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CAPM versus Consumption CAPM: a. Contrast the two models in words. 27 b. Explain why, in principle, the consumption CAPM is more satisfactory. c. Can you think of circumstances where the two models are essentially identical? 10.2. Consider an endowment economy identical to the one considered in the discussion of the consumption CAPM. Consider an option that entitles the owner to exercise the right to buy one unit of the asset one period in the future at a fixed price p*. The price p* is known at period t, whereas the option to buy is exercised at period t + 1. Suppose the representative agent's utility function is U (c) = ln(c). You are asked to price the option. Follow these steps: a. Write the value of the option at expiration as a function of the price of the underlying asset. b. Write the price of the asset at the later date as a function of the state at that date, , the total quantity of good available. c. Given the chosen utility function, write q(, c), the Arrow-Debreu price. d. Use Arrow-Debreu pricing to price the option. 10.3. Consider the setting described previously with the following modification: The asset in question requires the owner to buy the asset at a price p* (a forward contract). Price the security. 10.4. Consider the usual representative agent economy. Suppose that the representative agent has log utility, U (c) = ln(c). Define the wealth portfolio as a claim to all future dividends available for consumption in this economy. a. Show that the price of this wealth portfolio is proportional to consumption itself. 28 b. Show that the return on this portfolio is proportional to consumption growth. Hint: Think about the definition of the return on an asset to write down the return on the wealth portfolio. c. Finally, express the price of a cash flow paying off $100 at each date for the next two periods in terms of the return on the wealth portfolio. 10.5. Begin with the fundamental equation E[mRi] = 1 where Ri is the gross return on asset i, and m is the pricing kernel. Noting that this fundamental equation should hold for the gross returns on the riskless asset and on the market portfolio (RM) as well, obtain a CAPM-like expression. Define a functional relation between m and RM so that you will obtain precisely the CAPM expression

. The data is the same above. Now suppose instead that you are not in the oil business but can also rent the storage facility at the same cost. Can you take advantage of the current market conditions and the rental opportunity? If yes, please explain how (i.e., describe the actions you need to take). If not, briefly explain why. 10. The current price of silver is $13.50 per ounce. The storage costs are $0.10 per ounce per year payable quarterly at the beginning of each quarter and the interest rate is 5% APR compounded quarterly (1.25% per quarter). (a) Calculate the future price of silver for delivery in nine months. Assume that silver is held for investment only and that the convenience yield of holding silver is zero. (b) Suppose the actual price of the futures contract traded in the market is below the price you calculated in part (a). How would you construct a risk-free trading strategy to make money? What if the actual price is higher? To get full credit, say precisely what you will buy or sell, and how much money you will borrow or deposit into a bank account and for how long. 11. A pension plan currently has $50M in S&P 500 index and $50M in one-year zero-coupon bonds. Assume that the one-year interest rate is 6%. Assume that the current quote on the S&P 500 index is 1, 350, each futures contract is written on 250 units of the index and the dividend yield on the index is approximately 3% per year, i.e., $1, 000 invested in the index yields $30 in dividends at the end of the year. (a) Suppose you invest $1, 350 250 in one-year zero-coupon bonds and at the same time enter into a single futures contract on S&P 500 index with one year to maturity. Assume that in one year the index finishes at 1, 200. What is the total value of your position? How does this compare with buying 250 units of the index and holding them for a year? Assume that in one year the index finishes at 1, 400. Repeat the analysis. (b) If this plan decides to switch to a 70/30 stock/bond mix for a period of one year, how would you implement this strategy using S&P 500 futures? How many contracts with one year to maturity

12. Spot price for soybean meal is $152.70 per ton and the 12-month soybean-meal futures is traded at $148.00. The 1-year interest rate is 3%. (a) What is the net convenience yield on soybean meal for the 12 month period? (b) You need 1,000 tons of soybean meal in 12 months. How would you lock into a price today using the futures contracts? (The size for each soybean-meal futures contract is 100 tons.) 13. The spot price for smoked salmon is $5,000 per ton and its six-month futures price is $4,800. The monthly interest rate is .0025 (.25%). (a) What is the average monthly net convenience yield on smoked salmon for the next six months? (b) If you are a manager of Bread&Circus and need 10 tons of smoked salmon in six months. How can you avoid the risk in the price of smoked salmon over the next six months using futures? (c) Suppose that your net convenience yield for smoked salmon is 1.2%. How does this change your hedging strategy? 14. A wine wholesaler needs 100,000 gallons of Cheap Chardonnay for delivery in Boston in June 2007. A producer offers to deliver the wine at that time for $500, 000 paid now, in December 2006. The wholesaler can also buy Cheap Chardonnay futures contracts for June 2007. The current futures price is $51, 000 for each 10,000 gallon futures contract. The wholesaler is determined to lock in the cost of the 100,000 gallons needed in June. (a) The wholesaler considers the futures contract, but worries that the contract will not lock in her cost, because futures prices may fluctuate widely between now and June. Is her concern justified? Why or why not? (b) Do you recommend that the wholesaler pay the producer now or take a long position in Chardonnay futures? (Additional assumptions may be needed to answer. Make sure they are reasonable.) Explain briefly.