# Review Solutions FV = 4000*(1+.08/4) 5 = \$

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1 Review Solutions 1. Planning to use the money to finish your last year in school, you deposit \$4,000 into a savings account with a quoted annual interest rate (APR) of 8% and quarterly compounding. Fifteen months later, however, you decide to quit school to become a ski instructor, so you close the account. How much money will you receive when you close the account? FV = 4000*(1+.08/4) 5 = \$ You deposit \$100 today. You make no further deposits for the next 12 months, but deposit \$800 per month the following 12 months. How much will you have in 2 years if you earn interest at a quoted annual rate of 12%? FVA r=.01 t=12 = FV = 100*(1+.12/12) * FVA r=.01 t=12 = \$10, The company you work for will deposit \$200 at the end of each month into your retirement fund. Interest is compounded monthly. You plan to retire five years from now and estimate that you will need \$1,000 per month out of the account for the next 2 years. If the account pays 1% compounded monthly (i.e. quoted annual rate of 12%), how much do you need to put into the account each month in addition to your company's contribution in order to meet your financial objective? FVA r=.01 t=60 = A r=.01 t=24 = PV 60 = 1000*A r=.01 t=24 = 21, (200+x)*81.67 = 21, x=60.11 (4) You are an investment advisor who has been approached by a client for help on his financial strategy. He has \$250,000 in savings in the bank. He is 55 years old and expects to work for 10 more years, making \$100,000 a year. (He expects to make a return of 5% on his investments for the foreseeable future. You can ignore taxes.) a. Once he retires 10 years from now, he would like to be able to withdraw \$80,000 a year for the following 25 years. (His actuary tells him he will live to be 90 years old.) How much would he need in the bank 10 years from now to be able to do this? b. How much of his income would he need to save each year for the next 10 years to be able to afford these planned withdrawals (\$80,000 a year) after the tenth year? c. Assume that interest rates decline to 4%, ten years from now. By how much, if any, would your client have to lower his annual withdrawal, assuming that he still plans to withdraw cash each year for the next 25 years? a. A r=.05 t=25 = Amount needed in the bank to withdraw \$ 80,000 each year for 25 years = \$1,127,512 b. Future Value of Existing Savings in the Bank \$250,000* = \$407,224 Shortfall in Savings \$1,127,512 - \$407,224 = \$720,288 FVA r=.05 t=10 = Annual Savings needed to get FV of \$720,288 = \$57,265 c. If interest rates drop to 4% after the 10th year, A r=.04 t=25 =

2 Annuity based upon interest rate of 4% and PV of \$1,127,512 = \$72,174 The decline in the amount of withdrawal = \$80,000 - \$72,174 = \$7,826 (5) You are comparing houses in two towns. You have \$100,000 for a down payment, and 30-year mortgage rates are at 8%. Town 1 Town 2 Price of the house Annual property tax \$400,000 \$6,000 \$300,000 \$12,000 The houses are roughly equivalent. a. Estimate the total per year payments (mortgage and property taxes) you would have on each house. Which one is less expensive (based on the total dollar yearly payment)? b. Are mortgage payments and property taxes directly comparable? Why or why not? c. If property taxes are expected to grow 3% a year forever, which house is less expensive (assume property taxes are paid at the beginning of the year)? a. A r=.08/12 t=360 = Town 1 Town 2 Mortgage: \$300,000 \$200,000 Monthly Payment. \$2, \$1, Annual Payments \$26, \$17, Property Tax \$6,000 \$12,000 Total Payment \$32, \$29, b. Mortgage payments will end after 30 years. Property taxes are not only a perpetuity; they arc a growing perpetuity. Therefore, they are likely to be more onerous. c. If property taxes are expected to grow at 3% annually forever, PV of property taxes = Property tax * (1 +g) / (r -g) For Town 1, PV of property tax = \$6,000 + \$6000* 1.03/( ) = \$129,600 For Town 2, PV of property tax = \$12,000 + \$12000*1.03/( ) = \$259,200 To make the comparison, add these to the house prices, Cost of the Town 1 house \$400,000 + \$129,600 = \$529,600 Cost of the Town 2 house \$300,000 + \$259,200 = \$559,200 The Town 1 house is cheaper. (6) You are trying to assess the value of a small retail store that is for sale. The store generated a cash flow to its owner of \$100,000 in the most profitable year of operation and is expected to have growth of about 5% a year in perpetuity. If the rate of return required on this store is 10%, what is your assessment of the value of the store? What would the growth rate need to be to justify a price of \$2.5 million for this store? a. Value of Store = \$100,000 (l.05)/( )= \$2,100,000 b. Growth rate needed to justify a value of \$ 2.5 mi1lion: Solving for g, 100,000(1+g)/(.10-g) = 2,500,000 g = 5.77% (.05769)

3 (7) What is the value of 15-year corporate bonds, with a coupon rate of 9%, if current interest rates on similar bonds is 8%? How much would the value change if interest rates increased to 10%? Under what conditions will this bond trade at par (face value)? Value of 15-year corporate bond; 9% coupon rate; 8 % market interest rate Assuming coupons are paid semi-annually, A r=.04 t=30 = Value of Bond = 45*(1-1.04^(-30))/.04+ 1,000/1.04^30 = \$1, If market interest rates increase to 10%, A r=.05 t=30 = Value of Bond = 45*(1-1.05^(-30))/.05+1,000/1.05^30 = \$ The bonds will trade at par only if the market interest rate = coupon rate. (8) Chrysler bonds quoted on the New York Exchange include 12% coupon bonds maturing in 2003 and 12% coupon bonds maturing in Both are \$1,000 face value bonds and mature on March 1. The first bond was issued in 1988 and the second one in Interest rates are currently at the 8% level. a. Determine the current prices of the bonds (today is March 1, the price does not include the 3/1/2002 coupon payment). Assume coupon payments are made semi-annually. b. The Federal Reserve is expected to announce a cut in the discount rate, which will cause the level of interest rates to fall to 6%. Which bond will be more affected by the change in interest rates? a. P= 60/(1+.08/2) / = A r=.04 t=42 = P=60* / = b. bond 2, longer term. (9) 9.A. World Wide Pants (WWP), a well known entertainment company, has been experiencing financial difficulty and has omitted its annual dividend. The company is expected to resume dividend payments two years from now beginning at \$.50 per share; dividends are then expected to grow at 2% for 2 years, and then at 5% thereafter. What is the price of the stock today if you are given a monthly interest rate of 2%? EAR = 26.82% ; P=\$ B.George is considering purchasing stock and holding it for 3 years. The projected dividends (at a 5% growth rate) and market price are: D 1 = \$4.20; D 2 = \$ 4.41 ; D 3 = \$4.63; and P 3 = \$ His required rate of return, given the risk involved, is 10%. a. What is the maximum price George should pay for the stock? b. If the dividends for years 1 and 2 remain at \$4.20 and \$4.41, respectively, and are expected to grow at 5% per year to infinity, what would the stock price be at the end of the second year if George sold the stock then? (assume investors still require a 10% return for this stock)? What is

4 the total discounted present value (today) of the cash flows George receives if he holds the stock only 2 years? c. Is the value of the stock today dependent on how long George plans to hold it? Does its price today depend on whether he plans to hold the stock for 2 years, 3 years, or any other period of time? ; ; 84 ; no (10) What is the value of stock in a company that currently pays out \$1.00 per share in dividends and expects these dividends to grow 15% a year for the next five years and 6% a year forever after that? Assume that investors require a 12.5% return on stocks of equivalent risk. D 1 =1*1.15 ; growing annuity factor = [1-(1+g) T /(1+r) T ]/(r-g) = Value of Dividends during high growth period = \$ l.00(1.15)(1-1.15^5/1.125^5)/( ) = \$5.34 Expected Dividends in year 6 = \$1.00 (1.15)^5*1.06 = \$2.13 Expected Terminal Price \$2.13/( ) = \$32.77 Value of Stock = \$ \$32.77/1.125^5 = \$ \$ = \$23.53 (11) A company is considering building a hotel in Beijing, China. Because of the political risk involved, the company wants to have a small payback period. The cash flows are estimated to be as follows: Year Cash Flows 0-3,000, , , , , ,000 a. Find the payback period for the project. b. Find the discounted payback period if the discount rate is 10% a: the payback period = 5 years. b: the discounted payback period = 7 + some part of 8 th yr = 9.84 ; 9.84/ =.028 Yr CF PV Sum PV (12) Your company is considering a project that will bring in annual free cash flow in the amount of \$50,000 for 10 years. If the discount rate is 14%, what would be the maximum initial required investment below which the project would be accepted based on the NPV rule?

5 A r=.14 t=10 = The PV cash flows 50,000 * = \$260,805 This would be the maximum initial investment on the project so that the NPV would be greater than zero. (13) It is sometimes argued that net present value should be based on time-varying discount rates. As an example, consider a project with projected cash flows as follows: Year Cash Flows Appropriate discount rate 0-500, ,000 10% 2 350,000 12% What is the NPV of this project? The NPV = - 500, ,000/(1+10%) + 350,000/(1+12%)(1+10%) = -500, , ,090 = \$56,820 (14) A project is estimated to have the cash flows to firm as follows: Year Cash Flow 0-200, per year 25, ,000 a. What is the internal rate of return of this project? b. Should the project be accepted if the discount rate is 10%? c. Is the NPV positive or negative? a: The IRR 7.1 (.07096) b: The project should be rejected since IRR < discount rate (10 %). c: Then NPV = *A r=.1 t=9 + 75/ = - \$27,109. (A r=.1 t=9 = ) (15) There are two mutually exclusive projects with estimated cash flows to firm as follows: Year Project A Project B a. What is the IRR for Project A? b. What is the IRR for Project B? c. Which project should be accepted based on the IRR rule? d. What are the NPVs of Project A and Project B if the discount rate is 5%? Which project should be accepted based on the NPV rule? e. What are the NPVs of Project A and Project B if the discount rate is 8%? Which project should be accepted based on the NPV rule? f. Do the IRR rule and NPV rule always reach the same conclusion with regard to the

6 selection of two mutual exclusive projects? Which method is more consistent with the objective of corporate finance to maximize shareholders wealth? a: The IRR for Project A = 8.04% b: The IRR for Project B = 7.32% c: Project A is accepted based on the IRR rule since it has higher IRR. d: A r=.05 t=19 = Since the NPV of Project A = \$ and the NPV of Project B = \$ when the discount rate is 5%, it is obvious that the Project B should be accepted. e: A r=.08 t=19 = Since the NPV of Project A = \$1.63 and the NPV of Project B -\$ when the discount rate is 8%, then Project A should he accepted. f: In this example, the IRR rule finds the Project A superior to the Project B, However, the NPV rule may give different conclusions when the cost of capital changes. The NPV rule is more consistent with the objective of financial management to maximize the shareholders wealth.

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