3 Risk factor: movement of basis government yield curve
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Performance attribution
detail would be out of scope of this chapter, so just a few representatives
should be stated: equilibrium concepts – Vasicek (1977), Brennan and
Schwartz (1979; 1982) and Cox et al. (1985); no-arbitrage models – Ho and
Lee (1986), Black et al. (1990) and Hull and White (1990; 1993).
Also we do not deal with the principal components models in detail. One
shortcoming of this group of models is the huge number of required parameters: e.g. for three principal components and twelve times to maturity
thirty-six parameters are needed. These are the parameters which describe
the characteristic yield-curve movements for each of the three risk factors at
each of the twelve maturities. To mention a few, term structure models
based on principal components analysis can be found in Litterman and
Scheinkman (1991), Bliss (1997) and Esseghaier et al. (2004).
Spot rate models define the risk factors to be the changes of yields of
hypothetical zero-coupon bonds for specific times to maturity, i.e. the
changes of pre-defined spot rates. The number of the spot rates is a variable
within the modelling process – so the portfolio analyst has a huge degree of
freedom to specify the model the way that corresponds best with the
individual investment management attributes (as a popular reference, a spot
rate model is incorporated into the RiskMetrics methodology – see RiskMetrics Group 2006). The sensitivities to the changes of the defined spot
rates are known as ‘key rate durations’ (see Reitano 1991; Ho 1992); instead
of assuming a price sensitivity against the parallel change of the yield curve
(as the modified duration does), key rate durations treat the price P as a
function of N chosen spot rates – designated as the key rates r1, . . . ,rN:
P ¼ Pðr1 ; :::; rN Þ
ð6:22Þ
Key rate durations KRDi are partial durations which measure the sensitivity
of price changes of first order to the isolated changes of i various segments
on the government spot curve:
1 dP
8i 2 ½1; N
ð6:23Þ
KRDi ¼ À
P dri
Therefore every yield-curve movement can be represented as a vector of
changes of defined key rates (dr1, . . . ,drN). The relative price change is
approximated by
N
X
dP
%À
ðKRDi Á dri Þ
P
i¼1
ð6:24Þ
236
Marton, R. and Bourquin, H.
In practice, key rate durations are numerically calculated as follows:
KRDi ¼ À
1 Pi;up À Pi;down
P
2dri
ð6:25Þ
where Pi,up and Pi,down are the calculated model prices after shocking up and
down the diverse key rates. The key rate convexity KRCi,j for the simultaneous variation of the i-th and j-th key rate is given by (see Ho et al.
1996)
1 d 2 P
8i; j
ð6:26Þ
KRCi;j ¼
P dri Á drj
Formula (6.24) must therefore be extended to
N
X
dP
1X
ðKRDi Á dri Þ þ
ðKRCi;j Á dri Á drj Þ
%À
P
2 i;j
i¼1
ð6:27Þ
Summing up the key rate shocks should accumulate to a parallel curve
shock – this intuitively means that the exposure to a parallel change of the
yield curve comprises the exposures to various units of the curve. It is not
guaranteed that the sum of the key rate durations is equal to the modified
duration; but for instruments without cash flow uncertainties, which are the
usual case in risk-averse portfolios of public investors like central banks, the
difference is naturally small. For more complex products this difference can
be substantially bigger due to the non-linear cash flow structure.
A disadvantage of spot rate modelling approaches is that the characteristic yield-curve shifts are not defined to be continuous. This means that
certain interpolations of the interest rate changes are necessary to enable
applying the model to zero-coupon bonds with maturities that do not
correspond to the model-defined maturities – the bigger the number of risk
factors the more precise the model will be. A representative central bank
which successfully implemented a performance attribution framework based
on the key rate concept is the European Central Bank. The ECB model
consists of eighteen specified key rate maturities up to thirty years – this
large number along with a sophisticated cash flow mapping algorithm
reduces to a minimum any inaccuracies due to interpolations. The methodology of the ECB is briefly described in Section 5 of this chapter.
Functional models assume that the changes of interest rates, in particular
the government spot rates, are defined continuously. They belong to the
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Performance attribution
class of parsimonious models – this is due to the fact that these techniques
model the spot curve just by its first (mostly) three principal (i.e. orthogonal) components which together explain most of the variance of the historical values of government yield curves.12 The risk factors are represented
by the variations of the principal components during the analysis period:
parallel shift (level change), twist (slope change) and butterfly (curvature
change). Functional models can be divided into the following two categories:
polynomial models and exponential models.
In polynomial models the government spot rate rT–t,t for time to maturity
T – t as of time t is described by polynomials in ascending order of power,
where the general form is given by
rT Àt;t ¼ nt þ wt Á ðT À tÞ þ ft Á ðT À tÞ2 þ · · ·
ð6:28Þ
where nt, wt und ft are time-variable coefficients associated with the yieldcurve components level, slope and curvature.
The parallel shift PSdt in period dt is
PSdt ¼ atþdt À at
ð6:29Þ
where at stands for the average level of the spot curve at time t.
The twist TWdt in period dt is
TWdt ¼ ðbtþdt À bt Þ þ ðctþdt À ct Þ Á ðT À tÞ
ð6:30Þ
where bt þ ct · (T – t) represents the best linear approximation of the curve
at time t.
The butterfly BFdt in period dt is
BFdt ¼ ðdtþdt À dt Þþðetþdt À et Þ Á ðT À tÞþðftþdt À ft Þ Á ðT À tÞ2
ð6:31Þ
where dt þ et · (T – t) þ ft · (T – t)2 describes the best quadratic approximation of the curve at time t.
Exponential models on the other hand do not use polynomials but
exponentials to reconstruct the yield curve. As a benefit the yield curves can
be captured more accurately and so the resulting residual is smaller. The
approach by Nelson and Siegel (1987), used also in other chapters of this
12
As an example, for the US Treasury yield curve and the German government yield curve the explained original
variance by the first three principal components is about 95 per cent (first component: $75 per cent, second
component: $15 per cent and third component: $5 per cent).
238
Marton, R. and Bourquin, H.
book, is an exponential model which specifies a functional form of the spot
curve. The original motivation for this way of modelling was to cover the
entire range of observable shapes of the yield curves: a monotonous form,
humps on different positions of the curve and S-formations. The Nelson–
Siegel model has four parameters which are to be estimated: b 0, b1, b 2 and
s1. These coefficients identify three unique attributes: an asymptotic value,
the general shape of the curve and a humped or U-shape which combined
generate the Nelson–Siegel spot curve for a specific date.13
The spot rate rT–t,t is determined by
rT Àt;t ¼b 0;t þ b 1;t
þ b2;t
!
1 À e ÀðT ÀtÞ=s1;t
þ
Á
ðT À tÞ=s1;t
1 À e ÀðT ÀtÞ=s1;t
Á
À e ÀðT ÀtÞ=s1;t
ðT À tÞ=s1;t
!
ð6:32Þ
The optimal parameter values are those for which the resulting model prices
of the government securities (i.e. government bonds and eventually also
bills) match best the observed market prices at the same point of time.14
Regarding the model by Nelson and Siegel, the parameters b 0,t, b 1,t, b 2,t can
be interpreted as time-variable level, slope and curvature factors. Therefore
the variation of the model curve can be divided into the three principal
components parallel shift, twist and butterfly – every movement corresponds to the respective parameter b 0,t, b1,t or b 2,t. The parallel shift PSdt in
period dt is given by
PSdt ¼ b 0;tþdt À b 0;t
ð6:33Þ
The twist TWdt in period dt is covered by
TWdt ¼ b 1;tþdt
13
14
1 À e ÀðT ÀtÞ=s1;tþdt
Á
ðT À tÞ=s1;tþdt
!
Àb1;t
tþdt
1 À e ÀðT ÀtÞ=s1;t
Á
ðT À tÞ=s1;t
!
ð6:34Þ
t
An extension to the Nelson and Siegel (1987) method is the model by Svensson (1994). The difference between both
approaches is the functional form of the spot curve – the Svensson technique defines a second exponential
expression which specifies a further hump on the curve.
Actually there are two ways to define the objective function of the optimization problem: either by minimizing the
price errors or by minimizing the yield errors. As government bond prices are traded in the market it makes sense to
specify a loss function in terms of this variable which is directly observed in the market.
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Performance attribution
The butterfly BFdt in period dt is modelled by
BFdt ¼ b 2;tþdt
À b 2;t Á
1 À e ÀðT ÀtÞ=s1;t þdt
Á
À e ÀðT ÀtÞ=s1;tþdt
ðT À tÞ=s1;tþdt
1 À e ÀðT ÀtÞ=s1;t
À e ÀðT ÀtÞ=s1;t
ðT À tÞ=s1;t
!
tþdt
!
ð6:35Þ
t
Complementing the described partial motions of the yield curve, also the
sensitivities towards them can be derived from an exponential model (see
e.g. Willner 1996). Following the approach proposed by Kuberek,15 which is
a modification of the Nelson-Siegel technique, the price of a government
security i in continuous time can be represented in the following functional
form:
Pi;t ¼ f ðt; r; b0 ; b1 ; b 2 ; s1 Þ
i
Xh
ÀðT ÀtÞ=s1;t
þb 2;t Áðt=s1;t ÞÁe 1ÀðT ÀtÞ=s1;t Þ
CFi;T Àt;t Á e ÀðT ÀtÞÁðrT Àt;t þb0;t þb1;t Áe
¼
ð6:36Þ
8T Àt
The model-inherent factor durations of every instrument (and hence also
portfolio) can then be quantified analytically. The sensitivity to a parallel
shift Duri,PS,t is determined by
Duri;PS;t ¼
i
1 @Pi;t
1 Xh
¼À
ÀðT À tÞ Á CFi;T Àt;t Á e ÀðT ÀtÞÁrT Àt;t
Pi;t @b 0;t
Pi;t 8T Àt
ð6:37Þ
The sensitivity to a twist Duri,TW,t is calculated by
Duri;TW ;t ¼
i
1 @Pi;t
1 Xh
¼À
ÀðT À tÞ Á e ÀðT ÀtÞ=s1;t Á CFi;T Àt ;t Á e ÀðT ÀtÞÁrT Àt;t
Pi;t @b 1;t
Pi;t 8T Àt
ð6:38Þ
The sensitivity to a butterfly Duri,BF,t is given by
1 @Pi;t
Pi;t @b 2;t
i
À
Á
1 Xh
¼À
ÀðT À tÞ Á ðT À tÞ=s1;t Á e 1ÀðT ÀtÞ=s1;t Á CFi;T Àt;t Á e ÀðT Àt ÞÁrT Àt;t
Pi;t 8T Àt
Duri;BF;t ¼
ð6:39Þ
15
Kuberek, R. C. 1990, ‘Common factors in bond portfolio returns’, Wilshire Associates Inc. Internal Memo.
240
Marton, R. and Bourquin, H.
One major advantage of exponential functional models is the fact that they
only require very few parameters (times to payment of the cash flows and
the model beta factors) to be able to determine the corresponding spot rates.
Many central banks use exponential functions to construct the government
spot curve either by using the model by Nelson and Siegel (1987) or by
Svensson (1994) – for an overview see the study developed by the BIS (2005).
For the purpose of fixed-income performance attribution analysis,
exponential techniques are used most frequently when applying functional
models, because they produce better approximations of the yield curve
compared to the polynomial alternatives with comparable degree of
complexity. Thus, for example, polynomial modelling using a three-term
polynomial would only produce a quadratic approximation of the yield
curve, and this would lead to distorted results (mainly for short and long
maturities). As a reference, elaborations on the polynomial decomposition
of the yield curve can be found in Colin (2005, chapter 6) and Esseghaier
et al. (2004).
3.4 Risk factor: narrowing/widening of sector and euro country spreads
Alongside the movement of the government yield curve, the change of an
instrument’s yield to maturity is affected by the variation of the spread
against the basis government yield curve – see formula (6.20).16 When
analysing at portfolio level, at least two fundamental types of spreads can be
distinguished: sector and euro country spreads. Specifically in the case of
evaluating central bank portfolios it is advisable to separate these categories
and not to subsume them under the same expression, e.g. ‘credit spread’,
because the intentions behind the different types of spread positions can
vary. In euro portfolios, the German government yield curve could be
chosen as the reference yield curve; the differences between the non-German
government yield curves and the German government yield curve would
be designated as euro country spreads. In central banks the euro country
spread exposures (versus the benchmark) might not be taken as part of
portfolio management decisions, e.g. by decisions of an investment committee, and hence would not be treated as credit spread positions. On the
contrary, investments in non-government issues, like US agency bonds or
instruments issued by the Bank for International Settlements (BIS), which
16
Technically spoken, the spread could be interpreted as an option-adjusted spread (OAS), i.e. a constant spread to the
term structure based on an OAS model.
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Performance attribution
imply sector spread positioning (against the benchmark), are mostly due to
concrete strategic investment policy directives or tactical asset allocation
decisions.
To precisely quantify the sensitivity to a sector or euro country spread
change, the spread duration and not the modified duration should be used
as a measure. It specifies the amount by which the price of a risky (in terms
of deviating from the basis yield curve) interest rate-sensitive instrument
i changes in per cent due to a ceteris paribus parallel shift dsi,t of 100 basis
points of its spread. The numerical computation of the spread duration
Duri,spr,t is similar to the calculation of the option-adjusted duration – with
the difference that it is the spread that is shifted instead of the spot rate:
Duri;spr;t ¼ À
1 Pi;SpreadUp;t À Pi;SpreadDown;t
P i;t
2 Á dsi;t
ð6:40Þ
where Pi,SpreadUp,t and Pi,SpreadDown,t are the present values which result after
the upward and downward shocks of the spread, respectively. Consequently,
equation (6.21) for the calculation of the price (present value) of an
interest-sensitive instrument must be extended with respect to the influence
of the spread:
Pi;t ¼ f ðt; rT Àt;t ; si;t Þ ¼
X
CFi;T Àt;t
8T Àt
ð1 þ rT Àt;t þ si;t ÞT Àt
ð6:41Þ
where at time t: rTÀt,t is the government spot rate associated with the time to
cash flow payment T – t; si,t is the spread calculated for instrument i against
the government spot curve.
The total spread of an instrument’s yield versus the portfolio base currency-specific basis yield curve is normally to a great extent described by its
components sector and country spread – the residual term can be interpreted as the issue- or issuer-specific spread (whose change effects are
mostly explicitly or implicitly attributed to the category ‘selection effect’
within a performance attribution model).
4. Performance attribution models
In the previous section of this chapter the basis elements required to set up a
performance attribution model were derived: the general structure of a
multi-factor model and the return-driving risk factors for fixed-income