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2 Parameterizing the model: choice of risk factors

# 2 Parameterizing the model: choice of risk factors

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227

are defined explicitly and the extents of the risk factors are estimated via
regression analysis. The term ‘fundamental’ in this context stems from the
fact that these models have originally been developed for the analysis of
equity returns.
Differently, the second major category to identify the relevant risk factors
a portfolio is exposed to – the ‘implicit method’ – makes use of a statistical
technique called factor analysis. It comprises the set of statistical methods
which can be applied to summarize the information of a group of variables
with a reduced number of variables by minimizing the loss of information
due to the simplification. When designing risk factor models this technique
can be used to identify the factors required by the model (i.e. it can be used
to determine the explaining risk factors and the corresponding factor
factors – these have to be interpreted individually and be given an economic
meaning. Two variations of factor analysis prevail in practice: the maximum
likelihood method and principal components analysis (PCA). In particular,
PCA has been found to work well on yield curve changes, since in practice
all yield curve changes can be closely approximated using linear combinations of the first three eigenvectors from a PCA.
2.3 Fitting to practice: empirical multi-factor models
Empirical models have less restrictive assumptions than APT-related
approaches and do not use arbitrage theory. They do not assume that there
is a causal relationship between the asset returns and risk factors in every
period. But they postulate that the average investment returns (or risk
premia) can be directly decomposed with the help of the risk factors. So in
contrast to the APT only one regression step is required. For every asset i the
following model relationship is given:
EðRi Þ À RF ¼

K
X

ðbi;k Á Fk Þ þ ei

ð6:7Þ

k¼1

Specifically, passively oriented managers like central banks can use multi-risk
factor models to help keep the portfolio closely aligned with the benchmark
along all risk dimensions. This information is then incorporated into the
performance review process, where the returns achieved by a particular
strategy are weighed against the risk taken. The procedure of modelling asset

228

Marton, R. and Bourquin, H.

returns as applied within the framework of empirical models is found in
many modern performance attribution systems dealing with multi-factor
models – especially in the field of fixed-income attribution analysis.
To underline the importance of multi-factor models for risk management, in addition to performance attribution they can act as the building
blocks at other stages of the investment management process, such as risk
budgeting and portfolio and/or benchmark optimization (see e.g. Dynkin
and Hyman 2002; 2004; 2006). A multivariate risk model could also be
thought of being applied to the ideological ‘sister’ of performance attribution: risk attribution. Using exactly the same choice of risk factors it is
possible to quantify the portions of the absolute risk (e.g. volatility or VaR)
and the relative risk (e.g. tracking error or relative VaR) of a portfolio that
each risk factor and sector would contribute (in an ex ante sense) and
consequently risk concentrations could be easily identified (for the attribution of the forward-looking variance and tracking error, respectively, see e.g.
Mina 2002; Krishnamurthi 2004; Gre´goire 2006). Pairing both the absolute
and relative risk contributions with the absolute and relative (active) return
contributions, enables the implementation of a risk-adjusted performance
attribution (see among others Kophamel 2003 and Obeid 2004).
Theoretically, a multivariate fixed-income risk factor model that was
chosen to be adequate for the portfolio management process of a public
investor like a central bank (in terms of the selection of the risk factors)
could serve as the main investment control and evaluation module. Of
course, for a central bank this theoretical possibility would not necessarily
find practical application, because the strategies are (among other factors)
subject to policy constraints. But even then, risk factor models can contribute significantly to the investment management process in central banks,
by providing the ‘quantitative control centre’ of that process.

3. Fixed-income portfolios: risk factor derivation
In order to effectively employ fixed-income portfolio strategies that can
control interest rate risk and enhance returns, the portfolio managers must
understand the forces that drive bond markets. Focusing on central banks
this means that, to be able to effectively manage and analyse the foreign
currency reserves portfolios, it is of crucial importance that the portfolio
managers and analysts (i.e. front office and risk management) are familiar
with the specific risk factors to which the central bank portfolios are

229

exposed and that they understand how these factors influence the asset
returns of these portfolios.
The model price (i.e. the present value) of an interest rate-sensitive
instrument i, e.g. a bond, at time t, with deterministic cash flows (i.e.
without embedded options or prepayment facilities), is dependent on its
yield to maturity yi,t and on the analysis time t and is defined in discrete
time Pi,t,disc and continuous time Pi,t,cont, respectively, as follows:
Pi;t;disc ¼

X

CFi;T Àt;t

T Àt
8T Àt ð1 þ yi;t;disc Þ
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
discrete time

%

X

CFi;T Àt;t Á e ÀðT ÀtÞÁyi;t;cont ¼ Pi;t;cont

8T Àt

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

ð6:8Þ

continuous time

where for asset i: CFi,T–t,t is the future cash flow at time t with time to
payment T–t; yi,t,disc is the discrete-time version of the yield to maturity,
whereas yi,t,cont is its continuously compounded equivalent.
The determinants of the local-currency buy-and-hold return dP(t,y)/P
(i.e. without the impact of exchange rate appreciations or depreciations and
trading activities) of an interest rate-dependent instrument (and hence also
portfolio) without uncertain future cash flows are analytically derived by
total differentiation of the price as a function of the parameters time t and
yield to maturity y, and by normalizing by the price level P (for the derivation see e.g. Kang and Chen 2002, 42). Restricted to the differential terms
up to the second order, the analysis delivers:4
dPðt; yÞ 1 @P
1 @P
1 1 @2P
Á ðdtÞ2
%
Á dt þ
Á dy þ
P
P @t
P @y
2 P @t 2
!
1 @2P
1 @2P
1 @2P
2
Á dtdy þ
Á dydt
þ
Á ðdyÞ þ
P @t@y
P @y 2
P @y@t

ð6:9Þ

This means that the return on a fixed-income instrument is sensitive to the
linear change in time dt, the linear change of its yield dy, the quadratic
change in time (dt)2, the quadratic change of its yield (dy)2 and also crossproducts between the change in time and yield dtdy and dydt, respectively,
where higher order dependencies are ignored.
The most comprehensive way to determine the return contributions
induced by every risk factor is by so-called ‘pricing from first principles’.
This means that the model price of the instrument is determined via the
present value formula immediately after every ceteris paribus change of the
4

For the differential analysis the subscripts of the parameters were omitted.

230

Marton, R. and Bourquin, H.

considered risk factors. By applying total return formulae with respect to the
initial price of the instrument, the factor-specific contributions to the
instrument return can then be quantified. The main difficulties in terms of
practical application are the data requirements of the approach: first, all
instrument pricing algorithms must be available for the analysis, second, the
framework to be able to separately measure the impacts of the diverse risk
factors (see Burns and Chu 2005 for using an OAS framework for performance attribution analysis) and third (in connection with the second
point), a spot rate model would need to be implemented (e.g. following
Svensson 1994) to be able to accurately derive the spot rates required for the
factor-specific pricing.
Alternatively, return decomposition processing could be done by using an
approximate solution.5 This is the more pragmatic way, because it is relatively easy and quick to implement. Here it is assumed that the price level2
@2P
@2P
normalized partial derivatives P1 @@tP2 Á ðdtÞ2 , P1 @t@y
Á dtdy and P1 @y@t
Á dydt as of
formula (6.9) are equivalent to zero and hence could be neglected for the
purpose of performance attribution analysis. Therefore the following
intuitive relationship between the instrument return and its driving risk
factors remains:
dPðt; yÞ
1 @P
1 @P
1 @2P
Á ðdyÞ2
%
Á dt þ
Á dy þ
2
P
P
@t
P
@y
2P
@y
|ﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
time decay effect

ð6:10Þ

yield change effect

The identified return determinants due to the passage of time and caused by
the change of the yield to maturity are separately examined in the subsequent sections, whereof the yield change effect is further decomposed into
its influencing components; additionally, the accurate sensitivities against
the several risk factors are quantified.
3.1 Risk factor: passage of time
The first risk factor impact described by expression (6.10) represents the
return contribution solely due to the decay of time dt, i.e. shorter times to
5

Although approximate (also called perturbational) pricing is not as comprehensive as pricing from first principles, it
should not represent a serious problem, in view of other assumptions that are made when quantifying yield curve
movements. An advantage of this method is that the computations of the return and performance effects can be
processed very fast without the need of any detailed security pricing formulae.

231

cash flow maturities and therefore changing discount factors. It is to be
taken into consideration that the yield change in period dt does not have
any impact on the carry effect and therefore unchanged yield curves are
postulated as a prerequisite for its calculation. The precise carry return
Ri,carry (also sometimes called time return or calendar return) can at
instrument level be determined as follows:
Ri;carry ¼

Pi;tþdt À Pi;t
Pi;t

ð6:11Þ

whereof
Pi;t ¼

X

CFi;T Àt;t

8T Àt

ð1 þ yi;t ÞT Àt

ð6:12Þ

and
Pi;tþdt ¼

X

CFi;T ÀtÀdt;tþdt

8T ÀtÀdt

ð1 þ yi;t ÞT ÀtÀdt

ð6:13Þ

where for instrument i: Pi,tþdt is the model price at time tþdt; CFi,T–t–dt,tþdt
is a future cash flow with time to maturity T–t–dt at time tþdt; yi,t is the
discrete-time yield to maturity as of basis date t.
In an approximate (perturbational) model (which could methodically
also directly be applied to any sector level or to the total portfolio level) the
carry return on asset i is given by6
Ri;carry ¼ yi;t Á dt

ð6:14Þ

The approximate method does not enable one to disentangle the ordinary
income return (i.e. the return impact stemming from accrued interest and
coupon payments) from the roll-down return which combined would yield
the overall return attributable to the passage of time. This precise decomposition of the carry return would be feasible by pricing via the first
principles method and applying total return formulae.7

6
7

See e.g. Christensen and Sorensen 1994; Chance and Jordan 1996; Cubilie´ 2005, appendix C.
Ideally, an intelligent combination of the imprecise approximate solution and the resources- and time-consuming
approach via first principles should be found and implemented in particular for the derivation of the carry effect. The
ECB performance attribution methodology was designed in a way to overcome the disadvantages of both methods
(see Section 5).

232

Marton, R. and Bourquin, H.

To analytically derive the approximation representation of the price change
as a function of one single source, the widely used Taylor series expansion
technique is usually applied which delivers polynomial terms in ascending
order of power and so in descending explanatory order. The absolute price
change of an interest rate-sensitive instrument using the second-order Taylor
expansion rule with respect to time decay dt is approximated by (for the
Taylor series rule see among others Martellini et al. 2004, chapter 5; Fabozzi
et al. 2006, chapter 11)8
1
dPðtÞ ¼ P 00 ðtÞdt þ Á P 00 ðtÞðdtÞ2 þ oððdtÞ2 Þ
2

ð6:15Þ

where dP(t) is the price change solely caused by the change in time t; P’(t)
and P’(t) are the first and second derivatives of P with respect to the change
in time dt; and o((dt)2) is a term negligible compared to second order
terms.
For the relative price change formula (6.15) becomes
dPðtÞ P 0 ðtÞ
1 P 00
¼
dt þ Á ðtÞðdtÞ2 þ oððdtÞ2 Þ
P
P
2 P

ð6:16Þ

3.2 Risk factor: change of yield to maturity
To complement, the second return determinant of equation (6.10), the
change of the yield to maturity, is analyzed. Also for reasons of consistency
the Taylor expansion concept is used here – with the aim at deriving the
polynomials which are the explaining parameters of the price change due to
the yield change. The absolute price movement with respect to a yield
change is given by
1
dPðyÞ ¼ P 0 ðyÞdy þ Á P 00 ðyÞðdyÞ2 þ oððdyÞ2 Þ
2

ð6:17Þ

where the analogous notation as for equation (6.15) is valid.
The relative price change is then approximated by
dPðyÞ P 0 ðyÞ
1 P 00
¼
dy þ Á ðyÞðdyÞ2 þ oððdyÞ2 Þ
P
P
2 P
8

For the Taylor expansion analysis the subscripts of the parameters were dropped.

ð6:18Þ

233

Plugging in the well-known interest rate sensitivity measures modified
duration ModDur and convexity Conv, the yield change effect is decomposed into the components linear yield change effect and quadratic yield
change effect (i.e. convexity effect). The relative price change representation
is now given by
dPðyÞ
1
% ÀModDur Á dy þ Á Conv Á ðdyÞ2
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
P
2
linear yield
change effect

ð6:19Þ

convexity effect

The variation of the yield to maturity of an interest rate-sensitive instrument is mainly caused by the movement of the underlying basis yield
curve, which is usually the country-specific government yield curve.9
In case of pure government issues, the yield change is almost entirely
explained by the basis curve motions.10 But for credit risk-bearing
instruments (e.g. agency bonds or BIS instruments) there is also a residual
yield change implied by the change of the spread between the security’s
yield to maturity and the government spot curve. For practical reasons just
the linear yield change effect is broken down into the parts related to the
also be divided into these components, but as the convexity contribution
itself in a typical central bank portfolio environment is of minor dimension, the decomposition would not have any significant value added for the
desired attribution analysis. The relationship in formula (6.19) can be then
extended to11
dPðyÞ
1
ÀModDur Á ds þ Á Conv Á ðdyÞ2
% ÀModDur
Á dr |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
P
2
government yield
change effect

9

10

11

ð6:20Þ