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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

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    Established at the initiative of the Swiss Bankers' Association, the Swiss

    Finance Institute is a private foundation funded by the Swiss banks andSwiss Stock Exchange. It merges 3 existing foundations: the International

    Center FAME, the Swiss Banking School and the Stiftung "Banking and

    Finance" in Zurich. With its university partners, the Swiss Finance Institute

    pursues the objective of forming a competence center in banking and

    finance commensurate to the importance of the Swiss financial center. It will

    be active in research, doctoral training and executive education while also

    proposing activities fostering interactions between academia and the

    industry. The Swiss Finance Institute supports and promotes promising

    research projects in selected subject areas. It develops its activity in

    complete symbiosis with the NCCR FinRisk.

    The National Centre of Competence in Research Financial Valuation and

    Risk Management (FinRisk) was launched in 2001 by the Swiss National

    Science Foundation (SNSF). FinRisk constitutes an academic forum that

    fosters cutting-edge finance research, education of highly qualified finance

    specialists at the doctoral level and knowledge transfer between finance

    academics and practitioners. It is managed from the University of Zurich and

    includes various academic institutions from Geneva, Lausanne, Lugano,

    St.Gallen and Zurich. For more information see www.nccr-finrisk.ch .

    This paper can be downloaded without charge from the Swiss Finance

    Institute Research Paper Series hosted on the Social Science Research

    Network electronic library at:

    http://ssrn.com/abstract=1635291

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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

    mailto:[email protected]:[email protected]:[email protected]
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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

    2

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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),

    3

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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

    4

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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

    5

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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

    6

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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#bb24
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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.

    9

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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

    11

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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.)

    12

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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.)

    13

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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.

    12.

    2007

    02.

    06.

    2008

    02.

    12.

    2008

    USBD30Y

    USBD10Y

    USBDS5Y

    USBDS2Y

    20

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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

    29

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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.

    30

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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

    31

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    35

    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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