Question 3
At the beginning of 1995, General Motor’s pension funds were under- funded (i.e., the value of their liabilities exceeded the value of their assets) by $9.3 billion. Even though the company injected $10.4 billion during the year, at the end of 1995 the pension funds were still underfunded. Knowing that interest rates dropped during 1995, explain what must have happened. Describe a strategy with which GM could have hedged its interest rate risk and discuss the limitations of this strategy.

Question 4
Describe the “stop-loss” strategy for hedging a short call position. Use profit diagrams to explain how you can use this strategy to lock in the premium gained by selling the call option. What is the initial cost of setting up this strategy? Explain why this may not be an effective hedging strategy.

Section B
Answer ALL Questions
Question 1
Consider two defaultable 1-year loans with a principal of $1 million each. The probability of default on each loan is 2.5%. Assume that if one loan defaults, the other does not. Assume that in the event of default, the loan leads to a loss that can take any value between $0 and $1 million with equal probability, i.e., the probability that the loss is higher than $ 𝑥 million is 1 − 𝑥. If a loan does not default, it yields a profit equal to $20,000.
a) Compute the 1-year 98% Value at Risk (VaR) and Expected Shortfall (ES) of a single loan.

b) Compute the 1-year 98% VaR and ES for the portfolio of both loans.

c) Does the VaR and the ES satisfy the subadditivity property in this case?

Question 2
Bond A is a 4-year bond with a 10% coupon rate and Bond B is a 2-year bond with a 20% coupon rate. Both bonds have a face value of £100, and all coupons are paid annually, starting in year 1. Consider a portfolio that consists of one unit of Bond A and one unit of Bond B. The yield curve is flat at 𝑟 = 5%.
a) What is the Macaulay duration of this portfolio?

b) Compute the approximate percentage change in the value of the
portfolio if the interest rate increases by 200 basis points.

c) Suppose a pension fund holds this portfolio and is worried about fluctuations in its price. Explain how the fund can hedge its interest rate risk using 3-year zero-coupon bonds with a face value of £100.

Question 3
Consider two stocks, A and B, whose returns in a boom and a recession are given in the table below. Suppose that there is a 10% chance of a recession next year. How would you allocate money between these two stocks so as to minimize your risk? Are you able to perfectly eliminate your risk? Explain.