Solutions to Problems 1. The investor would earn income of $2.25 and a capital gain of $52.50 $45 =$7.50. The total gain is $9.75 or 21.7%. $8.25 on a stock that paid $3.75 in income and sold for $67.50. Part of the total dollar return includes a $4.50 capital gain, which is the difference between the proceeds of the sale and the original purchase price ($67.50 $63.00) of the stock. 3. a. Income: $ 2.70 b. Capital gain: $60.00 $50.00 $10.00 c. Total return: (1) In dollars: $2.70 $10.00 $12.70 (2) As a percentage of the initial investment: $12.70 0.25 or 25%. $50.00 5. a. Total return Income Capital gains (or losses) where: Capital gains (or losses) Ending price (1) (2) (3) (4) (5) (1) (2) (3) (4) Year Ending Price Beginning Price Capital Gain Current Income Total Return 2010 $32.50 $30.00 $2.50 $1.00 $3.50 2011 35.00 32.50 2.50 1.20 3.70 2012 33.00 35.00 2.00 1.30 0.70 20130 40.00 33.00 7.00 1.60 8.60 20141 45.00 40.00 5.00 1.75 6.75 b. Of course, there is no correct answer here, but one might forecast using the arithmetic average or the average one-year holding period return. i. The arithmetic average: $3.50 $3.70 $.70 $8.60 $6.75 $4.37 5 ii. The average holding period return (): Ending price Current income Total return (1) (2) (3) Year Total Return* Beginning Price (1) (2) 2010 $3.50 $30.00 11.7% 2011 3.70 32.50 11.4 2012 0.70 35.00 2.0 2013 8.60 33.00 26.1 2014 6.75 40.00 16.9 *From part a.
11.7 11.4 2.0 26.116.9 Average 12.8% 5 i. ii. Forecasts for: Based on Arithmetic Average Based on Average 2015 $4.37 ($45.00)* 0.128 $5.76 2016 $4.37 ($49.00)** 0.128 $6.27 End of 2010 price gain in original data For lack of information, we are assuming the 2011 return is $4.00 from capital gains and $1.76 from income. c. Students should be made aware of the fact that many other forecasts are possible. Other factors may be relevant here: Will the pattern of two good years followed by a bad one continue? Do future prospects seem bright? (We will discuss forecasting returns on specific investments in later chapters.) 7. a. Using the notation given in the chapter, the risk-free rate of return is: R f = r* + IP or 2% +3% b. The required returns for each investment are calculated as follows: r 1 r * r * IP RP i or R F RP i r A 2% 3% % 5% 4% 9% r B 2% 3% 6% 5% 6% 11% 9. Holding period return () Current income Ending price X Y $1.00 $1.20 $0 $2.30 $29.00 $30.00 $3.50 $30.00 $30.00 11.67% $0 $0 $0 $2.00 $56.00 $50.00 $8.00 50.00 $50.00 16% If the investments are held beyond a year, the capital gain (or loss) component would not be realized and would likely change. Assuming they are of equal risk, Investment Y would be preferred since it offers the higher return (16.00% for Y versus 11.67% for X). 11. ($50 ($1,000 $950))/$950 $100/$950 10.5%. 13. Using a present value interest factor of 4%:
15. $65 0.962 $62.53 $70 0.925 $64.75 $70 0.889 $62.23 $7,965 0.855 $6,810.08 $6,999.59 Initial Future Calculator Investment Investment Value Years Solution A $1,000 $1,200 5 3.71% B 10,000 20,000 7 10.41 C 400 2,000 20 8.38 D 3,000 4,000 6 4.91 E 5,500 25,000 30 5.18 17. The yield for these investments is the discount rate that results in the stream of income equaling the initial investment. The yield or internal rate of return for Investment A can easily be found with any financial calculator: N=5, PV=-8500, PMT=2500, FV =0, i=14.40% Most financial calculators can also solve for the internal rate of return of irregular cash flows, but the routines vary widely from device to another. The yields or internal rates of return for either A or B can be found quickly using Excel s internal rate of return formula =irr(range of cells containing cash flows). Be sure to include the initial investment as a negative cash flow. End of Year Investment A Investment B 0 (8,500.00) (9,500.00) 1 2,500.00 2,000.00 2 2,500.00 2,500.00 3 2,500.00 3,000.00 4 2,500.00 3,500.00 5 2,500.00 4,000.00 IRR 14.40% 15.36%
19. Again this problem is most easily solved using Excel s internal rate of return formula =irr(range of cash flows). The yield on this investment falls just short of the required 11%, and the NPV when discounted at 11% (=npv(.11,range of positive cash flows)-1000) is slightly negative, so the investment probably should not be accepted. End of Year Income Stream $ (1,000) 2014 140 2015 120 2016 100 2017 80 2018 60 2019 40 2020 1,220 IRR/Yield 10.87% NPV @ 11% 604.56% 21. 2013 2006 7 years Using a financial calculator, N=7, PV=-1.00, PMT=0, FV=2.21, i=12.00% ans Excel =rate(7,0,1.00,2.21) 23. a. Investment A, with returns that vary widely from 1% to 26% appears to be more risky than Investment B, whose returns vary from 8% to 16%. 2 b. s ( rr) n i1 Standard deviation CV Average return Investment A: (1) (2) (3) (4) Return Average (1) (2) (3) 2 Year r I Return, r r i r (r i r) 2 2006 19% 12% 7% 49% 2007 1 12 11 121 2008 10 12 2 4 2009 26 12 14 196 2010 4 12 8 64 434
434 SA 108.5 10.42% 10.42% CV.87 12.00% Investment B: (1) (2) (3) (4) Year Return Average (1) (2) (3) 2 r i Return, r r i r (r i r) 2 2006 8% 12% 4% 16% 2007 10 12 2 4 2008 12 12 0 0 2009 14 12 2 4 2010 16 12 4 16 40 40 SA 10 3.16% 3.16% CV.26 12.00% c. Investment A, with a standard deviation of 10.42, is considerably more risky than Investment B, whose standard deviation is 3.16. This confirms the conclusions reached in Part A.