obtained if the performance of the hedging portfolio would have been calculated on
the initial cheapesttodeliver bonds, instead of on the future contracts. The purpose
of this exhibit is to provide us with an attribution of the hedging error. In fact, the
difference between the SEI reported in Exhibit 1 (Exhibit 2) and the one reported in
Exhibit 3 for the traditional (erroradjusted) form of the tested models is an
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estimate of the impact on the hedging errors of elements which are specific to the
ts and are not reflected by the initial cheapesttodeliver bond.future contrac
Exhibit 3
Calculating the performance of hedging models based on the initial cheapestto
deliver bonds (December 1996 December 2008, 144 monthly observations).
CaseDescription PCA KRD DV
Traditional(1)
ErrorAdjusted(1)
Traditional(1)
ErrorAdjusted(1)
Traditional(1)
ErrorAdjusted(1)
ShortBullet 0.22% 0.06% 0.11% 0.08% 0.14% 0.08%
MediumBullet 0.25% 0.08% 0.13% 0.10% 0.16% 0.10%
LongBullet 0.35% 0.11% 0.36% 0.17% 0.41% 0.11%
Ladder
0.29%
0.08%
0.24%
0.12%
0.28%
0.08%
Barbell 0.32% 0.09% 0.35% 0.16% 0.39% 0.10%
Butterfly 0.37% 0.12% 0.50% 0.23% 0.55% 0.16%
Average 0.30% 0.09% 0.28% 0.14% 0.32% 0.10%
Note: (1) SEI (Standard Error of Immunization) represents the average absolute value
of the hedging error. The hedging error is the difference between the unexpected return
of the bond portfolio to be hedged and the unexpected return of the future portfolio. The
latter unexpected return has been calculated based on the cheapesttodeliver bonds
identified at the beginning of the hedging month and not like in the previous exhibits
on the quoted future prices.
Exhibit 3 highlights that these futurespecific elements explain a relevant
portion of the hedging errors for the erroradjusted models. For example, Exhibit 3
shows that using the initial cheapesttodeliver bonds as hedging vehicles would
have led the PCA model to an average hedging error of 0.09%. In Exhibit 2, we have
seen that using the futures would have led the same model to an average hedging
error of 0.19%. Accordingly, more than 50% of the latter error seems to be due to
futurespecific elements.
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Since they account for such a high proportion of the hedging errors, it makes
sense to analyze these futurespecific elements more in detail. We start from the
element we expect to play the least important role: a temporary mispricing between
the spot and future bond markets. Being bond futures and their cheapesttodeliver
bonds very liquid, we expect speculators to quite rapidly take advantage from this
kind of opportunities. Accordingly, we do not expect a sizeable mispricing to stay in
the market for too long and to significantly influence monthly returns.
A second element of this kind is represented by the difference in the time at
which spot and future prices are observed (5 pm for bonds, 2 pm for futures) and in
their meaning (mid price for bonds, closing price for futures). Also in this case, we
would normally not expect a sizeable impact of such differences on monthly returns.
Moreover, these differences are specific to our testing dataset, but would not affect a
reallife hedging problem. Accordingly, they probably make our hedging strategies
based o s t n bond future looking sligh ly worse than they really are.
The elements we expect to explain the vast majority of the futurespecific
hedging errors are actual changes in the cheapesttodeliver bonds and/or in the
value of the embedded options. The subsample analysis is likely to give us an
indication of the potential size of these effects, since we know that they are much
more relevant in the second subsample (going from March 2000 to May 2004) than
in the first subsample (going from December 1996 to February 2000) or in the
third one (going from June 2004 to December 2008). The next exhibit summarizes
the results of our subsample analysis.
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Exhibit 4
Alternative hedging models based on bond futures: subsample analysis.
CaseDescription PCA KRD DV
Traditional(1)
Error
Adjusted(1)
Traditional(1)
Error
Adjusted(1)
Traditional(1)
Error
Adjusted(1)
SubSample1 Dec.1996toFeb.2000
ShortBullet 15.10% 9.27% 14.29% 10.17% 16.62% 10.36%
MediumBullet 14.68% 8.70% 14.15% 9.76% 16.43% 9.60%
LongBullet 21.40% 10.15% 31.97% 13.10% 36.75% 9.36%
Ladder 18.55% 9.69% 24.75% 11.71% 28.79% 9.45%
Barbell 22.76% 10.73% 35.11% 13.83% 40.43% 10.09%
Butterfly 27.67% 12.38% 48.63% 17.71% 55.91% 12.63%
Average 20.03% 10.15% 28.15% 12.71% 32.49% 10.25%
SubSample2 March2000toMay2004
Short
Bullet
20.28%
13.31%
14.89%
12.35%
15.12%
12.89%
MediumBullet 18.36% 12.57% 14.16% 11.59% 14.08% 12.21%
LongBullet 16.70% 11.79% 12.33% 11.63% 13.71% 12.49%
Ladder 17.96% 12.35% 12.97% 11.61% 13.86% 12.38%
Barbell 18.23% 12.27% 13.02% 12.48% 14.80% 13.22%
Butterfly 18.61% 12.12% 13.55% 13.77% 15.95% 15.32%
Average 18.36% 12.40% 13.49% 12.24% 14.59% 13.09%
SubSample3 June2004toDec.2008
ShortBullet 18.67% 9.58% 13.10% 11.04% 13.98% 11.39%
MediumBullet 17.27% 9.07% 11.90% 9.68% 12.46% 10.32%
LongBullet
19.07%
8.19%
23.56%
12.83%
22.19%
8.55%
Ladder 18.81% 8.63% 19.43% 11.93% 18.88% 9.07%
Barbell 20.69% 8.59% 27.56% 14.65% 25.55% 9.18%
Butterfly 23.90% 9.13% 39.06% 19.45% 35.20% 10.86%
Average 19.74% 8.87% 22.44% 13.26% 21.38% 9.89%
Note: (1) SEI (Standard Error of Immunization) as a percentage of the standard
deviation of the unexpected return from the bond portfolio to be hedged. SEI represents
the average absolute value of the hedging error. The hedging error is the difference
between the unexpected return of the bond portfolio to be hedged and the unexpected
return of the future portfolio.
Our subsample analysis highlights the robustness of the erroradjustment.
Only in one case (the KRD hedging of the butterfly portfolio in the second sub
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sample) the traditional model performs better than the erroradjusted one.
Accordingly, the erroradjustment still works when the impact of the options to
deliver on the hedging portfolio is tangible, like in the second subsample.
Also the superiority of the erroradjusted PCA model over the corresponding
DV and KRD models is largely confirmed by the subsample analysis. On average
(even though not in any test case), the PCA outperforms the DV in every subsample
and the KRD in two of the three subsamples.
Let us now come to our initial question on the impact of changes in the
cheapesttodeliver bonds and/or in the value of the embedded options on the
hedging quality. Exhibit 4 does indeed highlight a worsening of this quality in the
second subsample for the two best models (the erroradjusted PCA and DV). The
size of this worsening is around 3% of the volatility we intended to hedge. For the
erroradjusted KRD model, the second subsample happens to be the one displaying
the best hedging performance, even though the results of this model are pretty
similar over all three subsamples.
Based on the latest observations, we can say that valuable options to deliver
have a limited impact on our erroradjusted models, at least in percentage terms.
Nevertheless, we should ask ourselves where the worsening displayed by the PCA
and DV models in the second subsample might be coming from. Does it come from
our simplified estimation of the futures sensitivity ignoring the options to deliver?
Or does it rather come from the fact that "thefuturepricenotonlydoesnotbehave
likeanyonebondornote,butbehavesinsteadlikeacomplexhybridofthebondsand
notes in the deliverable set.." (Burghardt et al. (2005)), a statement which is
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particularly true when market yields are close to the notional coupon? In other
words, is the slight worsening in the second subsample due to a weakness in our
modeling or to a weakness of futures as hedging instruments? We suspect the
second element to be the most relevant one: changes in the cheapesttodeliver
bonds during the hedging month are almost 5 times more frequent in the second
subsample than in the remaining subsamples, thus making the sensitivity of the
contracts intrinsically unstable. A formal and documented answer to this question is
eft to further research efforts.l
5 ConclusionsOur results highlight that traditional implementations of the models most
commonly used for hedging yield curve risk tend to lead to high exposure to model
errors and to sizable transaction costs, thus lowering the hedging quality and
making a clear ranking of the models difficult. In fact, the exposure to model errors
genera
.
ted by a certain model varies quite randomly across hedging problems.
As a consequence, including some mechanisms to control the exposure to
model errors is of paramount importance for a sound implementation of these
models. We presented the results of explicitly accounting for the variance of the
model errors displayed by each zero rate. We found out that the reduction in both
the hedging errors and the transaction costs is very substantial: the errors are
reduced on average by 46% for the PCA model, by 38% for the KRD model and by
49% for the DV model. If the costs of settingup the strategies are proportional to
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our squared weights statistics, the reduction in these costs would be circa 80% for
all thre me odels.
A further benefit of the error adjustment is to make the model ranking
consistent and often statistically significant: the erroradjusted PCA model
outperforms both alternative models on every single hedging case. To the best of
our knowledge, this result is new. We attribute it to the better quality of the interest
rate process underlying the PCA model, which explains the largest possible part of
the variance of yield curve shifts based on three orthogonal factors.
Finally, our study shows that bond futures can effectively be used to hedge
the yield curve risk of a bond portfolio. When erroradjusted models are applied,
only 10%12% of the risk to be hedged is left as a hedging error (gross of the effect
of the 3hour difference between spot and future endofday prices). This is still
more than what we would obtain by using bonds to hedge other bonds. However,
futures present other advantages, such as strongly reduced need of cash, higher
liquidit iy, and lower transact on costs.
All abovementioned results have been found to be robust to subsample
analysis. Particularly, since we applied a simplified estimation of futures sensitivity
ignoring the options to deliver, we checked this robustness on the second sub
period from March 2000 to May 2004, during which these options have been
particularly valuable. Even though we did notice a worsening in the hedging quality
of PCA and DV models for this subsample, this amounted to only 3% of the volatility
we intended to hedge.
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An important challenge we leave open to further research concerns the
development of hedging techniques reconciling an adjustment for model errors with
term structure models capable of properly assessing the sensitivity of the embedded
options to yield curve changes. A further improvement of the present paper could be
represented by the explicit consideration of futurespecific model errors in the
application of the error adjustment. Both developments should at least in theory
lead to further improvements in the hedging quality relatively to the results
eported in this paper.r
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ENDNOTES1The original formulation of duration relied on flat yield curves, but this restrictionwas overcome thanks to the formulation proposed by Fisher and Weil (1971). For
an extensive review of how the concept of duration was developed during the lastcentury, see Bierwag (1987).
2 Nawalkha et all. [2005] affirm that the DV model must be considered more robustand suitable for hedging purposes when time series of interest rate changes arenonstationary, since in this case the estimates of PCA models are highly instable.They also highlight that the KRD model leads to an arbitrary selection of the numberand maturity of the key rates and to implausible shapes for the yield curve shifts.
3 The first time option consists in the possibility for the short position to deliver atany time during the expiration month (generally speaking, early delivery ispreferable if the cost of financing exceeds the CTD coupon and vice versa). The
second time option the socalled end-of
-the
-month
option consists in thepossibility for the short position to deliver during the final business days of the
deliverable month after the invoice price has been locked in. The third time option isthe socalled wild-cardoption. It consists in the possibility for the short position tolock in the invoice price at 3 pm during the delivery month and make the delivery ifthe spot price falls below the established invoice price between 3 pm and 5 pm.
4 For the PCA model, this assumption is fulfilled by construction. Considering the
way how we estimated the error terms (t,DK) for the other models, this assumptionis also fulfilled by construction as far as the independency between the error termand the fitted value of the same zero rate is concerned. For models other than PCA,
t (t,D ) eshe independency between the error term K and the fitted value of zero ratwith maturity other than DK is a simplifying assumption.
5 The first condition ensures a good level of liquidity for the considered securities,while the second one allows us to avoid the complexity linked to the potentialirregularity of the first coupon payment and the third one leads to spread thes securities as evenly as possible within the elected range of maturities.
6 Given our testing approach, the maturity s is very short (ranging from one to threemonths). Now, the GDV method leads to much higher values of the sensitivityparameters clfor very shortterm rates than for any other rate. Since the portfolio tobe hedged displays a much lower sensitivity to changes in very shortterm ratesthan the future contracts, the GDV minimization procedure leads to wide longshortfuture positions having the goal of offsetting the high sensitivity of the zero rate ofmaturity s to the three risk factors.
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