Analytics of Risk Management III: Motivating Risk Measures

1. Risk Management Lecturer: Mr. Frank Lee Session 5 Analytics of Risk Management III: Motivating Risk Measures

2. Overview • Risk Measurement Applications, Scenario building and Simulations • JPM RiskMetrics • Historic or Back Simulations • Monte Carlo Simulations • Hedging and risk limits

3. Risk Management Application • Uncertainty or changes in value resulting from changes in the underlying parameter. Can be measured over periods as short as one day. • Usually measured in terms of ‘dollar’ exposure amount or as a relative amount against some benchmark • Find value at risk, e.g. market risk, interest rate risk, foreign exchange risk etc.

4. Application: Market Risk Measurement • Important in terms of: • Management information • Setting limits • Resource allocation • Performance evaluation • Regulation

5. Calculating Market Risk Exposure • Generally concerned with estimating potential loss under adverse circumstances. • Three major approaches of measurement • JP Morgan RiskMetrics (variance/covariance approach) • Historic or Back Simulation • Monte Carlo Simulation

6. JP Morgan RiskMetrics Model • Idea is to determine the daily earnings at risk = dollar value of position × price sensitivity × potential adverse move in yield or, DEAR = Dollar market value of position × Price volatility. • Can be stated as (-MD) × adverse daily yield move where, MD = D/(1+R) Modified duration = MacAulay duration/(1+R)

7. Confidence Intervals • If we assume that changes in the yield are normally distributed, we can construct confidence intervals around the projected DEAR. (Other distributions can be accommodated but normal is generally sufficient). • Assuming normality, 90% of the time the disturbance will be within 1.65 standard deviations of the mean. • Also, 98% of the time the disturbance will be within 2.33 standard deviations of the mean

8. Confidence Intervals: Example • Suppose that we are long in 7-year zero-coupon bonds and we define “bad” yield changes such that there is only 5% chance of the yield change being exceeded in either direction. Assuming normality, 90% of the time yield changes will be within 1.65 standard deviations of the mean. If the standard deviation is 10 basis points, this corresponds to 16.5 basis points. Concern is that yields will rise. Probability of yield increases greater than 16.5 basis points is 5%. *(suppose YTM=7.25%)

9. Confidence Intervals: Example • Price volatility = (-MD)  (Potential adverse change in yield) = (-6.527)  (0.00165) = -1.077% DEAR = Market value of position  (Price volatility) = ($1,000,000)  (.01077) = $10,770

10. Confidence Intervals: Example • To calculate the potential loss for more than one day: Market value at risk (VAR) = DEAR × N • Example: For a five-day period, VAR = $10,770 × 5 = $24,082

11. Foreign Exchange & Equities • In the case of Foreign Exchange, DEAR is computed in the same fashion we employed for interest rate risk. • For equities, if the portfolio is well diversified then DEAR = dollar value of position × stock market return volatility where the market return volatility is taken as 1.65 sM.

12. Aggregating DEAR Estimates • Cannot simply sum up individual DEARs. • In order to aggregate the DEARs from individual exposures we require the correlation matrix. • Three-asset case: DEAR portfolio = [DEARa2 + DEARb2 + DEARc2 + 2rab × DEARa × DEARb + 2rac × DEARa × DEARc + 2rbc × DEARb × DEARc]1/2

13. Historic or Back Simulation • Advantages: • Simplicity • Does not require normal distribution of returns (which is a critical assumption for RiskMetrics) • Does not need correlations or standard deviations of individual asset returns.

14. Historic or Back Simulation • Basic idea: Revalue portfolio based on actual prices (returns) on the assets that existed yesterday, the day before, etc. (usually previous 500 days). • Then calculate 5% worst-case (25th lowest value of 500 days) outcomes. • Only 5% of the outcomes were lower.

15. Estimation of VAR: Example • Convert today’s FX positions into dollar equivalents at today’s FX rates. • Measure sensitivity of each position • Calculate its delta. • Measure risk • Actual percentage changes in FX rates for each of past 500 days. • Rank days by risk from worst to best.

16. Weaknesses • Disadvantage: 500 observations is not very many from statistical standpoint. • Increasing number of observations by going back further in time is not desirable. • Could weight recent observations more heavily and go further back.

17. Monte Carlo Simulation • To overcome problem of limited number of observations, synthesize additional observations. • Perhaps 10,000 real and synthetic observations. • Employ historic covariance matrix and random number generator to synthesize observations. • Objective is to replicate the distribution of observed outcomes with synthetic data.

18. Monte Carlo Simulation Modeling Process • Step 1: Modeling the Project • Step 2: Specifying Probabilities • Step 3: Simulate the Results (e.g. cash flows, values etc.) • Monte Carlo simulation is conceptually simple, but is generally computationally more intensive than other methods.

19. Monte Carlo Simulation • The generic MC VaR calculation goes as follows: • Decide on N, the number of iterations to perform. • For each iteration: • Generate a random scenario of market moves using some market model. • Revalue the portfolio under the simulated market scenario. • Compute the portfolio profit or loss (PnL) under the simulated scenario. (i.e. subtract the current market value of the portfolio from the market value of the portfolio computed in the previous step). • Sort the resulting PnLs to give us the simulated PnL distribution for the portfolio. • VaR at a particular confidence level is calculated using the percentile function. For example, if we computed 5000 simulations, our estimate of the 95% percentile would correspond to the 250th largest loss, i.e. (1 - 0.95) * 5000. • Note that we can compute an error term associated with our estimate of VaR and this error will decrease as the number of iterations increases.

20. Monte Carlo Simulation • Monte Carlo simulation is generally used to compute VaR for portfolios containing securities with non-linear returns (e.g. options) since the computational effort required is non-trivial. • For portfolios without these complicated securities, such as a portfolio of stocks, the variance-covariance method is perfectly suitable and should probably be used instead. • MC VaR is subject to model risk if our market model is not correct.

21. Regulatory Models • BIS (including Federal Reserve) approach: • Market risk may be calculated using standard BIS model. • Specific risk charge. • General market risk charge. • Offsets. • Subject to regulatory permission, large banks may be allowed to use their internal models as the basis for determining capital requirements.

22. BIS Model • Specific risk charge: • Risk weights × absolute dollar values of long and short positions • General market risk charge: • reflect modified durations  expected interest rate shocks for each maturity • Vertical offsets: • Adjust for basis risk • Horizontal offsets within/between time zones

23. Large Banks: BIS versus RiskMetrics • In calculating DEAR, adverse change in rates defined as 99th percentile (rather than 95th under RiskMetrics) • Minimum holding period is 10 days (means that RiskMetrics’ daily DEAR multiplied by 10. • Capital charge will be higher of: • Previous day’s VAR (or DEAR  10) • Average Daily VAR over previous 60 days times a multiplication factor  3.

24. Websites Bank for International Settlements www.bis.org Federal Reserve www.federalreserve.gov Citigroup www.citigroup.com J.P.Morgan/Chase www.jpmorganchase.com Merrill Lynch www.merrilllynch.com RiskMetrics www.riskmetrics.com

25. Hedging and Derivatives

26. General idea of hedging Need to look for hedge that has opposite characteristic to underlying price risk Change in value Underlying risk Change in price Hedge position

27. Money Market Hedges • Locking in a Rate of Interest Loan in 3 Months Borrow Now Deposit for 3 Months • Locking in Exchange Rate Exchange £ for $ and Invest in US Money Market Now

28. Forwards and futures • Forward is agreement today to buy at future time but at price agreed today---OTC and counter-party risk • Futures contract is similar but in standard bundles on an organized exchange so risk is different and margining means that futures are like a string of daily forward contracts. FX or commodity contract cash

29. Hedging with Futures and Forwards - Difficulties • Asset Hedged may not be the same as that underlying the Futures Contract • Hedger may be uncertain as to when asset will be Bought or Sold • Hedge may have to be closed out with Futures contract well before Expiry Date • These problems give rise to Basis Risk

30. Basis Risk and Hedging Basis = Spot price of an - Futures price of asset to be hedged Contract Used Price Obtained with Short Hedge = S2 + F1 - F2 = F1 + b1 Price Paid for with Long Hedge = S2 + F1 - F2 = F1 + b1 Where Hedge Contract Different from Underlying Asset S2 + F1 - F2 = F1 + (S*2 - F2) + (S2 - S*2)

31. Optimal Hedge Ratios OHR -The ration of the size of the position taken in futures contract to the size of the exposure. Variance of Position Hedge Ratio h h*

32. SWAPs • Interest Rate - Fixed for Variable • Currency - Principle (Paid and Repaid) and Interest Payments

33. Black-Scholes Option Pricing Model OC = Ps[N(d1)] - S[N(d2)]e-rt OC- Call Option Price Ps - Stock Price N(d1) - Cumulative normal density function of (d1) S - Strike or Exercise price N(d2) - Cumulative normal density function of (d2) r - discount rate (90 day comm paper rate or risk free rate) t - time to maturity of option (as % of year) v - volatility - annualized standard deviation of daily returns

34. Options - Application Protective Put - Long stock and long put Long Stock Position Value Share Price

35. Options - Application Protective Put - Long stock and long put Long Put Position Value Share Price

36. Options - Application Protective Put - Long stock and long put Long Stock Protective Put Position Value Long Put Share Price

37. Options - Application Straddle - Long call and long put - Strategy for profiting from high volatility Long call Position Value Share Price

38. Options - Application Straddle - Long call and long put - Strategy for profiting from high volatility Long put Position Value Share Price

39. Options - Application Straddle - Long call and long put - Strategy for profiting from high volatility Straddle Position Value Share Price

40. Trading Strategies with Options • Vertical: same maturity, different exercise price • Horizontal: same ex. price, different maturity • Diagonal: different ex. price and different maturities • Bull Spread • Bear Spread • Butterfly Spreads • Calendar Spreads • Straddles • Strangles

41. Trading Strategies Involving Options

42. Trading Strategies involving Options • Single Option and a Stock strategies • Spreads • Combinations • Other Payoffs

43. Single option and a Stock • Writing a Covered Call (long stock and short call) • the long stock protects a trader from the payoff of the short call if there is a sharp rise in the stock price Long Stock Position Value Share Price Short Call

44. Single option and a Stock • Writing a Covered Call (long stock and short call) • the long stock protects a trader from the payoff of the short call if there is a sharp rise in the stock price Long Stock Covered Call Position Value Share Price Short Call

45. Single option and a Stock • Short stock and long call • Reverse of writing a covered call Short Stock Long Call Position Value Share Price

46. Single option and a Stock • Short stock and long call • Reverse of writing a covered call Short Stock Long Call Position Value Share Price

47. Single option and a Stock • Writing a Protective Put (buying a put and the stock itself) Long Stock Position Value Long Put Share Price

48. Single option and a Stock • Writing a Protective Put (buying a put and the stock itself) Long Stock Protective Put Position Value Long Put Share Price

49. Single option and a Stock • Short stock and short put • reverse of protective put Short Stock Short Put Position Value Share Price

50. Single option and a Stock • Short stock and short put • reverse of protective put Short Stock Short Put Position Value Share Price