Value At Risk

1. Value At Risk IEF 217a: Lecture Section 5 Fall 2002 Jorion Chapter 5

2. Outline • Computing VaR • Interpreting VaR • Time Scaling • Regulation and VaR • Jorion 3, 5.2.5-5.2.6 • Estimation errors

3. VaR Roadmap • Introduction • Methods • Reading: Linsmeier and Pearson • Easy example • Harder example: • Linsmeir and Pearson • Monte-carlo methods and even harder examples • Jorion

4. Value at Risk (VaR)History • Financial firms in the late 80’s used it for their trading portfolios • J. P. Morgan RiskMetrics, 1994 • Currently becoming: • Wide spread risk summary • Regulatory

5. Why VaR? • Risk summary number • Relatively simple • Relatively standardized • Give high level management risk in 1 number

6. What is VaR? • Would like to know maximum amount you stand to lose in portfolio • However, the max might too large • 5% VaR is the amount that you would lose such that 5% of outcomes will lose more

7. 5% VaR = 84: 100 Start ValueNormal Distribution (std = 10)

8. Value at Risk: Methods • Methods (Reading: Linsmeier and Pearson) • Historical • Delta Normal • Monte-carlo • Resampling

9. Historical • Use past data to build histograms • Method: • Gather historical prices/returns • Use this data to predict possible moves in the portfolio over desired horizon of interest

10. Easy Example • Portfolio: • $100 in the Dow Industrials • Perfect index tracking • Problem • What is the 5% and 1% VaR for 1 day in the future?

11. DataDow Industrials • dow.dat (data section on the web site) • File: • Column 1: Matlab date (days past 0/0/0) • Column 2: Dow Level • Column 3: NYSE Trading Volume (1000’s of shares)

12. Matlab and Data FilesKaplan: Appendix C • All data in matrix format • “Mostly” numerical • Two formats • Matlab format filename.mat • ASCII formats • Space separated • Excel (csv, common separated)

13. Loading and Saving • Load data • “load dow.dat” • Data is in matrix dow • Save data • ASCII • save -ascii filename dow • Matlab • save filename dow

14. Example: Load and plot dow data • Matlab: pltdow.m • Dates: • Matlab datestr function

15. Back to our problem • Find 1 day returns, and apply to our 100 portfolio • Matlab: histdvar.m

16. Value at Risk: Methods • Methods (Reading: Linsmeier and Pearson) • Historical • Delta Normal • Monte-carlo • Resampling

17. Delta Normal • Make key assumptions to get analytics • Normality • Linearization • Dow example: • Assume returns normal mean = m, std = s • 5% return = -1.64*s + m • 1% return = -2.32*s + m • Use these returns to find VaR • matlab: dnormdvar.m

18. Compare With Historical • Fatter tails • Plot Comparison: twodowh.m

19. Longer Horizon: 10 Days • Matlab: hist10d.m

20. Value at Risk: Methods • Methods (Reading: Linsmeier and Pearson) • Historical • Delta Normal • Monte-carlo • Resampling

21. Monte-Carlo VaR • Make assumptions about distributions • Simulate random variables • matlab: mcdow.m • Results similar to delta normal • Why? • More complicated portfolios and risk measures • Confidence intervals: mcdow2.m

22. Value at Risk: Methods • Methods (Reading: Linsmeier and Pearson) • Historical • Delta Normal • Monte-carlo • Resampling

23. Resampling (bootstrapping) • Historical/Monte-carlo hybrid • Also known as bootstrapping • We’ve done this already • data = [5 3 -6 9 0 4 6 ]; • sample(n,data); • Example • rsdow.m

24. VaR Roadmap • Introduction • Methods • Reading: Linsmeier and Pearson • Easy example • Harder example: • Linsmeir and Pearson • Monte-carlo methods and even harder examples • Jorion

25. Harder Example • Foreign currency forward contract • 91 day forward • 91 days in the future • Firm receives 10 million BP (British Pounds) • Delivers 15 million US $

26. Mark to Market Value(values in millions)

27. Risk Factors • Exchange rate ($/BP) • r(BP): British interest rate • r($): US interest rate • Assume: • ($/BP) = 1.5355 • r(BP) = 6% per year • r($) = 5.5% per year • Effective interest rate = (days to maturity/360)r

28. Find the 5%, 1 Day VaR • Very easy solution • Assume the interest rates are constant • Analyze VaR from changes in the exchange rate price on the portfolio

29. Mark to Market Value(current value in millions $)

30. Mark to Market Value(1 day future value) X = % daily change in exchange rate

31. X = ? • Historical • Normal • Montecarlo • Resampled

32. Historical • Data: bpday.dat • Columns • 1: Matlab date • 2: $/BP • 3: British interest rate (%/year) • 4: U.S. Interest rate (%/year)

33. BP Forward: Historical • Same as for Dow, but trickier valuation • Matlab: histbpvar1.m

34. BP Forward: Monte-Carlo • Matlab: mcbpvar1.m

35. BP Forward: Resampling • Matlab: rsbpvar1.m

36. Harder Problem • 3 Risk factors • Exchange rate • British interest rate • U.S. interest rate

37. 3 Risk Factors1 day ahead value

38. Daily VaR AssessmentHistorical • Historical VaR • Get percentage changes for • $/BP: x • r(BP): y • r($): z • Generate histograms • matlab: histbpvar2.m

39. Daily VaR AssessmentResample • Historical VaR • Get percentage changes for • $/BP: x • r(BP): y • r($): z • Resample from these • matlab: rsbpvar2.m

40. Resampling Question: • Assume independence? • Resampling technique differs • matlab: rsbpvar2.m

41. Risk Factors and Multivariate Problems • Value = f(x, y, z) • Assume random process for x, y, and z • Value(t+1) = f(x(t+1), y(t+1), z(t+1))

42. New Challenges • How do x, y, and z impact f()? • How do x, y, and z move together? • Covariance?

43. Delta Normal Issues • Life is more difficult for the pure table based delta normal method • It is now involves • Assume normal changes in x, y, z • Find linear approximations to f() • This involves partial derivatives which are often labeled with the Greek letter “delta” • This is where “delta normal” comes from • We will not cover this

44. Monte-carlo Method • Don’t need approximations for f() • Still need to know properties of x, y, z • Assume joint normal • Need covariance matrix • ie var(x), var(y), var(z) and • cov(x,y), cov(x,z), cov(y,z) • Next section, and Jorion