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

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A synthetic factor approach to the estimation of

value-at-risk of a portfolio of interest rate

swaps

Cindy I. Nikeer

a,*

, Robin D. Hewins

b,1

,

Richard B. Flavell

c,2

a

Algorithmics (UK) Limited, Ivory House, St. Katharine's Way, London E19AT, UK

b

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®edsyntheticfactors so that they may be used to

develop an innovative approach to the computation of Value-at-Risk (VaR) for a

JEL classi®cation:G13; G15; G21; G28

Keywords:Swaps; Value-at-Risk; Risk management

Journal of Banking & Finance 24 (2000) 1903±1932

www.elsevier.com/locate/econbase

*

Corresponding author. Paper written whilst a Ph.D. student at The Management School, Imperial College, currently at Algorithmics (UK) Limited. Tel.: 7553-2633; fax: +44-20-7481-3130.

E-mail addresses:cindyn@algorithmics.com (C.I. Nikeer), 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.}

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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$29.035 trillion in outstanding principal amount at 31 De-cember 1997.3 Of this amount, interest rate swaps account for US$ 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 de-velopment 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 theÔoriginalÕfactors because (i) unlike the original factors which are churned out by the principal com-ponents analysis of a computer package, thesynthetic 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 applyone 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/}

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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 _Ôexact_Õ parallel shift rather than a ``wavy line something like a parallel shift''. The twosyntheticfactors (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 dierent 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 dierent 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 in-formation 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 cur-rencies. 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 alter-native to a Monte Carlosimulation 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 com-putational 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 methodol-ogies, 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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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 eective 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 ser-vicing national debt. There is a very large body of literature on this topic but the two recent approaches that are of relevance here arecointegration analysis

andfactor analysis.

Researchers have used cointegration analysis to concentrate on developing a stochastic process type model for forecasting future interest rates for dif-ferent maturities and an adjustment of theexpectations hypothesisto take into account the cointegration of data, that is, the existence of a long-run equi-librium 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 dierent maturities which implies that the underlying variables are being driven by some common fundamentals, that is, some common factors.

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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 de-termine 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 curvatureof 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 Due 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 dierent temporal eects ± liquidity eects are short-lived whilst credit eects 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 dierences 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 dierences, 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 au-tocovariance (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 re-sults. We ®rst carried out the Dickey±Fuller test (Dickey and Fuller, 1981) which suggested that ®rst dierencing achieves stationarity. We further carried out the more sophisticated Dickey±Pantula (Dickey and Pantula, 1987) test which con®rmed that our data set isnotintegrated of order 2 butisintegrated of order 1 and thus ®rst dierencing is sucient to achieve stationarity. However, given that the Dickey±Fuller tests have been criticised for their

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failure to reject a unit root due to their low power against stationary alter-natives, we also carried out the KPSS unit root test (Kwiatkowski et al., 1992).4In the Dickey±Fuller test, the null hypothesis is non-stationarity and the alternative hypothesis is stationarity, whilst in the KPSS test the null hy-pothesis is stationarity and the alternative hyhy-pothesis 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 dierencing achieves stationarity. Thus using the changes in swap rates is appropriate for subsequent analyses.5Third, we developsynthetic versions of the original factors derived from principal com-ponents analysis so that the factors are more manageable for practical imple-mentation. Fourthly, we test the stability of thesyntheticfactors over dierent 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 be-tween observed variables (in this paper, 2±10 year6interest 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 uncor-related common factors.

The mathematical model for factor analysis is similar to a multiple regres-sion 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 factorsFj j1;. . .such that:

DYi

X

j

aijFj diUi fori2;. . .;10 years; 1

whereFjs are termed thecommonfactors,aijis theloadingof theith 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.

5

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}

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variableDYithat is not explained by the common factors anddiis theloading

on the unique factor.

A unique solution for theaijs 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 combi-nations of the variables as follows:

Fk

X

i

bkiXi; 2

whereFk represents thekth factor and bki is the correlation of the kth factor

with theith 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 cor-related variables into uncorcor-related factors or components. The corcor-related variables are decomposed into a linear combination of orthogonal components that best_Ôsummarises_Õthe 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 dierent 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 theÔtrueÕ2 year rates and so on. The use of zero-coupon rates is consistent with earlier studies (Litterman and Sche-inkman, 1988; Knez et al., 1994). Finally, we took the ®rst dierences of the zero-coupon rates; this is the data set used in subsequent analyses in this paper.

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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 dierent currencies.

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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 calledparallel shiftbased on all scores for years 2±10 falling generally in the range of 0.85±0.99. This suggests that rates for the dierent years move roughly in line with each other in a parallel manner. Aparallel shiftrepresents achange

in the level of interest rates. The second factor is calledrotationbecause 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 dif-ferent years are moving in a linear fashion and intersecting thex-axis at year 6 (or rotating at year 6). Arotationrepresents 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. Atwistrepresents 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 in-terest rate swap rates (Table 1). We now go on to modelsyntheticversions 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 dierence 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 dierences data set including the interpolated years 6, 8 and 9.

9

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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_Ôexact_Õ 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 anÔexactÕ rotation based on

Table 1

Summary of factor scoresa

Currency (weekly)b _{% of total variation} _{Total, 3 factors (%)}

Factor 1 Factor 2 Factor 3

USD 2±10 yr 95.0 3.6 0.7 99.3

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.

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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 ma-turities, 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^aas follows:

MinimiseSXYiÿa^

whereSis the squared deviation away from the ®tted constant change,a^,Yithe

value of the observed variables,Y the average of the observed variables, andn

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 cal-culation 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

syntheticparallel shift, columns (3)±(10) gives the 3 year residual variation and so on. The eect of this calculation is to remove a constant value from each of the observations for that week.

Our synthetic parallel shiftloadings,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}

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Table 2

Fitted synthetic factors

Swap rate maturity 2Y 3Y 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

Change in slope variable

)4 )3 )2 )1 0 1 2 3 4

Panel C: Calculation of ®tted rotation,b, from residual changes after ®rst^ syntheticfactor extracted

Swap rate maturity 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y b^

8 Jan 88 )0.066 )0.075 )0.021 0.015 0.008 0.001 0.023 0.046 0.069 0.016

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ni11; i2;. . .;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 steep-ness in the yield curve of each of the 10 currencies appeared to _Ôrotate_Õ

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roughly around a mid-maturity point. Our synthetic rotation, b^, was therefore given a pivot point of year 6. The explanatory variable, Xi, in this

linear regression is a _ÔChange in Slope_Õ 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 aÔResidual Changes in

Swap Rates_Õ variable (after extracting the synthetic parallel shift). Our ®tted slope change,b^, can now be calculated as follows:

^

b P10

i2XiÿXYiÿY

P10

i2XiÿX

2 : 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 anÔexactÕrotation 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 the oursyntheticrotation,b^, to the residual changes in swap rates for the week starting 8 January 1988. The synthetic rotation was then extracted from the residuals.

Our synthetic rotationloadings,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

se-lected 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

ni21:5ÿ0:25i; i2;. . .;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%

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and 98.6%. The synthetic factors are therefore considered an acceptable rep-resentation 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 prac-tical risk management tools than the original factors for the following reasons. First, they are easier to understand than the original factors. The synthetic

Table 3

Variation explained by synthetic factorsa

Currency Factors Original factors (%) (2±10 yr) Synthetic factors (%) (2±10 yr)

USD Factor 1 95.0 94.5

Factors 1&2 98.6 98.5

GBP Factor 1 91.0 89.9

Factors 1&2 97.9 97.5

DEM Factor 1 90.8 89.8

Factors 1&2 98.0 97.6

SWF Factor 1 93.1 91.9

Factors 1&2 97.6 97.0

ITL Factor 1 94.0 92.7

Factors 1&2 98.6 97.8

JPY Factor 1 91.7 90.5

Factors 1&2 97.8 97.1

NLG Factor 1 90.9 89.8

Factors 1&2 97.6 97.1

BEF Factor 1 88.6 86.3

Factors 1&2 97.1 96.5

FRF Factor 1 91.6 90.1

Factors 1&2 98.4 97.5

ESP Factor 1 91.3 90.0

Factors 1&2 97.7 95.9

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.

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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 aÔblack-boxÕwhich 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 _Ôexact_Õ parallel shift and rotation. The original factors, however, look_Ôlike_Õand can be interpreted as a parallel shift and ro-tation, 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 visuali-sable as ``squiggly lines somethingÔlikeÕa parallel shift or rotation.''

Third, the synthetic factors reduce the computational complexity of prac-tically 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 dierent time periods. The use of the original factors, on the other hand, would mean employing dierent factor loadings for the dif-ferent currencies and over dierent 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 sucient. 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 ap-proximation 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 man-agement purposes because it allows us to easily evaluate, say, one _Ôunit_Õ of

parallel shiftadded 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 eect on the other factors. Therefore, overall, we propose the use of thesynthetic factors in de-veloping risk management applications. One such application is in estimating VaR.

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4. Application of TSFs to VaR estimation

VaR for a given portfolio is expressed, for example, as ``$10 million over-night 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 cor-relation between each pair of returns:

r2_px2_ir2_i x2_jr_j22wiwjrirjqij; 7

wherer2

pis the volatility (variance) of the portfolio returns,xithe weight if the

ith asset/cash ¯ow in the portfolio,r2

i the variance of theith asset/cash ¯ow,

andqij is the correlation between the returns of theith andjth asset/cash ¯ow.

For a single asset, the VaR is calculated as follows:

VaRCIrd; 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, one-tailed, 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 anddis 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 (sayN) assets/cash ¯ows, the VaR formula becomes

VaRpVRVT_; ₉

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V CIridi. . .CIrN dN 10

and Ris an NN 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 dierent 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 inde-pendent normal random numbers must be transformed into random numbers that contain this speci®ed covariance structure. Third, a Cholesky decompo-sition (Jorion, 1997, pp. 242±243) of the covariance matrix is carried out to obtain

RAA0; 11

whereAis a lower triangular matrix with zeros in the upper right hand corner andA0_{is the transpose of}_A_.

Fourth, the independent normal random numbers are transformed into random numbers with a covariance structureRYiby calculating

YiAXi: 12

Fifth, recalling that Yi represents the simulated changein zero-coupon

in-terest rate swap rates, the new rates are now computed:

rit ri0Yi; 13

whereritis the new (simulated) rate one period ahead andri0is 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.

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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 dierent 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 calcu-lating VaR. Its limitation however is its computational intensity given that the portfolio is revalued under thousands of dierent 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 fol-lows:

Drit ptbtiÿi eit; 14

whererit is the zero-coupon interest rate swap rate, idenotes maturity,t the

date of the observation,_Drit the dierence between rit andritÿ1,ptthe parallel

shift for the zero-coupons betweentÿ1andt,bt the rotational movement,i

some pivot point for the rotation andeit 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 ofp_t andb_t, our synthetic parallel shift and rotation:

b_t CovDrit;i=Vari; 15

p_t EifDritg ÿbtEifiÿig: 16

It is interesting to note thatb_t does not depend uponi. This is because the rotation around one pivot pointi0 can always be described in terms of a ro-tation aroundi and a parallel shift i0ÿi.

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 port-folio whose valuePfrit. The value of the portfolio can thus be expressed

as:P P

wiri. If there is a change in the rates, over a given time periodt(say

daily), this will cause a change in P: DPtwidiDrit. We can write

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VartDPt

X

ij

didjEtfDritDrjtg ÿEtfDritg EtfDrjtg; 17

wheredidjare 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 theith andjth assets in calculating the sensitivity of the market value of the portfolio to changes in the underlying risk factors.

Substituting from Eq. (14) forDrit, we get:

VartDPt

X

ij

didjEfptbtiÿiptbtjÿig ÿEtptbti

ÿi Etptbtjÿi; 18

VartDPt Vartpt

X

ij

didjVartbt

X

ij

didjiÿijÿi

Covtpt;bt

X

ij

didjijÿ2i: 19

The covariance term Covtpt;btcan be written as

Covtpt;bt Etfptbtg ÿEtfptg Etfbtg 20

and substituting from Eq. (16) forpt, we get

Covtpt;bt CovtEifDritg;bt ÿVartbt Eifiÿig: 21

Sinceb

t does not depend on the location ofi0, this means that we can select any

value fori0_{, for example,}

CovtEifDritg;bt=Vartbt ÿEifiÿig 0: 22

i_{can be therefore be de®ned as}

iEifig ÿ CovtEifDritg;bt=Vartbt: 23

This means that Covp_t ;b_tis by construction set at zero (for orthogonal factors) where

p_t EifDritg ÿbtEifiÿig 24

and hence, from Eq. (19),

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VartDPt Vartpt

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:

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 in-terpolation 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

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vola-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 an-nual time periods from 30 June 1991 to 30 June 1998, using the same buc-keting 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 re-ceiving 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 topaying a ®xed rate of interest. We would thus expect some osetting of positive and negative cash ¯ows in the port-folio 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 in-terest, instead of the 3 year swap, with all else remaining the same as in Portfolio 1. Given the longer period of osetting cash ¯ows, a further re-duction 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 dierent 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. Port-folio 6 has the same swap maturities, amounts and rates as PortPort-folio 1 but dierent start dates and interest settlement frequencies. The resulting VaR estimates are lower that those of Portfolio 1 re¯ecting the more frequent interest settlements for the 3 and 5 year swaps and thus lower exposure of the portfolio. Finally, Portfolio 7, which consists of one hundred swap cash ¯ows, demonstrates that the TSF approach also produces results consistent with the Variance±Covariance and Monte Carlo approaches when a larger portfolio of cash ¯ows is used.

In general there are three main observations that can be made from these results. First, the VaR estimates produced by the TSF approach are broadly consistent with themagnitudeof values produced by the Variance±Covariance and Monte Carlo simulation approaches. Second, the TSF approach pro-duces VaR estimates which mirror the movements in VaR estimates of the

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Variance±Covariance and Monte Carlo simulation approaches across Port-folios 1±7. That is, when the VaR estimates of the latter two methods go up or down from portfolio to portfolio, the VaR estimate of the TSF approach moves in the same direction. A more rigorous statistical test with 100 random samples of data was undertaken by the authors and the results verify the conformity of estimates from the three main methods. Third, the TSF

Table 4

Comparison of VaR estimation using dierent methodologiesa

Hypothetical swap portfolios Two factor Variance± Covariance

Monte Carlo

Portfolio 1 $178,705 $180,943 $176,287

3 yr, $10 M receiving 8.0825% ®xed annually 5 yr, $20 M receiving 8.5976% ®xed annually 7 yr, $10 M receiving 8.9485% ®xed annually

Portfolio 2 $117,656 $122,049 $116,377

3 yr, $10 M paying 8.0825% ®xed annually 5 yr, $20 M receiving 8.5976% ®xed annually 7 yr, $10 M receiving 8.9485% ®xed annually

Portfolio 3 $67,390 $72,802 $73,177

3 yr, $10 M receiving 8.0825% ®xed annually 5 yr, $20 M receiving 8.5976% ®xed annually 7 yr, $10 M paying 8.9485% ®xed annually

Portfolio 4 $155,982 $157,051 $158,248

2 yr, $10 M receiving 8.0825% ®xed annually 4 yr, $20 M receiving 8.5976% ®xed annually 7 yr, $10 M receiving 8.9485% ®xed annually

Portfolio 5 £48,350 £56,434 £53,404

3 yr, £10 M receiving 11% ®xed semi-annual-ly. Start date: 15 April 1991

5 yr, £10 M receiving 11.5% ®xed semi-annually. Start date: 9 August 1990 7 yr, £20 M paying 12% ®xed annually. Start date: 21 January 1991

Portfolio 6 £145,599 £147,083 £140,625

3 yr, $10 M receiving 8.0825% ®xed semi-annually. Start date: 15 April 1991 5 yr, $20 M receiving 8.5976% ®xed semi-annually. Start date: 9 August 1990 7 yr, $10 M receiving 8.9485% ®xed annually. Start date: 21 January 1991

Portfolio 7 $106,971 $116,308 $112,754

100 randomly generated cash ¯ows over the period 30 June 1991 to 30 June 1998

a_{This compares the TSF VaR estimates versus the estimates produced by the traditional}

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method consistently produces a VaR estimate for these interest rate swap portfolios which is lower than that produced by the Variance±Covariance methodology. This result should be expected since the two factors are derived from the eigenvectors of the covariance of asset returns and therefore sum-marise the covariance matrix but do not contain all the information therein. However this slight understatement appears acceptable given the reduction in computational intensity achieved by the TSF approach.

4.5. Bene®ts of the TSF VaR measure

The TSF VaR approach provides the following key bene®ts versus the ex-isting methodologies. First, it simpli®es the computation of VaR versus a Variance±Covariance approach as it eliminates the need to calculate time-consuming and large covariance matrices between each pairs of securities in the portfolio. The TSF approach instead relies only on the determination of the common factors, parallel shift and rotation, to adequately summarise the majority of the variation in the term structure. The two factors thus reduce the dimensionality of the VaR computation. Additionally, it is computationally less intensive than a Monte Carlo simulation approach, albeit not as accurate, but this must be weighed against the costs of achieving greater accuracy. Second, the TSF approach decomposes the risk of the portfolio into two components, exposure to parallel shift and rotational movements in the yield curve so that its potential exposure to each component can be identi®ed. This enables the TSF approach to not only provide a ®ner reporting of market risk but also to be used as a tool for risk management as it allows the company to manage individual components, parallel shifts and rotations, and to limit its exposure to each factor depending on its risk pro®le. Third, the TSF approach takes into account a second order eect, by inclusion of a_Ôsteepening of the yield curve_Õrotationfactor, unlike the Variance±Covariance method which only takes into account a ®rst order eect. This not only increases the accuracy of the TSF approach but makes it potentially applicable to non-linear securities such as swaps and options, whereas the classical Variance±Covariance is limited to linear securities. Therefore the TSF VaR approach is particularly bene®cial to a portfolio of swaps although it may also be applied to a wider range of securities.

5. Conclusion

A key insight of this paper is the use of syntheticparallel shiftandrotation

factors, which are generalisable over a range of dierent time periods and currencies, to develop an innovative TSF approach to the estimation of VaR. We simulated a series of hypothetical portfolios of interest rate swaps and compared the VaR estimates obtained from using the new TSF approach

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versus the existing methodologies of Variance±Covariance and Monte Carlo simulation. The TSF methodology was found to produce estimates that were consistent with the traditional methodologies of Variance±Covariance and Monte Carlo simulation. The main bene®t of the TSF approach however is its simpli®cation of the computation of VaR.

Acknowledgements

We wish to thank Keith Cuthbertson, Stephen Hall, Nathan Joseph, Nigel Meade, David Miles and Constantine Thannasoulas for helpful comments. We also wish to thank participants at the Financial Risk Management Workshop held by the European Institute for Advanced Studies in Management in Brussels in March 1998, in particular Eric Peree, Richard Stapleton and Phillipe Jorion, for helpful comments. We also would like to thank conference participants at the European Finance Association (EFA) 25th Anniversary meeting held at INSEAD, Fontainebleau, France, in August 1998 for helpful comments. The authors, however, are solely responsible for the views expressed and any errors or omissions therein.

Appendix A

Table 5

Principal components results ± 2±10 yr zero-coupon dierenced (ZCDF) swap rates

Factor matrix Factor 1 Factor 2 Factor 3 Factor Eigenvalue Pct of Var US dollar: Number of observations: 461

ZCDF2Y 0.94210 0.30335 0.11583 1 8.55343 95.0 ZCDF3Y 0.96519 0.24357 0.04931 2 0.32462 3.6 ZCDF4Y 0.97917 0.14803 )0.05694 3 0.06306 _ _ _0.7

ZCDF5Y 0.98689 0.06814 )0.10037 99.3

ZCDF6Y 0.99519 )0.01824 )0.08994 ZCDF7Y 0.98536 )0.11355 )0.07781 ZCDF8Y 0.98606 )0.15837 0.01315 ZCDF9Y 0.97682 )0.20651 0.05592 ZCDF10Y 0.95590 )0.25644 0.12827 UK pound: Number of observations: 461

ZCDF5Y 0.97949 0.10556 )0.13051 98.9

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Table 5 (Continued)

Factor matrix Factor 1 Factor 2 Factor 3 Factor Eigenvalue Pct of Var German mark: Number of observations: 461

ZCDF5Y 0.97737 0.12980 )0.12806 99.1

ZCDF6Y 0.99463 )0.01176 )0.09137 ZCDF7Y 0.97570 )0.17015 )0.04832 ZCDF8Y 0.97150 )0.23005 0.00623 ZCDF9Y 0.95379 )0.29281 0.06382 ZCDF10Y 0.92121 )0.35540 0.12263 Swiss franc: Number of observations: 460

ZCDF5Y 0.97596 0.08628 )0.15068 98.6

ZCDF6Y 0.99291 )0.03455 )0.09141 ZCDF7Y 0.97411 )0.16691 )0.02382 ZCDF8Y 0.97959 )0.19043 0.02094 ZCDF9Y 0.97373 )0.21457 0.06761 ZCDF10Y 0.95559 )0.23803 0.11481 Italian lira: Number of observations: 302

ZCDF5Y 0.98984 0.03594 )0.12093 99.4

ZCDF6Y 0.99543 )0.05657 )0.06207 ZCDF7Y 0.98265 )0.15845 0.00018 ZCDF8Y 0.98338 )0.17433 0.03430 ZCDF9Y 0.97826 )0.19375 0.06952 ZCDF10Y 0.96674 )0.21572 0.10565 Japanese yen: Number of observations: 302

ZCDF5Y 0.97390 0.11334 )0.16194 99.0

ZCDF6Y 0.99220 )0.02626 )0.11379 ZCDF7Y 0.97051 )0.19064 )0.05265 ZCDF8Y 0.97238 )0.22557 0.00573 ZCDF9Y 0.96225 )0.26195 0.06898 ZCDF10Y 0.93791 )0.29793 0.13538 Dutch guilder: Number of observations: 277

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(see opposite for Appendix B)

Table 5 (Continued)

Factor matrix Factor 1 Factor 2 Factor 3 Factor Eigenvalue Pct of Var

ZCDF5Y 0.97431 0.11336 )0.15439 98.8

ZCDF6Y 0.99361 )0.01178 )0.09443 ZCDF7Y 0.97428 )0.15724 )0.02259 ZCDF8Y 0.97342 )0.21754 0.01971 ZCDF9Y 0.95708 )0.28094 0.06462 ZCDF10Y 0.92346 )0.34401 0.11037 Belgian franc: Number of observations: 277

ZCDF5Y 0.95721 0.18718 )0.16689 98.6

ZCDF6Y 0.98936 0.00962 )0.13956 ZCDF7Y 0.95979 )0.21068 )0.0980 ZCDF8Y 0.96082 )0.26715 )0.01691 ZCDF9Y 0.94200 )0.32702 0.07196 ZCDF10Y 0.89979 )0.38548 0.16448 French franc: Number of observations: 276

ZCDF5Y 0.98169 0.08957 )0.14604 99.3

ZCDF6Y 0.99457 )0.05264 )0.06783 ZCDF7Y 0.97132 )0.20876 0.01925 ZCDF8Y 0.97204 )0.22805 0.03746 ZCDF9Y 0.96593 )0.25003 0.05703 ZCDF10Y 0.95199 )0.27363 0.07767 Spanish peseta: Number of observations: 236

ZCDF2Y 0.87748 0.45895 0.01308 1 8.21301 91.3 ZCDF3Y 0.92454 0.36359 )0.00376 2 0.57647 6.4 ZCDF4Y 0.96693 0.22125 )0.02103 3 0.12668 _ _ _1.4 ZCDF5Y 0.97570 )0.07213 )0.16656 99.1 ZCDF6Y 0.97860 )0.16060 )0.12781

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Appendix B

Table 6

Illustration of cash ¯ows of threeParswap portfolio (Portfolio 1)

Year Dates Years Underlying

princi-USD swap1:3 yr Act/360.Swap rate(receive): 8.0825

0 01-Jan-91 )10,000,000.00 )10,000,000.00 1 )10,000,000.00

1 01-Jan-92 1.01389 10,000,000.00 819,479.10 7.500000 0.9293 761,568.19

2 01-Jan-93 1.01667 10,000,000.00 821,724.25 7.744110 0.8594 706,166.86

3 01-Jan-94 1.01389 10,000,000.00 10,819,479.10 8.082534 0.7886 8,532,264.95 Net present value 0.00

USD Swap2:5 yr Act/360.Swap rate(receive): 8.5976

0 01-Jan-91 )20,000,000.00 )20,000,000.00 1 )20,000,000.00

1 01-Jan-92 1.01389 20,000,000.00 1,743,408.44 7.500000 0.9293 1,620,205.33 2 01-Jan-93 1.01667 20,000,000.00 1,748,184.90 7.744110 0.8594 1,502,341.25 3 01-Jan-94 1.01389 20,000,000.00 1,743,408.44 8.082534 0.7886 1,374,855.72 4 01-Jan-95 1.01389 20,000,000.00 1,743,408.44 8.358463 0.7203 1,255,846.33 5 01-Jan-96 1.01389 20,000,000.00 21,743,408.44 8.597631 0.6552 14,246,751.37 Net present value 0.00

USD swap3:7 yr Act/360.Swap rate(receive): 8.9485

0 01-Jan-91 )10,000,000.00 )10,000,000.00 1 )10,000,000.00

1 01-Jan-92 1.01389 10,000,000.00 907,281.91 7.500000 0.9293 843,166.15

2 01-Jan-93 1.01667 10,000,000.00 909,767.61 7.744110 0.85937 781,828.86

3 01-Jan-94 1.01389 10,000,000.00 907,281.91 8.082534 0.7886 715,484.50

4 01-Jan-95 1.01389 10,000,000.00 907,281.91 8.358463 0.7203 653,551.18

5 01-Jan-96 1.01389 10,000,000.00 907,281.91 8.597631 0.6552 594,470.72

6 01-Jan-97 1.01667 10,000,000.00 909,767.61 8.768583 0.5953 541,582.01

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Appendix C

Table 7

Illustration of cash ¯ows of threeNon-Parswap portfolio (Portfolio 5)

Year Date Fraction of year Swap yield % pa Discount rates

0 06-Jun-91 1

Date Fraction of year Swap cash ¯ows Discount factor Discounted cash ¯ows Swap 1: 5 yr. Principal:)10,000,000.00

Last interest payment: 09-Feb-91; Swap rate: 11.50%; Receive ®xed

09-Aug-91 0.49589 570,273.97 0.9827 560,418.15

Swap 2: 7 yr. Principal: 20,000,000.00

Last interest payment: 21-Jan-91; Swap rate: 12.00%; Pay ®xed

21-Jan-91 ± 20,000,000.00 1 20,000,000.00 21-Jan-92 1.00000 )2,400,000.00 0.9382 )2,251,585.53 21-Jan-93 1.00274 )2,406,575.34 0.8473 )2,039,195.08 21-Jan-94 1.00000 )2,400,000.00 0.7643 )1,834,203.10 21-Jan-95 1.00000 )2,400,000.00 0.6876 )1,650,233.11 21-Jan-96 1.00000 )2,400,000.00 0.6191 )1,485,949.14 21-Jan-97 1.00274 )2,406,575.34 0.5563 )1,338,721.36 21-Jan-98 1.00000 )22,400,000.00 0.4985 )11,165,851.37 Newvalue )1,765,738.69 Oldvalue )1,765,738.69

Change ±

Swap 3: 3 year Principal:)10,000,000.00

Last interest payment: 15-Apr-91; Swap rate: 11.00%; Receive ®xed

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Table 7 (Continued)

Date Fraction of year Swap cash ¯ows Discount factor Discounted cash ¯ows 15-Apr-92 0.50137 551,506.85 0.9152 504,743.02 15-Oct-92 0.50137 551,506.85 0.8704 480,018.37 15-Apr-93 0.49863 548,493.15 0.8276 453,933.26 15-Oct-93 0.50137 551,506.85 0.7861 433,556.79 15-Apr-94 0.49863 10,548,493.15 0.7455 7,863,870.14 New value 268,118.67 Old value 268,118.67

Change ±

Total value of ®xed side of portfolio )838,069.99 Old value )838,069.99 Change in value ±

Bucketedcash ¯ows

Vertices Mapped cash ¯ows

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