Financial Markets and Valuation - Tutorial : SOLUTIONS Bonds, Stock Valuation & Capital Budgeting (*) denotes those problems to be covered in detail during the tutorial session Bonds Problem. (Ross, Westerfield & Jaffe) Consider a bond, which pays a $80 coupon annually and has a face value of $,000. Calculate the yield to maturity if the bond has a. 0 years remaining to maturity and it is sold at $,00. b. 0 years remaining to maturity and it is sold at $950. a. Since the bond sells at a premium, its yield is less than the coupon rate of 8%. PV =,00 <=> (80/y) * [ - /((y)^0)],000/[(y)^0] =,00 <=> y = 6.% b. Since the bond sells at a discount, its yield exceeds the coupon rate of 8%. PV = 950 <=> (80/y) * [ - /((y)^0)],000/[(y)^0] = 950 <=> y = 8.77% (*) Problem. (Ross, Westerfield & Jaffe) Available are three zero-coupon, $,000 face value bonds. All of these bonds are initially priced using an -percent interest rate. Bond A matures one year from today, bond B matures five years from today, and bond C matures 0 years from today. a. What is the current price of each bond? b. If the market rate of interest rises to 4%, what are the prices of these bonds? c. Which bond experienced the greatest percentage change in prices? a. PV (A, %) =,000/(0.) = 900.9 PV (B, %) =,000/[(0.)^5] = 593.45 PV (C, %) =,000/[(0.)^0] = 35.8 b. PV (A, 4%) =,000/(0.4) = 877.9 => % change in A = (877.9-900.9)/900.9 = -.63% PV (B, 4%) =,000/[(0.4)^5] = 59.37 => % change in B = (59.37-593.45)/593.45 = -.48% PV (C, 4%) =,000/[(0.4)^0] = 69.74 => % change in C = (69.74-35.8)/35.8 = -3.4% c. Bond C FMV/Tutorial Solutions/Sept.-Oct. 006
Problem 3. (Brealey and Myers) An 8-percent five-year bond yields 6%. a. If the yield remains unchanged, what will its price be one year hence? Assume annual coupon payments. b. What is the total return to an investor who held the bond over this year? We will find the price in percentages (the same as when assuming that the face value is $). a. Price today = 0.08 * A (5 years, 6%) / (.06)^5 = 0.08 * 4.4 0.7473 = 0.337 0.7473 =.0843 = 08.43% Price in one year = 0.08 * A(4 years, 6%) /(.06)^4 = 0.08 * 3.465 0.79 = 0.77 0.79 =.0693 = 06.93% b. Total annual return = (coupon plus difference in prices) / price today = (8% 06.93% - 08.43%) / 08.43% = 6.5% / 08.43% = 6% (*) Problem 4. (Ross, Westerfield & Jaffe) The one-year spot rate equals 0 percent and the two-year spot rate equals 8 percent. a. What should a 5-percent coupon two-year bond cost? b. What is the forward rate expected over year? a. P = $50 /.0 $,050 / (.08) = $45.45 $900. = $945.66 b. ( r )( ƒ ) = ( r ) (.0 ) ( ƒ ) = (.08 ) ƒ = 0.0604 (*) Problem 5. Your investment bank has just supplied you with the following term structure of spot rates (all annualized effective rates): Maturity Rate 3 months 3.5% 6 months 4% months 5% 8 months 5% 4 months 5% FMV/Tutorial Solutions/Sept.-Oct. 006
a. One broker calls to convince you to buy a bond with 8 months to maturity and face value of 000 that pays a semiannual coupon (stated rate) of 3%. He says he will sell you the bond for 00. Should you buy the bond? (Assume that the bond has just paid a coupon and ignore taxes). b. If market expectations about future interest rates are correct, at what price will the bond sell year from now? a. Coupon = 3%,000 = 30 (annual) Semiannual Coupon = 30 / = 5 (semiannual) 5 5 05 5 PV =.5 = r r r 0.04 05 ( ) ( ) ( ) ( ) ( 0.05) ( 0.05) 0,6m 0,m 0,8m 5.5 = 97.36 The price at which the broker wants to sell is too expensive. You shouldn t buy the bond. NOTE: The result was obvious, because a bond that pays a coupon of 3% when all market interest rates are above 3% must sell below par. Since the broker was asking a price above par, you shouldn t buy the bond. b. One year from now, the bond will have 6 months to maturity. The only payment left is the last coupon and the face value. If market expectations are correct, the 6-month rate prevailing one year from now equals today s forward rate between months and 8 months.,5.5 ( r ) = ( r ) ( f ) ( 0.05) = ( 0.05) ( f ) f 0. 05 = 0,8m 0,m m,8m m,8m m, 8m NOTE: This was again obvious! If the term structure is flat after months and the market expectations theory holds, the forward term structure is also constant. PV = 05 ( 0.05) = 0.5 9938 FMV/Tutorial Solutions/Sept.-Oct. 006 3
Stock Valuation Problem 6. (Ross, Westerfield & Jaffe) Suppose that a shareholder has just paid $50 per share for XYZ Company Stock. The stock will pay a dividend of $ per share in the upcoming year. This dividend is expected to grow at an annual rate of 0% for the indefinite future. The shareholder felt that she paid the fair price for the stock, given her assessment of XYZ s risks. What is the annual required rate of return of this shareholder? In this problem, you are given the current share price at $50. Dividends at t = is expected to be $, and growing indefinitely at 0% p.a. To find the annual required rate of return, we set up the solution as: Div Ρ = ( r g) 50 = 50 (r 0.) = r = 0.0 = 4% r 50 ( 0.) (*) Problem 7. (Ross, Westerfield & Jaffe) Brown, Inc. has just paid a $3 dividend per share of the common stock. The stock is currently being sold at $40. Investors expect that Brown s dividend will grow at a constant rate indefinitely. What growth rate is expected by investors if they require a 8% return on the stock? In this problem, we are given the current dividend (..just paid ) at t = 0 of $3 per share, and we expect the dividend to grow at a constant rate indefinitely. Therefore, we apply the constant growth model. The t= dividends are D = D ( g) = 3 ( g) I O P o Div at t = D0( g) = = ( r g) ( r g) 3 ( g) 40 = 40 ( 0.08 g ) = 3 ( g) ( 0.08 g) and then solving for g, g = 0.47% FMV/Tutorial Solutions/Sept.-Oct. 006 4
(*) Problem 8. (Ross, Westerfield & Jaffe) Whizzkids, Inc., is experiencing a period of rapid growth. Earnings and dividends are expected to grow at a rate of 8% during the next two years, 5% in the third year, and at a constant rate of 6% thereafter. Whizzkids last dividend, which has just been paid, was $.5. If the required rate of return on the stock is %, what is the price of the stock today? You are asked to compute the share price of a company that is expected to have stages of growth: an initial high growth period of about /3 years and a subsequent constant growth thereafter $.5 (.8) $.5 (.8) $.5(.8) (.5) Ρ O = 3. (.) (.).5 (.8) (.5)(.06) (0. 0.06) = $6.95 3 (.) Problem 9. Allen Inc. is expected to pay an equal amount of dividends at the end of the first two years. Thereafter the dividend will grow at a constant rate of 4% indefinitely. The stock is currently traded at $30. What is the expected dividend per share for the next year if the required rate of return is %? D = D and a growth rate g = 0.04 thereafter r = 0. Po = $30. Hence, we can set up the following equation: 30 = D. D (.) D (.04) ( 0. 0.04) ( ). 30 = D (.05357) D = $.49 (*) Problem 0. (Ross, Westerfield & Jaffe) California Electronics, Inc., expects to earn $00 million per year in perpetuity if it does not undertake any new projects. The firm has an opportunity that requires an investment of $5 million today and $5 million in one year. The new investment will begin to generate additional annual earnings of $0 million two years from today in perpetuity. The firm has 0 million shares of common stock outstanding, and the required rate of return on the common stock is 5 percent. (a) What is the price of a share of the stock if the firm does not undertake the new project? (b) What is the value of the growth opportunities resulting from the new project? FMV/Tutorial Solutions/Sept.-Oct. 006 5
(c) What is the price of a share of the stock if the firm undertakes the new project? Solution (a) The present value (PV) of the current earnings stream can be obtained using the perpetuity formula: CF/r. Then simply divide this PV by the number of shares outstanding to get the price per share. This implies that PV = $00/() = $666.67 million Price per share = 666.67/0 = $33.33 (b) The cash flows from the new project can be depicted as: Time (Years) 0 to infinity Cash flows -5-5 0 The NPV of the project is simply: NPV 5 0 = 5 = $38. (.5).5 6 million (c) The new share price can be obtained by adding the NPV of the new project to the firm s current PV and then dividing by the shares outstanding. This will be equal to: New Share Price = (38.6 666.67)/0 = $35.6 Capital Budgeting (*) Problem. (Ross, Westerfield & Jaffe) Dickinson Brothers, Inc., is considering investing in a machine to produce computer keyboards. The price of the machine is $400,000 and its economic life is 5 years. The machine is fully depreciated by the straight-line method. The machine will produce 0,000 units of keyboards each year. The price of the keyboard is $40 in the first year, and it will increase at 5% per year. The production costs per unit of the keyboard is $0 in the first year, and it will increase at 0% per year. Corporate tax rate for the company is 34%. If the appropriate discount rate is 5%, what is the NPV of the investment? FMV/Tutorial Solutions/Sept.-Oct. 006 6
$400,000 0 Straight Line Depreciations (5 yr life) per year = = $80, 000 5 t Sales Production Costs 0,000 units x $40 each 0,000 units x $0 each = $400,000 = $00,000 $400,000 x.05 $00,000 x.0 = $40,000 = $0,000 3 $400,000 x.05 $00,000 x.0 = $44,000 = $4,000 4 $400,000 x.05 3 $00,000 x.0 3 = 463,050 = $66,00 5 $400,000 x.05 4 $00,000 x.0 4 = $486,03 = $9,80 0 3 4 5 Sales Revenues $400,000 $40,000 $44,000 $463,050 $486,03 Production Costs ($00,000) ($0,000) ($4,000) ($66,00) ($9,80) Depreciation ($ 80,000) ($ 80,000) ($ 80,000) ($ 80,000) ($ 80,000) EBIT $0,000 $0,000 $9,000 $6,850 $3,383 Taxes (34%) -40,800-40,800-40,460-39,389-38,550 NOPLAT $ 79,00 $ 79,00 $ 78,540 $ 77,46 $ 74,833 Add back: Depreciation 80,000 80,000 80,000 80,000 80,000 CAPEX ($400,000) - - - - - Changes in NWC 0 0 0 0 0 0 Free Cash Flows ($400,000) $59,00 $59,00 $58,540 $57,46 $54,833 r = 5% NPV = $ 400,000 $59,00 $59,00 $58,540 3.5.5.5 $57,46 $54,833 = 4 5.5.5 = - $400,000 $38,435 $0,378 $04,43 $89,834 $76,979 = $9,869 FMV/Tutorial Solutions/Sept.-Oct. 006 7