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 14 Hedging Linear Risk Following P. Jorion 2001 Financial Risk Manager Handbook

3. Ch. 14, HandbookZvi Wiener slide 3 Hedging Taking positions that lower the risk profile of the portfolio. Static hedging Dynamic hedging

4. Ch. 14, HandbookZvi Wiener slide 4 Unit Hedging with Currencies A US exporter will receive Y125M in 7 months. The perfect hedge is to enter a 7-months forward contract. Such a contract is OTC and illiquid. Instead one can use traded futures. CME lists yen contract with face value Y12.5M and 9 months to maturity. Sell 10 contracts and revert in 7 months.

5. Ch. 14, HandbookZvi Wiener slide 5 Market data07mP&L time to maturity92 US interest rate6%6% Yen interest rate5%2% Spot Y/$125.00150.00 Futures Y/$124.07149.00

6. Ch. 14, HandbookZvi Wiener slide 6 Stacked hedge - to use a longer horizon and to revert the position at maturity. Strip hedge - rolling over short hedge.

7. Ch. 14, HandbookZvi Wiener slide 7 Basis Risk Basis risk arises when the characteristics of the futures contract differ from those of the underlying. For example quality of agricultural product, types of oil, Cheapest to Deliver bond, etc. Basis = Spot - Future

8. Ch. 14, HandbookZvi Wiener slide 8 Cross hedging Hedging with a correlated (but different) asset. In order to hedge an exposure to Norwegian Krone one can use Euro futures. Hedging a portfolio of stocks with index future.

9. Ch. 14, HandbookZvi Wiener slide 9 FRM-00, Question 78 What feature of cash and futures prices tend to make hedging possible? A. They always move together in the same direction and by the same amount. B. They move in opposite direction by the same amount. C. They tend to move together generally in the same direction and by the same amount. D. They move in the same direction by different amount.

10. Ch. 14, HandbookZvi Wiener slide 10 FRM-00, Question 78 What feature of cash and futures prices tend to make hedging possible? A. They always move together in the same direction and by the same amount. B. They move in opposite direction by the same amount. C. They tend to move together generally in the same direction and by the same amount. D. They move in the same direction by different amount.

11. Ch. 14, HandbookZvi Wiener slide 11 FRM-00, Question 17 Which statement is MOST correct? A. A portfolio of stocks can be fully hedged by purchasing a stock index futures contract. B. Speculators play an important role in the futures market by providing the liquidity that makes hedging possible and assuming the risk that hedgers are trying to eliminate. C. Someone generally using futures contract for hedging does not bear the basis risk. D. Cross hedging involves an additional source of basis risk because the asset being hedged is exactly the same as the asset underlying the futures.

12. Ch. 14, HandbookZvi Wiener slide 12 FRM-00, Question 17 Which statement is MOST correct? A. A portfolio of stocks can be fully hedged by purchasing a stock index futures contract. B. Speculators play an important role in the futures market by providing the liquidity that makes hedging possible and assuming the risk that hedgers are trying to eliminate. C. Someone generally using futures contract for hedging does not bear the basis risk. D. Cross hedging involves an additional source of basis risk because the asset being hedged is exactly the same as the asset underlying the futures.

13. Ch. 14, HandbookZvi Wiener slide 13 FRM-00, Question 79 Under which scenario is basis risk likely to exist? A. A hedge (which was initially matched to the maturity of the underlying) is lifted before expiration. B. The correlation of the underlying and the hedge vehicle is less than one and their volatilities are unequal. C. The underlying instrument and the hedge vehicle are dissimilar. D. All of the above.

14. Ch. 14, HandbookZvi Wiener slide 14 FRM-00, Question 79 Under which scenario is basis risk likely to exist? A. A hedge (which was initially matched to the maturity of the underlying) is lifted before expiration. B. The correlation of the underlying and the hedge vehicle is less than one and their volatilities are unequal. C. The underlying instrument and the hedge vehicle are dissimilar. D. All of the above.

15. Ch. 14, HandbookZvi Wiener slide 15 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

16. Ch. 14, HandbookZvi Wiener slide 16 The Optimal Hedge Ratio Minimum variance hedge ratio

17. Ch. 14, HandbookZvi Wiener slide 17 Hedge Ratio as Regression Coefficient The optimal amount can also be derived as the slope coefficient of a regression  s/s on  f/f:

18. Ch. 14, HandbookZvi Wiener slide 18 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!

19. Ch. 14, HandbookZvi Wiener slide 19 Optimal Hedge At the optimum the variance is

20. Ch. 14, HandbookZvi Wiener slide 20 FRM-99, Question 66 The hedge ratio is the ratio of the size of the position taken in the futures contract to the size of the exposure. Denote the standard deviation of change of spot price by  1, the standard deviation of change of future price by  2, the correlation between the changes in spot and futures prices by . What is the optimal hedge ratio? A. 1/  1 /  2 B. 1/  2 /  1 C.  1 /  2 D.  2 /  1

21. Ch. 14, HandbookZvi Wiener slide 21 FRM-99, Question 66 The hedge ratio is the ratio of the size of the position taken in the futures contract to the size of the exposure. Denote the standard deviation of change of spot price by  1, the standard deviation of change of future price by  2, the correlation between the changes in spot and futures prices by . What is the optimal hedge ratio? A. 1/  1 /  2 B. 1/  2 /  1 C.  1 /  2 D.  2 /  1

22. Ch. 14, HandbookZvi Wiener 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

23. Ch. 14, HandbookZvi Wiener 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

24. Ch. 14, HandbookZvi Wiener 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.

25. Ch. 14, HandbookZvi Wiener 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.

26. Ch. 14, HandbookZvi Wiener 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.

27. Ch. 14, HandbookZvi Wiener 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.

28. Ch. 14, HandbookZvi Wiener 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.

29. Ch. 14, HandbookZvi Wiener 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 )

30. Ch. 14, HandbookZvi Wiener 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

31. Ch. 14, HandbookZvi Wiener 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

32. Ch. 14, HandbookZvi Wiener slide 32 Duration Hedging Dollar duration

33. Ch. 14, HandbookZvi Wiener slide 33 Duration Hedging If we have a target duration D V * we can get it by using

34. Ch. 14, HandbookZvi Wiener 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.

35. Ch. 14, HandbookZvi Wiener 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

36. Ch. 14, HandbookZvi Wiener 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.

37. Ch. 14, HandbookZvi Wiener 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.

38. Ch. 14, HandbookZvi Wiener 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

39. Ch. 14, HandbookZvi Wiener 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

40. Ch. 14, HandbookZvi Wiener 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

41. Ch. 14, HandbookZvi Wiener 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

42. Ch. 14, HandbookZvi Wiener 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

43. Ch. 14, HandbookZvi Wiener 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

44. Ch. 14, HandbookZvi Wiener 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

45. Ch. 14, HandbookZvi Wiener slide 45 Beta Hedging  represents the systematic risk,  - the intercept (not a source of risk) and  - residual. A stock index futures contract

46. Ch. 14, HandbookZvi Wiener 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.

47. Ch. 14, HandbookZvi Wiener 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.

48. Ch. 14, HandbookZvi Wiener 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.

49. Ch. 14, HandbookZvi Wiener 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.

50. Ch. 14, HandbookZvi Wiener 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

51. Ch. 14, HandbookZvi Wiener slide 51 FRM-00, Question 93 The optimal hedge ratio is N = -1.8  $50,000,000/(0.623  $500,000)=289