Financial Risk Management Zvi Wiener mswienermscc huji ac

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

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

Portfolio of Random Variables Zvi Wiener Va. R-PJorion-Ch 7 -8 3

Portfolio of Random Variables Zvi Wiener Va. R-PJorion-Ch 7 -8 4

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 Zvi Wiener Va. R-PJorion-Ch 7 -8 5

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%. Zvi Wiener Va. R-PJorion-Ch 7 -8 6

Example 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% Zvi Wiener Va. R-PJorion-Ch 7 -8 7

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

with minor corrections Zvi Wiener Va. R-PJorion-Ch 7 -8 9

Marginal Va. R by currency. . . Zvi Wiener Va. R-PJorion-Ch 7 -8 with minor corrections 10

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

Incremental Va. R by Risk Type. . . Zvi Wiener Va. R-PJorion-Ch 7 -8 with minor corrections 12

Incremental Va. R by Currency. . Zvi Wiener Va. R-PJorion-Ch 7 -8 with minor corrections 13

Va. R decomposition Va. R Incremental Va. R Marginal Va. R Portfolio Va. R Component Va. R 100 Zvi Wiener Va. R-PJorion-Ch 7 -8 Position in asset A 14

Example of Va. R decomposition Currency Position Individual Marginal Component Contribution Va. R to Va. R in % CAD $2 M $165, 000 0. 0528 $105, 630 41% EUR $1 M $198, 000 0. 1521 $152, 108 59% Total $3 M $257, 738 100% Undiversified Diversified Zvi Wiener $363 K Va. R-PJorion-Ch 7 -8 15

Barings Example Long $7. 7 B Nikkei futures Short of $16 B JGB futures NK=5. 83%, JGB=1. 18%, =11. 4% Va. R 95%=1. 65 P = $835 M Va. R 99%=2. 33 P=$1. 18 M Actual loss was $1. 3 B Zvi Wiener Va. R-PJorion-Ch 7 -8 16

P. Jorion Handbook, Ch 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 17

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

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

P. Jorion Handbook, Ch 14 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! Zvi Wiener Va. R-PJorion-Ch 7 -8 20

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

P. Jorion Handbook, Ch 14 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 22

P. Jorion Handbook, Ch 14 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 23

P. Jorion Handbook, Ch 14 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. Zvi Wiener Va. R-PJorion-Ch 7 -8 24

P. Jorion Handbook, Ch 14 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. Zvi Wiener Va. R-PJorion-Ch 7 -8 25

P. Jorion Handbook, Ch 14 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. Zvi Wiener Va. R-PJorion-Ch 7 -8 26

P. Jorion Handbook, Ch 14 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. Zvi Wiener Va. R-PJorion-Ch 7 -8 27

P. Jorion Handbook, Ch 14 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. Zvi Wiener Va. R-PJorion-Ch 7 -8 28

P. Jorion Handbook, Ch 14 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% Zvi Wiener (= 0. 82432) Va. R-PJorion-Ch 7 -8 29

P. Jorion Handbook, Ch 14 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 30

P. Jorion Handbook, Ch 14 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 31

P. Jorion Handbook, Ch 14 Duration Hedging Dollar duration Zvi Wiener Va. R-PJorion-Ch 7 -8 32

P. Jorion Handbook, Ch 14 Duration Hedging If we have a target duration DV* we can get it by using Zvi Wiener Va. R-PJorion-Ch 7 -8 33

P. Jorion Handbook, Ch 14 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. Zvi Wiener Va. R-PJorion-Ch 7 -8 34

P. Jorion Handbook, Ch 14 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 35

P. Jorion Handbook, Ch 14 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. Zvi Wiener Va. R-PJorion-Ch 7 -8 36

P. Jorion Handbook, Ch 14 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. Zvi Wiener Va. R-PJorion-Ch 7 -8 37

P. Jorion Handbook, Ch 14 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 38

P. Jorion Handbook, Ch 14 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 39

P. Jorion Handbook, Ch 14 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 40

P. Jorion Handbook, Ch 14 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 41

P. Jorion Handbook, Ch 14 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 42

P. Jorion Handbook, Ch 14 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 43

P. Jorion Handbook, Ch 14 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 44

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

P. Jorion Handbook, Ch 14 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. Zvi Wiener Va. R-PJorion-Ch 7 -8 46

P. Jorion Handbook, Ch 14 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. Zvi Wiener Va. R-PJorion-Ch 7 -8 47

P. Jorion Handbook, Ch 14 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. Zvi Wiener Va. R-PJorion-Ch 7 -8 48

P. Jorion Handbook, Ch 14 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. Zvi Wiener Va. R-PJorion-Ch 7 -8 49

P. Jorion Handbook, Ch 14 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 Zvi Wiener Va. R-PJorion-Ch 7 -8 50

P. Jorion Handbook, Ch 14 FRM-00, Question 93 The optimal hedge ratio is N = -1. 8 $50, 000/(0. 623 $500, 000)=289 Zvi Wiener Va. R-PJorion-Ch 7 -8 51

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

Volatility Unobservable, time varying, clustering Moving average rt daily returns: Implied volatility (smile, smirk, etc. ) Zvi Wiener Va. R-PJorion-Ch 7 -8 53

GARCH Estimation Generalized Autoregressive heteroskedastic Heteroskedastic means time varying Zvi Wiener Va. R-PJorion-Ch 7 -8 54

EWMA Exponentially Weighted Moving Average - is decay factor Zvi Wiener Va. R-PJorion-Ch 7 -8 55

Home assignment Zvi Wiener Va. R-PJorion-Ch 7 -8 56

Va. R system Risk factors Portfolio Historical data positions Model Mapping Distribution of risk factors Va. R method Exposures Va. R Zvi Wiener Va. R-PJorion-Ch 7 -8 57

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