{ Value-at-Risk : any value for fund managers? }

1. { Value-at-Risk : any value for fund managers? } Dr. Jaco Maritz

2. Introduction • Suppose you are the managing director of a financial institution • One bright morning, you receive the following fax:

3. Introduction My sincere apologies for the predicament I have left you in. It was neither my intention nor aim for this to happen, but the pressure, both business and personal have become too much to bear and after receiving medical attention …..…. have affected my health to the extend that a breakdown is eminent. In light of this, I tender my resignation with immediate effect, and will contact you early next week to discuss the best course of action. Apologies

4. Introduction “The best course of action…” Nick Leeson – Barings Bank Loss : $ 1.3 billion Charge : fraud Sentence 6½ years in prison

5. Risk Management • During the 1990’s the investment community was shocked by a number of high profile financial disasters: • Metallgesellschaft (oil forward position that went sour) • The collapse of Barings Bank • Orange County (local government treasury) • Locally, we had the recent example of the Joint Municipal Pension Fund, where fund managers were heavily involved in agricultural futures speculation, and lost R 1.3 billion • Since these disasters occurred financial risk management has evolved into a sophisticated discipline • Internationally, banks have adopted risk management standards to strengthen the integrity of the financial system

6. Contents • Basic concepts of risk • Value-at-Risk • Calculation of Value-at-Risk • Parametric Value-at-Risk • Historical Value-at-Risk • Monte Carlo Simulation • Applications to fund management • Equity portfolio application • Domestic pension fund application (equities, bonds and cash) • Derivative structures • Global portfolios (currency risk) • Hedge funds • Relative VaR • VaR as predictor of risk • Extensions and refinements • Conclusions

7. Basic concepts of risk Traditional concepts in fund management • Risk as probability of underperformance • Fund volatility : Standard deviation of absolute returns – indication of fund members’ risk • Tracking error standard deviation : Std dev of returns relative to benchmark • Beta and duration are frequently used in equity and bond portfolio to quantify risk

8. Basic concepts of risk Tracking Error Standard deviation Volatility or Total Risk 2σT 2σ Investor's Risk Tracking Error Benchmark Risk Demonstration of Selection, Market Exposure and Timing 40 Fund 30 20 10 0 -40 -30 -20 -10 0 10 20 30 40 Relative Value-at-Risk -10 Value-at-Risk -20 -30 -40 Benchmark

9. Value-at-Risk Weakness of volatility measure • Indication of market fluctuations during normal market conditions • Based on historical performance – no consideration of what the market might do in future • A change in manager style can not be identified until enough data is collected • Risky holdings can not be identified immediately when they are entered into Definition of Value-at-Risk (VaR): • an estimate of the level of loss on a portfolio which is expected to be equaled or exceeded by a given, small probability We might say: • “We are x percent certain that we will not loose more than V money in the next N days” • Choice for x : 95% (1 out of 20), 99% (1 out of 100), 99.5% (1 out of 200), 99.9% (1 out of 1000) • Choice for N : 1 day, 10 days, 1 month (depends on expected period to unwind position)

10. Volatility or Total Risk 2 std dev 99% of returns are above this level Value-at-Risk Value-at-Risk

11. Value-at-Risk Properties of VaR • Universal measure, applicable to portfolios containing equities, bonds, cash, commodities, derivatives • VaR is based in current holdings • VaR analysis works by decomposing constituent risk exposures and aggregating the risks from each exposure across the fund Strengths of VaR • An attempt to put a single figure to the potential loss across different classes of securities • No need to assume that manager style stays consistent over time • VaR is the best measure available to estimate market risk in a forward-looking manner • VaR reacts fast to changes in market risk/volatilities in the market

12. Value-at-Risk Uses of VaR • Originally conceived in banking (treasury) environment • Calculated near end of trading of each day to monitor any risk limit breaches • Used by bank regulators to set Capital adequacy requirements • On trading desks VaR is frequently used to investigate impact of a deal on the book • Management of institution use VaR to set risk limits for each portfolio/trading desk

13. Calculation of Value-at-Risk Parametric Value-at-Risk • Follows Mean-Variance approach: • Standard deviation of portfolio return: • where σP is the standard deviation of portfolio returns or volatility, • w is the vector containing the weights of the different securities in the portfolio, • Ω is the covariance matrix • 99% percentile VaR = 2.326 σP (explain this better) Comments: • Assumption : Returns follows a normal distribution, thus this method is not suitable for derivatives • Straight forward calculation, follows same formulation as Markowitz theory • Positions might be designed to exploit their correlations. One might have to check whether small changes in correlation results in large VaR changes

14. Parametric VaR – Case study: Barings Bank Value-at-Risk = ασP = $835.16 million

15. Parametric VaR – Case study: Barings Bank • 95% VaR on Leeson’s book : $ 835.16 million • Shareholder capital of Barings bank : $ 695 million • Ultimate loss incurred : $ 1.3 billion • Barings sold to ING for: ₤ 1 Menu

16. Calculation of Value-at-Risk Historical simulation • Does not depend on any explicit distribution of security returns • Simulate possible returns on current portfolio using historical returns of constituent assets • Assume the future would be similar to the past Comments: • Powerful and simple to implement on a spreadsheet • One can smooth the distribution of return provided the actual distribution does not differ too much from normal • Method is robust and intuitive Menu

17. Weights vector (w) Historical simulation – three stock example • As an illustration of the historical simulation approach, consider a three stock portfolio, consisting of: • The first step is to generate historical returns for each assets Menu

18. NED AGL SOL Historical simulation – three stock example Menu

19. NED AGL SOL Historical simulation – three stock example Select the first date Determine one day portfolio return Select the next date -1.41% 2.80% Note: the portfolio returns are not cumulative! Menu

20. Historical simulation – three stock examplePortfolio returns Menu

21. Historical simulation – three stock examplePortfolio returns Menu

22. Historical simulation – three stock examplePortfolio returns Menu

23. Calculation of Value-at-Risk – Historical Simulation Historical returns/rates Full valuation Portfolio weights VaR Distribution of values Source: P. Jorion – Value at Risk

24. Normality of Returns • We investigated the normality of stock returns on the JSE • We found the following • None of the 165 largest shares exhibit normality on a daily basis • None of the 165 largest shares exhibit normality on a weekly basis • 62 shares exhibit normality on a monthly basis • 60 shares exhibit some evidence of normality on a monthly basis • 43 shares exhibit

25. Normality of returns “Fat” tails

26. Normality of returnsQQ – plot

27. Normality of returns

28. Normality of returns - daily

29. Normality of returns

30. Normality of returns

31. Normality of returns

32. Normality of returns

33. Normality of returns

34. Normality of returns

35. Normality of returns

36. Normality of returns

37. Some motivation for using historical VaR

38. Calculation of Value-at-Risk Monte Carlo simulation • Assume a joint distribution for the set of all risk factors • Generate a series of prices or rates for each risk factor • Perform full valuation for each security to determine portfolio valuation • From the portfolio values, determine VaR figures. Comments: • Computationally intensive – needs thousands of repetitions • Subject to specific stochastic model for the underlying risk factors • Also subject to pricing models • Potential onerous to develop from scratch Menu

39. Calculation of Value-at-Risk – Monte Carlo Simulation Historical/implied returns Stochastic model Model parameters Future rates/prices Securities model Full valuation Portfolio weights VaR Distribution of values

40. Applications of Value-at-Risk in the Fund Management environment

41. Applications of Value-at-Risk in the Fund Management environment • Equity portfolio application • Domestic pension fund application (equities, bonds and cash) • Derivative structures • Global portfolios (currency risk) • Hedge Funds Menu

42. Historical Value-at-Risk – Worked example Equity only test portfolio Menu

43. Historical Value-at-Risk – Worked example • Determine historical returns for each asset in the portfolio • Complete “bootstrap” procedure: • Select at random a date i from our set of historical returns • Determine a return vector for that date (ri) • Determine the portfolio return for the date i using the selected returns vector and the portfolio weights • Repeat 10000 times steps 1 and 2, each time keeping the portfolio return • From the set of portfolio returns, determine the following percentiles for 99% VaR: 1st percentile for 95% VaR 5th percentile Menu

44. Normal distribution having the same mean and standard deviation as the data Historical Value-at-Risk – Worked example “Fat tail” Menu

45. Historical Value-at-Risk – Worked example Results Menu

46. Parametric Value-at-Risk – Worked example • Determine covariance matrix (Ω) from the returns data • Determine the portfolio expected variance: • We assume the returns of the portfolio would follow a normal distribution with mean zero (μ = 0) and standard deviation σ. • The 99% confidence VaR is determined as • Where Φ-1 is the inverse standard normal distribution. Menu

47. Parametric Value-at-Risk – Worked example Results Menu

48. Normality of Portfolio returns? 99% Historical Simulation VaR 99% Parametric VaR Portfolio volatility Menu

49. Fixed Income Portfolios • Step 1 – Identifying exposures and cash flows • Step 2 – Mapping cash flows onto vertices • Step 3 – Computing Value-at-Risk Menu

50. Principal (1/3 each) Coupons Fixed Income Portfolios • Step 1 – Identifying exposures and cash flows • Fixed income securities can easily be represented as cash flows given their standard future stream of payments • The cash flow map shows the distribution over time of the current market value of all future streams (coupons + principals) Cash flows of the R153 3-tranch bond Menu