Directory UMM :Data Elmu:jurnal:J-a:Journal Of Banking And Finance:Vol24.Issue12.2001:

Journal of Banking & Finance 24 (2000) 1903±1932
www.elsevier.com/locate/econbase

A synthetic factor approach to the estimation of
value-at-risk of a portfolio of interest rate
swaps
Cindy I. Nikeer a,*, Robin D. Hewins
Richard B. Flavell c,2
b

b,1

,

a
Algorithmics (UK) Limited, Ivory House, St. Katharine's Way, London E19AT, UK
The Management School, Imperial College of Science, Technology and Medicine, 53 PrinceÕs Gate,
Exhibition Road, London SW7 2PG, UK
c
Lombard Risk Consultants Limited, 13th Floor, 21 New Fetter Lane, London EC4A 1AJ, UK

Received 4 December 1998; accepted 29 September 1999

Abstract
In this paper we decompose the interest rate swap yield curves of 10 major currencies
into their common factors and ®nd that the ®rst two factors, interpreted as parallel shift
and rotation, explain between 97.1% and 98.6% of the variation in the interest rate swap
rates across all 10 currencies. The main contribution of the paper however is that we
then model these two factors as simpli®ed synthetic factors so that they may be used to
develop an innovative approach to the computation of Value-at-Risk (VaR) for a
portfolio of interest rate swaps. Ó 2000 Elsevier Science B.V. All rights reserved.
JEL classi®cation: G13; G15; G21; G28
Keywords: Swaps; Value-at-Risk; Risk management

*

Corresponding author. Paper written whilst a Ph.D. student at The Management School,
Imperial College, currently at Algorithmics (UK) Limited. Tel.: +44-20-7553-2633; fax: +44-207481-3130.
E-mail addresses: cindyn@algorithmics.com (C.I. Nikeer), r.hewins@ic.ac.uk (R.D. Hewins),
RF@lombardrisk.com (R.B. Flavell).
1

Tel.: +44-171-594-9118.
2
Tel.: +44-171-353-5330.
0378-4266/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 4 2 6 6 ( 9 9 ) 0 0 1 1 9 - 3

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C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

1. Introduction
The swap market has been a signi®cant innovation in international
capital markets since the early 1980s. It has grown from virtually nothing
in 1980 to US$29.035 trillion in outstanding principal amount at 31 December 1997. 3 Of this amount, interest rate swaps account for US$ 22.291
trillion, 76.8%, of the outstanding notional principal. This rapid growth has
resulted in banks now holding substantial portfolios of these instruments
and there is thus a practical need to manage these risks. Whereas some
forms of factor analysis have already been used in an exploratory fashion to
explain asset prices (Litterman and Scheinkman, 1988; Steeley, 1990; Knez
et al., 1994), the motivation for our research is to contribute to the development of appropriate risk management tools for a portfolio of interest

rate swaps.
In this paper, we put forward an alternative approach for measuring the
Value-at-Risk (VaR) of a portfolio of interest rate swaps. This approach uses
factor analysis to decompose the interest rate swap yield curve of 10 major
currencies into two main factors which account for the majority of variation
in swap rates. Factor analysis is a multivariate statistical technique that is
used to uncover usually a smaller number of unobserved variables by
studying the covariation among a set of observed variables (Lewis-Beck,
1994). Whilst factor analysis is not itself a new statistical technique, it has
only relatively recently been applied to the ®nancial markets. Using one
speci®c form of factor analysis, principal components analysis, we ®nd that the
®rst two common factors, which we interpret as parallel shift and rotation,
account for between 97.1% and 98.6% of the variation in the swap yield
curves of the 10 currencies. However, in order to make these factors easier to
recognise and manage, and thus more useful for practical risk management
application, we go an important step further and model simpli®ed synthetic
versions of the factors. We propose that these synthetic factors are more
appropriate for risk management purposes than the ÔoriginalÕ factors because
(i) unlike the original factors which are churned out by the principal components analysis of a computer package, the synthetic factors are constructed
from ®rst principles, based on the interpretation of the original factors, (ii)

unlike the original factors which vary by currency and time period, the
synthetic factors allow us to apply one universal model of the factors across
currencies and time periods thereby reducing the computational requirements
and (iii) unlike the original factors which might be interpreted as squiggly

3
Obtained from ISDA Market Survey as at 31 December 1997, website http://www.isda.ogr/
d1.html. Total swaps include interest rate swaps, currency swaps and interest rate options.

C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

1905

lines ÔlikeÕ a parallel shift and rotation, the synthetic factors are designed to
have acceptably simple de®nitions and structures to represent an ÔexactÕ
parallel shift and rotation ± for example, it is easier for a trader to hedge
against an ÔexactÕ parallel shift rather than a ``wavy line something like a
parallel shift''. The two synthetic factors (TSFs) are found to explain between
95.9% and 98.5% of the variation in the interest rate swap yield curves for the
10 currencies and are thus acceptable approximations to the original factors.

They are also found to be reasonably stable over di€erent time periods. The
key contribution of this paper is the modelling and subsequent use of well de®ned, stable factors that may be universally applied over di€erent currencies and
time periods.
A further contribution of this paper is that the TSFs are then used to
derive an alternative methodology for the estimation of VaR of a portfolio of
interest rate swaps. VaR has become a key risk management concept for
banks and many corporations in recent years. VaR is a statistical estimate of
the maximum loss a portfolio can potentially incur over a given time period,
say one day, at a given con®dence level, say 95%. It is essentially a measure of
the volatility of a portfolio of securities to changes in market factors, such as
interest rates and exchange rates. A major limitation of the existing VaR
methodologies is their computational intensity because they depend on information on each asset within a portfolio. One study has aimed to reduce the
dimensions of the existing VaR methodologies, for a currency swap, by
adopting a multi-factor approach (Ho et al., 1998). In this paper, the factors
are assumed to be a two factor interest rate model for each of the currencies
represented in the swap and the foreign exchange rate between the two currencies. The factors are modelled using a binomial distribution methodology
developed previously by the same authors (Ho et al., 1995) to approximate
the joint probability distribution of the factors. This is essentially an alternative to a Monte Carlo simulation methodology to value the stochastic cash
¯ows of a swap at a given future date. In our paper, we derive an analytic
two factor approach to VaR estimation which can further reduce the computational intensity of the existing methodologies with little, if any, loss of

precision.
The remainder of the paper is organised as follows. In Section 2, we
outline the intuition behind factor analytic statistical techniques, the data
used and the results of the factor analysis of interest rate swap rates. In
Section 3, we model and test the goodness-of-®t of our synthetic versions of
the factors. In Section 4 we derive an alternative methodology for computing
VaR based on the TSFs. We also compare the VaR estimates produced by
this TSF VaR methodology with those produced by two existing methodologies, and highlight the bene®ts of the TSF VaR method. Section 5 concludes
the study by assessing the implications of our ®ndings and suggesting areas
for future research.

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C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

2. Factor analysis of interest rate swap rates
2.1. Theoretical framework
The theoretical foundation of our work lies in the modelling of the term
structure of interest rates. Yield curve modelling has attracted the attention of
academics and practitioners alike for many years because of its fundamental

importance to the study of ®nancial markets. It is important to traders and
portfolio managers because of the impact of yields on asset prices. A better
understanding of yield curve changes can thus lead to the development of more
e€ective hedging strategies or alternatively, generate pro®t opportunities. It is
also important to macroeconomists and policy makers because of the impact of
interest rates on monetary policy, the level of investment and the cost of servicing national debt. There is a very large body of literature on this topic but
the two recent approaches that are of relevance here are cointegration analysis
and factor analysis.
Researchers have used cointegration analysis to concentrate on developing
a stochastic process type model for forecasting future interest rates for different maturities and an adjustment of the expectations hypothesis to take into
account the cointegration of data, that is, the existence of a long-run equilibrium relationship between long and short interest rates (Choi and Wohar,
1995). The empirical results of cointegration analysis depend on the interest
rate market being analysed but it would seem that a common conclusion is
that cointegration is present within (and across) many markets. Other studies
have brought together, albeit in a small way, the two recent approaches of
cointegration analysis and factor analysis (Engsted and Tanggaard, 1992,
1994; Bradley and Lumpkin, 1992; Mougoue, 1992). The importance of these
®ndings is that they suggest the existence of a long-run equilibrium between
interest rates of di€erent maturities which implies that the underlying variables
are being driven by some common fundamentals, that is, some common

factors.
Factor analysis can be used to explain and interpret these common fundamentals. It builds upon earlier work involving duration analysis wherein one
can estimate how a change in the general level of interest rates a€ects the
prices of ®xed income securities. However in reality it is well documented that
yields do not only move in a parallel fashion. Hence a parallel shift only
partially explains the price changes in a portfolio of instruments at any one
point in time. The factor analytic approach is potentially more useful in that it
seeks to identify a few unobservable factors that can be ®tted by statistical
means to explain the majority of the return variability across the whole maturity spectrum of a market. Consequently, of critical importance to the factor

C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

1907

analytic approach is that the factors are deemed to be interpretable and
reasonably consistent over time.
The innovative research of Litterman and Scheinkman (1988), Steeley (1990)
and Knez et al. (1994) are the main references upon which the factor analytic
approach used in this paper is based. These papers use factor analysis to determine the common in¯uences on US government bond returns, US money
market returns and long-term UK government bonds respectively. They ®nd

that three factors, variously called level, steepness and curvature of the yield
curve, explained 96, 86 and 97%, respectively of the variability in the yield
curve. Litterman and Scheinkman (1988) and Knez et al. (1994) then go on to
use the common factors to hedge a portfolio of bonds. One ®nancial market
that has not to date been the subject of extensive empirical research is the
interest rate swap market. One exception is Due and Singleton (1997) in
which a multi-factor econometric model of the term structure of interest rate
swap yields is developed. This model incorporates both liquidity and credit
factors as key explanatory variables for the variation in interest rate swap
spreads over the past decade. However the authors argue that these factors
have di€erent temporal e€ects ± liquidity e€ects are short-lived whilst credit
e€ects are longer lasting.
2.1.1. Our study
In our study, we further decompose the interest rate swap yield curve into its
component factors in order to explain the variation in interest rate swap rates.
There are four main di€erences between our work and previous studies. First,
we analyse changes in swap rates rather than the levels of the swap rates
themselves. This is because we are interested in measuring and managing
changes in rates, irrespective of the overall level of rates. This distinction is
important as our objective is to develop ®nancial risk management tools to

protect against changes in portfolio value. A further rationale for the use of
changes, that is ®rst di€erences, in rates is that the levels of the swap rates are
found to be non-stationary. Non-stationarity implies the existence of stochastic
trends in the data. In a stationary time series, the mean, variance and autocovariance (at various lags) remain constant regardless of the time when they
are measured (Cuthbertson et al., 1992). We require stationarity of the data
before conducting factor analysis to eliminate the possibility of spurious results. We ®rst carried out the Dickey±Fuller test (Dickey and Fuller, 1981)
which suggested that ®rst di€erencing achieves stationarity. We further carried
out the more sophisticated Dickey±Pantula (Dickey and Pantula, 1987) test
which con®rmed that our data set is not integrated of order 2 but is integrated
of order 1 and thus ®rst di€erencing is sucient to achieve stationarity.
However, given that the Dickey±Fuller tests have been criticised for their

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C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

failure to reject a unit root due to their low power against stationary alternatives, we also carried out the KPSS unit root test (Kwiatkowski et al.,
1992). 4 In the Dickey±Fuller test, the null hypothesis is non-stationarity and
the alternative hypothesis is stationarity, whilst in the KPSS test the null hypothesis is stationarity and the alternative hypothesis is non-stationarity. Both
the Dickey±Fuller and KPSS statistics con®rm that the levels of our data series

are non-stationary and that ®rst di€erencing achieves stationarity. Thus using
the changes in swap rates is appropriate for subsequent analyses. 5 Third, we
develop synthetic versions of the original factors derived from principal components analysis so that the factors are more manageable for practical implementation. Fourthly, we test the stability of the synthetic factors over di€erent
time periods since we require stable factors if they are to be used for risk
management purposes. To our knowledge, these four innovations have not
been previously made.
2.2. Factor analysis methodology
Factor analysis is concerned with exploring the patterns of relationships
among a number of variables and assumes that observed variables are linear
combinations of underlying factors. Given the existence of correlations between observed variables (in this paper, 2±10 year 6 interest rate swap rates for
various currencies), factor analysis is used to determine whether the observed
correlations can be explained by a smaller number of unobserved and uncorrelated common factors.
The mathematical model for factor analysis is similar to a multiple regression equation. In a generalised form we hypothesise that the observed values of
the changes in 2±10 year zero-coupon swap rates, DYi , are linear combinations
of the unobservable factors Fj …j ˆ 1; . . .† such that:
X
aij Fj ‡


‡ di Ui for i ˆ 2; . . . ; 10 years;
…1†
DYi ˆ
j

where Fj s are termed the common factors, aij is the loading of the ith variable on
the jth factor, Ui is the unique factor which represents that component of

4

We are grateful to an anonymous referee for suggesting this procedure to us.
It should be noted that a Bayesian approach to unit root testing has also been proposed in the
literature (for example, Sims, 1988). However, in this paper our aim is to put forward appropriate
risk management tools for practical implementation. We therefore propose that the classical
approach of Dickey±Fuller and KPSS remain the mostly convenient and widely used tests for the
practitioner to adopt.
6
We use 2±10 year rates as 1 year swap rates are not quoted (as at the time of obtaining the data
from Datastream).
5

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1909

variable DYi that is not explained by the common factors and di is the loading
on the unique factor.
A unique solution for the aij s can be found if we impose the condition that
the factors have a mean of zero and a variance of one and are uncorrelated
with each other. The factors themselves can also be de®ned as linear combinations of the variables as follows:
X
bki Xi ;
…2†
Fk ˆ
i

where Fk represents the kth factor and bki is the correlation of the kth factor
with the ith variable.
The speci®c form of factor analysis that is used in this paper is principal
components analysis. Principal components analysis transforms a set of correlated variables into uncorrelated factors or components. The correlated
variables are decomposed into a linear combination of orthogonal components
that best ÔsummarisesÕ the data points. The principal components are chosen so
that the ®rst one explains the greatest variation in the original data, the second
one explains the most variation amongst data points orthogonal to the ®rst and
so on. The principal components have the same variability as the original data
set.
2.3. Data
Weekly and daily interest rate swap rates (2, 3, 4, 5, 7, 10 year) for 10
currencies were obtained from Datastream. 7 Data for 6, 8 and 9 year swap
rates were derived using linear interpolation. This method of interpolation has
been found to produce results which are not signi®cantly di€erent from other
interpolation methods (Flavell, 1991 p. 54; Cossin and Pirotte, 1997, p. 1354).
We then extracted an implied yield curve from the interest rate swap rates by
deriving the zero-coupon rates. We derived zero-coupon rates as they represent
the 2, 3 year, and so on, interest rate swap rate without any intervening coupon
payments and thus can be regarded as the ÔtrueÕ 2 year rates and so on. The use
of zero-coupon rates is consistent with earlier studies (Litterman and Scheinkman, 1988; Knez et al., 1994). Finally, we took the ®rst di€erences of the
zero-coupon rates; this is the data set used in subsequent analyses in this paper.

7
The period covered ranged as follows: 1 January 1988 to 1 November 1996 (US Dollar, UK
Pound, German Mark), 15 January 1988 to 1 November 1996 (Swiss Franc), 11 January 1991 to 18
October 1996 (Japanese Yen, Italian Lira), 5 July 1991 to 11 October 1996 (French Franc), 5 July
1991 to 18 October 1996 (Belgian Franc, Dutch Guilder) and 10 April 1992 to 11 October 1996
(Spanish Peseta). The start dates vary according to the availability of the rates for the di€erent
currencies.

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C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

2.4. Principal component analysis results
We used SPSS, a statistical software package, to perform the principal
components analysis. On extracting the common factors, a distinct pattern is
clearly discernible across all 10 currencies (see Appendix A). The ®rst factor is
called parallel shift based on all scores for years 2±10 falling generally in the
range of 0.85±0.99. This suggests that rates for the di€erent years move roughly
in line with each other in a parallel manner. A parallel shift represents a change
in the level of interest rates. The second factor is called rotation because scores
display a general pattern of negative values from years 2 to 5 and then positive
values from years 6 to 10 (or vice versa). This suggests that the rates for different years are moving in a linear fashion and intersecting the x-axis at year 6
(or rotating at year 6). A rotation represents a change in the steepness of the
swap yield curve. The third factor is called a twist because scores generally
display a positive±negative±positive pattern across all currencies. This suggests
a factor in the form of a quadratic or bilinear type curve. A twist represents a
change in the curvature of the swap yield curve.
Table 1 shows that the ®rst factor is the most signi®cant in terms of
explanatory power and explains on average 91.8% of the variation in swap
rates across all 10 currencies (2±10 year data). The second and third factors
explain on average 6.2% and 1.1% of the variation, respectively. 8 For all
currencies, the three factors explain an average of 99.1% for 2±10 year
data. 9
These ®ndings imply that the changes in the swap yield curve may be
modelled by modelling the main common factors. Given that the explanatory
power of the third factor appears very small (approximately 1%), hereafter we
will only be considering the two main factors, parallel shift and rotation. These
®rst two factors account for between 97.1% and 98.6% of the variation in interest rate swap rates (Table 1). We now go on to model synthetic versions of
these factors.

8
Table 1 compares the results when the interpolated data is included (2±10 year) and excluded
(excl 6, 8 and 9 years). For completeness, we also analysed the original swap rates, with and without
the interpolated years, and found a factor structure consistent with that outlined in Table 1. These
results suggest that there is no signi®cant di€erence between the factor structure or the proportion
of variation explained when the interpolated data is included or not. Hereafter we will thus use the
full 2±10 year zero-coupon ®rst di€erences data set including the interpolated years 6, 8 and 9.
9
We then tested the robustness of the data by also analysing daily data and found that these
results displayed a similar pattern. Additionally, we split the data into six monthly groups and
carried out the principal component analysis on each group. Each group was found to display a
consistent three factor structure in line with the results in Section 2. These results suggest that the
three factor structure is reliable over di€erent time periods in explaining the majority of the
variation in swap rates.

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1911

Table 1
Summary of factor scoresa
Currency (weekly)b

% of total variation
Factor 1

Factor 2

Factor 3

USD 2±10 yr
USD excl 6, 8 and 9 yr
GBP 2±10 yr
GBP excl 6, 8 and 9 yr
DEM 2±10 yr
DEM excl 6, 8 and 9 yr
SWF 2±10 yr
SWF excl 6, 8 and 9 yr
ITL 2±10 yr
ITL excl 6, 8 and 9 yr
JPY 2±10 yr
JPY excl 6, 8 and 9 yr
NLG 2±10 yr
NLG excl 6, 8 and 9 yr
BEF 2±10 yr
BEF excl 6, 8 and 9 yr
FRF 2±10 yr
FRF excl 6, 8 and 9 yr
ESP 2±10 yr
ESP excl 6, 8 and 9 yr
Avg. factors ± 2±10 yr
Avg. factors ± ex. 6, 8 and 9 yr

95.0
94.4
91.0
89.9
90.8
89.7
93.1
92.0
94.0
93.0
91.7
90.7
90.9
89.6
88.6
87.6
91.6
90.5
91.3
90.1
91.8
90.8

3.6
3.9
6.9
7.3
7.2
7.6
4.5
4.8
4.6
5.2
6.1
6.4
6.7
7.2
8.5
8.6
6.8
7.3
6.4
7.0
6.2
6.5

0.7
0.9
1.0
1.3
1.1
1.4
1.0
1.3
0.8
1.1
1.2
1.5
1.2
1.5
1.5
1.9
0.9
1.2
1.4
1.5
1.1
1.3

Total, 3 factors (%)
99.3
99.2
98.9
98.5
99.1
98.7
98.6
98.1
99.4
99.3
99.0
98.6
98.8
98.3
98.6
98.1
99.3
99.0
99.1
98.6
99.1
98.6

a
This table shows the proportion of variation in the interest rate swap yield curve explained by the
®rst three factors for each of the ten currencies. For each currency there are two sets of results e.g.
``USD 2±10 yr'' represents 2±10 yr interest rate swap rates (including interpolated data for years 6,8
and 9). ``USD excl 6, 8 and 9 yr'' represents 2±10 yr swap rates excluding years 6,8 and 9 which are
the interpolated data.
b
Currency codes: USD ± United States dollar; GBP ± UK pound sterling, DEM ± German mark,
SWF ± Swiss franc, ITL ± Italian lira, JPY ± Japanese yen, NLG ± Dutch guilder, BEF ± Belgian
franc, FRF ± French franc, ESP ± Spanish peseta.

3. Modelling of synthetic factors
In this section, we ®t the synthetic parallel shift and rotation movements
based on the data itself. So, having identi®ed from principal components
analysis a ®rst, parallel form of shift factor ± that is, the maturity structure
of data seems to be mainly a function of a parallel upward or downward
movement in the rates across all maturities, we then ®tted an ÔexactÕ parallel
shift (hereafter called a synthetic parallel shift) based on the data. Once this
®rst synthetic parallel shift factor has been ®tted to and extracted from the
data we proceeded to ®t and extract the second factor. Again, having
identi®ed from principal components analysis that this sequential second
factor depicts a type of rotation, we then ®tted an ÔexactÕ rotation based on

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C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

the data. The residuals left after ®tting and extracting the TSFs were then
tested to determine the proportion of variation explained by the synthetic
factors.
3.1. Synthetic parallel shift
A parallel shift was modelled as a constant change in rates across all maturities, a. Our ®tted constant change, a^, was determined by minimising the
squared deviation of each of the observed variables, changes in 2±10 year swap
rates, from a^ as follows:
X
X
X
oS
ˆ0)
Yi ˆ
a^ ˆ n^
a ) a^
…Yi ÿ a^†2 )
o^
a
1X
Yi ˆ Y ;
ˆ
n

Minimise S ˆ

…3†

where S is the squared deviation away from the ®tted constant change, a^, Yi the
value of the observed variables, Y the average of the observed variables, and n
is the number of variables.
Thus, Eq. (3) shows that by calculating the average of the observed changes
in 2±10 year swap rates for each given week/day and using this as our ®tted
constant change, we thereby ®t an ÔexactÕ parallel shift to the data. This calculation is shown in Table 2 (Panel A) for the week starting 8 January 1988.
Fig. 1 (Panel A) further graphically illustrates the ®tting of our synthetic
parallel shift, a^, to the changes in swap rates for the week starting 8 January
1988. This calculation can also be viewed as ®nding the constant in a linear
regression.
The synthetic parallel shift was then extracted from the original data set by
subtracting the average weekly parallel shift value from each observed value, 2,
3 year and so on, for that week. Following the example in Table 2 (Panel A),
columns (2)±(10) gives the 2 year residual variation in rates after extracting the
synthetic parallel shift, columns (3)±(10) gives the 3 year residual variation and
so on. The e€ect of this calculation is to remove a constant value from each of
the observations for that week.
Our synthetic parallel shift loadings, ni1 , can thus generally be represented by
a value of, say, 10 1 for each of the observed variables to indicate that, in
relative terms, each variable moves by the same amount when the level of swap
rates change:

10
It is only the relative value of the loadings that is important in describing the form of the ®tted
factors, not the absolute value of the loadings.

Swap rate maturity

4Y

5Y

6Y

7Y

8Y

9Y

10Y

a^ (average)

Panel A: Calculation of ®tted parallel shift (^
a)
8 Jan 88
0.073
0.064

0.118

0.155

0.148

0.141

0.162

0.185

0.208

0.139

Panel B: Change in slope at each maturity
Swap rate maturity
2Y
3Y

4Y

5Y

6Y

7Y

8Y

9Y

10Y

)2

)1

0

1

2

3

4

^ from residual changes after ®rst synthetic factor extracted
Panel C: Calculation of ®tted rotation, b,
Swap rate maturity
2Y
3Y
4Y
5Y
6Y
7Y
8Y

9Y

10Y

b^

8 Jan 88

0.046

0.069

0.016

Change in slope
variable

2Y

)4

)0.066

3Y

)3

)0.075

)0.021

0.015

0.008

0.001

0.023

C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

Table 2
Fitted synthetic factors

1913

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C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

Fig. 1. Illustration of synthetic factors: Panel A: synthetic parallel shift (this represents a graphical
illustration of the observed changes in swap rates and our ®tted parallel shift for the week of 8
January 1988); Panel B: synthetic rotation (this represents a graphical illustration of the residual
changes in swap rates, after extracting the synthetic parallel shift, and our ®tted rotation for the
week of 8 January 1988).

ni1 ˆ 1;

i ˆ 2; . . . ; 10 years:

…4†

The residuals were then subjected to principal components analysis to verify
whether they still displayed the structure of the remaining factor, rotation. This
was found to be the case for all currencies.
3.2. Synthetic rotation
A rotation was next modelled as a regression line representing a change
in the slope, b, of the yield curve. From the factor loadings obtained from
principal components analysis, outlined in Section 2, the change in steepness in the yield curve of each of the 10 currencies appeared to ÔrotateÕ

C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

1915

^ was
roughly around a mid-maturity point. Our synthetic rotation, b,
therefore given a pivot point of year 6. The explanatory variable, Xi , in this
linear regression is a ÔChange in SlopeÕ variable which at year 6 has a value
of 0, and at all other maturities has a value equivalent to the distance of
the individual maturity point from the rotation point of year 6 (Table 2
(Panel B)).
The dependent variable, Yi , in this linear regression is a ÔResidual Changes in
Swap RatesÕ variable (after extracting the synthetic parallel shift). Our ®tted
^ can now be calculated as follows:
slope change, b,
b^ ˆ

P10



iˆ2 …Xi ÿ X †…Yi ÿ
P10
 2
iˆ2 …Xi ÿ X †

Y †

:

…5†

Given that our ®tted rotation is forced to go through the origin at year 6, X
and Y in Eq. (5) are equal to 0. By using b^ as our ®tted slope change, we
thereby ®t an ÔexactÕ rotation to the data.
This calculation is shown once again for the week starting 8 January 1988 in
Table 2 (Panel C). Fig. 1 (Panel B) further graphically illustrates the ®tting of
^ to the residual changes in swap rates for the week
the our synthetic rotation, b,
starting 8 January 1988. The synthetic rotation was then extracted from the
residuals.
Our synthetic rotation loadings, ni2 , can thus generally be represented by a
value of 1 for year 2, )1 for year 10 and 0 for year 6. These values are selected to synthesise a change in the slope of the yield curve with the short
(year 2) and long (year 10) rates moving in opposite direction and a pivot
point of 0 at year 6. These values produce synthetic factor loadings of the
form
ni2 ˆ 1:5 ÿ 0:25i;

i ˆ 2; . . . ; 10 years:

…6†

The new residuals, after extracting the synthetic parallel shift and rotation
factors, were then subjected to principal component analysis to test whether
they still displayed the structure of the last remaining factor, twist. This was
again found to be the case for all currencies.
3.3. Percentage variation explained by the TSFs
Having extracted the TSFs from the original data, we then examined the
residuals to determine proportion of variation in the interest rate swap rates
explained by the synthetic factors. Table 3 illustrates that the TSFs explain
between 95.9% and 98.5% of the variation in swap rates, which closely matches
the proportion of variance explained by the original factors of between 97.1%

1916

C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

Table 3
Variation explained by synthetic factorsa
Currency

Factors

Original factors (%) (2±10 yr)

Synthetic factors (%) (2±10 yr)

USD

Factor 1
Factors 1&2
Factor 1
Factors 1&2
Factor 1
Factors 1&2
Factor 1
Factors 1&2
Factor 1
Factors 1&2
Factor 1
Factors 1&2
Factor 1
Factors 1&2
Factor 1
Factors 1&2
Factor 1
Factors 1&2
Factor 1
Factors 1&2

95.0
98.6
91.0
97.9
90.8
98.0
93.1
97.6
94.0
98.6
91.7
97.8
90.9
97.6
88.6
97.1
91.6
98.4
91.3
97.7

94.5
98.5
89.9
97.5
89.8
97.6
91.9
97.0
92.7
97.8
90.5
97.1
89.8
97.1
86.3
96.5
90.1
97.5
90.0
95.9

GBP
DEM
SWF
ITL
JPY
NLG
BEF
FRF
ESP

a
This table shows the proportion of variation explained by our modelled synthetic factors as
compared with the true (non-synthetic) factors. The latter represents the results obtained from the
principal components decomposition of the interest rate swap rates (2±10 yr) as summarised in
Table 1.

and 98.6%. The synthetic factors are therefore considered an acceptable representation of the two original factors. 11
3.4. Implications of empirical ®ndings
The implication of these ®ndings is that the TSFs may be used in risk
management applications to represent changes in the swap yield curve. We
propose that the synthetic factors are more appropriate for developing practical risk management tools than the original factors for the following reasons.
First, they are easier to understand than the original factors. The synthetic
11
We further tested the consistency of our synthetic factors by segmenting the data for each
currency into six monthly groups and ®tting the synthetic factors to each of these sub-groups. We
then compared the proportion of variation explained by the two factors for each of the sub-groups
with the proportion of variation explained for the full period of data as well as the standard
deviation of the six monthly results. These results showed that the two synthetic factors explain a
consistently high level of variation in the swap rates for the six monthly groups and thus appear to
be reasonably stable over di€erent time periods.

C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

1917

factors are constructed from ®rst principles and are thus transparent and make
clear what the factors represent. On the other hand, the risk managers may
perceive principal components analysis to be a Ôblack-boxÕ which mysteriously
churns out statistical factor loadings. The synthetic factors therefore aid the
understanding and implementation of the factors.
Second, the synthetic factors are more tangible and less conceptual than the
original factors. The synthetic factors are designed to have clearly de®ned
structures which represent an ÔexactÕ parallel shift and rotation. The original
factors, however, look ÔlikeÕ and can be interpreted as a parallel shift and rotation, but are not exactly a parallel shift and rotation. This clear de®nition of
the factors is important for risk management purposes because it allows, for
example, a trader to more clearly visualise the form of the factors and thus to
hedge against a precise parallel shift or rotation, rather than factors visualisable as ``squiggly lines something ÔlikeÕ a parallel shift or rotation.''
Third, the synthetic factors reduce the computational complexity of practically utilising the factors for risk management purposes. The modelling of
synthetic factors allows us to apply one model of the factors across the 10
currencies and across di€erent time periods. The use of the original factors, on
the other hand, would mean employing di€erent factor loadings for the different currencies and over di€erent time periods. Thus, the exact form of the
parallel shift and rotation factors would vary from currency to currency and
from one time period to another. For exploratory purposes, this would be
sucient. However, for practical risk management application of the factors,
there is a need to go one step further to create simply constructed synthetic
factors that may be universally applied across markets and time periods. The
synthetic factors thus reduce the computational burden and hence make the
factors more manageable.
It should be noted that the trade-o€ in obtaining the above bene®ts is that
the synthetic factors approximate the original factors and hence there is some
loss of information and precision. However, we have shown that this approximation is acceptably good, and furthermore that the synthetic factors are
stable over time. It should be emphasised that the synthetic factors also
maintain the orthogonality of the original factors; this makes the factors
addditive and statistically independent. Additivity is important for risk management purposes because it allows us to easily evaluate, say, one ÔunitÕ of
parallel shift added to an existing portfolio of positions without worrying about
the existing parallel shift positions. This feature thereby facilitates the VaR
computations. Statistical independence is important for risk management
purposes because it allows us to manage the factors separately, say to hedge
against a parallel shift without having to think about its e€ect on the other
factors. Therefore, overall, we propose the use of the synthetic factors in developing risk management applications. One such application is in estimating
VaR.

1918

C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

4. Application of TSFs to VaR estimation
VaR for a given portfolio is expressed, for example, as ``$10 million overnight with a 5% probability''. This indicates that there is a 5% probability that
the portfolio value will decline by more than $10 million overnight. The two
main methods of estimating VaR that are used as a reference point in this study
are: (i) Variance±Covariance and (ii) Monte Carlo Simulation. We use these
two methodologies because they are appropriate for instruments that are linear
in nature, such as interest rate swaps.
4.1. Variance±Covariance methodology
This approach expresses the volatility of a portfolio of assets as a function of
the variance of the return of each instrument in the portfolio and of the correlation between each pair of returns:
r2p ˆ x2i r2i ‡ x2j r2j ‡ 2wi wj ri rj qij ;

…7†

where r2p is the volatility (variance) of the portfolio returns, xi the weight if the
ith asset/cash ¯ow in the portfolio, r2i the variance of the ith asset/cash ¯ow,
and qij is the correlation between the returns of the ith and jth asset/cash ¯ow.
For a single asset, the VaR is calculated as follows:
VaR ˆ CI
r
d;

…8†

where CI is the number of standard deviations for a given con®dence level (for
example, 95% con®dence interval represents 1.65 standard deviations, onetailed, since we are interested in losses to the portfolio, not gains), r the
standard deviation of the risk factor, for example the interest rate for a ®xed
income instrument and d is here de®ned as the sensitivity of the asset value to a
1% change in the risk factor. In the case where the asset generates multiple cash
¯ows, as in an interest rate swap, this de®nition also incorporates the weighting
of the assets/ cash ¯ows in calculating the sensitivity of the market value of the
asset to changes in the underlying risk factors. The VaR thus represents the
maximum negative change in value (loss) of the portfolio at, say, a 95% con®dence level.
For a portfolio consisting of more than two (say N) assets/cash ¯ows, the
VaR formula becomes
VaR ˆ

p
VRV T ;

where V is a …1  N † vector,

…9†

C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

V ˆ ‰…CI
ri
di † . . . …CI
rN
dN †Š

1919

…10†

and R is an N  N correlation matrix of the returns on the underlying cash
¯ows resulting from the portfolio of assets.
The advantage of this methodology is that the formula for its calculation is
relatively straightforward. However, its main limitation is its dependence upon
the calculation of large covariance matrices between each security within the
portfolio.
4.2. Monte Carlo simulation
This approach is based on approximating the behaviour of ®nancial prices
by using computer simulations to generate random price paths. A variety of
di€erent scenarios for the portfolio value on the target date is then simulated.
The VaR is read o€ directly from the distribution of simulated portfolio values.
A Monte Carlo simulation is performed by carrying out six main steps. First, a
series of independent normal random numbers ( Xi ) is generated for each of the
risk factors. Second, a desired covariance matrix, R, is determined, for example
based on historical data. This is because the covariance matrix of the independent normal random numbers must be transformed into random numbers
that contain this speci®ed covariance structure. Third, a Cholesky decomposition (Jorion, 1997, pp. 242±243) of the covariance matrix is carried out to
obtain
R ˆ AA0 ;

…11†

where A is a lower triangular matrix with zeros in the upper right hand corner
and A0 is the transpose of A.
Fourth, the independent normal random numbers are transformed into
random numbers with a covariance structure R…Yi † by calculating
Y i ˆ A  Xi :

…12†

Fifth, recalling that Yi represents the simulated change in zero-coupon interest rate swap rates, the new rates are now computed:
rit ˆ ri0 ‡ Yi ;

…13†

where rit is the new (simulated) rate one period ahead and ri0 is this periodÕs rate.
Finally, the portfolio is then repeatedly revalued for each of the simulated
scenarios and the simulated portfolio returns ranked from lowest to highest.

1920

C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

The VaR ®gure is read o€ at the required percentile, for example, at the ®fth
percentile.
The main advantage of this methodology is its ability to simulate many
di€erent scenarios of rates with a desired distribution and covariance structure.
This makes it ¯exible to potentially take into account a wide range of risks such
as price and volatility. This approach is thus a very powerful tool for calculating VaR. Its limitation however is its computational intensity given that the
portfolio is revalued under thousands of di€erent trials.

4.3. A TSF approach to VaR
We have demonstrated in the previous section that the ®rst two synthetic
factors account for the majority of the variation in interest rate swap prices.
Also recalling that the changes in swap rates can be expressed as a linear
combination of the factors, we can now express the synthetic factors as follows:
Drit ˆ pt ‡ bt …i ÿ i † ‡ eit ;

…14†

where rit is the zero-coupon interest rate swap rate, i denotes maturity, t the
date of the observation, Drit the di€erence between rit and ritÿ1 , pt the parallel
shift for the zero-coupons between t ÿ 1and t, bt the rotational movement, i
some pivot point for the rotation and eit is the unexplained variation in the
changes in zero-coupon swap rates.
By directly forming a regression of Eq. (14) we can thus solve for optimal
values of pt and bt , our synthetic parallel shift and rotation:
bt ˆ Cov…Drit ; i†=Var…i†;

…15†

pt ˆ Ei fDrit g ÿ bt
Ei fi ÿ i g:

…16†

It is interesting to note that bt does not depend upon i . This is because the
rotation around one pivot point i0 can always be described in terms of a rotation around i and a parallel shift …i0 ÿ i †.
Based on Eq. (14) we can now derive a VaR methodology based on parallel
and rotational movements in the yield curve, rather than on changes in the
value of each individual security within a portfolio. Suppose we have a portfolio whose
P value P ˆ f …rit †. The value of the portfolio can thus be expressed
as: P ˆ
wi ri . If there is a change in the rates, over a given time period t (say
daily), this will cause a change in P: DPt ˆ wi di Drit . We can write approximately the variance of DPt as

C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

1921

X
di dj ‰Et fDrit
Drjt g ÿ Et fDrit g
Et fDrjt gŠ;

…17†

Vart …DPt † ˆ

ij

where di dj are here de®ned as the change in market value of the portfolio given
a 1% change in the ith and jth zero-coupon swap rates, respectively. This
de®nition incorporates the weighting of the ith and jth assets in calculating the
sensitivity of the market value of the portfolio to changes in the underlying risk
factors.
Substituting from Eq. (14) for Drit , we get:
Vart …DPt † ˆ

X
di dj ‰Ef…pt ‡ bt …i ÿ i ††…pt ‡ bt …j ÿ i ††g ÿ Et …pt bt …i
ij

ÿ i ††
Et …pt bt …j ÿ i ††Š;
Vart …DPt † ˆ Vart …pt †

‡

X
X
di dj ‡ Vart …bt †

di dj …i ÿ i †…j ÿ i †
ij

Covt …pt ; bt †

…18†

ij




X
ij

di dj …i ‡ j ÿ 2i †:

…19†

The covariance term Covt …pt ; bt † can be written as
Covt …pt ; bt † ˆ Et fpt
bt g ÿ Et fpt g
Et fbt g

…20†

and substituting from Eq. (16) for pt , we get
Covt …pt ; bt † ˆ Covt …Ei fDrit g; bt † ÿ Vart …bt †
Ei fi ÿ i g:

…21†

Since bt does not depend on the location of i0 , this means that we can select any
value for i0 , for example,
Covt …Ei fDrit g; bt †=Vart …bt † ÿ Ei fi ÿ i g ˆ 0:

…22†

i can be therefore be de®ned as
i ˆ Ei fig ÿ ‰Covt …Ei fDrit g; bt †=Vart …bt †Š:

…23†

This means that Cov…pt ; bt † is by construction set at zero (for orthogonal
factors) where
pt ˆ Ei fDrit g ÿ bt
Ei fi ÿ i g
and hence, from Eq. (19),

…24†

1922

C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

Vart …DPt † ˆ Vart …pt †


X
ij

di dj ‡ Vart …bt †


X
ij

di dj …i ÿ i †…j ÿ i †:

…25†

Eq. (25) states that the variance of a portfolio can be expressed as a function
of independent parallel and rotational movements in the yield curve. Thus a
TSF approach to VaR can be derived, for example at the 95% con®dence as
follows:
VaRdaily ˆ 1:65

s
X
X
di dj …i ÿ i †…j ÿ i †:
di dj ‡ Vart …bt †

Vart …pt †

ij

ij

…26†
4.4. Comparison of TSF vs traditional VaR methodologies
In order to assess the TSF VaR model, we now develop a progressively more
complex series of hypothetical interest rate swap portfolios and compare the
VaR estimates of the TSF approach with those produced by the traditional
methodologies. The common features of Portfolios 1±4 are:
(i) each swap starts on 1 January 1991 and interest on each swaps is payable
annually on the same date;
(ii) an actual/360 day basis is used for the computation of number of days
per year;
(iii) each swap is denominated in US dollars.
As an illustration, the cash ¯ows of Portfolio 1, discounted by the respective
zero-coupon interest rate swap rates, are shown in Appendix B. Each of the
cash ¯ow streams (1, year 2 years, etc.) can be considered a zero-coupon bond.
In our calculations, for Drit , we used daily US dollar data for the period 1
January 1988±25 October 1996.
The common features of Portfolios 5 and 6 are:
(i) each portfolio is valued at 6 June 1991,
(ii) the start dates, as well as the interest settlement dates, of the three swaps
in these portfolios vary,
(iii) the 5 and 3 year swaps are ®xed on a semi-annual basis whilst the 7 year
swap is ®xed on an annual basis,
(iv) each swap is denominated in UK pounds.
The cash ¯ows of Portfolio 5 are illustrated in Appendix C. Linear interpolation was used to determine the discount factors of time periods for
which swap rates were not quoted (Flavell, 1991). Cash ¯ow buckets were
created to map diverse timings of cash ¯ows into standardised time periods,
here annual periods from 6 June 1991 to 6 June 1998. The bucketing
methodology employed was based on matching the market value, the vola-

C.I. Nikeer et al. / Journal of Banking & Finance 24 (2000) 1903±1932

1923

tility and the sign of the actual cash ¯ows with the bucketed cash ¯ows
(Longerstaey and Spencer, 1996). For Drit in these calculations, we again
used daily UK pound sterling data for the period 1 January 1988 to 25
October 1996.
Portfolio 7 consists of a more realistic portfolio of one hundred swap cash
¯ows. Individual swap cash ¯ows are once again mapped onto bucketed annual time periods from 30 June 1991 to 30 June 1998, using the same bucketing methodology as in Portfolios 5±8. The portfolio is valued at 30 June
1991.
The results of the three approaches to VaR estimation for Portfolios 1±7
are summarised in Table 4. Portfolio 1 consists of three par swaps each receiving a ®xed rate of interest annually. The TSF VaR estimate closely
matches the estimates produced by the Variance±Covariance and Monte
Carlo methodologies. In Portfolio 2, the only change made to Portfolio 1 is
that the 3 year swap is changed to paying a ®xed rate of interest. We would
thus expect some o€setting of positive and negative cash ¯ows in the portfolio and therefore a reduced VaR exposure. The resulting VaR estimates
indicate that this is the case and the TSF VaR estimate once again closely
follows the Variance±Covariance and Monte Carlo estimates. As a further
test, in Portfolio 3 the 7 year swap is changed to paying a ®xed rate of interest, instead of the 3 year swap, with all else remaining the same as in
Portfolio 1. Given the longer period of o€setting cash ¯ows, a further reduction in VaR exposure versus Portfolio 2 would be expected. This is again
found to be the case. In Portfolio 4, two changes are made to Portfolio 1 ±
the 3 year swap is changed to a 2 year swap and the 5 year swap is changed
to a 4 year swap. Once again the resulting VaR estimates re¯ect the expected
reduced exposure. Portfolio 5 consists of a more complex portfolio of interest
rate swaps. The swaps have di€erent start dates and make annual as well as
semi-annual interest settlements. The TSF VaR estimate is again found to be
consistent with the Variance±Covariance and Monte Carlo estimates. Portfolio 6 has the same swap maturities, amounts and rates as Portfolio 1 b