risk t u t orial How to Estimate and Calibrate Analytical VaR for Interest Rate Risk These absolute one-week interest rate changes are specified for each maturity bucket and for each principal component or scenario — the first scenario being a proportional shift of the yield curve, the second scenario a rotation of the term structure and the third scenario a change of the curvature. Combinations of the scenarios are applied to the interest rate sensitivities. In the remainder of this article, we will apply our interest rate scenarios to the bp sensitivities illustrated in Figure 1 (below). What is principal component analysis (PCA)? How can a firm use PCA to calculate analytical VaR for interest rate risk? And how does analytical VaR compare with Monte Carlo VaR? By Marco Folpmers and Niels Eweg Interest Rate Scenarios Derived by Principal Component Analysis Principal Component Analysis (PCA) is applied to highly correlated data series in order to reduce dimensionality. The underlying idea is that highly correlated time-series data, such as weekly observations of the term structure, can be summarized into a small number of principal components. As is explained in the appendix “PCA 101” (see pgs. 4950), term structure fluctuations can often be captured in three meaningful components; hence, the “effective dimensionality” is three rather than the 18 maturity buckets of the original data set (up to a given confidence level, which in our case is 96.3%). The results of the PCA decomposition, as described in the appendix, are three sets of independent 99% interest rate shocks ∆Rij (i = 1, …, 3, j = 1, …, 18 for all 18 maturities). The three sets of shocks capture three generic movements in 46 RISK PROFESSIONAL J U N E 2 0 0 9 1.5 the term structure: a parallel change, a rotation and a change of the curvature. In Table 1 (below), we present the outcomes in basis points (bp) per year, stressed at a 99% level. We conclude that the parallel shift contained in the first principal component is roughly consistent with the 200 bp shock scenario proposed by CEPS.1 However, the combination of the three principal components allows a more realistic scenario that includes a rotation and a change of the curvature of the term structure. 0.5 0.0 Euro -0.5 PC 1 PC 2 PC 3 Maturity PC 1 PC 2 PC 3 3M 26 -152 48 7Y 234 7 -36 6M 81 -138 77 8Y 228 23 -26 9M 120 -124 44 9Y 224 36 -17 1Y 151 -118 14 10Y 220 48 -9 2Y 221 -95 -42 12Y 212 63 7 3Y 241 -67 -64 15Y 201 81 31 4Y 247 -49 -62 20Y 189 103 61 5Y 243 -30 -55 25Y 179 112 76 6Y 239 -11 -46 30Y 173 120 87 www.garp.com -1.0 -1.5 -2.0 -2.5 Table 1 Eigenvector Values for Principal Components 1 through 3 in Basis Points per Year (Stressed to the 99% Confidence Level) Maturity x 104 Sensitivities: change in EUR per 1 bp increase in interest rate 1.0 -3.0 -3.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Maturity bucket Our stylized example shows a typical pattern in which the bank is funded with the help of short-term liquidity, whereas its assets have longer maturities (an increase in interest rate implies a decrease in present value of the assets). We denote these key rate durations (KRDs) as KRDj , j =1,...,18. The sum of the KRDs is equal to € - 220,000. This means that a 1 bp parallel increase in the term structure leads to a €220,000 decline in market value. The one-week 99% VaR (with bp equal to 10-4) 2 equals: 3 Charts by Desiree Nunez ince analytical value-at-risk (VaR) is easier to understand and to apply than Monte Carlo VaR, the hunt for analytical VaR methodologies is worthwhile, particularly if they can compete with the complexities of Monte Carlo modeling. This article demonstrates how analytical interest rate risk VaR can be estimated using principal component analysis (PCA). As we will show below, the analytical VaR can be backtested with the help of historical simulation. This means that relevant term-structure scenarios can be developed quickly and conservatively. These scenarios can be used for risk measurement, such as VaR estimation, but also for the immunization/hedging of fixed income portfolios. VaR 1week (0.99)= √ { 18 ∑ ∑ i=1 j=1 2 { S Figure 1 Basis Point Sensitivities (∆Rij • KRDj/bp) Note that a Euclidean distance in cubic space is calculated www.garp.com 18 (square root of the sum of three squares), which is justified since the principal components are independent (see the appendix). For our example, the one-week 99% VaR equals €4.3 million. Calibration and Back-Testing of the PCA-based Interest Rate Risk VaR Whereas the normal approximation illustrated earlier is often used, the principal components for the weekly growth rates exhibit fat tails. A more conservative approach would be to use a t-distribution. The parameters for standard deviation and degrees of freedom can be derived by fitting t-distributions to the principal component scores using maximum likelihood estimation (MLE). Instead of multiplying by a z-score when calculating the stressed principal components, we multiply with a score from the inverse t-distribution.3 We derive the degrees of freedom parameter needed from a MLE estimation of a fitted t-distribution applied to the principal component scores. The degrees of freedom for the fitted-t distribution of the principal component scores are relatively low ( [ 7.1, 2.4, 2.3 ] for the first three principal components), signaling a clear departure from normality because of fat tails. In Figure 2 (below), we compare the VaR calculated with the help of the normal and t-approximation. We conclude that, in our example, fat tails are relevant for confidence levels above 97%. Figure 2 VaR, according to Normal and t-approximation 8 PCA normal approx x 106 PCA t-approx 7 I Week Value at Risk (Euro) risk t u t orial 6 5 4 3 2 90 91 92 93 94 95 96 Confidence level 97 98 99 100 J U N E 2 0 0 9 RISK PROFESSIONAL 47 risk t u t orial Below the 97% confidence level, the normal approximation has higher VaR outcomes since the standard deviations of the MLE-fitted t-distributions are slightly lower than the standard deviations from the PCA procedure (which are equal to the square roots of the eigenvalues, as shown in the appendix). The 99.9% 1-week VaR that we estimate with the help of the t-distribution equals €7.6 million. This result can be compared to the VaR that we calculated with the help of a back-test. This back-test is performed using a historical simulation in which we select the 99.9th historical decline in market value due to a weekly change in the term structure. The historical 99.9% weekly VaR that we estimate equals €7.7 million, which is very close to the €7.6 million that we estimate with the help of the PCA approach using t-distribution scaling. At the same time, this back-test illustrates the necessity of using the t-distribution scaling instead of the normal approximation for an adequate VaR level. { 18 ∑ ∑ 2 { 3 Immune Portfolios (∆R • KRDj/bp) outcomes of the PCA are notijonly useful for VaR esVaR 1weekThe (0.99)= timation, but also for the establishment of portfolios that are j=1 i=1 immune to √ interest rate shock. An immune portfolio is defined as a non-trivial mix of KRDs, determined in such a way that the VaR equals zero.4 We consider a portfolio immunized if VaR = 0. For this constraint to hold, the sum of the products ∆Rij • KRDj needs to be zero for each principal component; hence, the portfolio is immune if and only if This is a set of three linear equations in four unknowns. A possible solution approach is to set Yj=4=1. An example with maturities {1,6,11,16} results in the following equations: [ 1.98 18.73 17.79 14.75 -11.86 -5.24 1.78 8.02 3.73 -4.99 -2.05 4.78 Figure 3 Example of a Basis Point Sensitivity Profile for an Immunized Portfolio RISK PROFESSIONAL J U N E 2 0 0 9 x 104 Immunized Sensitivity Profile 1.5 1.0 j=1 48 =a• -0.44 1.67 -2.53 1 where a is an arbitrary multiplicative factor. This method of finding the KRDs can be applied to all groups of four maturities. The complete portfolio now is a linear combination of the solutions of the complete set of groups of maturities. An example of an immunized portfolio is given in Figure 3 (below). (∆Rij • KRDj) = 0 Q•y=0 [ ][] [ ][] KRD1 KRD6 KRD11 KRD16 2.0 0.5 Euro for all principal components i = {1,2,3}. Since the system consists of three equations with 18 variables (KRDj), it cannot be solved directly. We therefore suggest a piecewise solution in which the equation above is solved for groups of four maturities ( ˆj= {1,2,3,4}) at a time. We construct a matrix Q with elements ∆ Rij (dimension [3 * 4]) and a column vector y with elements KRDj (dimension [4 * 1]). The problem can now be stated as follows: ] KRD1 0 • 10 * KRD6 = 0 0 KRD11 1 -4 Note that, although the VaR of the aforementioned portfolio equals zero, the sum of the KRDs equals € - 4,095. This is the change in market value if all interest rates increase simultaneously with one basis point. Although this portfolio is not equal to a portfolio with duration = 0, it is more effective in reducing the interest rate risk. In contrast to the durationmatching portfolio, it is immune for variations described by the first three principal components (96.3% of the total variance, as shown in the appendix). This gives the following KRDs: 18 ∑ risk t u t orial 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 Closing Thoughts This article demonstrates how an analytical VaR can be derived for interest rate risk. With the help of a standard procedure from multivariate statistics, (principal component analysis) coherent interest rate scenarios can be computed that take into account the co-movement of interest rate changes across maturities. This VaR procedure has existed for approximately 12 years (since Golub & Tilman coined the term “principal component duration” in 1997) and is integrated in some commercial software applications for the management of interest rate risk. It is much superior to single duration values that assume a parallel shift of the term structure (e.g., the standard 200 basis point shock proposed by CEPS). We demonstrated the usefulness of the PCA method for constructing immunized portfolios, and we have also shown that a multiplier from a normal distribution may underestimate the VaR for the higher confidence levels. (This is easily remedied by using parameters from a MLE-fitted t-distribution.) With the help of this feature, we think that analytical VaR methods for interest rate risk VaR remain interesting for a straightforward and quick computation, as an alternative to Monte Carlo simulation. Appendix: Principal Component Analysis 101 We start with a dataset, R, containing for each week t the euro zero-coupon interest rate for maturity bucket j (j=1, ...,18). In our example, R contains the term structure data of a sixyear history ending on December 31, 2008. For the PCA, the input data consists of the weekly growth rates of the interest rates (so X1t = RR –1) for the 18 maturity buckets after normalization to z-scores (zero mean, unit standard deviation). This data-matrix is referred to as XT = [X1,..., X18]. The first principal component is found with the help of a linear transformation using coefficients contained in a 1t 1t-1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Maturity bucket www.garp.com www.garp.com [18 * 1 ] vector a1, as follows: Y1 = aT1X The vector a1 is chosen so that Y1 has the largest possible variance, subject to the constraint that aT1 a1 =1. The second principal component is found by choosing a2 so that Y2 has the largest possible variance, subject to the constraints that aT2a2 =1 and a2Ta1 =0, or Y2YT1=0.(The principal components are orthogonal: i.e., they are independent and, hence, uncorrelated). Similarly, Y3, ..., Y18 are derived, so as to be orthogonal and to have decreasing variance. We will not go into the constrained optimization procedure to derive the a- vectors and the principal components (this procedure is covered in standard textbooks on multivariate statistics5), but we will only point out that this procedure is based on a (singular value) decomposition of the [ 18 * 18 ] covariance matrix6 and that it delivers the a vectors that are called eigenvectors, along with the variances of the principal components 1, ..., 18, which are called eigenvalues. The sum of these lambdas are equal to the sum of the variances of the original X vectors. Since we used normalized X vectors, the sum of the variances of the original X vectors and, hence, the sum of the lambdas, equals 18. The optimization procedure outlined above warrants that the eigenvalues 1, ..., 18 are decreasing in value. Since it is convenient to claim that the first m principal components account for n percent of the variance, it is useful to calculate the proportion ∑mi=1 i / ∑18 . Since we use normalized raw i=1 i data, the formula can be simplified to ∑mi=1 i/18. When applying the PCA to yield curve data, often the following interpretation (which is an empirical finding rather than a mathematical truth) can be given to the first three principal components:7 (1) the first principal component is highly correlated with the rates of all maturities and the correlations are of the same sign. So the first principal component is a factor that shifts the whole term structure (“level”); (2) the second principal component is negatively correlated with short-maturity rates and positively correlated with long-maturity series. This allows for an interpretation of the second principal component as a factor that tilts the term structure (“steepness”); and (3) the third principal component is positively correlated with short- and long-maturity rates and negatively correlated with intermediate-maturity rates, and is therefore a factor that effects the curvature. The reduction of dimensionality consists of the modeling J U N E 2 0 0 9 RISK PROFESSIONAL 49 risk t u t orial of the entire 18-dimensional term structure with the help of the first three principal components. For the computation of the interest rate risk VaR, a “shocked” term structure will be derived from stressed values of the first three principal components. The [18 * 18] matrix of eigenvectors is denoted by A, where A = [a1, ..., a18]. The principal components are denoted by Y, where Y = ATX. Since we want to apply a shock to the three main principal components and convert these shocked values to a stressed term structure across 18 maturities, we need to incorporate the inverse transformation (from principal components to the normalized weekly growth rates8), using the following formula: X = AY. Figure 4 Eigenvectors 1 through 3 Eigenvectors 1, 2 and 3 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 Table 2 Symbols Used for PCA Symbol Interpretation -0.5 Dimension ROriginal dataset, weekly term structure for 18 maturity buckets; 6 year history ending on December 2008 [ Nr of weeks * 18 ] XWeekly growth rates for 18 maturity buckets [ 18 * Nr of weeks ] A Eigenvectors [ 18 * 18 ] Y Principal components of X [ 18 * Nr of weeks ] i,i=1,...,18Eigenvalues, or variances of the principal components [ 18 * 1 ] For our example, we use weekly growth rates of interest rate across 18 buckets (3M, 6M, 9M, 1Y, 2Y, 3Y, 4Y, 5Y, 6Y, 7Y, 8Y, 9Y, 10Y, 12Y, 15Y, 20Y, 25Y, 30Y), from January 1, 2003, until December 31, 2008 (six years). The X matrix comprises the normalized growth rates across these 18 buckets and 313 weeks. The first three principal components account for 96.3% of the total variance. This proves that the dimensionality can be reduced to three while losing only a small fraction of the original total variance. The 96.3% variance is sufficient for many risk applications. The first three eigenvectors are shown in Figure 4 (above). 50 RISK PROFESSIONAL J U N E 2 0 0 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Maturity bucket Our decomposition of the term structure into three principal components shows the typical patterns outlined above. The blue line illustrates the first eigenvector and shows a fairly stable value across maturity buckets. Hence, the first principal component captures proportional increases and decreases of the term structure. The green line illustrates the second eigenvector and depicts the typical increasing trend of the term structure. The red line (third eigenvector) accounts for the curvature of the yield curve. A 99% VaR is calculated with the help of stressed values of the principal components. The stressed values of the three principal components are equal to strPC = a • √ • G (0.99), i = 1,...,3, j = 1,..., 18 ij i ij for a one-week 99% VaR with G(.) the inverse standard normal distribution. The result is one vector for each principal component (i) containing stressed values for the normalized one-week growth rates for the 18 maturities (j). Note that a normal distribution is assumed since we multiply with a zscore. This inherent normality assumption is discussed the “calibration and back testing” section of the article. The vectors have to be multiplied by the standard deviations of the growth rates for all maturities, and, subsequently, by the last interest rates in the dataset (term structure of December 31, 2008 = T) in order to convert the stressed growth rates to stressed absolute interest rate changes, as follows: ∆R =strPC • ( ∆R ) • R , i = 1,...,3, j = 1,..., 18 ij ij j T,j www.garp.com risk t u t orial FOOTNOTES REFERENCES 1. CEPS, 2006, 11. Agca, S. “The performance of alternative interest rate risk measures and immunization strategies under a Heath-Jarrow-Morton framework.” Virginia Polytechnic Institute and State University (published thesis, March 2002). 2. Note that we implement a delta-normal approach. More precise alternatives are: delta-gamma (correcting for convexity) and full revaluation. The delta-normal approach follows Golub & Tilman, 2000. 3. See Fiori & Iannotti (see references), 2006, p.12, about the t-distribution as a valid fat-tailed distribution for stressing the eigenvectors. CEPS, CP11, March 2006. Consultation paper on technical aspects of the management of interest rate risk arising from non-trading activities and concentration risk under the supervisory review process. 4. “Non-trivial” excludes the solution of zero KRDs for all maturity buckets. Chatfield, C. and A. Collins. “Introduction to Multivariate Analysis,” Chapman & Hall,1981. 5. See, e.g., Chatfield & Collins, 1981, Ch. 4. Fiori, R. and S. Iannotti. “Scenario based principal component value-at-risk: an application to Italian banks’ interest rate profile,” Banca d’Italia, September 2006. 6. Since we use normalized X vectors, the decomposition is applied to the correlation matrix of the data before normalization. Some statistical software packages use an option when calling the PCA procedure to distinguish between a PCA based on the covariance matrix or a PCA based on the correlation matrix. A PCA based on the correlation matrix is equivalent to a PCA based on the covariance matrix if the raw data have been normalized first. For the decomposition of interest rate data, it is generally recommended to use the correlation matrix (or use normalized raw data) rather than the covariance matrix (see Loretan, 1997, p. 26). The singular value decomposition of the variance matrix ∑ delivers the decomposition ∑ = U U1, with the matrix with the eigenvalues on the main diagonal and zeros elsewhere. The matrix U contains the eigenvectors. V V 7. See Loretan (1997, pg. 28), Golub & Tilman (2000, pg. 94), Rachev (e.a., 2007, pg. 453) and Fiori & Iannotti. 8. Note that A is an orthogonal matrix whose transpose is its inverse. Golub, W. and L. Tilman. “Risk Management: Approaches for Fixed Income Markets,” John Wiley & Sons, 2000. Loretan, M. “Generating market risk scenarios using principal component analysis: methodological and practical considerations,” Federal Reserve Board, March 1997. McNeil, A.J., R. Frey and P. Embrechts. « Quantitative Risk Management: Concepts, Techniques, Tools, » Princeton, 2005. Rachev, S.T., S. Mittnik, F.J. Fabozzi, S.M. Focardi and T. Jasic. “Financial Econometrics,” John Wiley & Sons, 2007. M. Weiskopf. “Immunization of fixed income portfolios.” (published thesis, 2009). Marco Folpmers (PhD, FRM) is the leader of the financial risk management service line at Capgemini Consulting in the Netherlands. He can be reached at [email protected] Niels Eweg (M.Sc., FRM) works as a structured finance and asset and liability management consultant at Capgemini. He can be reached at [email protected] 52 RISK PROFESSIONAL J U N E 2 0 0 9 www.garp.com

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