Generating Daily Changes in Market Variables using Multivariate Mixture-of-Normals

1. Generating Daily Changes in Market Variables using Multivariate Mixture-of-Normals Jin Wang Valdosta State University Michael R. Taaffe University of Minnesota December 10, 2001 Winter Simulation Conference 2001

2. Outline • Introduction • Fat Tails and Skewness • Generating Mixtures of Normal Variates • Portfolio Value-at-Risk • Fitting Mixture of Normal Distributions

3. Summary We describe an efficient analytical Monte Carlo method for generating daily changes using a multivariate mixture of normal distributions with arbitrary covariance matrix. The main purpose of this method is to transform (linearly) a multivariate normal with an input covariance matrix into the desired multivariate mixture of normal distributions. The input covariance matrix is derive analytically. Any linear combination of mixtures of normal distributions can be shown to be a mixture of normal distributions.

4. Fat Tails and Skewness • The normal distribution, the most commonly used model of daily changes in market variables, has skewness = 0, and kurtosis = 3; however, some market-change variables, such as exchange rates, exhibit significant positive kurtosis and negative skewness. • Mixture of normals is a more general and flexible family of distributions for fitting market daily changes. This family of distributions can accurately model departures from kurtosis = 3 and skewness = 0.

5. Kurtosis and Skewness • Skewness, , is a measure of asymmetry • Kurtosis, , is a measure of how fat the tails of a distribution

6. Mixture of Normals The cdf of a mixture of k normal random variables X is: where is the N(0,1) cdf, and the pdf is:

7. Example Mixture of 2 normals Normal

8. Generating Mixtures of Normal Variates Univariate Case Proposition 1: If and and U are independent then, is a mixture of k normals, where is the indicator function.

9. Generating Mixtures of Normal Variates ALGORITHM 1: • Generate Y from the N(0,1). • Generate U from the U(0,1). • Return

10. Generating Mixtures of Normals Multivariate Case: Assume is a random vector, where the marginals are each univariate mixtures of normals having pdf and having covariance matrix

11. Generating Mixtures of Normals Let where are iid U(0,1) and , where the are N(0,1) and having covariance matrix And let

12. Generating Mixtures of Normals Proposition 2: If are independent, then is a multivariate mixture of normals having mean:

13. Generating Mixtures of Normals Proposition 2: If are independent, then is a multivariate mixture of normals having

14. Generating Mixtures of Normals Proposition 3: Under all assumptions of proposition 2, can be derived in terms of where

15. Generating Mixtures of Normals Proposition 4: (Cholesky Decomposition) For any covariance matrix , if where the are iid N(0,1), there exists a unique lower triangular matrix , such that and is MVN having covariance and can be recursively computed via

16. Generating Mixtures of Normals ALGORITHM 2 • Calculate , where • Calculate where

17. ALGORITHM 2– cont’d 3. Generate Z 4. Generate U 5.Calculate Y, where 6. Return X, where

18. Portfolio VaR Suppose that X is a random vector of portfolio returns. For any set of weights , where is the weight on asset , the total return on portfolio is

19. Portfolio VaR: Confidence Intervals Precisely, , the VaR at the 100(1- )% confidence level for portfolio for a specified period of time is the solution of:

20. Portfolio VaR Proposition 5:Assume X is a random vector of market daily returns and ‘s are marginally mixtures of normals. For any portfolio, , the return is a univariate mixture of normals whose cdf is: Where

21. Portfolio VaR Proposition 6:Under all assumptions of Proposition 5, the VaR of a portfolio is the solution to:

22. Fitting Mixtures of Normals • Fitting the parameters of the mixture of normal distributions is one of the oldest estimation problems in the statistical literature • Method of Moments • Method of Maximum Likelihood and the EM Algorithm • Details in the paper [2001]