Exercise Session #1 Suggested Solutions

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1 JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Date: 3/10/2017 Exercise Session #1 Suggested Solutions Problem The continuously compounded interest rate is 12%. a You invest $1,000 at this rate. What is the investment worth after five years? b What is the P V of $5 million to be received in eight years? c What is the P V of a continuous stream of cash flows, amounting to $2,000 per year, starting immediately and continuing for 15 years? Solution: a F V = Ce rt = e = $1, b P V = Ce rt = 5e = $1.914 million. c P V = C r 1 2, 000 e rt = e = $13, 912. Problem Halcyon Lines is considering the purchase of a new bulk carrier for $8 million. The forecasted revenues are $5 million a year and operating costs are $4 million. A major refit costing $2 million will be required after both the fifth and tenth years. After 15 years, the ship is expected to be sold for scrap at $1.5 million. If the discount rate is 8%, what is the ship s NP V? Solution: We can break this down into several different cash flows, such that the sum of these separate cash flows is the total cash flow. Then, the sum of the present values of the separate cash flows is the present value of the entire project. Cost of the ship is $8 million. P V = $8 million. Revenue is $5 million per year, operating expenses are $4 million. Thus, operating cash flow is $1 million per year for 15 years. $1 million P V = = $8.559 million Major refits cost $2 million each, and will occur at times t = 5 and t = 10. P V = $2 million $2 million = $2.288 million Sale for scrap brings in revenue of $1.5 million at t = 15. P V = $1.5 million = $0.473 million. 1

2 Adding these present values gives the net present value of the entire project: N P V = $8 million$8.559 million $2.288 million$0.473 million = $1.256 million. Problem David and Helen Zhang are saving to buy a boat at the end of five years. If the boat costs $20,000 and they can earn 10% a year on their savings, how much do they need to put aside at the end of years 1 through 5? Solution: Assume that Zhangs will put aside the same amount each year. One approach to solving this problem is to find the present value of the cost of the boat and then equate that to the present value of the money saved. From this equation, we can solve for the amount to be put aside each year. P V boat = P V savings = Because P V savings must equal P V boat: Annual savings 0.1 Annual savings = $20, 000 = $12, Annual savings $12, 418 1/ / = $12, 418. = $3, 276. Another approach is to use the future value of an annuity formula: Annual savings Annual savings = $3, 276. = $20, 000. Problem If the interest rate is 7%, what is the value of the following three investments? a An investment that offers you $100 a year in perpetuity with the payment at the end of each year. b A similar investment with the payment at the beginning of each year. c A similar investment with the payment spread evenly over each year. Solution: a This is the usual perpetuity, and hence: P V = C r = $ = $1, b This is worth the PV of stream a plus the immediate payment of $100: P V = $100 $1, = $1,

3 c The continuously compounded equivalent to a 7% annually compounded rate is approximately 6.77%, because: e = Thus: P V = C r = $100 = $1, Note that the pattern of payments in part b is more valuable than the pattern of payments in part c. It is preferable to receive cash flows at the start of every year than to spread the receipt of cash evenly over the year; with the former pattern of payment, you receive the cash more quickly. Problem A leasing contract calls for an immediate payment of $100,000 and nine subsequent $100,000 semi annual payments at six month intervals. What is the P V of these payments if the annual discount rate is 8%? Solution: Because the cash flows occur every six months, we first need to calculate the equivalent semi annual rate. Thus, 1.08 = 1 r /2 2 r = 7.85% semi annually compounded annual discount rate. Therefore, the rate for six months is 7.85 /2 = 3.925%: P V = $100, 000 $100, = $846, 081. Problem Dear Financial Adviser, My spouse and I are each 62 and hope to retire in three years. After retirement we will receive $7,500 per month after taxes from our employers pension plans and $1,500 per month after taxes from Social Security. Unfortunately our monthly living expenses are $15,000. Our social obligations preclude further economies. We have $1,000,000 invested in a high grade, tax free municipal bond mutual fund. The return on the fund is 3.5% per year. We plan to make annual withdrawals from the mutual fund to cover the difference between our pension and Social Security income and our living expenses. How many years before we run out of money? Sincerely, Luxury Challenged Marblehead, MA You can assume that the withdrawals one per year will sit in a checking account no interest. The couple will use the account to cover the monthly shortfalls. Solution: In three years, the balance in the mutual fund will be: F V = $, = $1, 108, 718. The monthly shortfall will be: $15, 000 $7, 500 $1, 500 = $6, 000. Annual withdrawals from the mutual fund will be: $6, = $72,

4 Assume the first annual withdrawal occurs three years from today, when the balance in the mutual fund will be $1,108,718. Treating the withdrawals as an annuity due a level stream of payments starting immediately, we solve for t as follows: P V = C r. r 1 r t $1, 108, 718 = As a result, we find that t = 22.5 years. $72, t Problem A six year government bond makes annual coupon payments of 5% on $1,000 and offers a yield of 3% annually compounded. Suppose that one year later the bond still yields 3%. What return has the bondholder earned over the 12 month period? Now suppose that the bond yields 2% at the end of the year. What return would the bondholder earn in this case? Solution: Purchase price for a 6 year government bond with 5 percent annual coupon: P V = 50 = $1, Price one year later yield = 3%: P V = Rate of return = Price one year later yield = 2%: P V = = $1, $50 $1, $1, $1, = $1, = 0.03 or 3%. Rate of return = $50 $1, $1, $1, = or 7.49%. Problem A 6% six year bond yields 12% and a 10% six year bond yields 8%. The face value of each bond is $1,000. Calculate the six year spot rate. Assume annual coupon payments. Hint: What would be your cash flows if you bought % bonds? Solution: The key here is to find a combination of these two bonds i.e., a portfolio of bonds that has a cash flow only at t = 6. Then, knowing the price of the portfolio and the cash flow at t = 6, we can calculate the 6 year spot rate. We begin by specifying the cash flows of each bond and using these and their yields to calculate their current prices: 4

5 Investment Yield C 1 C 5 C 6 Price 6% bond 12% ,060 $ = % bond 8% ,100 $1, = From the cash flows in years one through five, we can see that buying two 6% bonds produces the same annual payments as buying 1.2 of the 10% bonds. To see the value of a cash flow only in year six, consider the portfolio of two 6% bonds minus % bonds. This portfolio costs: 2 $ $1, = $ The cash flow for this portfolio is equal to zero for years one through five and, for year 6, is equal to: 2 $1, $1, 100 = $800. Thus, $ = $800 1 r 6 6 r 6 = or 26.45%. Problem Find the arbitrage opportunity opportunities?. Assume for simplicity that coupons are paid annually. In each case the face value of the bond is $1,000. Bond Maturity years Coupon, $ Price, $ A B C , D E , F , G Solution: Arbitrage opportunities can be identified by finding situations where the implied forward rates or spot rates are different. We begin with the shortest term bond, Bond G, which has a two year maturity. Since G is a zero coupon bond, we determine the two year spot rate directly by finding the yield for Bond G. The yield is 9.5 percent $834 = $1,000 /1r 2 2, so the implied two year spot rate r 2 is 9.5 percent. Using the same approach for Bond A, we find that the three year spot rate r 3 is 10 percent $751.3 = $1,000 /1r 3 3. Next we use Bonds B and D to find the four year spot rate. The following position in these bonds provides a cash payoff only in year four: a long position in two of Bond B and a short position in Bond D. Cash flows for this position are: 2 $842.3 $ = $ today, 5

6 2 $50 $100 = $0 in years 1, 2, and 3, 2 $1, 050 $1, 100 = $ in year 4. We determine the four year spot rate from this position as follows: $ = $ 1 r 4 4 r 4 = or 9.17%. Next, we use r 2, r 3 and r 4 with one of the four year coupon bonds to determine r 1. For Bond C: $1, = $120 $120 1 r $120 $1, = $120 $ r 1 r 1 = or 38.67%. Now, in order to determine whether arbitrage opportunities exist, we use these spot rates to value the remaining two four year bonds. This produces the following results: for Bond B, the present value is $854.55, and for Bond D, the present value is $1, Since neither of these values equals the current market price of the respective bonds, arbitrage opportunities exist. Similarly, the spot rates derived above produce the following values for the three year bonds: $1, for Bond E and $ for Bond F. Problem a What spot and forward rates are embedded in the following Treasury bonds? The face value of Treasury bond is $100. The price of one year strip is $ Assume for simplicity that bonds make only annual payments. Hint: Can you devise a mixture of long and short positions in these bonds that gives a cash payoff only in year 2? In year 3? Maturity years Coupon $ Price $ b A three year bond with a coupon $4 is selling at $95. Is there a profit opportunity here? If so, how would you take advantage of it? Solution: a We make use of the one year Treasury bill information in order to determine the one year spot rate as follows: $93.46 = $100 1 r 1 r 1 = 0.07 or 7%. The following position provides a cash payoff only in year two: Cash flows for this position are: a long position in twenty five two year bonds and a short position in one one year Treasury bill. 25 $94.92 $93.46 = $2, today, 6

7 25 $4 $100 = $0 in year 1, 25 $104 = $2, 600 in year 2. We determine the two year spot rate from this position as follows: $2, = The forward rate f 2 is computed as follows: $2, r 2 2 r 2 = or 6.8%. f 2 = 1 r = = or 6.6%. 1 r The following position provides a cash payoff only in year three: a long position in thirteen three year bond and a short position equal to a package consisting of a one year Treasury bill and a two year bond. Cash flows for this position are: 13 $ $93.46 $94.92 = $1, today, 13 $8 $100 $4 = $0 in year 1, 13 $8 $104 = $0 in year $108 = $1, 404 in year 3. We determine the three year spot rate from this position as follows: $1, = The forward rate f 3 is computed as follows: $1, r 3 3 r 3 = or 6.6%. f 3 = 1 r r 2 1 = = or 6.2% b We make use of the spot and forward rates to calculate the price of the three year bond with a coupon $4: P = $4 $4 1 r 1 1 r 1 1 f 2 $104 1 r 1 1 f 2 1 f 3 = = $ $ $ = $ The actual price of the bond $95 is significantly greater than the price deduced using the spot and forward rates embedded in the prices of the other bonds $ Hence, a profit opportunity exists. In order to take advantage of this opportunity, one should sell the 4 percent coupon bond short and, for example, purchase the 8 percent coupon bond. 7