Financial Risk Management Zvi Wiener Following P Jorion

Financial Risk Management Zvi Wiener Following P. Jorion, Financial Risk Manager Handbook http: //pluto. huji. ac. il/~mswiener/zvi. html FRM 972 -2 -588 -3049

Chapter 14 Hedging Linear Risk Following P. Jorion 2001 Financial Risk Manager Handbook http: //pluto. huji. ac. il/~mswiener/zvi. html FRM 972 -2 -588 -3049

Hedging Taking positions that lower the risk profile of the portfolio. • Static hedging • Dynamic hedging Ch. 14, Handbook Zvi Wiener 3

Unit Hedging with Currencies A US exporter will receive Y 125 M 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 Y 12. 5 M and 9 months to maturity. Sell 10 contracts and revert in 7 months. Ch. 14, Handbook Zvi Wiener 4

Market data time to maturity US interest rate Yen interest rate Spot Y/$ Futures Y/$ Ch. 14, Handbook 0 9 6% 5% 125. 00 124. 07 Zvi Wiener 7 m 2 6% 2% 150. 00 149. 00 P&L 5

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

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 Ch. 14, Handbook Zvi Wiener 7

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

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. Ch. 14, Handbook Zvi Wiener 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. Ch. 14, Handbook Zvi Wiener 10

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. Ch. 14, Handbook Zvi Wiener 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. Ch. 14, Handbook Zvi Wiener 12

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. Ch. 14, Handbook Zvi Wiener 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. Ch. 14, Handbook Zvi Wiener 14

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 Ch. 14, Handbook Zvi Wiener 15

The Optimal Hedge Ratio Minimum variance hedge ratio Ch. 14, Handbook Zvi Wiener 16

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

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! Ch. 14, Handbook Zvi Wiener 18

Optimal Hedge At the optimum the variance is Ch. 14, Handbook Zvi Wiener 19

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/ 2 B. 1/ 2/ 1 C. 1/ 2 D. 2/ 1 Ch. 14, Handbook Zvi Wiener 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/ 2 B. 1/ 2/ 1 C. 1/ 2 D. 2/ 1 Ch. 14, Handbook Zvi Wiener 21

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 Ch. 14, Handbook Zvi Wiener 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 Ch. 14, Handbook Zvi Wiener 23

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. Ch. 14, Handbook Zvi Wiener 24

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. Ch. 14, Handbook Zvi Wiener 25

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. Ch. 14, Handbook Zvi Wiener 26

Solution The notional is Qs=$2, 770, 000, the SD in $ is ( s/s)s. Qs=0. 2117 $277 10, 000 = $586, 409 the SD of one futures contract is ( f/f)f. Qf=0. 1859 $0. 6903 42, 000 = $5, 390 with a futures notional f. Qf = $0. 6903 42, 000 = $28, 993. Ch. 14, Handbook Zvi Wiener 27

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 Qs s/Qf f = 89. 7, or 90 contracts. Ch. 14, Handbook Zvi Wiener 28

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% Ch. 14, Handbook (= 0. 82432) Zvi Wiener 29

FRM-99, Question 67 In the early 90 s, Metallgesellshaft, a German oil company, suffered a loss of $1. 33 B in their hedging program. They rolled over short dated futures to hedge long term exposure created through their longterm 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 Ch. 14, Handbook Zvi Wiener 30

FRM-99, Question 67 In the early 90 s, Metallgesellshaft, a German oil company, suffered a loss of $1. 33 B in their hedging program. They rolled over short dated futures to hedge long term exposure created through their longterm 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 Ch. 14, Handbook Zvi Wiener 31

Duration Hedging Dollar duration Ch. 14, Handbook Zvi Wiener 32

Duration Hedging If we have a target duration DV* we can get it by using Ch. 14, Handbook Zvi Wiener 33

Example 1 A portfolio manager has a bond portfolio worth $10 M 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. Ch. 14, Handbook Zvi Wiener 34

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 Ch. 14, Handbook Zvi Wiener 35

Example 2 On February 2, a corporate treasurer wants to hedge a July 17 issue of $5 M of CP with a maturity of 180 days, leading to anticipated proceeds of $4. 52 M. The September Eurodollar futures trades at 92, and has a notional amount of $1 M. Compute a. The current dollar value of the futures contract. b. The number of futures to buy/sell for optimal hedge. Ch. 14, Handbook Zvi Wiener 36

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. Ch. 14, Handbook Zvi Wiener 37

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 0. 01%=$25 Ch. 14, Handbook Zvi Wiener 38

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 Ch. 14, Handbook Zvi Wiener 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 Ch. 14, Handbook Zvi Wiener 40

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 Ch. 14, Handbook Zvi Wiener 41

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 Ch. 14, Handbook Zvi Wiener 42

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

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 $200 M 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 Ch. 14, Handbook Zvi Wiener 44

Beta Hedging represents the systematic risk, - the intercept (not a source of risk) and - residual. A stock index futures contract Ch. 14, Handbook Zvi Wiener 45

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. Ch. 14, Handbook Zvi Wiener 46

Example A portfolio manager holds a stock portfolio worth $10 M, with a beta of 1. 5 relative to S&P 500. 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. Ch. 14, Handbook Zvi Wiener 47

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. Ch. 14, Handbook Zvi Wiener 48

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. 52)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. Ch. 14, Handbook Zvi Wiener 49

FRM-00, Question 93 A fund manages an equity portfolio worth $50 M 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. 5 M. How many options contracts are needed to hedge the portfolio? A. 169 B. 289 C. 306 D. 321 Ch. 14, Handbook Zvi Wiener 50

FRM-00, Question 93 The optimal hedge ratio is N = -1. 8 $50, 000/(0. 623 $500, 000)=289 Ch. 14, Handbook Zvi Wiener 51