1. RM http://pluto.mscc.huji.ac.il/~mswiener/zvi.html HUJI-03 Zvi Wiener mswiener@mscc.huji.ac.il 02-588-3049 Financial Risk Management

2. RM http://pluto.mscc.huji.ac.il/~mswiener/zvi.html HUJI-03 Following P. Jorion, Value at Risk, McGraw-Hill Chapter 7 Portfolio Risk, Analytical Methods Financial Risk Management

3. Zvi WienerVaR-PJorion-Ch 7-8 slide 3 Portfolio of Random Variables

4. Zvi WienerVaR-PJorion-Ch 7-8 slide 4 Portfolio of Random Variables

5. Zvi WienerVaR-PJorion-Ch 7-8 slide 5 Product of Random Variables Credit loss derives from the product of the probability of default and the loss given default. When X 1 and X 2 are independent

6. Zvi WienerVaR-PJorion-Ch 7-8 slide 6 Transformation of Random Variables Consider a zero coupon bond If r=6% and T=10 years, V = $55.84, we wish to estimate the probability that the bond price falls below $50. This corresponds to the yield 7.178%.

7. Zvi WienerVaR-PJorion-Ch 7-8 slide 7 The probability of this event can be derived from the distribution of yields. Assume that yields change are normally distributed with mean zero and volatility 0.8%. Then the probability of this change is 7.06% Example

8. Zvi WienerVaR-PJorion-Ch 7-8 slide 8 Marginal VaR How risk sensitive is my portfolio to increase in size of each position? - calculate VaR for the entire portfolio VaR P =X - increase position A by one unit (say 1% of the portfolio) - calculate VaR of the new portfolio: VaR Pa = Y - incremental risk contribution to the portfolio by A: Z = X-Y i.e. Marginal VaR of A is Z = X-Y Marginal VaR can be Negative; what does this mean...?

9. Zvi WienerVaR-PJorion-Ch 7-8 slide 9 with minor corrections

10. Zvi WienerVaR-PJorion-Ch 7-8 slide 10 Marginal VaR by currency..... with minor corrections

11. Zvi WienerVaR-PJorion-Ch 7-8 slide 11 Incremental VaR Risk contribution of each position in my portfolio. - calculate VaR for the entire portfolio VaR P = X - remove A from the portfolio - calculate VaR of the portfolio without A: VaR P-A = Y - Risk contribution to the portfolio by A: Z = X-Y i.e. Incremental VaR of A is Z = X-Y Incremental VaR can be Negative; what does this mean...?

12. Zvi WienerVaR-PJorion-Ch 7-8 slide 12 Incremental VaR by Risk Type... with minor corrections

13. Zvi WienerVaR-PJorion-Ch 7-8 slide 13 Incremental VaR by Currency.... with minor corrections

14. Zvi WienerVaR-PJorion-Ch 7-8 slide 14 VaR decomposition Position in asset A VaR 100 Portfolio VaR Incremental VaR Marginal VaR Component VaR

15. Zvi WienerVaR-PJorion-Ch 7-8 slide 15 Example of VaR decomposition Currency Position Individual Marginal ComponentContribution VaR VaR VaRto VaR in % CAD $2M $165,000 0.0528 $105,63041% EUR $1M $198,000 0.1521 $152,10859% Total $3M Undiversified $363K Diversified $257,738100%

16. Zvi WienerVaR-PJorion-Ch 7-8 slide 16 Barings Example Long $7.7B Nikkei futures Short of $16B JGB futures  NK =5.83%,  JGB =1.18%,  =11.4% VaR 95% =1.65  P = $835M VaR 99% =2.33  P =$1.18B Actual loss was $1.3B

17. Zvi WienerVaR-PJorion-Ch 7-8 slide 17 The Optimal Hedge Ratio  S - change in $ value of the inventory  F - change in $ value of the one futures N - number of futures you buy/sell P. Jorion Handbook, Ch 14

18. Zvi WienerVaR-PJorion-Ch 7-8 slide 18 The Optimal Hedge Ratio Minimum variance hedge ratio P. Jorion Handbook, Ch 14

19. Zvi WienerVaR-PJorion-Ch 7-8 slide 19 Hedge Ratio as Regression Coefficient The optimal amount can also be derived as the slope coefficient of a regression  s/s on  f/f: P. Jorion Handbook, Ch 14

20. Zvi WienerVaR-PJorion-Ch 7-8 slide 20 Optimal Hedge One can measure the quality of the optimal hedge ratio in terms of the amount by which we have decreased the variance of the original portfolio. If R is low the hedge is not effective! P. Jorion Handbook, Ch 14

21. Zvi WienerVaR-PJorion-Ch 7-8 slide 21 Optimal Hedge At the optimum the variance is P. Jorion Handbook, Ch 14

22. Zvi WienerVaR-PJorion-Ch 7-8 slide 22 FRM-99, Question 66 The hedge ratio is the ratio of derivatives to a spot position (vice versa) that achieves an objective such as minimizing or eliminating risk. Suppose that the standard deviation of quarterly changes in the price of a commodity is 0.57, the standard deviation of quarterly changes in the price of a futures contract on the commodity is 0.85, and the correlation between the two changes is 0.3876. What is the optimal hedge ratio for a three-month contract? A. 0.1893 B. 0.2135 C. 0.2381 D. 0.2599 P. Jorion Handbook, Ch 14

23. Zvi WienerVaR-PJorion-Ch 7-8 slide 23 FRM-99, Question 66 The hedge ratio is the ratio of derivatives to a spot position (vice versa) that achieves an objective such as minimizing or eliminating risk. Suppose that the standard deviation of quarterly changes in the price of a commodity is 0.57, the standard deviation of quarterly changes in the price of a futures contract on the commodity is 0.85, and the correlation between the two changes is 0.3876. What is the optimal hedge ratio for a three-month contract? A. 0.1893 B. 0.2135 C. 0.2381 D. 0.2599 P. Jorion Handbook, Ch 14

24. Zvi WienerVaR-PJorion-Ch 7-8 slide 24 Example Airline company needs to purchase 10,000 tons of jet fuel in 3 months. One can use heating oil futures traded on NYMEX. Notional for each contract is 42,000 gallons. We need to check whether this hedge can be efficient. P. Jorion Handbook, Ch 14

25. Zvi WienerVaR-PJorion-Ch 7-8 slide 25 Example Spot price of jet fuel $277/ton. Futures price of heating oil $0.6903/gallon. The standard deviation of jet fuel price rate of changes over 3 months is 21.17%, that of futures 18.59%, and the correlation is 0.8243. P. Jorion Handbook, Ch 14

26. Zvi WienerVaR-PJorion-Ch 7-8 slide 26 Compute The notional and standard deviation f the unhedged fuel cost in $. The optimal number of futures contracts to buy/sell, rounded to the closest integer. The standard deviation of the hedged fuel cost in dollars. P. Jorion Handbook, Ch 14

27. Zvi WienerVaR-PJorion-Ch 7-8 slide 27 Solution The notional is Qs=$2,770,000, the SD in $ is  (  s/s)sQ s =0.2117  $277  10,000 = $586,409 the SD of one futures contract is  (  f/f)fQ f =0.1859  $0.6903  42,000 = $5,390 with a futures notional fQ f = $0.6903  42,000 = $28,993. P. Jorion Handbook, Ch 14

28. Zvi WienerVaR-PJorion-Ch 7-8 slide 28 Solution The cash position corresponds to a liability (payment), hence we have to buy futures as a protection.  sf = 0.8243  0.2117/0.1859 = 0.9387  sf = 0.8243  0.2117  0.1859 = 0.03244 The optimal hedge ratio is N* =  sf Q s  s/Q f  f = 89.7, or 90 contracts. P. Jorion Handbook, Ch 14

29. Zvi WienerVaR-PJorion-Ch 7-8 slide 29 Solution  2 unhedged = ($586,409) 2 = 343,875,515,281 -  2 SF /  2 F = -(2,605,268,452/5,390) 2  hedged = $331,997 The hedge has reduced the SD from $586,409 to $331,997. R 2 = 67.95%(= 0.8243 2 ) P. Jorion Handbook, Ch 14

30. Zvi WienerVaR-PJorion-Ch 7-8 slide 30 FRM-99, Question 67 In the early 90s, Metallgesellshaft, a German oil company, suffered a loss of $1.33B in their hedging program. They rolled over short dated futures to hedge long term exposure created through their long- term fixed price contracts to sell heating oil and gasoline to their customers. After a time, they abandoned the hedge because of large negative cashflow. The cashflow pressure was due to the fact that MG had to hedge its exposure by: A. Short futures and there was a decline in oil price B. Long futures and there was a decline in oil price C. Short futures and there was an increase in oil price D. Long futures and there was an increase in oil price P. Jorion Handbook, Ch 14

31. Zvi WienerVaR-PJorion-Ch 7-8 slide 31 FRM-99, Question 67 In the early 90s, Metallgesellshaft, a German oil company, suffered a loss of $1.33B in their hedging program. They rolled over short dated futures to hedge long term exposure created through their long- term fixed price contracts to sell heating oil and gasoline to their customers. After a time, they abandoned the hedge because of large negative cashflow. The cashflow pressure was due to the fact that MG had to hedge its exposure by: A. Short futures and there was a decline in oil price B. Long futures and there was a decline in oil price C. Short futures and there was an increase in oil price D. Long futures and there was an increase in oil price P. Jorion Handbook, Ch 14

32. Zvi WienerVaR-PJorion-Ch 7-8 slide 32 Duration Hedging Dollar duration P. Jorion Handbook, Ch 14

33. Zvi WienerVaR-PJorion-Ch 7-8 slide 33 Duration Hedging If we have a target duration D V * we can get it by using P. Jorion Handbook, Ch 14

34. Zvi WienerVaR-PJorion-Ch 7-8 slide 34 Example 1 A portfolio manager has a bond portfolio worth $10M with a modified duration of 6.8 years, to be hedged for 3 months. The current futures prices is 93-02, with a notional of $100,000. We assume that the duration can be measured by CTD, which is 9.2 years. Compute: a. The notional of the futures contract b.The number of contracts to by/sell for optimal protection. P. Jorion Handbook, Ch 14

35. Zvi WienerVaR-PJorion-Ch 7-8 slide 35 Example 1 The notional is: (93+2/32)/100  $100,000 =$93,062.5 The optimal number to sell is: Note that DVBP of the futures is 9.2  $93,062  0.01%=$85 P. Jorion Handbook, Ch 14

36. Zvi WienerVaR-PJorion-Ch 7-8 slide 36 Example 2 On February 2, a corporate treasurer wants to hedge a July 17 issue of $5M of CP with a maturity of 180 days, leading to anticipated proceeds of $4.52M. The September Eurodollar futures trades at 92, and has a notional amount of $1M. Compute a. The current dollar value of the futures contract. b. The number of futures to buy/sell for optimal hedge. P. Jorion Handbook, Ch 14

37. Zvi WienerVaR-PJorion-Ch 7-8 slide 37 Example 2 The current dollar value is given by $10,000  (100-0.25(100-92)) = $980,000 Note that duration of futures is 3 months, since this contract refers to 3-month LIBOR. P. Jorion Handbook, Ch 14

38. Zvi WienerVaR-PJorion-Ch 7-8 slide 38 Example 2 If Rates increase, the cost of borrowing will be higher. We need to offset this by a gain, or a short position in the futures. The optimal number of contracts is: Note that DVBP of the futures is 0.25  $1,000,000  0.01%=$25 P. Jorion Handbook, Ch 14

39. Zvi WienerVaR-PJorion-Ch 7-8 slide 39 FRM-00, Question 73 What assumptions does a duration-based hedging scheme make about the way in which interest rates move? A. All interest rates change by the same amount B. A small parallel shift in the yield curve C. Any parallel shift in the term structure D. Interest rates movements are highly correlated P. Jorion Handbook, Ch 14

40. Zvi WienerVaR-PJorion-Ch 7-8 slide 40 FRM-00, Question 73 What assumptions does a duration-based hedging scheme make about the way in which interest rates move? A. All interest rates change by the same amount B. A small parallel shift in the yield curve C. Any parallel shift in the term structure D. Interest rates movements are highly correlated P. Jorion Handbook, Ch 14

41. Zvi WienerVaR-PJorion-Ch 7-8 slide 41 FRM-99, Question 61 If all spot interest rates are increased by one basis point, a value of a portfolio of swaps will increase by $1,100. How many Eurodollar futures contracts are needed to hedge the portfolio? A. 44 B. 22 C. 11 D. 1100 P. Jorion Handbook, Ch 14

42. Zvi WienerVaR-PJorion-Ch 7-8 slide 42 FRM-99, Question 61 The DVBP of the portfolio is $1,100. The DVBP of the futures is $25. Hence the ratio is 1100/25 = 44 P. Jorion Handbook, Ch 14

43. Zvi WienerVaR-PJorion-Ch 7-8 slide 43 FRM-99, Question 109 Roughly how many 3-month LIBOR Eurodollar futures contracts are needed to hedge a position in a $200M, 5 year, receive fixed swap? A. Short 250 B. Short 3,200 C. Short 40,000 D. Long 250 P. Jorion Handbook, Ch 14

44. Zvi WienerVaR-PJorion-Ch 7-8 slide 44 FRM-99, Question 109 The dollar duration of a 5-year 6% par bond is about 4.3 years. Hence the DVBP of the fixed leg is about $200M  4.3  0.01%=$86,000. The floating leg has short duration - small impact decreasing the DVBP of the fixed leg. DVBP of futures is $25. Hence the ratio is 86,000/25 = 3,440. Answer A P. Jorion Handbook, Ch 14

45. Zvi WienerVaR-PJorion-Ch 7-8 slide 45 Beta Hedging  represents the systematic risk,  - the intercept (not a source of risk) and  - residual. A stock index futures contract P. Jorion Handbook, Ch 14

46. Zvi WienerVaR-PJorion-Ch 7-8 slide 46 Beta Hedging The optimal N is The optimal hedge with a stock index futures is given by beta of the cash position times its value divided by the notional of the futures contract. P. Jorion Handbook, Ch 14

47. Zvi WienerVaR-PJorion-Ch 7-8 slide 47 Example A portfolio manager holds a stock portfolio worth $10M, with a beta of 1.5 relative to S&P500. The current S&P index futures price is 1400, with a multiplier of $250. Compute: a. The notional of the futures contract b. The optimal number of contracts for hedge. P. Jorion Handbook, Ch 14

48. Zvi WienerVaR-PJorion-Ch 7-8 slide 48 Example The notional of the futures contract is $250  1,400 = $350,000 The optimal number of contracts for hedge is The quality of the hedge will depend on the size of the residual risk in the portfolio. P. Jorion Handbook, Ch 14

49. Zvi WienerVaR-PJorion-Ch 7-8 slide 49 A typical US stock has correlation of 50% with S&P. Using the regression effectiveness we find that the volatility of the hedged portfolio is still about (1-0.5 2 ) 0.5 = 87% of the unhedged volatility for a typical stock. If we wish to hedge an industry index with S&P futures, the correlation is about 75% and the unhedged volatility is 66% of its original level. The lower number shows that stock market hedging is more effective for diversified portfolios. P. Jorion Handbook, Ch 14

50. Zvi WienerVaR-PJorion-Ch 7-8 slide 50 FRM-00, Question 93 A fund manages an equity portfolio worth $50M with a beta of 1.8. Assume that there exists an index call option contract with a delta of 0.623 and a value of $0.5M. How many options contracts are needed to hedge the portfolio? A. 169 B. 289 C. 306 D. 321 P. Jorion Handbook, Ch 14

51. Zvi WienerVaR-PJorion-Ch 7-8 slide 51 FRM-00, Question 93 The optimal hedge ratio is N = -1.8  $50,000,000/(0.623  $500,000)=289 P. Jorion Handbook, Ch 14

52. RM http://pluto.mscc.huji.ac.il/~mswiener/zvi.html HUJI-03 Following P. Jorion, Value at Risk, McGraw-Hill Chapter 8 Forecasting Risks and Correlations Financial Risk Management

53. Zvi WienerVaR-PJorion-Ch 7-8 slide 53 Volatility Unobservable, time varying, clustering Moving average r t daily returns: Implied volatility (smile, smirk, etc.)

54. Zvi WienerVaR-PJorion-Ch 7-8 slide 54 GARCH Estimation Generalized Autoregressive heteroskedastic Heteroskedastic means time varying

55. Zvi WienerVaR-PJorion-Ch 7-8 slide 55 EWMA Exponentially Weighted Moving Average - is decay factor

56. Zvi WienerVaR-PJorion-Ch 7-8 slide 56 Home assignment

57. Zvi WienerVaR-PJorion-Ch 7-8 slide 57 VaR system Risk factors Historical data Model Distribution of risk factors VaR method Portfolio positions Mapping Exposures VaR

58. Zvi WienerVaR-PJorion-Ch 7-8 slide 58 Ideas Monte Carlo for financial assets Stress testing VaR – OG Collar example ESOP hedging Swaps + Credit Derivatives Linkage Your personal financial Risk