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4 Risk factor: narrowing/widening of sector and euro country spreads

4 Risk factor: narrowing/widening of sector and euro country spreads

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241

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

242

Marton, R. and Bourquin, H.

portfolios typically managed in central banks. We now provide a method to
attribute the value added arising from the active investment decisions in a
way, so that the senior management, the portfolio management (front
office) and the risk management (middle office) are aware of the sources of
this active return, which should consequently improve the transparency of
the investment management process. A suitable model must fulfil specific
requirements:
1. The appropriate performance attribution model which is to be chosen
must be in accordance with the active investment decision processes,
primarily with respect to the factor-specific relative positioning versus
the benchmark. This is most probably the crucial point when carrying
out the model specification or verification. It is common knowledge that
the local-currency buy-and-hold return (i.e. without the impact of
trading activities) of an interest rate-sensitive instrument is mainly
driven by the impacts of the decay of time and the change of its yield. But
how to best disentangle those factors to be in conformity with the
investment strategies and to be able to measure the distinct success of
each of them in excess to the benchmark return?
2. The variable which is aimed at being explained as accurately as possible by
the model components is the active return (i.e. the out- or underperformance) versus the benchmark based on the concept of the TWRR17 – so the
incorporation of solely market risk factors of course does not sufficiently
cover the range of the performance determinants. As parts of the TWRR
are caused by holding instruments in the portfolio that perform better
or worse than the average market changes (based on the incorporated
yield curves) would induce, also an instrument selection effect must be
part of the analysis. Additionally, dealing with better or worse transaction
prices than quoted on the market at portfolio valuation time has an
impact on the TWRR and therefore the trading skills of the portfolio
managers themselves also act as an explanatory variable of the attribution
model.
3. The performance attribution reports are to be tailored for the objective
classes of recipients, i.e. senior management, portfolio management, risk
management, etc. The classes determine the level of detail reported; for
example, whether reporting at individual security level is necessary or
desired. The design of the model is dependent on the needs of the
clients – the resulting decision is of significant influence on the model
17

See Chapter 5 for the determination of the time-weighted rate of return.

243

Performance attribution

building: defining and implementing an attribution model just for the
total portfolio level is usually easier than following a bottom-up approach
from security level upwards.
4. As described in the subsequent sections there are different ways to
process attribution analysis within a single period and multiple periods.
But which technique is the most suitable in the individual case? The
model must in any case guarantee mathematical precision without
causing technically (i.e. methodically) induced residuals. Central banks
are rather passive investors versus the benchmarks and so naturally the
active return which is to be explained by the attribution model is rather
small. Using an imprecise model could therefore easily lead to a
dominance of the residual effect which is of course a result to be avoided.
Additionally, the results of the attribution analysis must be intuitive to
the recipients of the reports – no ‘exotic’ (i.e. non-standard) calculation
concepts are to be used.
Basically, two ways of breaking down the active return into its determining
contributions are prevalent: arithmetically and geometrically – in the
first case the decomposition is processed additively and in the second case
it is done multiplicatively. In each of these cases the model must be
able to quantify the performance contributions for single-currency and
multi-currency portfolios (for the latter the currency effects must be supplemented). A further considerable component is the time factor as the
single-period attribution effects must be correctly linked over time. As a
reference, a good overview of the different types of attribution models is
given by Bacon (2004).
In principle, the formal conversion of return decomposition models into
performance attribution models is done as follows. The decomposition
model for the return RP of portfolio P is defined by (for the notation see the
previous Section 2 on multi-factor return decomposition models)
RP ¼

K
X

ðbP;k Á Fk Þ þ eP

ð6:42Þ

k¼1

The representation of the performance attribution model, which explains
the active portfolio return ARP, (assuming an additive model) is given by
ARP ¼ RP À RB ¼

K À
X
k¼1

Á
ðbP;k À bB;k Þ Á Fk þ eP

ð6:43Þ

244

Marton, R. and Bourquin, H.

where bP,k is the sensitivity of portfolio P versus the k-th risk factor, bB,k is
the sensitivity of benchmark B versus the k-th risk factor and Fk is the
magnitude of the k-th risk factor.
At this point the fundamental differences between empirical return
decomposition models (as described in Section 2.3) and performance attribution models should be emphasized: in equation (6.42) the buy-and-hold
return is the dependent variable and so the risk factors are solely market risk
factors as well as a residual or idiosyncratic component representing the
instrument selection return, whereas in equation (6.43) the dependent
variable is the performance determined via the method of the time-weighted
rate of return and so the market risk factors and the security selection effect
are extended by a determinant which represents the intraday trading
contribution.
Performance attribution models can be applied to various levels within
the portfolio and the benchmark, beginning from security level, across
all possible sector levels, up to total portfolio level. The transition of a
sector-level model to a portfolio model (i.e. the conversion of sector-level
performance contributions into portfolio-level performance contributions)
is done by market value-weighting the determinants of the active return
ARP:
ARP ¼ RP À RB ¼

N X
K À
X

Á
ðbP;i;k Á wP;i À bB;i;k Á wB;i Þ Á Fk þ ei

ð6:44Þ

i¼1 k¼1

where wP,i and wB,i are the market value weights of the i-th sector within
portfolio P and benchmark B, respectively; bP,i,k and bB,i,k are the sensitivities of the i-th sector of portfolio P and benchmark B, respectively, versus
the k-th risk factor.
The flexible structure of the model allows the influences on aggregate
(e.g. total portfolio) performance to be reported along the dimensions of
risk factor categories and also sector classes. The contribution PCk related to
the k-th factor across all N sectors within the portfolio and benchmark to
the active return is determined by
PCk ¼

N
X

ðbP;i;k Á wP;i À bB;i;k Á wB;i Þ Á Fk

ð6:45Þ

i¼1

The contribution PCi related to the i-th sector across all K risk factors of the
attribution model to the performance is then given by

245

Performance attribution

PCi ¼

K
X

ðbP;i;k Á wP;i À bB;i;k Á wB;i Þ Á Fk

ð6:46Þ

k¼1

4.1 Fundamental types of performance attribution models
In an arithmetical (additive) performance attribution model, in every single
period the following theorem must be satisfied: the sum of the individual
performance contributions at a given level must equal the active return
ARadd at this level:
AR add ¼

N X
K
X

PCi;k ¼ RP À RB

ð6:47Þ

i¼1 k¼1

where PCi,k is the performance contribution related to the k-th risk factor
and the i-th sector; RP is the return on portfolio P; RB is the return on
benchmark B; N is the number of sectors within the portfolio and the
benchmark; K is the number of risk factors within the model.
For a single-currency portfolio whose local-currency return is not converted
into another currency, all K return drivers are represented by local risk factors.
But in case of a portfolio comprising more than one currency, the local returns
of the assets are to be transformed into the portfolio base currency in order to
obtain a reasonable portfolio return measure. As central banks and other
public investors are global financial players that invest the foreign reserves
across diverse currencies, a single-currency attribution model is not sufficient
to explain the (active) returns on aggregate portfolios in base currency.
This implies for the desired attribution model that currency effects
affecting the portfolio return and performance additionally would have to
join the local determinants:
RP;Base ¼ RCP;i;currency þ

N X
K
X

RCP;i;k;local

ð6:48Þ

i¼1 k¼1

where RP,Base is the return on portfolio P in base currency; RCP,i,currency is the
contribution to the portfolio base currency return stemming from the
variation of the exchange rate of the base currency versus the local currency
of the i-th sector of portfolio P; RCi,k,local is the contribution of the i-th
sector of portfolio P to the notional local-currency portfolio return, with
respect to the local risk factor k.

246

Marton, R. and Bourquin, H.

The local return of a currency-homogeneous sector (where ‘sector’ can
therefore also stand for an asset) and the appreciation or depreciation of the
exchange rate of the base currency relative to the local currency of the sector
on a given day are linked multiplicatively to get the return in base currency
(see formula [5.6]), while, on the contrary, arithmetical attribution models
decompose the (base currency) return additively (the same is therefore valid
for the active return). Caused by these methodically heterogeneous procedures for return calculation and return decomposition, an interaction
effect (i.e. an intra-temporal cross product) arises which should be visualized separately in the attribution reports, because it is model-induced and
not intended by any active investment decisions. At portfolio level the
interaction effect IP is given by
IP ¼

N Â
X

ðwP;i À wB;i Þ Á ðRP;i;local À RB;i;local Þ Á Ri;xchÀrate

Ã

ð6:49Þ

i¼1

where wP,i and wB,i are the market value weights of the i-th sector within
portfolio P and benchmark B, respectively; RP,i,local and RB,i,local are the local
returns of the i-th sector within portfolio P and benchmark B, respectively;
Ri,xch-rate is the movement of the exchange rate of the base currency versus
the local currency of the i-th sector.
A first simple, intuitive approach to determine the currency contribution
CYP,i of the i-th sector to the performance of a multi-currency portfolio P
relative to a benchmark B could be defined as follows:
CYP;i ¼ wP;i Á ðRP;i;base À RP;i;local Þ À wB;i Á ðRB;i;base À RB;i;local Þ

ð6:50Þ

As this method does not explicitly incorporate the effect of currency
hedging it is only of restricted applicability for typical central bank portfolios. In the following paragraphs two alternative theoretical attribution
techniques which explicitly include the impact of hedging are sketched and
subsequently a pragmatic solution is introduced.
In the Ankrim and Hensel (1994) approach the currency return is defined to
consist of two components – the unpredictable ‘currency surprise’ and the
anticipatable interest rate differential (the forward premium) between the
corresponding countries, respectively. By adopting the models by Brinson and
Fachler (1985) and Brinson et al. (1986) the performance contributions
resulting from asset allocation and instrument selection decisions as well as
from the interaction between those categories are derived and the contributions attributable to the currency surprise and the forward premia are added.

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Performance attribution

Alternatively, the method by Karnosky and Singer (1994) incorporates
continuous-time returns18 and treats forward premia as so called ‘return
premia’ above the local risk-free rates. Again, the local asset allocation,
instrument selection and interaction concepts are used and the currency
effect (with a separate contribution originating from currency forwards) is
added (see the articles by Laker 2003; 2005 as exemplary evaluations of the
Karnosky–Singer model). Although both of the mentioned multi-currency
models already exist for more than a decade, the portfolio managers who
use it in daily practice represent a minority. This is probably due to the fact
that these two approaches are too complex and academic to be applied in a
practical environment.
To overcome the methodical obstacles and interpretational disadvantages
which are prevalent within both of the above-described approaches, a
pragmatic way to disentangle the currency hedging effect from the overall
currency effect is presented. This scheme is also suitable for the attribution
analysis of foreign reserves portfolios of central banks and other public
wealth managers. The currency impact CYP,i of the i-th sector on the
portfolio performance can be broken down by
CYP;i ¼ ðwP;i À wB;i Þ Á Ri;xchÀrate
¼ ðwP;i; invested À wB;i; invested Þ Á Ri;xchÀrate
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
invested currency exposure effect

ð6:51Þ

þ ðwP;i; hedged À wB;i; hedged Þ Á Ri;xchÀrate
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
hedged currency exposure effect

where wP,i,invested and wB,i,invested are the invested weights, and wP,i,hedged and
wB,i,hedged are the hedged weights of the i-th sector within portfolio P and
benchmark B, respectively.
To be able to determine the hedged weights, each currency-dependent
derivative instrument within the portfolio, e.g. a currency forward, must be
split into its two currency sides – long and short. The currency-specific
contribution to the hedged weights stemming from each relevant instrument is its long market value divided by the total portfolio market value
and its short market value divided by the total portfolio market value,
respectively. Subsequently, for every i-th sector within the portfolio, the
sum of the currency-specific hedged weights contributions and the sum of
18

Continuous-time returns were applied to enable the simple addition and subtraction of returns.

248

Marton, R. and Bourquin, H.

the currency-specific invested weights contributions (e.g. from bonds) must
be compared with their benchmark equivalents to obtain the hedged currency exposures and invested currency exposures of the i-th sector needed
for the attribution analysis.
Contrary to the arithmetical attribution model, where the active return is
quantified as the difference between the portfolio and the benchmark
return, in a geometric attribution model it is based on the factorized quotient of both returns. Burnie et al. (1998) and Menchero (2000b) describe
such models for single-currency portfolios while Buhl et al. (2000) present
a geometric attribution model for multi-currency portfolios. An explicit
comparison of arithmetic and geometric models based on case studies can
be found in Wong (2003).
In the multi-period case it is a fundamental law that the result of linking
the single-period performance contributions over the entire period must be
equal to the total-period active return. In additive models, however, the
model-inherent problem exists that the sum of the active returns (and
correspondingly the sum of the risk factor-specific performance effects) of
every single period does not equal the active return of the total period.19
Different methods of arithmetic multi-period attribution analysis attempt
to solve the inter-temporal residual-generating problem in different ways
(see e.g. Carino 1999; Kirievsky and Kirievsky 2000; Mirabelli 2000; Davies
and Laker 2001; Campisi 2002; Menchero 2000a; 2004). An algorithm which
should be explicitly mentioned is the recursive periodization method that
was first published by Frongello (2002a) and then also confirmed by
Bonafede et al. (2002). This approach completely satisfies the requirements
on a linking algorithm as defined by Carino (1999, 6) and Frongello (2002a,
13) – the mathematical proof for the residual-free compounding of the
single-period contributions was presented by Frongello (2002b). As a
central bank reference, the suggested performance attribution model of
Danmarks Nationalbank also uses this linking technique (see Danmarks
Nationalbank 2004, appendix D).20

19

20

Also replacing the summation of the single-period effects by the multiplication of the factorized effects (in analogy to
the geometric compounding of discrete returns over time) would not lead to correct total-period results.
For the sake of completeness we also want to point at an alternative approach of how to accurately link performance
contributions over time. When carrying out the attribution analysis based on absolute (i.e. nominal) currency units
instead of relative (i.e. percentage) figures, the performance effects for the total period can simply be achieved by
summing up the single period contributions. This is exactly the way how the compounding over time is done within
the ECB attribution framework which is described in Section 5.

249

Performance attribution

4.2 Fixed-income performance attribution models
For central bank currency reserves portfolios, i.e. interest rate-sensitive
portfolios, the categories of asset allocation, security selection and interaction taken from the classical equity-oriented attribution models are not
adequate to mirror the investment process (an equivalent statement in
general terms can be found in Spaulding 2003, 74). In the context of a fixedincome portfolio there are at least five independent ways to deviate from the
benchmark in terms of risk factor exposures: duration position (e.g. by
trading futures contracts), term structure / yield position (e.g. by preferring
specific maturity buckets), country weighting (e.g. by overweighting highyield government bond markets), sector weighting / credit position (e.g. by
investing in market segments like agency bonds or Pfandbriefe) and
instrument selection. An accurate performance attribution model for a
central bank should be able to measure the distinct contributions of each
type of active investment decisions that is relevant for its individual portfolio management process. To study, Colin (2005) provides an introduction into the field of fixed-income attribution analysis and also gives an
overview of the relevant concepts in a compact form; see also Buchholz
et al. (2004).
In this section we will concentrate on additive attribution models as they
seem more adequate than their geometric alternatives for the investment
analysis practice, in particular related to central banks and other public
investors. This is, among others, due to the fact that the resulting performance contributions are more intuitive to understand from a methodical
point of view. The basic structure of a fixed-income performance attribution model – independent of the aggregation level and the analysis period –
can be represented by
ARbase ¼ PCmarketÀrisk
þ PCselection;intraday;residual

ð6:52Þ

where ARbase is the active time-weighted rate of return in base currency;
PCmarketÀrisk is the contribution to active return related to market risk
factors; PCselection,intraday,residual is the portion of the performance which is
unexplained by the applied market risk factor model and which usually is
due to instrument selection, intraday trading activities and the real model
residual term.

250

Marton, R. and Bourquin, H.

The model can then take the form of
ARbase ¼ PCcarry þ PCgovtYldChg þ PCspreadChg þ PCconvexity
þ PCcurrencyÀinvested þ PCcurrencyÀhedged
þ PCselection;intraday;residual

ð6:53Þ

where the market risk factor contributions to the performance are divided into
local and currency effects and can be classified as follows: carry effect PCcarry;
government yield change effect PCgovtYldChg; spread change effect PCspreadChg;
convexity effect PCconvexity; invested currency exposure effect PCcurrency-invested;
hedged currency exposure effect PCcurrency-hedged.
The following passage introduces an example of an explicit performance
attribution proposal which (among other alternatives) can be thought of
being appropriate for the investment process of fixed-income central bank
currency reserves portfolios. The evaluation of the government yield change
effect is based on the concept of parsimonious functional models (see
Section 3.3) which derive a defined number of the principal components of
the entire government yield curve motion, representing the government
return-driving parameters. By explicitly modelling the unique movements
parallel shift, twist and butterfly, the isolated contributions of typical central
bank term structure positioning strategies versus the benchmark, like flatteners, steepeners or butterfly trades, can be clearly quantified.21 In publications on fixed-income attribution analysis, the basis government yield
change effect is regularly broken down into those three partial motions (see
e.g. Ramaswamy 2001; Cubilie´ 2005; Murira and Sierra 2006). As a reference
publication by a central bank, the performance attribution proposal outlined by Danmarks Nationalbank (2004, appendix D) decomposes the
government curve movement effect into the impacts originating from the
parallel shift and from the variation of the curve shape (additionally it
disentangles the sector-specific spread change contribution from the
instrument-specific spread change effect).
Applying a perturbational technique (see Section 3) to our illustrative
example, the risk factor-related representation of the performance attribution model at portfolio level is defined as follows:22
21

22

In literature sometimes the parallel shift effect is designated as ‘duration effect’ and the combined twist and butterfly
effect is called ‘yield curve reshaping effect’.
The duration against the parallel shift, twist and butterfly could either be a modified duration or an option-adjusted
duration. The most appropriate measure with respect to the diverse instrument types should be used; so in case of
portfolios with e.g. callable bonds the option-adjusted duration would be a more accurate measure than the
modified duration.

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Performance attribution

ARP;base ¼

8
ðyP;i Á wP;i À yB;i Á wB;i Þ Á dt
>
>
>
>
>
>
þðDurP;i;PS Á wP;i À DurB;i;PS Á wB;i Þ Á ðÀPSÞ
>
>
>
>
>
þðDurP;i;TW Á wP;i À DurB;i;TW Á wB;i Þ Á ðÀTW Þ
>
>
>
>
>
>
þðDurP;i;BF Á wP;i À DurB;i;BF Á wB;i Þ Á ðÀBFÞ
>
>
>
>
>
N < þðDur
Á w À Dur
Á w Þ Á ðÀds
X
P;i;sector

i¼1

P;i

B;i;sector

B;i

sector Þ

>
þðDurP;i;country;euro Á wP;i À DurB;i;country;euro Á wB;i Þ Á ðÀdscountry;euro Þ
>
>
>
>
>
> þ1=2 Á ðConvP;i Á wP;i À ConvB;i Á wB;i Þ Á ðdyÞ2
>
>
>
>
>
>
þðwP;i;invested À wB;i;invested Þ Á Ri;xchÀrate
>
>
>
>
>
>
þðwP;i;hedged À wB;i;hedged Þ Á Ri;xchÀrate
>
>
>
:
þeselection;intraday;residual

ð6:54Þ
where for the i-th of N sectors within portfolio P with weighting wP,i: yP,i is
the yield to maturity; DurP,i,PS is the duration against a 100 basis point basis
curve parallel shift PS; DurP,i,TW is the duration towards a 100 basis point
basis curve twist TW; DurP,i,BF is the duration with respect to a 100 basis
point basis curve butterfly BF; DurP,i,sector is the duration related to a 100
basis point change of the spread between the yield of credit instruments and
the basis curve dssector; DurP,i,country,euro is the duration versus a 100 basis
point tightening or widening of the spread between the yield of eurodenominated government instruments and the basis curve dscountry,euro;
ConvP,i is the convexity of the price/yield relationship; wP,i,invested is the
weight of the absolute invested currency exposure and wP,i,hedged is the weight
of the absolute hedged currency exposure, respectively, towards the appreciation or depreciation of the exchange rate of the portfolio base currency
versus the local currency of the considered sector Ri,xchÀrate; eselection,intraday,
residual is the remaining fraction of the active return. The analogous notation is
valid for sector i within benchmark B.
Equation (6.54) can be rewritten as
ARP;base ¼ PCP;carry
þ PCP;PS þ PCP;TW þ PCP;BF þ PCP;sector þ PCP;country;euro
þ PCP;convexity þ PCP;currencyÀinvested þ PCP;currencyÀhedged
þ PCP;selection;intraday;residual
ð6:55Þ
where compared with equation (6.53): PCP,govtYldChg is split into the components impacted by the parallel shifts PCi,PS, twists PCi,TW and butterflies