OF UNIVERSITY TORONTO Joseph L. Rotman School of Management Oct. 21, 2008 RSM332 Ezer/Kan/Florence Pomorski/Zhou
MID-TERM EXAMINATION
DURATION - 2 hours Aid Allowed: Silent electronic calculator and one 1-sided 8 1 11 crib sheet 2 Name: Circle the section that you are registered in: Ezer Kan (Mon.) Pomorski (Tue.) Zhou Florence (10a.m.12p.m.) Kan (Tue. 9a.m.11a.m.) Pomorski (Wed. 2p.m.4p.m.) Florence (12p.m.2p.m.) Kan (Tue. 1p.m.3p.m.) Pomorski (Wed. 4p.m.6p.m.) Student Number:
Instructions
1. Write all your answers on the examination paper. 2. Answer ve out of six questions. Each question is worth 20 marks. Do not answer all six questions! In the table below, cross out the question that you choose not to answer. Question 1 2 3 4 5 6 Marks
Total
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1. Consider two consumers, Mr. C. E. Oh and Mrs. O. Ner. Mr. Ohs utility function 1 1 1 3 is UO (C0 , C1 ) = C02 C12 and Mrs. Ners utility function is UN (C0 , C1 ) = C04 C14 . They each receive labor income of $1,000 today and nothing next period. Moreover, there is a factory: a production opportunity that transforms input invested 1 today, I0 , into the output of f (I0 ) = 30I02 next period. A capital market is not available. (a) Mr. Oh is managing the factory. What is the optimal investment implied by his own utility function and income? (5 points) (b) Mrs. Ner is the owner of the factory. If she invests exactly as what Mr. Oh suggested in part (a), what will be her utility from consumption today and next period? (2 points) (c) If Mrs. Ner instead chose the optimal level of investment according to her own utility function and income, how much would she invest? What would be her utility? (4 points) (d) We want to compensate Mrs. Ner for having to separate ownership and control. Suppose that Mrs. Ner lets Mr. Oh tell her how much to invest (as in part (a)). By how much would her consumption today, C0 , need to go up (keeping C1 at the same level as in part (b)) so that her utility is the same as in part (c)? (4 points) (e) Now suppose that there is a capital market that Mr. Oh and Mrs. Ner can lend and borrow at the interest rate of 3.33%. Again, let Mr. Oh manage the factory and tell Mrs. Ner how much to invest. (Mr. Oh will only tell Mrs. Ner what I0 should be. Mrs. Ner will choose how much to lend or borrow and how much to consume.) Suppose we want to compensate Mrs. Ner for the separation of ownership and control. By how much would her consumption today need to go up for her to attain the same level of utility as she would if she chose I0 according to her own preferences? (5 points) 2. (a) You plan to make six equal annual deposits of $5,000. The rst deposit will be made one year from now. A bank oers two savings accounts with interest rates as follows: Account A: stated annual rate of 12% with semi-annual compounding Account B: stated annual rate of 11.75% with continuous compounding Which account would you choose? Why? If you keep the deposits in the chosen account, how much money will you have 20 years from now? (6 points) (b) Your brother plans to enter university next year and now asks for your suggestion on student loans. In order to pay the tuition, he plans to borrow $25,000 a year from now, $20,000 two years from now, another $20,000 three years from now, and $30,000 in four years from today. He plans to work after graduation and pay back his loan in 15 years (with the rst payment to be paid six years from now, and the last payment to be paid 20 years from now). He expects his salary to increase by 4% each year, so he plans to increase his annual repayment at the same rate. Assume the interest rate is 10%/year, compounded annually. 2
(i) What is the balance of his student loan upon his graduation (i.e., ve years from today)? (4 points) (ii) What is the amount of your brothers rst payment for his student loan? (5 points) (iii) In your brothers last payment, how much of it is for interest and how much of it is for principal repayment? (5 points) 3. (a) John, who just celebrated his 25th birthday, always fancied the number 3. He therefore decided that on his 30th he birthday will begin saving for retirement by placing $3,000 into a retirement savings account. He plans to continue to save $3,000 every 3 years thereafter. (i.e.,$3,000 at age 30, $3,000 at age 33, $3,000 at age 36, etc, with the last saving occurring on his 63rd birthday). Beginning on his 66th birthday, John intends to draw an equal payment 3 times a year (i.e., every 4 months) forever, with the rst payment to be drawn on his 66th birthday. Suppose the stated annual rate is 8% per year, compounded quarterly. What is the periodic amount that John can withdraw after his retirement? (12 points) (b) Someone oers you a security which pays $100 at the end of the rst year but the payment declines by $1 every year afterward until it reaches zero (i.e., it pays $99 at the end of the second year, $98 at the end of the third year, all the way to $1 at the end of the 100th year) If the annually compounded interest rate is 10% per year, what is the fair price of such a security? (8 points) 4. (a) You are a bond trader for an investment bank. A client calls you up and asks for the quote of an annuity which pays o $1,000 each year in the next two years. Although no such annuity exists, you notice that a 10% coupon bond with face value of $1,000 and matures next year costs $1,016.76. You also nd a 5% coupon bond with face value $1,000 and matures in two years costs $929.22. The coupons on both bonds are paid on an annual basis. Show how you can create the annuity for your client and at what price should you quote him for you to breakeven? (10 points) (b) Someone shows you a term structure of spot interest rates with the 1-year spot rate is 4%/year, the 2-year spot rate is 10%/year, and the 3-year spot rate is 6%/year. Compute the implied forward rates for the second year and third year. (4 points) (c) What is wrong with this term structure of interest rates in part (b)? If this is indeed the actual term structure of interest rates, show how you can make an arbitrage prot. Assume 1-year to 3-year pure discount bonds are available and they are priced based on the spot interest rates. (6 points) 5. The current term structure is at: the spot interest rates for all maturities are the same and equal to 4%/year. (a) What is the annual coupon payment of a bond that trades at par and has the yield to maturity equal to the interest rate, 4%? The bond has a face value of $1,000 and 3
pays annual coupons. (4 points) (b) What is the price of an annual coupon bond with a face value of $1,000, a coupon rate of 10%, and 30 years to maturity? (4 points) For parts (c) and (d), we assume the term structure of interest rates are not at. The one year spot rate, r1 , is 6.01%/year, the two-year and the three-year spot rates, r2 and r3 , are both 5%/year. (c) What are the forward interest rates for the second and the third year? (4 points) (d) Consider a zero-coupon bond with the face value of $1,000 and three years to maturity. You are planning to buy this bond in one year and hold it for two years (until its maturity). If the expectations hypothesis holds, what holding period return do you expect to realize on this transaction? (Compute the holding period return over the entire two year holding period.) (8 points) 6. Consider an all-equity rm with 50,000 shares outstanding and a required rate of return of 12%/year. (a) What is the total value of the rm if it expects to earn $25,000/year in perpetuity, starting in 1 year from today, if it does not take on any new projects. (5 points) (b) The rm has an unusual investment opportunity, which requires it to invest $10,000 today, and make 9 more annual investments, at the end of each year for 9 more years, each investment decreasing by 15% from the previous year. In return, the investments expect to generate an additional $15,000 at the end of the 10th year, with additional annual cash ows growing after that, in perpetuity at 5%/year. What is the value of the investment opportunity? (10 points) (c) What is the value of the rm, and the price per share of the rm, with the investment opportunity? (5 points)
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