(S1) Soluções da Primeira Avaliação

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Jody Thornton
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1 Professor: Victor Filipe Monitor: Christiam Miguel EPGE-FGV Graduação em Ciências Econômicas Finanças Corporativas Setembro 2000 (S) Soluções da Primeira Avaliação Question (2.5 points). Casper has $200,000 available to support consumption in periods 0 (now) and (next year). He wants to consume exactly the same amount in each period. The interest rate is 8%. There is no risk. a. How much should he invest, and how much can he consume in each period? Draw a graph to illustrate your answer. () Answer. Let x be the amount that Casper should invest now. Then ($200, 000 x) is the amount he will consume now, and (.08 x) is the amount he will consume next year. Since Casper wants to consume exactly the same amount each period, x should satisfy the equation $200, 000 x =.08 x. Solving, we find that x = $96, so that Casper should invest $96,53.85 now, he should spend ($200, 000 $96, 53.85) = $03, now, and he should spend (.08 $96, 53.85) = $03, next year. b. Suppose Casper is given an opportunity to invest up to $200,000 at 0% risk-free. The interest rate stays at 8%. What should he do, and how much can he consume in each period? Complete the previous graph to illustrate your answer. () Answer. Since Casper can invest $200,000 at 0% risk-free, he can consume as much as ($200, 000.0) = $220, 000 next year. The present value of this $220,000 is: ($220, 000/.08) = $203, Therefore Casper can consume as much as $203, now by first investing $200,000 at 0% and then borrowing, at the 8% rate, against the $220,000 available next year. If we use the $203, as the available consumption now, and again let x be the amount that Casper should invest now, we can then solve the following for x: $203, x =.08 x and obtain x = $97, Therefore, Casper should invest $97, now at 8%, he should spend ($203, $97, ) = $05, now, and he should spend the amount ($97, ) = $05, next year. Note that this approach leads to the result that Casper borrows $203, at 8% and then invests $97, at 8%. We could simply say that he should borrow ($203, $97, ) = $05, at 8% against the $220,000 available next year. This is the amount that he will consume now.

2 c. What is the NPV of the opportunity in (b)? (0.5) Answer. The NPV of the opportunity in (b) is: $203, $200, 000 = $3, Question 2 (2.5 points). Compost Science, Inc. (CSI), is in the business of converting Boston s sewage sludge into fertilizer. The business is not in itself very profitable. However, to induce CSI to remain in business, the Metropolitan District Commission (MDC) has agreed to pay whatever amount is necessary to yield CSI a 0% book return on equity. At the end of the year, CSI is expected to pay a 4$ dividend. It has been reinvesting 40% of earnings and growing at 4% a year. a. Suppose CSI continues on this growth trend. What is the expected long-run rate of return from purchasing the stock at $00? What part of the $00 price is attributable to the present value of growth opportunities? () Answer. Here we can apply the standard growing perpetuity formula with DIV = $4, g = 0.04 and P 0 = $00: r = DIV + g = $ = 0.08 = 8% P 0 $00 The $4 dividend is 60 percent of earnings, implying that EPS = 4/0.6 = $6.67. Also: P 0 = EPS r Therefore PVGO = $ PVGO i.e., $00 = $ PVGO. b. Now the MDC announces a plan for CSI to treat Cambridge s sewage. CSI s plant will, therefore, be expanded gradually over five years. This means that CSI will have to reinvest 80% of its earnings. What will be CSI s stock price once this announcement is made (and its consequences for CSI are known)? (.5) Answer. DIV will decrease to 0.20 $6.67 = $.33. However, by plowing back 80 percent of earnings, CSI will grow by 8 percent per year for 5 years. Thus 2

3 Year t DIV t Note that DIV 6 increases sharply as EPS t the firm switches back to a 60 percent payout policy. Forecasted stock price in year 5 is Therefore, CSI s stock price will increase to P 5 = DIV 6 r g = $ = $47 P 0 = $ $ $ $.68 $.8 + $ = $06.2 Question 3 ( point). A six-year government bond (face value of $,000) makes annual coupon payments of 5% 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 2-month period? Now suppose that the bond yields 2% at the end of the year. What return would the bondholder earn in this case? Answer. Purchase price for a 6-year government bond with 5% annual coupon: PV 0 = $ , (.03) 6 + = $, (.03) 6 Price on year later with yield of 3% PV = $ , (.03) 5 + = $, (.03) 5 The rate of return is $50 + ($, $, 08.34) = 3% $, Price on year later with yield of 2% PV = $ , (.02) 5 + = $, 4.40 (.03) 2 The rate of return is $50 + ($, 4.40 $, 08.34) = 7.49% $, Question 4 ( point). As winner of a 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. 3

4 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? Answer. a. PV = $00, 000 b. PV = 80, 000/(.2) 5 = $02, c. PV = $, 400/0.2 = $95, 500 d. The present value is PV = $9, 000 e. PV = $6, 500 ( ) = $92, (.2) = $07, Question 5 ( point). You can form a portfolio of two assets, A and B, whose returns have the following characteristics: If you demand an expected return of 2%, what are the portfolio weights? What is the portfolio s standard deviation? Answer. Denote by r the expected return of the portfolio. We have r = x A r A + x B r B where x A is the proportion of wealth invested in asset A and x b is the proportion of wealth invested in asset B. Using x B = x A we get implying that x A = 0.60 and x B = The variance σ 2 of the portfolio is defined by This yields 0.2 = x A ( x A ) 0.5 σ 2 = x 2 A σ2 A + x 2 B σ2 B + 2(x Ax b ρ AB σ A σ N ) σ 2 = (0.60) 2 (20 2 ) + ( ) (40 2 ) + 2(0.60)(0.40)(0.50)(20)(40) = 592 and the standard deviation is σ = 592 = 24.33%. Question 6 ( point). The Treasury bill rate is 4%, and the expected return on the market portfolio is 2%. Using the Capital Asset Pricing Model: (a) Draw a graph showing how the expected return varies with beta. (0.20) 4

5 (b) What is the risk premium on the market? (0.20) Answer. The market risk premium is r m r f = = 0.08 or 8% (c) What is the required return on an investment with a beta of.5? (0.20) Answer. Using the security market line: r = r f + β(r m r f ) = [.5 ( )] = 0.6 or 6% (d) If an investment with a beta of 0.8 offers an expected return of 9.8%, does it have a positive NPV? (0.20) Answer. For any investment, we can find the opportunity cost of capital using the security market line. With β = 0.8, the opportunity cost of capital is: r = r f + β(r m r f ) = [0.8 ( )] = 0.04 or 0.4% The opportunity cost of capital is 0.4% and the investment is expected to earn 9.8%. Therefore, the investment has a negative NPV. (e) If the market expects a return of.2% from stock X, what is its beta? (0.20) Answer. Again, we use the security market line: r = r f + β(r m r f ) 0.2 = [β ( )] yielding β = 0.9. Question 7 (2 points). A project has the following forecasted cash flows: 5

6 The estimated project beta is.5. The market return r m is 6%, and the risk-free rate r f is 7%. (a) Estimate the opportunity cost of capital and the project s PV (using the same rate to discount each cash flow). () Answer. Using the security market line, we find the cost of capital r = [.5 ( )] = or 20.5% Therefore PV = = (b) What are the certainty-equivalent cash flow to the expected cash flow in each year? (0.5) Answer. CEQ = = CEQ 2 = 60 = CEQ 3 = 50 = (c) What is the ratio of the certainty-equivalent cash flow to the expected cash flow in each year? (0.25) Answer. a = = a 2 = = a 3 = = (d) Explain why this ratio declines. (0.25) 6

7 Answer. Using a constant risk-adjusted discount rate is equivalent to assuming that a t decreases at a constant compounded rate. Indeed, we have by definition 2 CEQ t ( + r f ) t = C t ( + r t )( + r t )... ( + r ) where r t is the risk-adjusted discount rate to compute the value at date t of a cash flow at date t. This implies that a t = + r f + r t + r f + r t... + r f + r Therefore If t r t is constant, we get that a t = + r f + r t a t a t a t = + r f + r < 2 Here we assume that the risk-free rate is constant. 7

Midterm Review. P resent value = P V =

Midterm Review. P resent value = P V = JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Midterm Review F uture value of $100 = $100 (1 + r) t Suppose that you will receive a cash flow of C t dollars at the end of