1. KYIV SCHOOL OF ECONOMICS Financial Econometrics (2nd part): Introduction to Financial Time Series May 2011 Instructor: Maksym Obrizan Lecture notes III # 2. In this lecture: Review the basic ideas behind Value at Risk (VaR) calculations based on various time series models: 1. RiskMetrics 2. Econometric models 3. Quantile models 4. Extreme value theory Various types of risk in financial time series: Credit risk, liquidity risk and market risk # 3. Value at Risk (VaR) is primarily concerned with market risk Value at Risk (VaR) – Basic idea: # 4. One way to think about VaR is as of a maximal loss associated with a rare (or extraordinary) event Definitions of long and short financial positions: A long financial position is – A short financial position is –

2. # 5. VaR under a probabilistic framework The VaR of a long position over the time horizon h with probability p is # 6. VaR defined on slide #5 typically assumes a negative value when p is small VaR is concerned with tail behavior of the CDF F h (x) # 7. The definition on slide #5 continues to apply to a short position if one uses the distribution of -∆V t (h) # 8. NOTES

3. # 9. For a known univariate CDF F h (x) and probability p one can simply use the pth quantile # 10. Calculation of VaR # 11. RiskMetrics TM Developed by J.P. Morgan RiskMetrics assumes that # 12. In addition, RiskMetrics is built on an IGARCH(1,1) process without a drift It can be shown that the conditional distribution of r t [k] is

4. # 13. Thus, under this special IGARCH(1,1) model the conditional variance of r t [k] is proportional to the time horizon k For the continuously compounded (i.e. log) returns # 14. … and for a k-day horizon is Thus, under RiskMetrics we have VaR(k) = √k VaR This rule is referred to the square root of time rule in VaR calculation # 15. Example# 16. Cont’d

5. # 17. The main advantage of RiskMetrics – it’s simplicity In addition, many stocks have non-zero means of a return. For example, # 18. In this case, the distribution of k-period return is The 5% quantile used in k-period horizon VaR calculation is then # 19. VaR with multiple positions Define ρ ij - the cross-correlation coefficient between the two returns (i and j) Then VaR can be generalized to m positions as # 20. NOTES

6. # 21. VaR based on a general time series model Consider the log return of r t of a financial asset # 22. The error term ε t is often assumed to be normal or a standardized Student-t distribution For a normal distribution obtain the 5% quantile of a distribution for VaR calculations as # 23. For a standardized Student-t distribution the quantile is Observe that if q is the pth quantile of a Student-t distribution with v degrees of freedom then Is the pth quantile of a standardized Student-t distribution with v degrees of freedom # 24. Thus, the 1-period horizon VaR at time t is

7. # 25. Example based on a standard normal ε t # 26. Cont’d # 27. Example based on a standardized Student-t ε t # 28. Cont’d

8. # 29. Quantile estimation – This method makes no specific distributional assumption Use: -Empirical quantile directly -Quantile regression # 30. Quantile and order statistics For example, r (1) and r (n) are the sample min and the sample max # 31. Based on the asymptotic result one can use r (h) to estimate the quantile x p where h = np # 32. Then the quantile x p can be estimated by

9. # 33. Check yourself Daily log returns of Intel stock with 6,329 observations VaR of a long position of $10 mln? # 34. NOTES # 35. Pros and Cons of Empirical Quantile: “+” “-” Assumes that the distribution of return r t does not change (i.e. loss cannot be greater than the historical loss – not true!) CONCLUSION: # 36. Quantile regression In practical applications, some explanatory variables may be used to facilitate model building

10. # 37. Quantile regression: choose β to minimize# 38. Familiar estimator: Least Absolute Deviations (LAD) Minimizes the sum of absolute deviations (OLS: sum of squared deviations) Basic idea of quantile regression: Quantile regression estimator is available in Stata # 39. NOTES# 40. NOTES