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A1. (a) Find the moment generating function of a random variable x ~ N(11,02). (b) Find E|x*] (c) For 0 and o = 1, find E[X|X > 0]. A3. If you play roulette 100 times, betting $100 on black each time, what is the probability of winning at least $1000, and what is the probability of losing at least $1000? (Note: Roulette has 18 red slots, 18 black slots, and 2 green slots.) A4. Suppose that in any given time period a certain stock is equally likely to go up 1 unit or down 1 unit, and that the outcomes of different periods are independent. Let X be the cumulative amount the stock goes up in the first two periods, and let Y be the cumulative amount it goes up in the first three periods. Find the correlation between x and Y. C1. Suppose you have agreed to a bank loan of $120000, for which the bank charges no fees but 2 points. The quoted interest rate is .5% per month. You are required to pay only the accumulated interest each month for the next 36 months, at which point you must make a balloon payment of the still-owed $120000. What is the effective interest rate of this loan? C2. The yield of a semiannual coupon bond with 6% coupon rate and 30 months to maturity is 9%. What are the price, duration, and convexity of the bond? Find the approximate price of the bond if the yield increases by 50 basis points. C3. Assume that the continuously compounded zero rate curve is t re(0,t) = 0.02 - 0.01 1+t² (a) find the instantaneous interest rate curve. (b) compute the corresponding annually compounded zero rate curve. (c) compute the corresponding semiannually compounded zero rate curve. C4. Consider two cash flow streams, where each will return the ith payment after i years: (100, 140, 131) and (90, 160, 120). Is it possible to tell which cash flow stream is preferable without knowing the interest rate? C4. Use bootstrapping to obtain a continuously compounded zero rate curve given that the price of a 6-month T-bill is 97.5, the price of a 12-month T-bill is 100, and the price of a 20- month T-bond with a 4.5% coupon rate is Use your zero-rate curve to price a 18-month T-bond with a 3% coupon rate. The Treasury bonds pay semiannual coupons. C5. Consider a bond portfolio worth $50mil with DV01 equal to $10,000 and dollar convexity equal to $400mil. (a) Assume that the yield curve moves up by fifty basis points. What is the new value of your bond portfolio? (b) The following bonds are available for trading: Principal Value Duration Convexity Bond 1 100 104 2.5 10 Bond 2 100 106 3 9 How do you immunize the portfolio? What is the new value of the immunized portfolio if the yield curve moves up by fifty basis points? E4. Assume that the risk-free interest rates are zero. Consider an asset with spot price 50. The values of the following options on this asset are given below: Option Type Strike Maturity Value Put 45 8 months $1.5 Put 56 8 months $7.25 Call 45 8 months $7.5 Call 56 8 months $2.25 (a) Synthesize a 45-56 bull spread using only the call options. Draw the P&L diagram at maturity of the bull spread. (b) Synthesize a 45-56 bull spread using only the put options. Draw the P&L diagram at maturity of the bull spread. (c) Synthesize a 45-56 bear spread using only the call options. Draw the P&L diagram at maturity of the bull spread. F1. You suspect that Delta airlines will merge with Northwest airlines in the coming month. Delta stock is trading at $0.85. There is a 60% chance that the merger will occur, in which case, the stock will be worth $1.2. There is a 40% chance that the merger will not occur, in which case, the stock will continue its downward plunge to $0.3. If r = 0.02, what is the value of a call with strike price K = $1 and maturity one month, i.e. T = 1/12? If you were to sell one thousand of these calls, explain precisely how much of the money you would keep in cash and how much you would invest in the stock to be perfectly hedged in one month. H3. Show that the approximate formula 1 2 e derived in the notes is actually an exact formula connecting the Gamma and the Theta of a European call option on a non-dividend paying asset, when interest rate is zero.

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Actuarial Science Questions
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