Robust Monte Carlo Simulation for Approximate Covariance Matrices ...

Probabilistic Constrained Optimization: Methodology and Applications (S. P. Uryasev, Editor), pp. 166-178

c 2000 Kluwer Academic Publishers

Robust Monte Carlo Simulation for Approximate Covariance Matrices and VaR Analyses1 Alexander Kreinin ([email protected]) Algorithmics Inc., 185 Spadina Ave., Toronto, Ontario, Canada, M5T 2C6 Alexander Levin ([email protected]) Bank of Montreal, Global Treasury Group, First Canadian Place, Toronto, Ontario, M5X 1A1, Canada

Abstract Value at Risk (VaR) analysis plays very important role in modern Financial Risk Management. The are two very popular approaches to portfolio VaR estimation: approximate analytical approach and Monte Carlo simulation. Both of them face some technical diculties steaming from statistical estimation of covariance matrix decribing the distribution of the risk factors. In this paper we develop a new robust method of generating scenarios in a space of risk factors consistent with a given matrix of correlations containing possible small negative eigenvalues, and nd an estimate for a change in VaR. Namely, we prove that the modi ed VaR of a portfolio VaR0 satis es the inequality g 2 , VaR2 j  K
; where  is the maximum of the absolute values of negjVaR ative eigenvalues of the approximate covariance matrix and K is an explicitly expressed constant, closely related to the market value of the portfolio. Keywords : Value{at{Risk, risk management

1

This paper was prepared when Alexander Levin was with Risk Lab, University of Toronto.

166

1 Introduction Accurate and ecient Value-at-Risk (VaR) analysis is an important part of modern risk management strategy. This strategy requires that nancial institutions implement a risk management engine to compute market risk of their full portfolio. Computation of the VaR of a large portfolio represents a serious numerical problem. There are three major methodologies currently implemented in the risk management systems. The rst methodology is based on an approximate analytical approach using a linear approximation of the portfolio pricing function. The second approach is based on Monte Carlo simulation of the underlying risk factors. These methodologies are based on a log-normal model of risk factors joint behavior that requires estimation of the covariance matrix of the risk factors. The third methodology is historical simulation. Historical simulation draws scenarios from the observable discrete historical changes in the risk factors during a speci ed period of time. This method can be combined with some randomization technique, but direct implementation does not require any model of risk factor changes. Historical simulation is not considered in this paper. The number of underlying risk factors aecting the portfolio value may reach several hundreds or even thousands. In this case the estimation problem becomes ill{posed since the rounding procedure may aect computation of the elements of the covariance matrix and, in particular, the sign of the eigenvalues of the matrix. If an eigenvalue of the covariance matrix is negative, direct application of the Cholesky decomposition required for Monte Carlo simulation is impossible. In this paper we consider two questions related to the Value at Risk analysis and Monte Carlo scenario generation based on Risk Metrics methodology:  The algorithm of modi cation of an arbitrary non{positive de nite approximate covariance matrix;  Bounds on VaR if the covariance matrix is modi ed to be positive semi{de nite. Let the present value of the portfolio at time t = 0 be V (0). The portfolio present value depends on risk factors, that are random variables, and therefore, at time t = T the value will be dierent because of random changes of the risk factors. We say that the value at risk at time t of the portfolio is V, if

Pr fV (0) , V (t) < Vg = ; 0 <  < 1: The parameter  is called con dence probability corresponding to the value at risk V . The VaR analysis is based on the two major issues:  the model of the market risk factor behavior.  the statistical data to de ne the parameters of the model. 167

In [2] the following model was suggested to describe the movements of market parameters. The short term dynamics of the risk factors is driven by the equation Rt = Rt,1
e ; where the random vector of logreturns  has a joint normal distribution. Based on this model, the following approximate formula to compute VaR was suggested [2]: p VaR() = W t
C
W
x ; (1) where W is a vector of so{called Daily Earning at Risk, C is a correlation matrix of the logreturns of risk factors, and x is a quantile of a standard normal distribution. The notation W t is always used for transposed vector. The problem of VaR computation is related both to analytical approximation and to the Monte Carlo simulation. The Monte Carlo scenario generation algorithm, suggested in [2] uses Cholesky decomposition [1]. This transformation is applicable only to the positive de nite covariance matrix C. In theory, if a proper statistical estimation is applied to obtain the elements of the covariance matrix, then C must be symmetric positive de nite. But in practice, because of computational and rounding errors, the eigenvalues of the covariance matrix might become zero or negative. Another important issue is the statistical procedure itself. If the number of observations used for estimation of C is small enough with respect to the size of the matrix, then the result will be singular or almost singular matrix Ce , with the eigenvalues very close to zero. Again, rounding of the estimation of the elements of the matrix C most likely, might lead to the negative eigenvalues. In this case, Cholesky decomposition can not be performed. This problem has been discussed in the literature. In [8] it is suggested to adjust the user{de ned matrix so that it is positive semi{de nite and remains as close as possible to the original matrix. The idea developed in [8] is to nd an lower{triangular matrix A to minimize the weighted quadratic distance min kC , A
AT k; A between the original matrix C and A
AT . It is proved that the Gauss{Newton method converges to the solution under some regularity conditions. Another approach to the problem was suggested by Davenport and Iman in [4]. They perform a spectral decomposition of the correlation matrix and replace negative eigenvalues with a small positive numbers. This method was also used in [3] to handle symmetric matrices with the negative eigenvalues. In this paper we develop a new robust algorithm (the Method of the Minimal Symmetric Pseudoinverse Operator) which is stable with respect to perturbations of the covariance matrix. Our approach is close to that presented in [4], but diers in the way we replace the negative and small positive eigenvalues. 168

This paper is arranged as follows. In section 2 we show that the statistical VaR estimation is a stable problem in the following sense. If the portfolio value function V (
) is computed with an error not exceeding " than the Value{at{Risk will be computed with the error bounded by the same quantity ". In section 3 the Method of the Minimal Symmetric Pseudoinverse Operator is described and the properties of the Pseudoinverse Operator are studied. It is shown that the modi ed matrix obtained after application of the Method of the Minimal Symmetric Pseudoinverse Operator has minimal deviation from the original matrix. The advantage of our approach is that we avoid to solve the minimization problem representing signi cant diculties with respect to convergence properties. Bounds on the Value{at{Risk based on the "Minimal Symmetric Modi cation" are considered in section 4. The bounds improve those obtained in [3]. The nal section contains concluding remarks and some numerical results.

2 Stability of VaR computation In this section we prove the following result about VaR. Suppose that the distribution of the portfolio value is to be estimated by generating n scenarios and by approximating the portfolio distribution with the empirical one. Let us assume, for simlicity, that the value of the portfolio Wi , (i = 1; 2; : : : ; n) is positive on every scenario and that the estimator of the portfolio value admits a uniform error bound

jVi , Wi j  " i = 1; 2; : : : ; n; W i

(2)

where Vi is the estimation of the portfolio value on ith scenario, Wi is the "true" value of the portfolio and " is a small positive number. We introduce the vectors ~ = (W1; : : : ; Wn) and call them the approximating vector and V~ = (V1; : : : ; Vn) and W the vector of "true" values, correspondingly.

Proposition 1 The estimation of Value{at{Risk corresponding to probability  > 0, V (), admits the same error bound as the portfolio value. Namely, if W () is the VaR estimation based on "true" portfolio values Wi , (i = 1; 2; : : : ; n), then

jW () , V ()j  ": W ()

(3)

~ Proof: The VaR estimations are obtained from the vector of observations V~ and W as follows. Consider rst the vector V~ . Let V~  = (V[1]; : : : ; V[n]) be a new vector with the coordinates of V~ but reordered in ascending order: V[1]  V[2]  V[k]  : : :  V[n]: 169

Denote by m() = b
nc. Then

V () = V[m()]

(4)

~ . In this case The same construction can be applied to the vector of "true" values W the VaR correspon ng to the probability  is given by W () = W[m()] : Let us prove that the approximate VaR estimation and the estimation based on the "true" portfolio values satisfy the inequality jW () , V ()j  " (5) W () Inequality (5) is intuitively appealling but still need some justi cation because the permutation (V ) of the vector V and the permutation (W ) of the vector W resulting in monotonic ordering of the coordinates of V and W may be dierent. The permutations (V ) and (W ) can be represented as a product of the transpositions of the adjacent components . Let us show that if two elements of the vector V , say, Vi and Vi+1 are to be transposed but the corresponding elements of the true vector W follow in the right order, then after transposition the coordinates of the new vector satisfy inequality (2). Indeed, let us suppose that

Wi  Wi+1; but

Vi  Vi+1: Then to nd VaR the coordinates i and i + 1 of the vector V will be transposed. We want to prove that inequality (2) implies that jVk , Wk+1j  "; Wk+1 and at the same time jVk+1 , Wk j  ": W We have from (2)

k

Wk
(1 , ")  Vk  Wk
(1 + ")  Wk+1
(1 + "): At the same time,

Wk+1
(1 , ")  Vk+1  Vk  Wk
(1 + ")  Wk+1
(1 + "): 170

Thus, we nd that Vk belongs to the interval [Wk+1
(1 , "); Wk+1
(1 + ")]. The same reasoning is applicable to Vk+1 and leads to the conclusion that Vk+1 2 [Wk
(1 , "); Wk
(1+ ")]. Therefore inequality (2) will be satis ed for the vectors V and W after the transposition was applied. Then we derive that inequality (2) is ful lled after any nite number of transpositions, and, therefore after application of the permutations (V ) and (W ). This proposition shows that the VaR estimation problem is well{posed problem.

3 The method of the minimal symmetric pseudoinverse operator Before setting the numerical method, we will calculate the least estimation of the error of the covariance matrix C when the approximate covariance matrix Ce has small negative eigenvalues. We will measure perturbations of matrices using a spectral norm (see [7]) kAxk ; kAk = max x6=0 kxk

where k
k is the usual Euclidean norm in a vector space. The spectral norm of a symmetric matrix A satis es the relation

j(Ax; x)j ; kAk = max x6=0 (x; x)

(6)

where (
;
) denotes inner product of vectors. Obviously, the spectral norm in this case is equal to the maximum of absolute values of the matrix eigenvalues.

Lemma 1 The error h of an approximation of any covariance matrix C by the symmetric matrix Ce which has at least one negative eigenvalue, satis es the inequality: h = kCe , Ck  ; (7) where  is the maximum of absolute values of the negative eigenvalues of Ce . Proof. Any covariance matrix C is symmetric semi-positive de nite: (Cx; x)  0 for all vectors x. Let the eigenvector x of the matrix Ce correspond to the negative eigenvalue , < 0; j,j = . Then we have from (6):

j((C , Ce )x; x)j h = kC , Ce k = max x6=0 (x; x) e  j((C ,(x C; )xx); x )j = (Cx ; x(x) ,; x(),x ; x)  :     171

In accordance with Lemma 1, if there is no any additional information about the error " of the covariance matrix Ce ,  can be used as the least estimate of ". The proposed Method of the Minimal Symmetric Pseudoinverse Operator is a modi cation of the Method of the Minimal Pseudoinverse Operator [5] for the case of symmetric matrices. For the known approximate covariance matrix Ce and the upper estimate h   of its accuracy let us consider the following "Class of h{equivalent symmetric matrices" h(Ce ) = fB : kB , Ce k  h; B t = B g: (8) The unknown exact covariance matrix C belongs to the class h(Ce ). For each matrix B , B 2 h(Ce ) we denote by B + the (unique) pseudoinverse matrix [6], [7]. Now we nd the stable semi-positive de nite approximation of the covariance matrix C and explicit formula for matrix A in the representation C = A
Ct (9) using the following Theorem 1 There exists a semi-positive de nite matrix Ch 2 h(Ce ) which has the minimal norm of its pseudoinverse matrix over all matrices from the class h(Ce ): Ch 2 h(Ce ) : kCh+k = min kB +k: (10)

e) B2h (C

Proof. It is well known, that there exists the spectral factorization of the symmetric matrix Ce of the form [7, 6]: Ce = O

Ot; where O is an orthogonal matrix (O,1 = Ot), and  is a diagonal matrix with the eigenvalues 1  2  : : :  n of the matrix Ce on the diagonal

0 BB 1 2 =B @ :::

1 CC CA

n (under our assumptions, n = , < 0; jnj =  ). This factorization could be eectively done using Householder algorithm to tridiagonolize the matrix Ce and then applying QR algorithm for diagonalization (for more details see [1, 7]). Then we de ne the matrix Ch as Ch = O
e
Ot; (11) where 0f 1  1 B CC f2 e = BBB CC ; (12) ::: @ A fn 172

( >h f i = !(i); i = 1; : : : ; n; !() =  +0 h ifif jjjj  h

In fact, we replace by zeros the eigenvalues whose absolute values are less or equal to h and add h to the other eigenvalues. It follows from (11) that Ch is symmetric and satis es the relation kCh , Ck = h: (13) Therefore, Ch 2 h (Ce ). The pseudoinverse matrix C+h is given by formula

C+h = O
e +
Ot; e + = diag((f1); : : : ; (fn)); ( ,1 6 0 () =  if  =

0 if  = 0 To nish the proof of Theorem we have to show, that kB +k  kC+h k for all matrices B 2 h(Ce ). It follows from (11) - (13) and Wielandt-Homan inequality for symmetric perturbations [7], that the eigenvalues satisfy the inequality max j (B ) , j (Ce )j  kB , Ce k  h: 0j n j

(14)

The triangle inequality for a spectral norm implies the bound on the error and prove the convergence of the method.

Corollary 1 The upper bound on the error satis es the relation kCh , Ck  kCh , Ce k + kCe , Ck  2h:

(15)

Let us calculate the matrix A from (9) in the form

Ah = D
Ot; Ath = O
D; (16) 0q 1 f  BB 1 q CC f BB CC 2 D=B B@ CA ::: q C f n Note that if the dimension of the problem is large, then many (positive or negative) of eigenvalues of Ce are very close to zero. In the Method of the Least Symmetric Pseudoinverse Operator we assign them zero value. This transformation leads to a signi cant reduction of the dimension of the risk factors space. Moreover, this method gives the optimal compression of the dimension with respect to a given level of accuracy h of the approximate covariance matrix Ce :

Theorem 2 The rank of the matrix Ch (and matrix Ah) is the least over ranks of all matrices B from the class h (Ce ). 173

Proof of the theorem immediately follows from the de nition of the matrix Ch and Wielandt-Homan inequality (14) for the eigenvalues of the matrix Ce which satisfy inequalities j > h. Importance of existense of optimal solution to the problem of regularization of empirical covariance matrix becomes clear if the scenario generation problem involves invertion of the matrix. Such a problem arises in the case of generation of conditional scenarios. Suppose that we would like to generate a conditional distribution of the components of a vector given the values of some of the coordinates. We assume that this vector has a joint normal distribution. It is natural to represent the vector X in the form X = (X (1) ; X (2)); where X (1) and X (2) are the random vectors of an arbitrary nite dimension. Let (i) = EX (i), be the vector of expected values of the coordinates of X (i) and CovX (i) = ii , be the covariance matrix of the coordinates of corresponding vector (i = 1; 2). Denote by   12 = Cov X (1) ; X (2) :   We assume that the vector X = X (1) ; X (2) has joint normal distribution N (; ), where the vector    = (1) ; (2) ; and the matrix !   11 12 =   :

Obviously,

21

22

21 = t12 ; where Dt denotes the transposed matrix D. The problem of interest is to nd the conditional distribution of the vector X (2) given the values of the coordinates of X (1) . The solution to this problem (see [9]) is given by Proposition 2 The conditional distribution of the vector X (2) is a normal distribution   L X (2) jX (1) = x(1) = N (m; B) ; where the vector m and the matrices A and B are de ned by the relations m = (2) + A(x(1) , (1) ); A = 21
11 ,1 ; (17) and B = 22 , 21
11 ,1
12 : (18) Obviously, computation of the parameters of the conditional distribution in (17) and (18) requires nding reciprocal matrix. 174

4 Bounds on the Value{at{Risk The natural question to ask is how the changes in the covariance matrix aect the RiskMetrics VaR. It is not dicult to give such an estimate for a \linear" position. For any \linear" position X we have [2]

VaRX = MVX
X
Xa where MV is the market value,  is the sensitivity and Xa is the maximal adverse movement per $ of the market value with a given probability p (usually 0.95). For p = 0:95, Xa is approximately 1:65  , where  is standard deviation of the position value. We will denote this coecient by q. Thus

VaRX = MVX
X
q


(19)

For a portfolio P comprised of n \linear\ positions X1 ; : : : ; Xn

q VaRP = V
CX
V t ;

where CX is the correlation matrix and

V = (VaRX1 ; : : : ; VaRXn ) However, our estimate is for the covariance matrix and therefore we have to use the following equality which follows from the associative property of matrix multiplication and (19): VaR2P = V
CX
V t = V
C
V t; where V = q
(MVX1
X1 ; : : : ; MVXn
Xn ) Suppose now that the covariance matrix C has been modi ed and let  be the estimate of its perturbation. Then for the modi ed matrix Ch we have

VaR0P 2 = V
Ch
V t Therefore, from (15) and from (6) we obtain

jVaR0P 2 , VaR2P j = jV
(C , Ch)
V tj = j((C , Ch)V ; V )j  2(V ; V ) Finally,

X

jVaR0P 2 , VaR2P j  2
q2
( MVXi
Xi )2 i

Thus, we proved the following 175

(20)

Proposition 3 For a portfolio P with assets X1; : : : ; Xn there is an estimate for a change in the VaR introduced by making the matrix of the correlations semi-positive de nite:

X

jVaR0P 2 , VaR2P j  2
q2
( MVXi
Xi )2 i

Note, that if VaR0P and VaRP are close, then VaR0P + VaRP  2VaR0P and we have that X jVaR0P , VaRP j  
q2
( MVXi
Xi )2=VaR0P : i

Also, if all the sensitivities are equal to one, then the inequality (20) simpli es to

jVaR0P2 , VaR2P j  2
q2
MVP2 Therefore, we developed the stable numerical method of modi cation of the nonpositive de nite covariance matrix and calculated the explicit estimate of the error of VaR.

5 Examples In this section we describe several examples of how the modi cation of the eigenvalues of the covariance matrix of the risk factors aect the changes in correlations and covariances. In the rst example, the risk factor space contains 41 risk factor. The key rates of the interest rate curves IRGBP, IRITL, IRUSD, FXITL and FXGBP comprise the risk factor space. The covariance matrix is read from the RM data les on 19 March 1998. The covariance matrix is not positive{de nite. It has 3 negative eigenvalues. The minimal eigenvalue is ,4:65
10,8, the maximal eigenvalue is 8:0
10,4. The modi cation of the eigenvalues leads to changes in the elements of the correlation matrix. The average change of the correlations is 1:1
10,6; the maximal change in the correlations is 4:10
10,4. In the rst example the number of negative eigenvalues did not exceed 10% of the total number of the eigenvalues. If the number of risk factors grow the number of negative eigenvalues may comprise up to 15 and even 20%. The second example illustrates the modi cation of the covariance matrix when the number of risk factors is 83. The risk factors are the key rates of the interest rate curves DEM, FRF, ITL, USD, JPY, as well as foreign exchange rates between USD and DEM, FRF, ITL and JPY. In this case we have 13 negative eigenvalues, or 15:6%. The average change of correlations is 8:57
10,5; the maximal change of correlations is 3:95
10,5. The covariance matrix is read from the data set posted on 26 September 1996. 176

The method behaved well in all tests with the RiskMetricsTM data. In one of the tests, 50 eigenvalues were modi ed in the 190  190 covariance matrix. However, the maximal error in the matrix of correlations (it is more convenient to measure errors in the matrix of correlations as opposed to the matrix of covariations) was about 0:002 while the average error was 0:0001. The maximal relative error in the variations is 0:18% and the average about 0:03%. Smaller matrices usually produce even less error.

6 Conclusion The statistical procedures applied to estimate the covariance structure of the risk factor space lead to unavoidable errors in computation of the eigenvalues. The regularization technique based on the method of the minimal symmetric pseudo{inverse operator allows to nd optimal solution to the problem. We believe that the method discussed above can be used by practitioners either for Monte Carlo simulation of the market risk scenarios or for quasi{analytical estimation of the Value{at{Risk when the information about covariance matrix is not precise. The regularization algorithm admits small relative error in the Value{at{Risk estimation justi ed by the theoretical upper bound.

Acknowledgements

The authors are very grateful to Leonid Merkoulovitch, Dan Rosen and Michael Zerbs for fruitful discussions and suggestions.

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[7] Parlett B.N. The Symmetric Eigenvalue Problem. Englewood Clis, N.J.: Prentice-Hall, 1980. [8] Lurie P.M., Goldberg M.S. An Approximate Method for Sampling Correlated Random Variables from Partially Speci ed Distribution. Management Science, 1998, vol. 44, 2, 203{218. [9] Mood A., Graybill A. Introduction to the Theory of Statistics., McGraw{Hill Inc., New York, 1963.

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