Name:_________________



Name:_________________

First Midterm Exam

MBAC 6060

Fall 2004

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. There are 4 full 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 a numerical answer. You have an hour and fifteen minutes which suggests you should budget about 15 minutes per question so as to have time to check your answers at the end. Assume all interest rates are given on a stated annual basis unless otherwise explicitly identified.

(1) (35 points)

(a) If the interest rate is 7.5% what is the present value of $8,600 to be received one year from today? One month from today?

The stated annual rate is 7.5% so the present value of $86,000 received in one year is $86,000/1.075 = $8.000. When the $86,000 is to be received in one month we need to use the stated annual rate to find the monthly rate (.075/12 = .00625) so the present value is $86,000/1.00625 = 8,546.58

(b) If the interest rate is 9% what will be the ending balance in a savings account that compounds interest quarterly if you invest $10,000 for 18 months? If you invest for one year?

With a stated annual rate of 9% the quarterly rate is .09/12 = .0225. Then if you will invest $10,000 for 18 months or 6 quarters the future value will be $11,428.25 = $10,000(1.0225)6. If you invest for a year this is 4 quarters so the future value is (1.0225)4($10,000) = 10,930.83 and importantly is not $10,900 = $10,000(1.09).

(c) What is the present value of receiving $11,000 one year from now, $24,200 two years from now, and $39,930 three years from now if the interest rate is 10%. Show that having the present value of this cash flow stream available to invest today can generate exactly the same cash flow stream over these three years.

The present value is $60,000 = $11,000/(1.1) + $24,200/(1.1)2 + $39,930/(1.1)3.

Now note that if you have $60,000 today and can invest at 10% in one year you will have $66,000. Making a payment of $11,000 leaves $55,000 invested after 1 year. Investing that for a year at 10% gives you $60,500 and making a $24,200 payment leaves $36,3000 after 2 years. Investing this for a third year provides $39,930 at the end of the third year just satisfying the final payment so there is nothing left.

(d) One year from now your Uncle Ralph has promised to give you $90,000. While this pending gift represents all your existing wealth, being a student in this class you realize that you don’t have to wait till next year to spend money. If the interest rate is 4% what is the present value of the gift? If you want to spend the same dollar amount today and in one year how much can you spend today?

Can work with present values or future values on this problem. The present value of the gift is $90,000/1.04 = 86,538.46. Now you want to be able to spend the same dollar amount now and in one year. That means the present value of the two expenditures must just equal the $86,538.46. Thus $86,538.46 = X + X/1.04 solving for X = $44,117.65. Note for me to spend $44,117.65 now I borrow this amount and then must payback $45,882.35 next year leaving $90,000 - $45,882.35 = $44,117,65 for me to spend from the gift after repayment of the loan.

(e) What is the current value of an 8 year annuity whose first payment of $100 arrives one year from today, if the subsequent payments all grow at 6% per year and the interest rate is 4%? What is the value if the $100 dollars is instead the first payment of a perpetual cash flow stream that grows at 6% per year?

Using the annuity formula (which is valid) you find the PV is $823.05. A perpetuity growing at a rate larger than the current discount rate has infinite present value.

(2) (35 points) The current term structure of interest rates is described as follows. For one and two year investments the interest rate is 5%. For three year investments the interest rate is 5.5%. For four and five year investments the interest rate is 6%.

(a) You are thinking of buying a standard level coupon bond which has semi-annual coupons, the first arrives in exactly six months. The coupon rate is reported as 5%, the bond has 5 years till maturity and a face value of $1,000. What should you be willing to pay for this bond?

You value the bond by simply discounting the payments at the appropriate interest rates, which happen to differ here according to the given term structure of interest rates. Also recall the interest rates are on a stated annual basis and you need to use semiannual compounding. Discounting the coupons and face properly the value is 959.042. Properly means that you discount the first 4 coupons at 2.5%, the next two at 2.75% and the final coupons and the face at 3% per semiannual period. So for example the last term is 1025/(1.03)10 and the fifth coupon has a present value of 25/(1.0275)5, etc.

(b) For the bond given in part (a) what is its yield to maturity if the bond is currently trading at the value you calculated in part (a)?

Taking the calculated 959.042 as the price the yield to maturity equation tells us what constant discount rate sets the PV of the payments equal to this price. That rate is 5.96% when put on a stated annual basis.

(c) A level coupon bond with a yield to maturity of 8%, 18 and a half years to maturity, and a face value of $1,000 currently sells at par. What is its coupon rate?

Trivial it must be 8% if the bond sells at par and the YTM is 8%.

(3) (25 points) You are planning for your retirement. You will begin making annual withdrawals from your savings account twenty one years from today. You anticipate that this first withdrawal must be equal to $500,000 in order to keep you in the manner to which you have become accustom. Being squeamish about such things you are unwilling to contemplate death so your plan includes making the same withdrawal each year forever. Prior to your retirement your savings must also be used to finance your only child’s college education. Your child (whom you now refer to as Lucifer) will enter college three years from today and the first of four annual tuition payments, each in the amount of $25,000, must be made on that day. The interest rate at which you can borrow and lend is 7% and compounding in the account is done monthly.

(a) You currently have $10,000. Starting two years from now you will be able to make deposits into the account of $20,000 each year for 5 years. Seven years from now how much must you deposit into the account in order to fulfill your obligations if you plan to make annual deposits, the final one occurring twenty years from now, and plan for the annual deposits to grow at a 3% rate each year? What will be the size of the last deposit you make in twenty years?

First note that we need to use an effective annual discount rate for all the discounting. This is (1 + .07/12)12 – 1 = .0723 = 7.23%. Now we take the present value of the two obligations. First the perpetual withdrawals starting in year 21 have a time 20 value of $500,000/.0723 = $6,915,629.32 and so a present value of $6,915,629.32/(1.0723)20 = $1,712,007.20, a pittance. The college tuition payments comprise an annuity with four annual payments of $25,000 and an effective rate again of 7.23%. The annuity formula gives you a value of $84,242.45 which is a time two value. Discounting this back for two years at the 7.23% gives $73,265.31. The present value of the total obligations for the account is the sum of these two numbers. $1,712,007.20 + $73,265.31 = $1,785,272.51. We need to ensure that the present value of our deposits equals this amount. First we already have $10,000 in the account this is of course a present value of $10,000. Next we have, beginning at time 2, five annual deposits of $20,000. The value the annuity equation gives us for this set of deposits is $81,501.41, a time 1 value with a present value of $76,006.16. The sum of these two contributions is $86,006.16. The obligations still are larger than the contributions by $1,785,272.51 - $86.006.16 = 1,699,266.35. We now need to make sure that the present value of the growing annuity (which is the form of the rest of the contributions is exactly this large. Set the PV of the growing annuity equal to this amount and solve for the first deposit. The first deposit must be $253,645.32 and the last will be $253,645.32(1.03)13 = 372,486.71.

(b) Will you be able to pay for your child’s college education using your savings account? Does the answer to this question alter your response to part (a)?

Yes you can and no it does not and it also would not if you couldn’t cover the tuition from the account. The present value of any loan would be zero so it would not alter the deposits you need to make into the account.

(4) (30 points) What is the price per share?

(a) Dimeff Inc. just paid out dividends totaling $6 million dollars to the owners of their 10,000,000 shares of stock. For the next 7 years this total dividend payment is expected to grow 9% per year. After that time the total dividend payment is expected to grow by 1% per year forever. If the interest rate is 14% what is the price per share of Dimeff Inc. stock?

Take the total dividend and divide by the number of shares to find a $0.6 dividend per share was just paid. The time 1 dividend will be 9% higher than that or $0.654. Now the value of a growing annuity for 7 years, with this initial dividend is $3.52437, this is the present value of the first 7 dividends paid. Starting in year 8 you have a growing perpetuity. The first payment is $1.10779 per share. Using the growing perpetuity formula gives a value of $8.52146, but this is a time 7 value. Discounting this for 7 years at 14% gives $3.40549. This is the present value of the dividend stream from year 8 onward. The sum of these two values represents the present value of the total stream of dividends: $3.52437 + $3.40549 = $6.93 per share.

(b) TrailRidge Inc. is expected to have dividends of $3 per share paid at the end of each year forever. A stated annual interest rate of 12% is appropriate, however, available instruments in the capital markets all compound interest quarterly.

Here we have a level perpetuity to value but the only thing is that the opportunity cost of capital is based upon quarterly compounding. A stated 12% becomes an effective annual 12.55% = (1 + 12/4)4 – 1. At the effective rate the value per share is given as P = $3/(.1255) = $23.90, remember the coupons are annual not quarterly.

(c) Firm X has 100 shares outstanding, no current liabilities, yet has enough risk that a 20% discount rate is appropriate. At the end of each of the next five years X is expected to generate after tax operating cash flows of $25,000 of which (for the first four years) $5,000 must be reinvested for maintenance of existing machinery to enable the generation of the following year’s cash flow. At the end of the five years due to the increased competition and the lack of continued investment there will be no value left in the firm. At the beginning of the next year X is planning to sell a AA coupon bond in order to raise $3,000 in extra cash. The bond is expected to sell at par with a coupon rate of 8%. The maturity of the bond is 3 years.

Here we are valuing the operations of this firm. There are no growth options so NPVGO is zero. No existing liabilities so PVL is zero. The capital market is competitive so NPVF is zero and all the firm’s value is in PVA. This is equal to $20,000 in each of the first four years and $25,000 in the fifth. Discounting at the 20% rate gives $61,821.63 in total, divided by 100 shares gives a price per share of $618.22. The competitive capital markets assumption means that all the nonsense about the planned bond sale is a zero NPV transaction and does not change current value at all so there is no need to waste time considering it.

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