Homework #1 Suggested Solutions

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1 JEM034 Corporate Finance Winter Semester 207/208 Instructor: Olga Bychkova Problem. 2.9 Homework # Suggested Solutions a The cost of a new automobile is $0,000. If the interest rate is 5%, how much would you have to set aside now to provide this sum in five years? b You have to pay $2,000 a year in school fees at the end of each of the next six years. If the interest rate is 8%, how much do you need to set aside today to cover these bills? c You have invested $60,476 at 8%. After paying the above school fees, how much would remain at the end of the six years? C $0, 000 a P V = = = $7, assuming the cost of the car does not + r t.05 5 appreciate over those five years. b You need to set aside $2,000 6 year annuity factor = = $2, 000 = $55, c At the end of 6 years you would have.08 6 $60, 476 $55, 476 = $7, 934. Problem A factory costs $800,000. You reckon that it will produce an inflow after operating costs of $70,000 a year for 0 years. If the opportunity cost of capital is 4%, what is the net present value of the factory? What will the factory be worth at the end of five years? The present value of the 0 year stream of cash inflows is: $70, 000 P V = = $886, Thus: NP V = $800, $886, = $86, At the end of five years, the factory s value will be the present value of the five remaining $70,000 cash flows: $70, 000 P V = = $583, Problem Mike Polanski is 30 years of age and his salary next year will be $40,000. Mike forecasts that his salary will increase at a steady rate of 5% per annum until his retirement at age 60.

2 a If the discount rate is 8%, what is the PV of these future salary payments? b If Mike saves 5% of his salary each year and invests these savings at an interest rate of 8%, how much will he have saved by age 60? c If Mike plans to spend these savings in even amounts over the subsequent 20 years, how much can he spend each year? a P V = C + gt $40, 000 =.0530 = $760, r g + r t b P V salary 0.05 = $38, F uture value = $38, = $382, c P V = C. r + r t $382, = C $382, C = = $38, /0.08 / Problem As winner of a breakfast cereal competition, you can choose one of the following prizes: a $00,000 now. b $80,000 at the end of five years. c $,400 a year forever. d $9,000 for each of 0 years. e $6,500 next year and increasing thereafter by 5% a year forever. If the interest rate is 2%, which is the most valuable prize? a P V = $00, 000. $80, 000 b P V = = $02, $, 400 c P V = = $95, $9, 000 d P V = = $07, $6, 500 e P V = = $92, Prize d is the most valuable because it has the highest present value. Problem Kangaroo Autos is offering free credit on a new $0,000 car. You pay $,000 down and then $300 a month for the next 30 months. Turtle Motors next door does not offer free credit but will give you $,000 off the list price. If the rate of interest is 0% a year, about 0.83% a month which company is offering the better deal? 2

3 The fact that Kangaroo Autos is offering free credit tells us what the cash payments are; it does not change the fact that money has time value. A 0% annual rate of interest is equivalent to a monthly rate of 0.83%: r monthly = r annual 2 = 0. 2 = = 0.83%. The present value of the payments to Kangaroo Autos is: $, $ = $8, A car from Turtle Motors costs $9,000 cash. Therefore, Kangaroo Autos offers the better deal, i.e., the lower present value of cost. Problem Which would you prefer? a An investment paying interest of 2% compounded annually. b An investment paying annual interest of.7% compounded semi annually. c An investment paying.5% compounded continuously. Work out the value of each of these investments after, 5, and 20 years. Assume the amount invested is one dollar. Let A represent the investment at 2%, compounded annually. Let B represent the investment at.7%, compounded semi annually. Let C represent the investment at.5%, compounded continuously. After one year: F V A = $ = $.200. After five years: After twenty years: The preferred investment is C. F V B = $ = $.204. F V C = $ e 0.5 = $.29. F V A = $ = $ F V B = $ = $ F V C = $ e = $.777. F V A = $ = $ F V B = $ = $ F V C = $ e = $ Problem Several years ago The Wall Street Journal reported that the winner of the Massachusetts State Lottery prize had the misfortune to be both bankrupt and in prison for fraud. The prize was $9,420,73, to be paid in 9 equal annual installments. There were 20 installments, but the winner had already received the first payment. The bankruptcy court judge ruled that the prize should be sold off to the highest bidder and the proceeds used to pay off the creditors. 3

4 a If the interest rate was 8%, how much would you have been prepared to bid for the prize? b Enhance Reinsurance Company was reported to have offered $4.2 million. Use Excel to find the return that the company was looking for. a Each installment is: $9,420,73 /9 = $495, 827. P V = $495, b If ERC is willing to pay $4.2 million, then: $4, 200, 000 =.08 9 $495, 827 r = $4, 76, r 9 Using Excel or a financial calculator, we find that r = 9.8%. Problem You estimate that by the time you retire in 35 years, you will have accumulated savings of $2 million. If the interest rate is 8% and you live 5 years after retirement, what annual level of expenditure will those savings support? Unfortunately, inflation will eat into the value of your retirement income. Assume a 4% inflation rate and work out a spending program for your retirement that will allow you to increase your expenditure in line with inflation. This is an annuity problem with the present value of the annuity equal to $2 million as of your retirement date, and the interest rate equal to 8% with 5 time periods. Thus, your annual level of expenditure C is determined as follows: P V = C r $2, 000, 000 = C 0.08 C = $2, 000, 000 /0.08 / r t = $233, 659. With an inflation rate of 4% per year, we will still accumulate $2 million as of our retirement date. However, because we want to spend a constant amount per year in real terms R, constant for all t, the nominal amount C t must increase each year. Therefore, we end up with growing annuity, where growth rate is equal to inflation rate: R.045 = $2, 000, R = $77, 952. Alternatively, consider that the real rate is /+0.04 = Then, redoing the steps above using the real rate gives a real cash flow equal to: C = $2, 000, 000 / / = $77, 952.

5 Thus C = $77, = $85, 070, C 2 = $85, = $92, 473, etc. Problem Your firm s geologists have discovered a small oil field in New York s Westchester County. The field is forecasted to produce a cash flow of C = $2 million in the first year. You estimate that you could earn an expected return of r = 2% from investing in stocks with a similar degree of risk to your oil field. Therefore, 2% is the opportunity cost of capital. What is the present value? The answer, of course, depends on what happens to the cash flows after the first year. Calculate present value for the following cases: a The cash flows are forecasted to continue forever, with no expected growth or decline. b The cash flows are forecasted to continue for 20 years only, with no expected growth or decline during that period. c The cash flows are forecasted to continue forever, increasing by 3% per year because of inflation. d The cash flows are forecasted to continue for 20 years only, increasing by 3% per year because of inflation. a P V = 2 = $6.667 million. 0.2 b P V = $2 0.2 = $4.939 million c P V = = $ million $2 d P V = = $8.06 million Problem The following statements are true. Explain why. a If a bond s coupon rate is higher than its yield to maturity, then the bond will sell for more than face value. b If a bond s coupon rate is lower than its yield to maturity, then the bond s price will increase over its remaining maturity. a If the coupon rate is higher than the yield, then investors must be expecting a decline in the capital value of the bond over its remaining life. Thus, the bond s price must be greater than its face value. Mathematically, P = cf V + r + cf V + r + + cf V 2 + r + F V n + r = cf V n r + F V + r n + r, n where P = price, F V = face value, c = coupon rate, r = yield to maturity. Therefore, c = r P = F V ; c > r P > F V ; c < r P < F V. 5

6 b Conversely, if the yield is greater than the coupon, the price will be below face value and it will rise over the remaining life of the bond. Problem. 3.6 Which comes first in the market for U.S. Treasury bonds: a Spot interest rates or yields to maturity? b Bond prices or yields to maturity? a Spot interest rates. Yield to maturity is a complicated average of the separate spot rates of interest. b Bond prices. The bond price is determined by the bond s cash flows and the spot rates of interest. Once you know the bond price and the bond s cash flows, it is possible to calculate the yield to maturity. Problem You have estimated spot rates as follows: r = 5%, r 2 = 5.4%, r 3 = 5.7%, r 4 = 5.9%, r 5 = 6%. a What are the discount factors for each date that is, the present value of $ paid in year t? b Calculate the P V of the following bonds assuming annual coupons: i 5%, two year bond; ii 5%, five year bond; and iii 0%, five year bond. The face value of each bond is $,000. c Show explicitely and explain intuitively why the yield to maturity on the 0% bond is less than that on the 5% bond. d What should be the yield to maturity on a five year zero coupon bond? e Show that the correct yield to maturity on a five year annuity is 5.75%. f Explain intuitively why the yield on the five year bonds described in part c must lie between the yield on a five year zero coupon bond and a five year annuity. a Year Discount factor /.05 = / = / = / = /.06 5 = b i 5%, two year bond: ii 5%, five year bond: P V = $50 $, = $ P V = $ $ $ $50 $, = $

7 iii 0%, five year bond: P V = $ $ $ $00 $, 00 + = $, c First, we calculate the yield for each of the two bonds. For the 5% bond, this means solving for y in the following equation: For the 0% bond: $ = $50 + y + $50 + y + $ y + $50 $, y 4 + y. 5 y = or 5.964%. $, 7.43 = $00 + y + $00 + y + $ y + $00 $, y 4 + y. 5 y = or 5.937%. The yield depends upon both the coupon payment and the spot rate at the time of the coupon payment. The 0% bond has a slightly greater proportion of its total payments coming earlier, when interest rates are low, than does the 5% bond. Thus, the yield of the 0% bond is slightly lower. d The yield to maturity on a five year zero coupon bond is the five year spot rate, here 6%. e First, we find the price of the five year annuity, assuming that the annual payment is $: P V = $.05 + $ $ $ $ 4.06 = $ Now we find the yield to maturity for this annuity: $4.247 = $ + y + $ + y 2 + $ + y 3 + $ + y 4 + $ + y 5. y = or 5.745%. f The yield on the five year note lies between the yield on a five year zero coupon bond and the yield on a 5 year annuity because the cash flows of the Treasury bond lie between the cash flows of these other two financial instruments during a period of rising interest rates. That is, the annuity has fixed, equal payments, the zero coupon bond has one payment at the end, and the bond s payments are a combination of these. Problem The formula for the duration of a perpetual bond that makes an equal payment each year in perpetuity is +yield /yield. If each bond yields 5%, which has the longer duration a perpetual bond or a 5 year zero coupon bond? What if the yield is 0%? The duration of a perpetual bond is +yield /yield. The duration of a perpetual bond with a yield of 5% is: D 5 = = 2 years. 7

8 The duration of a perpetual bond yielding 0% is: D 0 =. 0. = years. Because the duration of a zero coupon bond is equal to its maturity, the 5 year zero coupon bond has a duration of 5 years. Thus, comparing the 5% perpetual bond and the zero coupon bond, the 5% perpetual bond has a longer duration. Comparing the 0% perpetual bond and the 5 year zero coupon bond, the zero coupon bond has a longer duration. 8