Financial Risk Management

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

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

Portfolio of Random Variables VaR-PJorion-Ch 7-8

Product of Random Variables • Credit loss derives from the product of the probability of default and the loss given default. When X1 and X2 are independent VaR-PJorion-Ch 7-8

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%. VaR-PJorion-Ch 7-8

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% VaR-PJorion-Ch 7-8

Marginal VaR • How risk sensitive is my portfolio to increase in size of each position? • - calculate VaR for the entire portfolio VaRP=X • - increase position A by one unit (say 1% of the portfolio) • - calculate VaR of the new portfolio: VaRPa= 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...? VaR-PJorion-Ch 7-8

with minor corrections VaR-PJorion-Ch 7-8

Marginal VaR by currency..... with minor corrections VaR-PJorion-Ch 7-8

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

Incremental VaR by Risk Type... with minor corrections VaR-PJorion-Ch 7-8

Incremental VaR by Currency.... with minor corrections VaR-PJorion-Ch 7-8

VaR decomposition VaR Incremental VaR Marginal VaR Portfolio VaR Component VaR Position in asset A 100 VaR-PJorion-Ch 7-8

Example of VaR decomposition Currency Position Individual Marginal Component Contribution VaR VaR VaR to VaR in % CAD $2M $165,000 0.0528 $105,630 41% EUR $1M $198,000 0.1521 $152,108 59% Total $3M Undiversified $363K Diversified $257,738 100% VaR-PJorion-Ch 7-8

Barings Example • Long $7.7B Nikkei futures • Short of $16B JGB futures • NK=5.83%, JGB=1.18%, =11.4% VaR95%=1.65P = $835M VaR99%=2.33 P=$1.18B Actual loss was $1.3B VaR-PJorion-Ch 7-8

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 VaR-PJorion-Ch 7-8

P. Jorion Handbook, Ch 14 The Optimal Hedge Ratio Minimum variance hedge ratio VaR-PJorion-Ch 7-8

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: VaR-PJorion-Ch 7-8

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! VaR-PJorion-Ch 7-8

P. Jorion Handbook, Ch 14 Optimal Hedge • At the optimum the variance is VaR-PJorion-Ch 7-8

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 VaR-PJorion-Ch 7-8

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. VaR-PJorion-Ch 7-8

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. VaR-PJorion-Ch 7-8

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. VaR-PJorion-Ch 7-8

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

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. VaR-PJorion-Ch 7-8

P. Jorion Handbook, Ch 14 Solution • 2unhedged = ($586,409)2 = 343,875,515,281 • - 2SF/ 2F = -(2,605,268,452/5,390)2 • hedged = $331,997 • The hedge has reduced the SD from $586,409 to $331,997. • R2 = 67.95% (= 0.82432) VaR-PJorion-Ch 7-8

P. Jorion Handbook, Ch 14 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 VaR-PJorion-Ch 7-8

P. Jorion Handbook, Ch 14 Dollar duration Duration Hedging VaR-PJorion-Ch 7-8

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

P. Jorion Handbook, Ch 14 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. VaR-PJorion-Ch 7-8

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 VaR-PJorion-Ch 7-8

P. Jorion Handbook, Ch 14 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. VaR-PJorion-Ch 7-8

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. VaR-PJorion-Ch 7-8

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$1,000,0000.01%=$25 VaR-PJorion-Ch 7-8

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 VaR-PJorion-Ch 7-8

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 VaR-PJorion-Ch 7-8

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 VaR-PJorion-Ch 7-8

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 $200M, 5 year, receive fixed swap? • A. Short 250 • B. Short 3,200 • C. Short 40,000 • D. Long 250 VaR-PJorion-Ch 7-8

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 • $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 VaR-PJorion-Ch 7-8

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 VaR-PJorion-Ch 7-8

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. VaR-PJorion-Ch 7-8

P. Jorion Handbook, Ch 14 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. VaR-PJorion-Ch 7-8

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. VaR-PJorion-Ch 7-8

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. VaR-PJorion-Ch 7-8

P. Jorion Handbook, Ch 14 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 VaR-PJorion-Ch 7-8

Financial Risk Management