Chapter 10 Discussion Questions 10-1. The valuation of financial assets is based on the required rate of return to security holders. This, in turn, becomes the cost of financing (capital) to the corporation. 10-2. The valuation of a financial asset is equal to the present value of future cash flows. 10-3. Because BCE, Inc. has less risk than Air Canada, BCE, Inc. has relatively high returns and a strong market position; the latter firms have had financial difficulties. 10-4. The three factors that influence the demanded rate of return are: a. The real rate of return b. The inflation premium c. The risk premium 10-5. The real rate of return is the financial rent received by investors for giving up use of their funds, above inflation and without a risk premium. 10-6. If inflationary expectations increase, the yield to maturity (required rate of return) will increase. This will mean a lower bond price. 10-7. The longer the time period remaining to maturity, the greater the impact of a difference between the rate the bond is paying and the current yield to maturity (required rate of return). For example, a two percent ($20) differential is not very significant for one year, but very significant for 20 years. In the latter case, it will have a much greater effect on the bond price. 10-8. The valuation models represent the complex real world in a simplified manner. As such they are incomplete. If we can keep the other components of the model constant, then the model will hinge on the investor’s required or expected rate of return. Three components of an investor’s required rate of return have been suggested. If a change in an investor’s required return can be fully captured, the model will work. For practical purposes holding other factors constant and fully capturing changes in investor expectations and psychology proves difficult. 10-9. The three adjustments in going from annual to semiannual bond analysis are: 1. Divide the annual interest rate by two. 2. Multiply the number of years by two. 3. Divide the annual yield to maturity by two. 10-10. The longer the life of an investment, the greater the impact of a change in the required rate of return. Since preferred stock has a perpetual life, the impact is likely to be at a maximum. 10-11. The no-growth pattern for common stock is similar to the dividend on preferred stock. 10-12. To go from Formula (10-7) to Formula (10-8): The firm must have a constant growth rate (g). The discount rate (k e ) must exceed the growth rate (g). 10-13. The two components that make up the required return on common stock are: D a. Dividend yield = 1 P0 b. The growth rate (g). This actually represents the anticipated growth in dividends, earnings, and stock price over the long term. 10-14. The price-earnings ratio is influenced by the earnings and sales growth of the firm, the risk (or volatility in performance), the debt-equity structure of the firm, the dividend policy, the quality of management, and a number of other factors. Firms that have bright expectations for the future tend to trade at high P/E ratios while the opposite is true of low P/E firms. 10-15. The higher the firm’s K e , the lower will be the price-earnings ratio, assuming all other things remain equal. This makes sense because a relatively high K e implies a higher level of risk. The greater the estimate of dividend growth (g), the higher the price-earnings ratio because of buoyancy of future expectations. 10-16. A supernormal growth pattern is represented by very rapid growth in the early years of a company or industry that eventually levels off to more normal growth. The supernormal growth pattern is often experienced by firms in emerging industries, such as in the early days of electronics or microcomputers. 10-17. In valuing a firm with no cash dividend, one approach is to assume that at some point in the future a cash dividend will be paid, perhaps a liquidating dividend. You can then take the present value of future cash dividends. A second approach is to take the present value of future earnings as well as a future anticipated stock price. The discount rate applied to future earnings is generally higher than the discount rate applied to future dividends. Internet Resources and Questions 1. www.bankofcanada.ca/rates/interest-rates/ www.pfin.ca/canadianfixedincome 2. www.tmx.com www.reuters.com/finance/stocks Problems 10-1. Wild Rose Company a. 6 percent yield to maturity Present value of interest payments PV A = A × PV IFA (n = 20, %I/Y = 6) (Appendix D) PV A = 90 × 11.470 = $1,032.30 Present value of principal payment at maturity PV = FV × PV IF (n = 20, %I/Y = 6) (Appendix B) PV = $1,000 × 0.312 = $312 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond $1,032.30 312.00 $1,344.30 Calculator: PMT = $90 Compute: PV =? FV = $1,000 n = 20 %I/Y = 6% PV = $1,344.10 b. 8 percent yield to maturity PV A = A × PV IFA (n = 20, %I/Y = 8) (Appendix D) PV A = $90 × 9.818 = $883.62 PV = FV × PV IF (n = 20, %I/Y = 8) (Appendix B) PV = $1,000 × 0.215 = $215 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond $ 883.62 215.00 $1,098.62 Calculator: PMT = $90 Compute: PV =? FV = $1,000 n = 20 %I/Y = 8% PV = $1,098.18 c. 12 percent yield to maturity PV A = A × PV IFA (n = 20, %I/Y = 12) (Appendix D) PV A = $90 × 7.469 = $672.21 PV = FV × PV IF (n = 20, %I/Y = 12) (Appendix B) PV = $1,000 × 0.104 = $104.00 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond Calculator: Compute: 10-2. PV =? FV = $1,000 n = 20 %I/Y = 12% PV = $775.92 $672.21 104.00 $776.21 PMT = $90 Midland Oil a. 7 percent yield to maturity Present value of interest payments PV A = A × PV IFA (n = 25, %I/Y = 7) (Appendix D) PV A = $80 × 11.654 = $932.32 Present value of principal payment at maturity PV = FV × PV IF (n = 25, %I/Y = 7) (Appendix B) PV = $1,000 × 0.184 = $184.00 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond $ 932.32 184.00 $1,116.32 Calculator: PMT = $80 Compute: PV =? FV = $1,000 n = 25 %I/Y = 7% PV = $1,116.54 b. 10 percent yield to maturity PV A = A × PV IFA (n = 25, %I/Y = 10) (Appendix D) PV A = $80 × 9.077 = $726.16 PV = FV × PV IF (n = 25, %I/Y = 10) (Appendix B) PV = $1,000 × 0.092 = $92.00 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond Calculator: Compute: PV =? FV = $1,000 n = 25 %I/Y = 10% PV = $818.46 $726.16 92.00 $818.16 PMT = $80 c. 13 percent yield to maturity PV A = A × PV IFA (N = 25, %I/Y = 13) (Appendix D) PV A = $80 × 7.330 = $586.40 PV = FV × PV IF (N = 25, %I/Y = 13) (Appendix B) PV = $1,000 × 0.47 = $47.00 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond Calculator: Compute: PV =? FV = $1,000 N = 25 %I/Y = 13% PV = $633.50 $586.40 47.00 $633.40 PMT = $80 10-3. Exodus Limousine Company a. 5 percent yield to maturity Present value of interest payments PV A = A × PV IFA (N = 50, %I/Y = 5) (Appendix D) PV A = $100 × 18.256 = $1,825.60 Present value of principal payment at maturity PV = FV × PV IF (N = 50, %I/Y = 5) (Appendix B) PV = $1,000 × 0.087 = $87.00 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond $1,825.60 87.00 $1,912.60 Calculator: PMT = $100 Compute: PV =? FV = $1,000 N = 50 %I/Y = 5% PV = $1,912.80 b. 15 percent yield to maturity Present value of interest payments PV A = A × PV IFA (N = 50, %I/Y = 15) (Appendix D) PV A = $100 × 6.661 = $666.10 Present value of principal payment at maturity PV = FV × PV IF (N = 50, %I/Y = 15) (Appendix B) PV = $1,000 × 0.001 = $1.00 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond Calculator: Compute: PV =? FV = $1,000 N = 50 %I/Y = 15% PV = $666.97 $666.10 1.00 $667.10 PMT = $100 10-4. Exodus Limousine Company (Continued) The principal payment of $1.00 represents: of the bonds value. 10-5. Applied Software a. 30 years to maturity Present value of interest payments PV A = A × PV IFA (N = 30, %I/Y = 7) (Appendix D) PV A = $120 × 12.409 = $1,489.08 Present value of principal payment at maturity PV = FV × PV IF (N = 30, %I/Y = 7) (Appendix B) PV = $1,000 × 0.131 = $131.00 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond $1,489.08 131.00 $1,620.08 Calculator: PMT = $120 Compute: PV =? FV = $1,000 N = 30 %I/Y = 7% PV = $1,620.45 b. 15 years to maturity PV A = A × PV IFA (N = 15, %I/Y = 7) (Appendix D) PV A = $120 × 9.108 = $1,092.96 PV = FV × PV IF (N = 15, %I/Y = 7) (Appendix B) PV = $1,000 × 0.362 = $362.00 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond $1,092.96 362.00 $1,454.96 Calculator: PMT = $120 Compute: PV =? FV = $1,000 N = 15 %I/Y = 7% PV = $1,455.40 c. 1 year to maturity PV A = A × PV IFA (N = 1, %I/Y = 7) (Appendix D) PV A = $120 × 0.935 = $112.20 PV = FV × PV IF (N = 1, %I/Y = 7) (Appendix B) PV = $1,000 × 0.935 = $935.00 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond $ 112.20 935.00 $1,047.20 Calculator: PMT = $120 PV =? FV = $1,000 N=1 %I/Y = 7% Compute: PV = $1,046.73 10-6. Victoria Telephone Company a. 30 years to maturity Present value of interest payments PV A = A × PV IFA (N = 30, %I/Y = 8) (Appendix D) PV A = $50 × 11.258 = $562.90 Present value of principal payment at maturity PV = FV × PV IF (N = 30, %I/Y = 8) (Appendix B) PV = $1,000 × 0.099 = $99.00 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond Calculator: Compute: PV =? FV = $1,000 N = 30 %I/Y = 8% PV = $662.27 $562.90 99.00 $661.90 PMT = $50 b. 15 years to maturity PV A = A × PV IFA (N = 15, %I/Y = 8) (Appendix D) PV A = $50 × 8.559 = $427.95 PV = FV × PV IF (N = 15, %I/Y = 8) (Appendix B) PV = $1,000 × 0.315 = $315.00 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond Calculator: Compute: PV =? FV = $1,000 N = 15 %I/Y = 8% PV = $743.22 $427.95 315.00 $742.95 PMT = $50 c. 1 year to maturity PV A = A × PV IFA (N = 1, %I/Y = 8) (Appendix D) PV A = $50 × 0.926 = $46.30 PV = FV × PV IF (N = 1, %I/Y = 8) (Appendix B) PV = $1,000 × 0.926 = $926.00 Total present value: Present value of interest payments Present value of principal payments Total present value or price of the bond Calculator: PV =? N=1 Compute: PV = $972.22 10-7. FV = $1,000 %I/Y = 8% $ 46.30 926.00 $972.30 PMT = $50 Victoria Telephone Company (continued) 5% bond, $1,000 par (maturity) value Bond value $1,000 $900 $800 $700 Assumes 8% yield to maturity (YTM) $600 $500 30 25 20 15 Time to maturity 10 5 0 As the time to maturity becomes less and less, the importance of the difference between the interest rate the bond pays and the yield to maturity becomes less significant. Therefore, the bond trades closer to par value. 10-8. Ron Rhodes-Golden Years Recreation Corporation PV A = A × PV IFA (N = 18 × 2 = 36, %I/Y = 11/ 2 = 5.5) (Appendix D) PV A = $65 × 15.536 = $1,009.84 PV = FV × PV IF (N = 36, %I/Y = 5.5) (Appendix B) PV = $1,000 × 0.146 = $146.00 Total present value: Present value of interest payments $1,009.84 Present value of principal payments 146.00 Total present value or price of the bond $1,155.84 Calculator: Compute: PV =? FV = $1,000 N = 18 × 2 = 36 PV = $1,155.36 PMT = $65 (130/ 2) %I/Y = 11/ 2 = 5.5% Broker’s price is at $1,170 is too high compared to $1,155.36 value. 10-9. Vinny Cartier Company First compute the new required rate of return (yield to maturity). Real rate of return Inflation premium Risk premium Total return 3% 3 2 8% Then use this value to find the price of the bond: Present value of interest payments PV A = A × PV IFA (N = 15 × 2 = 30, %I/Y = 8/ 2 = 4%) (App. D) PV A = $60 × 17.292 = $1,037.52 Present value of principal payment at maturity PV = FV × PV IF (N = 30, %I/Y = 4%) (Appendix B) PV = $1,000 × 0.308 = $308.00 $1,037.52 308.00 $1,345.52 Calculator: Compute: 10-10. PV =? FV = $1,000 N = 15 × 2 = 30 PV = $1,345.84 PMT = $60 ($120/ 2) %I/Y = 8/ 2 = 4% Martin Shipping Lines First compute the new required rate of return (yield to maturity). Real rate of return Inflation premium Risk premium Total return 2% 2 4 8% Then use this value to find the price of the bond. Present value of interest payments PV A = A × PV IFA (N = 20 × 2 = 40, %I/Y = 8/ 2 = 4%) (App. D) PV A = $50 × 19.793 = $989.65 Present value of principal payment at maturity PV = FV × PV IF (N = 40, %I/Y = 4%) (Appendix B) PV = $1,000 × 0.208 = $208.00 $ 989.65 208.00 $1,197.65 Calculator: Compute: PV =? FV = $1,000 N = 20 × 2 = 40 PV = $1,197.93 PMT = $50 ($100/ 2) %I/Y = 8/ 2 = 4% 10-11. Lance Whittingham IV-Leisure Time Corporation a. Current price of the bonds PV A = A × PV IFA (N = 16 × 2 = 32, %I/Y = 10/ 2 = 5%) (Appendix D) PV A = $20 × 15.803 = $316.06 PV = FV × PV IF (N = 32, %I/Y = 5%) (Appendix B) PV = $1,000 × 0.210 = $210.00 $316.06 + 210.00 $526.06 Calculator: Compute: PV =? FV = $1,000 N = 16 × 2 = 32 PV = $525.92 b. Percent increase at maturity Maturity value Current price Dollar increase Percentage increase = PMT = $20 ($40/ 2) %I/Y = 10/ 2 = 5% $1,000.00 – 525.92 $ 474.08 $474.08 = 0.9014 = 90.14% $525.92 c. Compound rate of growth The bond will grow by 90.14 percent over 16 years. Using Appendix A, the future value of $1, we see the growth rate is between 4 and 5 percent (4.02 percent based on interpolation). Calculator: Compute: PV = $525.92 FV = $1,000 N = 16 %I/Y =? %I/Y = 4.10% PMT = 0 10-12. Coleman Manufacturing Company Approximate yield to maturity is represented by Y'. Principal payment - Price of the bond Annual interest payment + Number of years to maturity Y1 = 0.6 (Price of the bond ) + 0.4 (Principal payment ) $1,000 − $850 $90 + $90 + $15 10 = = = 0.1154 = 11.54% 0.6 ($850 ) + 0.4 ($1,000 ) $510 + $400 Calculator: Compute: 10-13. PV = $850 FV = $1,000 PMT = $45 ($90/2) n = 10 × 2 = 20 %I/Y =? %I/Y = 5.785 × 2 = 11.57% Tyler Food Chain Approximate yield to maturity is represented by Y'. Principal payment - Price of the bond Annual interest payment + Number of years to maturity Y1 = 0.6 (Price of the bond ) + 0.4 (Principal payment ) $1,000 − $1,080 $125 + $125 − $4 20 = = = 0.1155 = 11.55% 0.6 ($1,080 ) + 0.4 ($1,000 ) $648 + $400 Calculator: ($125/2) Compute: PV = $1,080 FV = $1,000 PMT N = 20 × 2 = 40 %I/Y =? %I/Y = 5.736 × 2 = 11.47% = $62.50 10-14. Pia Cloe - Miette Music Company 12%/2 = 6% semiannual interest rate 6% × $1,000 = $60 semiannual interest 15 × 2 = 30 number of periods (n) 10%/ 2 = 5% yield to maturity expressed on a semiannual basis PV A = A × PV IFA (N = 30, %I/Y = 5%) (Appendix D) PV A = $60 × 15.372 = $922.32 PV = FV × PV IF (N = 30, %I/Y = 5%) (Appendix B) PV = $1,000 × 0.231 = $231.00 Present value of interest payments $ 922.32 Present value of principal payment 231.00 Total present value or price of the bond $1,153.32 Calculator: Compute: PV =? FV = $1,000 N = 15 × 2 = 30 PV = $1,153.72 PMT = $60 ($120/2) %I/Y = 10/ 2 = 5% 10-15. Granny’s Karate and Judo School a. Present Value of Interest Payments PV A = A × PV IFA (N = 40, %I/Y = 6%) (Appendix D) PV A = $45 × 15.046 = $677.07 Present Value of Principal Payment at Maturity PV = FV × PV IF (N = 40, %I/Y = 6%) (Appendix B) PV = $1,000 × 0.097 = $97.00 $677.07 97.00 $774.07 Calculator: Compute: PV =? FV = $1,000 N = 20 × 2 = 40 PV = $774.31 PMT = $45 ($90/2) %I/Y = 12/ 2 = 6% b. PV A = A × PV IFA (N = 20, %I/Y = 4%) (Appendix D) PV A = $45 × 13.590 = $611.55 PV = FV × FV IF (N = 20, %I/Y = 4%) (Appendix B) PV = $1,000 × .456 = $456 $ 611.55 456.00 $1,067.55 Calculator: Compute: PV =? FV = $1,000 N = 10 × 2 = 20 PV = $1,067.95 PMT = $45 ($90/2) %I/Y = 8/ 2 = 4% c. Principal payment - Price of the bond Number of years to maturity Y1 = 0.6 (Price of the bond ) + 0.4 (Principal payment ) $1,000 − $900 $90 + $90 + $10 10 = = = 0.1064 = 10.64% 0.6 ($900 ) + 0.4 ($1,000 ) $540 + $400 Annual interest payment + Calculator: Compute: 10-16. PV = $900 FV = $1,000 PMT = $45 ($90/2) N = 10 × 2 = 20 %I/Y = ? %I/Y = 5.325 × 2 = 10.65 Fraser Bonds Approximate yield to maturity is represented by Y'. Principal payment - Price of the bond Annual interest payment + Number of years to maturity Y1 = 0.6 (Price of the bond ) + 0.4 (Principal payment ) $1,000 − $1,218 $95 + $95 − $22 10 = = = 0.0645 = 6.45% 0.6 ($1,218) + 0.4 ($1,000 ) $731 + $400 Calculator: PV = $1,218 FV = $1,000 PMT = $47.50 ($95/2) N = 10 × 2 = 20 %I/Y =? Compute: %I/Y = 3.251 × 2 = 6.50% 10-17. Bond Relationships a. Calculator: Compute: Calculator: Compute: Calculator: Compute: Calculator: Compute: PV =? FV = $1,000 N=1×2=2 PV = $1,019.42 PMT = $30 ($60/2) %I/Y = 4/ 2 = 2% PV =? FV = $1,000 N=1×2=2 PV = $981.14 PMT = $30 ($60/2) %I/Y = 8/ 2 = 4% PV =? FV = $1,000 N = 20 × 2 = 40 PV = $802.07 PMT = $30 ($60/2) %I/Y = 8/ 2 = 4% PV =? FV = $1,000 N = 20 × 2 = 40 PV = $1,273.55 PMT = $30 ($60/2) %I/Y = 4/ 2 = 2% b. Price and yield are inversely related. c. Bond prices change more for longer terms, for a given yield change. 10-18. Hilton Chocolate Company Pp = 10-19. or : a. Original price Kp = $6.00 = $75 0.08 Rocket Rods Pp = 10-20. Dp Dp Kp = $2.00 = $50 0.04 = $0.50 = $50 0.01 X-Tech Pp = Dp Kp = $5.00 = $100 0.05 b. Current value Pp = Dp Kp = $5.00 = $41.67 0.12 c. The price of preferred stock will increase as yields decline. Since preferred stock is a fixed income security, its price is inversely related to yields as would be true with bond prices. The present value of an income stream has a higher present value as the discount rate declines, and a lower present value as the discount rate increases. 10-21. Quaid Brothers Corporation Kp = 10-22. Dp Pp = $12 = 0.1111 = 11.11% $108 Frontier Corporation Kp = Dp Pp or : 10-23. = $4 = 0.08 = 8.00% $50 = $1.00 = 0.02 × 4 = 0.08 = 8.00% $50 Stagnant Iron and Steel Co. P0 = 10-24. D1 $4.20 = = $35.00 K e − g 0.12 − 0.0 Allied Coal P0 = D1 $3.40 $3.40 = = = $56.67 K e − g 0.14 − 0.08 .06 10-25. P0 = Adventure Tours D1 $0.40 × 4 $1.6 $0.40 = = = $40.00 or = = $40.00 K e − g 0.10 − 0.06 .04 0.025 − 0.015 10-26. Friedman Steel Company a. P0 = D1 $1.50 $1.50 = = = $30.00 K e − g 0.10 − 0.05 .05 b. P0 = D1 $1.50 $1.50 = = = $21.43 K e − g 0.12 − 0.05 .07 c. P0 = D1 $1.50 $1.50 = = = $50.00 K e − g 0.10 − 0.07 .03 d. P0 = D1 $2.00 $2.00 = = = $40.00 K e − g 0.10 − 0.05 .05 10-27. D1 Fleming Corporation = D 0 (1 + g) = $4.00(1.08) = $4.32 P0 = 10-28. D1 $4.32 $4.32 = = = $86.40 K e − g 0.13 − 0.08 .05 Rick’s Department Store a. Earnings have been growing at a rate of 5 percent per year. (Actual 4.99%) 2008 (Base Period) 2009 $4.20/ 4.00 2010 $4.41/ 4.20 2011 $4.63/ 4.41 2012 $4.86/ 4.63 5% growth 5% growth 5% growth 5% growth The projected EPS for 2013 is $5.10 ($4.86 × 1.05) b. Dividend for 2013 represent 40% of earnings or $2.04 ($5.10 × 40%) This is the value for D 1 . K e (required rate of return) is 13% and the growth rate is 5%. D1 $2.04 $2.04 P0 (2006 ) = = = = $25.50 K e − g 0.13 − 0.05 .08 10-29. A Firm Ke = 10-30. Ke = D1 $4.00 +g= + 0.05 = 0.08 + 0.05 = 0.1300 = 13.00% P0 $50.00 Another Firm D1 $0.30 × 4 +g= + 0.07 = 0.03 + 0.07 = 0.1000 = 10.00% P0 $40.00 10-31. a. K e = Still Another Firm D1 $1.50 +g= + 0.09 = 0.0250 + 0.09 = 0.1150 = 11.50% P0 $60.00 b. If the dividend payment increases, the dividend yield (D 1 /P o ) will go up, and the required rate of return (K e ) will also go up. c. If the expected growth rate (g) increases, the required rate of return (K e ) will go up. d. If the stock price increases, the dividend yield (D 1 /P o ) will go down, and the required rate of return (K e ) will also go down. 10-32. a. D 1 D2 D3 Hunter Petroleum Corporation = $2.00 × (1.05) = = $2.10 × (1.05) = = $2.205 × (1.05) = b. Dividends D1 D2 D3 $2.10 $2.205 $2.315 PV of Dividends @ 12% = $2.10 = $2.205 = $2.315 $1.875 1.757 1.648 $5.280 c. D 4 = $2.315 × (1.05) = $2.430 P3 = D4 $2.43 $2.43 = = = $34.71 K e − g 0.12 − 0.05 .07 d. PV of P 3 for n = 3, %I/Y = 12% Calculator: PV = $24.71 PV = $34.71 × .712 = $24.716 e. answer to part b (PV of dividends) answer to part d (PV of P 3 ) current value of the stock f. P0 = D1 $2.10 $2.10 = = = $30.00 K e − g 0.12 − 0.05 .07 $ 5.28 24.72 $30.00 10-33. A Common Share If a $3.00 dividend has just been paid this is D 0 . D1 = $3.00 (1.06) = $3.18 P0 = D1 $3.18 $3.18 = = = $26.50 K e − g 0.18 − 0.06 .12 10-34. A Bond PV A = A × PV IFA (N = 9 × 2 =18, %I/Y = 8/ 2 = 4%) (Appendix D) = $120/2 × 12.659 = $759.54 PV = FV × PV IF (N = 18, %I/Y = 4%) (Appendix B) = $1,000 × .494 = $494.00 $ 759.54 494.00 $1,253.54 Calculator: Compute: PV =? FV = $1,000 N = 18 (9 × 2) PV = $1,253.19 10-35. PMT = $60 ($120/2) %I/Y = 4% (8%/2) Bali Lottery This is a perpetuity. The annual stipend can be considered D p . Pp = Dp Kp = $50,000 = $555,555.55 0.09 10-36. A Bond PV A = A × PV IFA (N = 7, i% = 9%) (Appendix D) = ($7,000 × 0.12) × 5.033 = $840 × 5.033 = $4,228 PV = FV × PV IF (N = 7, %I/Y = 9%) (Appendix B) = $7,000 × 0.547 = $3,829 $4,228 3,829 $8,057 Calculator: Compute: 10-37. PV =? FV = $7,000 N=7 %I/Y = 9% PV = $8,056.92 PMT = $840 Jingle Bell PV A = A × PV IFA (N = 14, i% = 8%) (Appendix D) = $80 × 8.244 = $659.52 PV = FV × PV IF (N = 14, %I/Y = 8%) (Appendix B) = $1,000 × 0.340 = $340.00 $ 659.52 340.00 $ 999.52 Calculator: PV =? FV = $1,000 PMT = $80 N = 14 %I/Y = 8% Compute: PV = $1,000.00 If YTM = coupon rate bond will sell at par. 10-38. A Preferred Share Pp = 10-39. Dp Kp = $10.00 = $166.67 0.06 A Common Share P0 = D1 $2.30 $2.30 = = = $15.33 K e − g 0.22 − 0.07 .15 10-40. Annual Annuity A = PV A / PV IFA (N = 10, i% = 9%) (Appendix D) = $100,000/ 6.418 = $15,581.18 Calculator: Compute: 10-41. PV = $100,000 FV =0 PMT =? N = 10 %I/Y = 9% PMT = $15,582.01 A Preferred Share Pp = 10-42. Dp Kp = $6.00 = $54.55 0.11 Batman Company P0 = D1 $1.75 $1.75 = = = $13.46 K e − g 0.18 − 0.05 .13 10-43. Mark Spence PVIF = PV $20 = = 0.463 FV $43.18 Appendix B, N = 10, IF PV = .463 We find %I/Y = 8% Calculator: Compute: 10-44. PV = $20 FV = $43.18 N = 10 %I/Y =? %I/Y = 8% PMT = 0 Strip Bond PV = FV × PV IF (N = 15, %I/Y = 9%) (Appendix B) = $10,000 × 0.275 = $2,750 Calculator: Compute: 10-45. PV =? FV = $10,000 N = 15 %I/Y = 9% PV = $2,745.38 PMT = 0 A Life Payment Current market yield Perpetuity formula = Pp = 1% 3% 2% 6% Dp Kp = inflation real rate of return risk premium $50,000 = $833,333 0.06 10-46. Blue Jay Ltd. PV A = A × PV IFA [N = 18 (9 × 2), i% = 4% (8%/2)] (Appendix D) = $60 ($120/ 2) × 12.659 = $759.54 PV = FV × PV IF (N = 18, %I/Y = 4%) (Appendix B) = $1,000 × 0.494 = $494.00 $ 759.54 494.00 $1,253.54 Calculator: Compute: 10-47. PV =? FV = $1,000 N = 18 (9 × 2) PV = $1,253.19 PMT = $60 ($120/2) %I/Y = 4% (8%/ 2) Royal Alberta Bonds FV A = A × FV IFA (N = 7, %I/Y = 11) (Appendix C) = $110 × 9.783 = $1,076.13 Calculator: Compute: PV = 0 FV =? PMT = $110 N=7 %I/Y = 11% FV = $1,076.16 Amount now available: Calculator: Compute: $1,076.16 1,075.00 2,151.16 PV = $1,000 FV = $2,151.16 PMT = 0 N=7 %I/Y =? %I/Y = 11.56% With tables: PV $1,000 = = 0.465 FV $2,151.16 Appendix B, n = 7, PV IF = .465; between 11% and 12% PVIF = PV IF at 11% = PV IF at 12% = Difference = .482 .452 .030 PV IF at 11% = PV IF designated = Difference = .482 .465 .017 .017 % i = 0.11 + 0.01 = 0.11 + 0.01 (.57 ) = 0.1157 = 11.57% . 030 10-48. Wapiti College a. Perpetuity formula = Pp = Dp Kp = $6,000 = $54,545.45 0.11 b. If the endowment won’t commence for five years, $54,545.45 is required at the beginning of the fifth year or four years from now. PV = FV × PV IF (N = 4, i = 11%) (Appendix B) = $54,545.45 × 0.659 = $35,945.45 Calculator: Compute: PV =? FV = $54,545.45 N=4 %I/Y = 11% PV = $35,930.78 PMT = 0 Comprehensive Problem 10-49. Dunning Chemical Valuation D1 $1.30 $1.30 a. P0 = = = = $32.50 K e − g 0.14 − 0.10 .04 b. Share valuation based on components Growth rate = 10% Discount rate = 14% D1 = 1.30 = $1.30 D2 = 1.30(1.10) = 1.43 D3 = 1.43(1.10) = 1.57 D4 P.V. of Dividends $ 1.14 1.10 1.06 $3.30 1.57(1.10) = $1.73 D1 $1.73 $1.73 P3 = = = = $43.25 K e − g 0.14 − 0.10 .04 Total share value c. = $29.19 $32.49 Average P/E ratio of five chemical firms: = (15 + 18 + 7 + 19 + 21)/ 5 = 80/ 5 =16 Share price = P/E × EPS = 16 × $2.10 = $33.60 The share price using the P/E ratio approach is slightly higher than the value using the dividend valuation model approach. d. Weighted P/E: = [(15 × 0.15) + (18 × 0.15) + (7 × 0.15) + (19 × 0.15) + (21 × 0.40)] = 17.25 Share price e. Share price (d) Share price (c) = P/E × EPS $36.23 33.60 = 17.25 × $2.10 = $36.23 $ 2.63 % change in price = $2.63 = 0.0783 = 7.83% $33.60 MINI CASE Gilbert Enterprises This case examines valuation concepts from both a theoretical dividend valuation model approach and a price-earnings ratio approach. Because an initial period of supernormal growth is assumed, a review of Appendix 10B is necessary for the case. The case also makes strong use of ratios as part of the comparative P/E ratio analysis and shows how ratios influence valuation. Determine the share price using the valuation of a supernormal growth firm. Discount rate = 10% P.V. of Dividends D0 D1 D2 D3 D 4 to D ∞ = = = = = $1.20 1.20(1.15) = $1.38 1.38(1.15) = 1.59 1.59(1.15) = 1.83 P3 D1 $1.83 (1.06 ) P0 = = = $48.50 K e − g 0.10 − 0.06 P0 $1.25 1.31 1.37 36.42 $40.35 Because the stock is only selling in the market for $35.25, it appears to be undervalued. Gilbert Enterprises has the second lowest P/E ratio of the four firms. Based on the financial information provided in Figure 1, this does not appear to be appropriate. First of all, Gilbert Enterprises has the fastest growth rate in earnings per share of any of the four firms. Furthermore, the growth is expected to accelerate to 15 percent over the next three years (as explained earlier in the case). Gilbert Enterprises also has the second highest return on shareholder’s equity. Only Reliance Parts has a higher return, but its return is achieved solely as a result of its high debt ratio of 68 percent. As we learned in Chapter 3, it is possible to generate a high return on equity using debt, but still have relatively low profitability. In fact, Reliance Parts has the lowest return on total assets of any firm in the industry. In evaluating debt utilization as a separate item, Gilbert Enterprises once again looks attractive with a debt to total assets ratio of 33 percent. Only Standard Auto has a lower ratio. We get further insight by evaluating market value to book value as well as market value to replacement value. In terms of market value to book value, Gilbert Enterprises appears to be overvalued relative to other firms in the industry. Its’ market value to book value ratio is 2.15 ($35.25/ $16.40). For the other three firms, the ratios are more conservative. Gilbert Enterprises Reliance Parts Standard Auto Allied Motors Market Value $35.25 70.50 24.25 46.75 Book Value $16.40 50.25 19.50 50.75 Market Value to Book Value 2.15 1.40 1.24 0.92 But keep in mind that book value is a relatively meaningless concept because it is based on historical cost. A more meaningful analysis relates market value to replacement value. In this instance, we see that Gilbert Enterprises is the most conservatively valued of the four firms. Gilbert Enterprises Reliance Parts Standard Auto Allied Motors Market Value Replacement Value $35.25 $43.50 Market Value to Replacement Value .81 70.50 24.25 46.75 68.75 26.00 37.50 1.03 .93 1.25 What about dividends? In terms of dividend yield, only Standard Auto provides a higher return to its shareholders. In summarizing the variables under consideration, it appears that Gilbert Enterprises may be undervalued relative to its competitors. While it has the second lowest P/E ratio, it has the fastest growth rate in earnings per share, the highest return on assets, and the lowest ratio of market value to replacement value. The average P/E ratio for the four firms in the industry is 18.3. A strong case can be made that Gilbert Enterprises belongs at least at that level. Our analysis reveals that the firm may undervalued. Albert Roth should seriously consider recommending that the firm repurchase part of its shares in the marketplace. There are two possible caveats. One is that the market tends to be efficient in the pricing of securities so that one could possibly argue that there is some missing information that justifies Gilbert Enterprises’ relatively low valuation. While an extended discussion of this point goes beyond the scope of this case, it probably should be brought up. Second is even if the stock is undervalued in the marketplace, the management of Gilbert Enterprises must make sure this is the best possible use of its funds. While the justification for a repurchase decision is not covered until Chapter 18 of the text, the instructor should at least make mention of the alternative uses of funds that must be considered in a stock repurchase decision. Appendix 10A Problems 10A-1. Rocky Corporation Since the bond is trading below par value at $1,177, we can assume the yield to maturity must be above the quoted interest rate of 10 percent. (The yield to maturity would be 10 percent at a bond value of $1,000.) As a first approximation, we will try 9 percent. Present value of interest payments PV A = A × PV IFA (N = 9, %I/Y = 9%) (Appendix D) PV A = $100 × 5.995 = $599.50 Present value of principal payment at maturity PV = FV × PV IF (N = 9, %I/Y = 9%) (Appendix B) = $1,000 × 0.460 = $460.00 $599.50 460.00 $1059.50 The discount rate of 9 percent gives us too high a present value in comparison to the bond price of $1,177. So we next use a higher discount rate of 7 percent. Present value of interest payments PV A = A × PV IFA (N = 9, %I/Y = 7%) (Appendix D) = $100 × 6.515 = $651.50 Present value of principal payment at maturity PV = FV × PV IF (N = 9, %I/Y = 7%) (Appendix B) = $1,000 × .544 = $544.00 $ 651.50 544.00 $1,195.50 The discount rate of 7 percent provides a value lower than the price of the bond. The actual value for the bond must fall between 7% and 9%. Using interpolation, the answer is: $1,195.50 – 1,059.50 $ 136.00 7% + Calculator: Compute: 10A-2. PV at 7% PV at 9% $1,195.50 -1,177.00 $ 18.50 PV at 9% Bond price $18.50 (1% ) = 7% + 0.14 (1% ) = 7.14% $136.00 PV = $1,177 FV = $1,000 N=9 %I/Y =? %I/Y = 7.25% PMT = $100 Boris Corporation Since the bond is trading below par value at $841, we can assume the yield to maturity must be above the quoted interest rate of 4 %. (The yield to maturity would be 4 % at a bond value of $1,000.) As a first approximation, we will try 6 % or 3 % semiannually. The semiannual payment is $20 ($40/ 2). Present value of interest payments PV A = A × PV IFA (N = 11 × 2 = 22, %I/Y = 6/ 2 = 3%) (Appendix D) PV A = $20 × 15.937 = $318.74 Present value of principal payment at maturity PV = FV × PV IF (N = 22, %I/Y = 3%) (Appendix B) = $1,000 × 0.522 = $522 $ 318.74 522.00 $840.74 The discount rate of 6% gives us too low a present value in comparison to the bond price of $841, but it is close.. So we next use a lower discount rate of 5% or 2.5% semiannually. Present value of interest payments PV A = A × PV IFA (N = 22, %I/Y = 2.5%) (Appendix D) = $20 × 16.765 = $335.30 Present value of principal payment at maturity PV = FV × PV IF (N = 22, %I/Y = 2.5%) (Appendix B) = $1,000 × 0.581 = $581 $335.30 581.00 $916.30 The discount rate of 5 percent provides a value higher than the price of the bond. The actual value for the bond must fall between 5% and 6%. Using interpolation, the answer is: $916.30 – 840.74 $ 75.56 5% + Calculator: Compute: PV at 5% PV at 6% $916.30 –841.00 $ 75.30 PV at 5% Bond price $75.30 (1% ) = 5% + 0.997 (1% ) = 5.997% $75.56 PV = $841 FV = $1,000 N = 22 %I/Y =? %I/Y = 2.997 × 2 = 5.995% PMT = $20 Appendix 10B Problems 10B-1. a. D 1 D2 D3 McMillan Corporation $2.40(1.25) = $3.00(1.25) = $3.75(1.25) = b. Supernormal dividends $3.00 $3.75 $4.69 Present value of dividends during the supernormal growth period at K e = 14% (Appendix B) D 1 = $3.00 D 2 = $3.75 D 3 = $4.69 $2.63 2.89 3.17 $8.69 c. D 4 = D 3 (1.06) = $4.69 (1.06) = $4.97 D4 $4.97 $4.97 P3 = = = = $62.13 K e − g 0.14 − 0.06 .08 d. PV of P 3 for n = 3, %I/Y = 14% $62.13 PV = $41.94 e. Answer to part b (PV of dividends) Answer to part d (PV of P 3 ) Current value of the stock $ 8.69 41.94 $50.63 10B-2. Ninja Co. Discount rate = 25% P.V. of Dividends D1 = $5.00 $4.00 D2 = 5.00(1.09) = $5.45 3.49 D3 = 5.45(1.09) = 5.94 3.04 5.94(1.09) = 6.48 2.65 D4 = P4 D 5 to D ∞ = D5 $6.48 (1.06 ) P4 = = = $36.12 14.79 K e − g 0.25 − 0.06 $27.97 P0 10B-3. Slowdown Ltd. Discount rate = 17% P.V. of Dividends D0 = $1.00 – 1.00(1.25) = $1.25 1.07 D1 = D2 = 1.25(1.25) = 1.56 1.14 D3 = 1.56(1.25) = 1.95 1.22 D 4 to D ∞ = P3 D4 $1.95 P3 = = = $11.47 7.16 K e − g 0.17 − 0.00 $10.59 P0 10B-4. Aero Inc. Discount rate = 17% P.V. of Dividends D1 = $2.25 $1.92 D2 = 2.25 1.64 D3 = 2.25 1.40 2.25 1.20 D4 = P4 D 5 to D ∞ = D5 $2.25 (1.08) P4 = = = $27.00 14.41 K e − g 0.17 − 0.08 $ 20.57 P0 10B-5. Ides Ltd. Discount rate = 12% P.V. of Dividends D0 = $0.97 – D1 = 1.09 0.97 1.30 1.04 D2 = P2 D 3 to D ∞ = D3 $1.30(1.07) P2 = = = $27.82 22.18 Ke − g 0.12 − .07 $24.19 P0 Solution Manual for Foundations of Financial Management Stanley B. Block, Geoffrey A. Hirt, Bartley Danielsen, Doug Short, Michael Perretta 9780071320566, 9781259268892, 9781259261015
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