1. http://pluto.huji.ac.il/~mswiener/zvi.html 972-2-588-3049 FRM Zvi Wiener Following P. Jorion, Financial Risk Manager Handbook Financial Risk Management

2. http://pluto.huji.ac.il/~mswiener/zvi.html 972-2-588-3049 FRM Chapter 17 VaR Methods Following P. Jorion 2001 Financial Risk Manager Handbook

3. Ch. 17, HandbookZvi Wiener slide 3 Risk Factors There are many bonds, stocks and currencies. The idea is to choose a small set of relevant economic factors and to map everything on these factors. Exchange rates Interest rates (for each maturity and indexation) Spreads Stock indices

4. Ch. 17, HandbookZvi Wiener slide 4 How to measure VaR Historical Simulations Variance-Covariance Monte Carlo Analytical Methods Parametric versus non-parametric approaches

5. Ch. 17, HandbookZvi Wiener slide 5 Historical Simulations Fix current portfolio. Pretend that market changes are similar to those observed in the past. Calculate P&L (profit-loss). Find the lowest quantile.

6. Ch. 17, HandbookZvi Wiener slide 6 Example 4.00 4.20 4.10 4.15 Assume we have $1 and our main currency is SHEKEL. Today $1=4.30. Historical data: 4.30*4.20/4.00 = 4.515 4.30*4.20/4.20 = 4.30 4.30*4.10/4.20 = 4.198 4.30*4.15/4.10 = 4.352 P&L 0.215 0 -0.112 0.052

7. Ch. 17, HandbookZvi Wiener slide 7 today USD NIS 2000 100 -120 2001 200 100 2002-300 -20 2003 20 30

8. Ch. 17, HandbookZvi Wiener slide 8 today Changes in IR USD: +1%+1% +1% +1% NIS: +1% 0% -1% -1%

9. Ch. 17, HandbookZvi Wiener slide 9 Returns year 1% of worst cases

10. Ch. 17, HandbookZvi Wiener slide 10 Profit/Loss VaR 1% VaR 1%

11. Ch. 17, HandbookZvi Wiener slide 11 Variance Covariance Means and covariances of market factors Mean and standard deviation of the portfolio Delta or Delta-Gamma approximation VaR 1% =  P – 2.33  P Based on the normality assumption!

12. Ch. 17, HandbookZvi Wiener slide 12  Variance-Covariance 2.33   -2.33  1%

13. Ch. 17, HandbookZvi Wiener slide 13 Monte Carlo

14. Ch. 17, HandbookZvi Wiener slide 14 Monte Carlo Distribution of market factors Simulation of a large number of events P&L for each scenario Order the results VaR = lowest quantile

15. Ch. 17, HandbookZvi Wiener slide 15 Monte Carlo Simulation

16. Ch. 17, HandbookZvi Wiener slide 16 Weights Since old observations can be less relevant, there is a technique that assigns decreasing weights to older observations. Typically the decrease is exponential. See RiskMetrics Technical Document for details.

17. Ch. 17, HandbookZvi Wiener slide 17 Stock Portfolio Single risk factor or multiple factors Degree of diversification Tracking error Rare events

18. Ch. 17, HandbookZvi Wiener slide 18 Bond Portfolio Duration Convexity Partial duration Key rate duration OAS, OAD Principal component analysis

19. Ch. 17, HandbookZvi Wiener slide 19 Options and other derivatives Greeks Full valuation Credit and legal aspects Collateral as a cushion Hedging strategies Liquidity aspects

20. Ch. 17, HandbookZvi Wiener slide 20 Credit Portfolio rating, scoring credit derivatives reinsurance probability of default recovery ratio

21. Ch. 17, HandbookZvi Wiener slide 21 Reporting Division of VaR by business units, areas of activity, counterparty, currency. Performance measurement - RAROC (Risk Adjusted Return On Capital).

22. Ch. 17, HandbookZvi Wiener slide 22 Backtesting Verification of Risk Management models. Comparison if the model’s forecast VaR with the actual outcome - P&L. Exception occurs when actual loss exceeds VaR. After exception - explanation and action.

23. Ch. 17, HandbookZvi Wiener slide 23 Backtesting Green zone - up to 4 exceptions Yellow zone - 5-9 exceptions Red zone - 10 exceptions or more OK increasing k intervention

24. Ch. 17, HandbookZvi Wiener slide 24 Stress Designed to estimate potential losses in abnormal markets. Extreme events Fat tails Central questions: How much we can lose in a certain scenario? What event could cause a big loss?

25. Ch. 17, HandbookZvi Wiener slide 25 Local Valuation Simple approach based on linear approximation. Full Valuation Requires repricing of assets.

26. Ch. 17, HandbookZvi Wiener slide 26 Delta-Gamma Method The valuation is still local (the bond is priced only at current rates).

27. Ch. 17, HandbookZvi Wiener slide 27 FRM-97, Question 13 An institution has a fixed income desk and an exotic options desk. Four risk reports were produced, each with a different methodology. With all four methodologies readily available, which of the following would you use to allocate capital? A. Simulation applied to both desks. B. Delta-Normal applied to both desks. C. Delta-Gamma for the exotic options desk and the delta-normal for the fixed income desk. D. Delta-Gamma applied to both desks.

28. Ch. 17, HandbookZvi Wiener slide 28 An institution has a fixed income desk and an exotic options desk. Four risk reports were produced, each with a different methodology. With all four methodologies readily available, which of the following would you use to allocate capital? A. Simulation applied to both desks. B. Delta-Normal applied to both desks. C. Delta-Gamma for the exotic options desk and the delta-normal for the fixed income desk. D. Delta-Gamma applied to both desks. FRM-97, Question 13 Bad question!

29. Ch. 17, HandbookZvi Wiener slide 29 Mapping Replacing the instruments in the portfolio by positions in a limited number of risk factors. Then these positions are aggregated in a portfolio.

30. Ch. 17, HandbookZvi Wiener slide 30 Delta-Normal method Assumes linear exposures risk factors are jointly normally distributed The portfolio variance is Forecast of the covariance matrix for the horizon

31. Ch. 17, HandbookZvi Wiener slide 31 Delta-normalHistor.MC Valuationlinearfullfull Distributionnormalactualgeneral Extreme eventslow prob.recent possible Ease of comput.Yesintermed.No CommunicabilityEasyEasyDifficult VaR precisionBaddependsgood Major pitallsnonlinearityunstablemodel fat tails risk

32. Ch. 17, HandbookZvi Wiener slide 32 FRM-97, Question 12 Delta-Normal, Historical-Simulations, and MC are various methods available to compute VaR. If underlying returns are normally distributed, then the: A. DN VaR will be identical to HS VaR. B. DN VaR will be identical to MC VaR. C. MC VaR will approach DN VaR as the number of simulations increases. D. MC VaR will be identical to HS VaR.

33. Ch. 17, HandbookZvi Wiener slide 33 FRM-97, Question 12 Delta-Normal, Historical-Simulations, and MC are various methods available to compute VaR. If underlying returns are normally distributed, then the: A. DN VaR will be identical to HS VaR. B. DN VaR will be identical to MC VaR. C. MC VaR will approach DN VaR as the number of simulations increases. D. MC VaR will be identical to HS VaR.

34. Ch. 17, HandbookZvi Wiener slide 34 FRM-98, Question 6 Which VaR methodology is least effective for measuring options risks? A. Variance-covariance approach. B. Delta-Gamma. C. Historical Simulations. D. Monte Carlo.

35. Ch. 17, HandbookZvi Wiener slide 35 FRM-98, Question 6 Which VaR methodology is least effective for measuring options risks? A. Variance-covariance approach. B. Delta-Gamma. C. Historical Simulations. D. Monte Carlo.

36. Ch. 17, HandbookZvi Wiener slide 36 FRM-99, Questions 15, 90 The VaR of one asset is 300 and the VaR of another one is 500. If the correlation between changes in asset prices is 1/15, what is the combined VaR? A. 525 B. 775 C. 600 D. 700

37. Ch. 17, HandbookZvi Wiener slide 37 FRM-99, Questions 15, 90

38. Ch. 17, HandbookZvi Wiener slide 38 Example On Dec 31, 1998 we have a forward contract to buy 10M GBP in exchange for delivering $16.5M in 3 months. S t - current spot price of GBP in USD F t - current forward price K - purchase price set in contract f t - current value of the contract r t - USD risk-free rate, r t * - GBP risk-free rate  - time to maturity

39. Ch. 17, HandbookZvi Wiener slide 39

40. Ch. 17, HandbookZvi Wiener slide 40 The forward contract is equivalent to a long position of SP* on the spot rate a long position of SP* in the foreign bill a short position of KP in the domestic bill

41. Ch. 17, HandbookZvi Wiener slide 41 On the valuation date we have S = 1.6595, r = 4.9375%, r* = 5.9688% V t = $93,581 - the current value of the contract