Name:______
Midterm Exam
MBAC 6060 - Eve
Fall 2006
This exam will serve as the answer sheet. You should have enough room; however, if you require more space in which to write your answers I have additional paper at the front of the room. There are 8full problems (some with multiple parts) on this exam; be sure you are aware of them all. If you would like to have the possibility of partial credit for any of the questions, be sure to show how you developed the answers rather than simply reporting anumerical answer. You have two hours for this exam. Assume all interest rates are given on a stated annual basis and that compounding is done annually unless otherwise explicitly stated for a given problem.
(1) (10 points) You are able to borrow or lend from your bank at a rate of 6%.
(a) What is a promise of receiving $425 in two years worth to you today?
$425/(1.06)2 = $378.25
(b) What is a promise of receiving $750 in twelve years worth to you today?
$750/(1.06)12 = $372.73
(c) Explain why these are the answers to (a) and (b).
Simply because this is the amount you would need to put in the bank at 6% in order to generate the stated amounts at the stated points in time. For example, if you put $186.36 in the bank today you would receive $375 in twelve years; thus, receiving $375 in twelve years must have a present value of $186.36.
(2) (15 points) You have the opportunity to buy a machine that costs $290,000 today. It will generate cash flow of $85,000 at the end of each year for the next five years. The machine requires start up costs of $15,000 today and maintenance costs of $10,000 at the beginnings of years 2, 3, 4, and 5. If you can borrow and lend at 12% should you buy this machine? Why? What should you do if the appropriate discount rate is 8%?
Only thing to this problem is the timing of the cash flows. The $10k costs are at times 1 thru 4. So: NPV(@12%) = -$290,000 -$15,000 +($85,000-$10,000)/1.12 + ($75,000)/1.122 + ($75,000)/1.123 + ($75,000)/1.124 + ($85,000)/1.125 = -$28,967.52 This means you should not invest since to do so would be equivalent to throwing away $28,967.52.
NPV(@8%) = -$295,000 -$15,000 +($85,000-$10,000)/1.08 + ($75,000)/1.082 + ($75,000)/1.083 + ($75,000)/1.084 + ($85,000)/1.085 = $1,259.08 The answer is now yes.
(3) (15 points) You wish to buy a new BMW four years from now. The BMW M5 you have your eye on will cost $214,000 at that time. You currently have $59,000. What annual rate of return must you receive on this investment for each of the four years if your savings is to generate enough for you to buy the car?
Think about the problem as $59,000(1+r)4 = $214,000 then r = ($214,000/$59,000)1/4 – 1. The return would need to be 38.004%. Fat chance, think Yugo.
(4) (20 points) The bank offers to borrow and lend at an 8% stated annual basis where the interest is to be compounded monthly. Find
(a) The future value in 5 years of $2,100 invested today.
Calculate the monthly interest rate from the stated annual rate by dividing by 12 then: $2,100(1+.00667)60 = $3,129.30 rounding may account for small differences from this value
(b) The present value of $50,000 that will be received in 6 ½ years.
$50,000/(1.00667)78 = $29,769.68
(c) What is the present value of an annuity that will make 7 annual payments to you where the first payment of $100 will be received in one year and the payments grow at a rate of 2% per year?
Here you need to use the growing annuity formula. The difficulty is that since the alternative investment (the bank) offers monthly compounding we need to use the effective annual rate to value the growing annual annuity. The answer using the growing annuity formula with r = .083, g = .02, T = 7, and C = $100 is $543.87.
(d) If the compounding were done quarterly instead of monthly would the answers to (a) and (b) be larger or smaller?
The answer to (a) would be smaller and the answer to (b) would be larger.
(5) (15 points) Pricing Equity
(a) Ahlburg Inc. paid its annual dividend yesterday in the amount of $3.00 per share. The dividends are expected to grow at a constant rate of 4% per year forever and the market requires a 17% return on the company’s stock. What is today’s price per share of Ahlburg stock? What will the price of Ahlburg stock be in 5 years?
Value the stock as a growing perpetuity where the first payment received in a year is $3.12 so $3.12/(0.17 – 0.04) = $24.00, note the $3 dividend paid yesterday is immaterial for today’s price of the stock (only cash flows to be received by the purchaser affect the price. In five years we will have just missed the year five dividend but we still have a perpetuity growing at 4%. The important point is that the payment to be received at year six will be $3.80 so the value will be $3.80/0.13 = $29.23.
(b) Blob Inc. is expected to pay a dividend of $1.00 per share in one year. It plans to pay dividends annually. The market expects its dividends to grow at 50% per year for the three years after the $1.00 is paid. After that the expectation is for dividend growth to be 2% per year. What is the current price of a share of Blob stock if the market requires a 15% return on the stock?
Here we have a stock that can be valued as a growing perpetuity for its “terminal value” and we have a four year growing annuity for the first four years of its life. The dividend in year 5 is expected to be $1(1.5)3(1.02) = $3.4425, so the time 4 (why time 4?) value of the growing perpetuity is $3.4425/(0.15 – 0.02) = $26.48. Discount this to time zero to find $15.14. Now add the four years of rapid growth discounting each at 15% which is $5.41 for a total price of $20.55.
(6) (20 points) Bond Pricing (Assume semi-annual compounding for all parts of this question.)
(a) What is the current price of a pure discount bond that matures in 17 years and has a face value of $1,000 if the current interest rate is 12%?
The answer can be found as $1,000/(1.06)34 = $137.91.
(b) What is the current price of a standard $1,000 face value coupon bond that will mature in 9 ½ years and has a coupon interest rate of 6.4% if the spot rate for all investment horizons is 9% per year?
Here we use the fact that the bond can be seen as a zero coupon bond and an annuity. The annuity formula, using r = 4.5%, C = $32, and T = 19 gives $402.99. The value of the return of the $1,000 face value in 9 ½ years is $1,000/(1.045)19 = $433.30. Therefore the bond is worth $433.30 + $402.99 = $836.29 it trades at a discount because the coupon interest rate is lower than the spot rate.
(7) (20 points) Your firm is evaluating the following R&D project. You are certain that an investment of $1 Million per year for the next 4 years (payments made at the beginning of each of the 4 years) will enable you to bring a new, more efficient manufacturing process on line. The efficiency gain will enable your firm to realize a cost saving of $1 Million per year in perpetuity. Due to engineering and implementation lags the cost savings is not expected to begin to be realized until 6 years from today. You estimate that a discount rate of 16% is appropriate for evaluating the investment.
(a) What is the NPV of the investment?
A perpetuity of $1 Million per year at a discount rate of 16% has a value of $6.25 Million but this is a value at the end of year 5. The present value of this is $6.25 M/(1.16)5 = $2.976 Million. The present value of the initial costs equals: $1 M + $1 M/(1.16) + $1 M/(1.16)2 + $1 M/(1.16)3 = $3.246 Million. Therefore the NPV is $2.976 M – $3.246 M = -$0.27 M or about - $270,000.
(b) Interpret this number.
Undertaking this investment is equivalent to flushing $270,000 in cash. More precisely, firm value would decrease by exactly this amount if the project were undertaken. Recall the net present value is today’s dollar value of the change in wealth from making the investment.
(c) What estimate is the NPV very sensitive to?
One obvious thing for this setup is the discount rate. The benefits are very heavily discounted and the costs relatively lightly. If the discount rate is a little lower, the value is much higher. For example the NPV at 15% (a relative reduction of about 6%) is $3.315 M - $3.283 M = +$32,000.
(8)(10 points) A ten year zero coupon bond with a face value of $1,000 is being issued today by Ralph Inc. The ten year spot rate is currently 6%.
(a) What is the present value of the future payment associated with the bond?
Recall that the bond markets use semi-annual discounting so the present value of this bond is given as PV = $1,000/(1.03)20 = $553.68
(b) What will Ralph be able to sell this bond for? Explain your answer.
Ralph should be able to sell the bond for $553.68 due to competition in the financial markets. Consider the answer to problem 7b. If the price of the bond were less than $553.68 then it would have a positive net present value and so represent a wealth creating opportunity for any investor who just buys it. As such for any price below this present value there will be high demand for the bond driving its price up. For any price above the present value of the bond’s payment purchasing the bond is a negative NPV opportunity and so no investor would be interested in purchasing it, driving the price down.