Solution manual financial management 10e by keown chapter 06

CHAPTER Risk and Rates of Return CHAPTER ORIENTATION This chapter introduces the concepts that underlie the valuation of securities and their rates of return We are specifically concerned with common stock, preferred stock, and bonds We also look at the concept of the investor's expected rate of return on an investment CHAPTER OUTLINE I II III The relationship between risk and rates of return A Data have been compiled by Ibbotson and Sinquefield on the actual returns for various portfolios of securities from 1926-2002 B The following portfolios were studied Common stocks of small firms Common stocks of large companies Long-term corporate bonds Long-term U.S government bonds U.S Treasury bills C Investors historically have received greater returns for greater risk-taking with the exception of the U.S government bonds D The only portfolio with returns consistently exceeding the inflation rate has been common stocks Effects of Inflation on Rates of Return A When a rate of interest is quoted, it is generally the nominal or, observed rate The real rate of interest represents the rate of increase in actual purchasing power, after adjusting for inflation B Consequently, the nominal rate of interest is equal to the sum of the real rate of interest, the inflation rate, and the product of the real rate and the inflation rate Term Structure of Interest Rates 144 The relationship between a debt security’s rate of return and the length of time until the debt matures is known as the term structure of interest rates or the yield to maturity IV Expected Return A The expected benefits or returns to be received from an investment come in the form of the cash flows the investment generates B Conventionally, we measure the expected cash flow, X , as follows: X = XiP(Xi) where N = the number of possible states of the economy Xi = the cash flow in the ith state of the economy P(Xi) = V the probability of the ith cash flow Riskiness of the cash flows A Risk can be defined as the possible variation in cash flow about an expected cash flow B Statistically, risk may be measured by the standard deviation about the expected cash flow C Risk and diversification Total variability can be divided into: a The variability of returns unique to the security (diversifiable or unsystematic risk) b The risk related to market movements (nondiversifiable or systematic risk) By diversifying, the investor can eliminate the "unique" security risk The systematic risk, however, cannot be diversified away The market rewards diversification We can lower risk without sacrificing expected return, and/or we can increase expected return without having to assume more risk Diversifying among different kinds of assets is called asset allocation Compared to diversification within the different asset classes, the benefits received are far greater through effective asset allocation Risk and being patient a An investor in common stocks must often wait longer to earn the higher returns than those provided by bonds b The capital markets reward us not just for diversifying, but also for being patient The returns tend to converge toward the average as we lengthen our holding period 145 VI The characteristic line tells us the average movement in a firm's stock price in response to a movement in the general market, such as the stock market The slope of the characteristic line, which has come to be called beta, is a measure of a stock's systematic or market risk The slope of the line is merely the ratio of the "rise" of the line relative to the "run" of the line If a security's beta equals one, a 10 percent increase (decrease) in market returns will produce on average a 10 percent increase (decrease) in security returns A security having a higher beta is more volatile and thus more risky than a security having a lower beta value A portfolio's beta is equal to the average of the betas of the stocks in the portfolio Required rate of return A The required rate of return is the minimum rate necessary to compensate an investor for accepting the risk he or she associates with the purchase and ownership of an asset B Two factors determine the required rate of return for the investor: C The risk-free rate of interest which recognizes the time value of money The risk premium which considers the riskiness (variability of returns) of the asset and the investor's attitude toward risk Capital asset pricing model-CAPM The required rate of return for a given security can be expressed as = + beta x or kj = krf + βj (km - krf) Security market line a Graphically illustrates the CAPM b Designates the risk-return trade-off existing in the market, where risk is defined in terms of beta according to the CAPM equation 146 ANSWERS TO END-OF-CHAPTER QUESTIONS 6-1 Data have been compiled by Ibbotson and Sinquefield on the actual returns for the following portfolios of securities from 1926-2002 U.S Treasury bills U.S government bonds Corporate bonds Common stocks for large firms Common stocks for small firms Investors historically have received greater returns for greater risk-taking with the exception of the U.S government bonds Also, the only portfolio with returns consistently exceeding the inflation rate has been common stocks 6.2 When a rate of interest is quoted, it is generally the nominal or, observed rate The real rate of interest represents the rate of increase in actual purchasing power, after adjusting for inflation Consequently, the nominal rate of interest is equal to the sum of the real rate of interest, the inflation rate, and the product of the real rate and the inflation rate 6-3 The relationship between a debt security’s rate of return and the length of time until the debt matures is known as the term structure of interest rates or the yield to maturity In most cases, longer terms to maturity command higher returns or yields 6-4 (a) The investor's required rate of return is the minimum rate of return necessary to attract an investor to purchase or hold a security (b) Risk is the potential variability in returns on an investment Thus, the greater the uncertainty as to the exact outcome, the greater is the risk Risk may be measured in terms of the standard deviation or by the variance term, which is simply the standard deviation squared (c) A large standard deviation of the returns indicates greater riskiness associated with an investment However, whether the standard deviation is large relative to the returns has to be examined with respect to other investment opportunities Alternatively, probability analysis is a meaningful approach to capture greater understanding of the significance of a standard deviation figure However, we have chosen not to incorporate such an analysis into our explanation of the valuation process (a) Unique risk is the variability in a firm's stock price that is associated with the specific firm and not the result of some broader influence An employee strike is an example of a company-unique influence (b) Systematic risk is the variability in a firm's stock price that is the result of general influences within the industry or resulting from overall market or economic influences A general change in interest rates charged by banks is an example of systematic risk 6-5 147 6-6 Beta indicates the responsiveness of a security's returns to changes in the market returns Beta is multiplied by the market risk premium and added to the risk-free rate of return to calculate a required rate of return 6-7 The security market line is a graphical representation of the risk-return trade-off that exists in the market The line indicates the minimum acceptable rate of return for investors given the level of risk Since the security market line results from actual market transactions, the relationship not only represents the risk-return preferences of investors in the market but also represents the investors' available opportunity set 6-8 The beta for a portfolio is equal to the weighted average of the individual stock betas, weighted by the percentage invested in each stock 6-9 If a stock has a great amount of variability about its characteristic line (the graph of the stock's returns against the market's returns), then it has a high amount of unsystematic or company-unique risk If, however, the stock's returns closely follow the market movements, then there is little unsystematic risk SOLUTIONS TO END-OF-CHAPTER PROBLEMS Solutions to Problems Set A 6-1A krf = 045 + 073 + (.045 x 073) krf = 1213 or 12.13% = nominal rate of interest 6-2A krf = 064 + 038 + (.064 x 038) krf = 1044 or 10.44% = nominal rate of interest 148 6-3A (A) Probability P(ki) 15 30 40 15 (B) Return (ki) -1% (A) x (B) Expected Return k -.15% 0.60% 1.20% 1.20% 2.85% 2 k=  Weighted Deviation (ki - k )2P(ki) 2.223% 0.217% 0.009% 3.978% = 6.427% = 2.535% No, Pritchard should not invest in the security The level of risk is excessive for a return which is less than the rate offered on treasury bills 6-4A Common Stock A: (A) Probability P(ki) 0.3 0.4 0.3 (B) Return (ki) 11% 15 19 (A) x (B) Expected Return k 3.3% 6.0 5.7 2 k = 15.0%  Weighted Deviation (ki - k )2P(ki) 4.8% 0.0 4.8 = 9.6% = 3.10% Common Stock B (A) Probability P(ki) 0.2 0.3 0.3 0.2 (B) Return (ki) -5% 14 22 (A) x (B) Weighted Expected Return Deviation (ki - k )2P(ki) k -1.0% 41.472% 1.8 3.468 4.2 6.348 4.4 31.752 2 = 83.04% k = 9.4%  = 9.11% Common Stock A is better It has a higher expected return with less risk 149 6-5A Common Stock A: (A) Probability P(ki) 0.2 0.5 0.3 (B) Return (ki) - 2% 18 27 (A) x (B) Weighted Expected Return Deviation (ki - k )2P(ki) k -0.4% 69.9% 9.0 0.8 8.1 31.8 = 16.7%  = 102.5% k  = 10.12% Common Stock B: (A) Probability P(ki) 0.1 0.3 0.4 0.2 (B) Return (ki) 4% 10 15 k (A) x (B) Weighted Expected Return Deviation (ki - k )2P(ki) k 0.4% 2.704% 1.8 3.072 4.0 0.256 3.0 6.728 = 9.2%  = 12.76%  = Common Stock A k = 16.7%  = 10.12% 3.57% Common Stock B k = 9.2%  = 3.57% We cannot say which investment is "better." It would depend on the investor's attitude toward the risk-return tradeoff 6-6A (a) = + Beta = % + 1.2 (16% - 6%) = 18% (b) The 18 percent "fair rate" compensates the investor for the time value of money and for assuming risk However, only nondiversifiable risk is being considered, which is appropriate 6-7A Eye balling the characteristic line for the problem, the rise relative to the run is about 0.5 That is, when the S & P 500 return is eight percent Aram's expected return would be about four percent Thus, the beta is also approximately 0.5 (4 ÷ 8) 150 6-8A A B C D 6-9A.`= + + + + + 6.75% 6.75% 6.75% 6.75% + (12% (12% (12% (12% - x x x x x 6.75%) 6.75%) 6.75%) 6.75%) Beta = 1.50 = 0.82 = 0.60 = 1.15 = 14.63% 11.06% 9.90% 12.79% (Market Return - Risk-Free Rate) X Beta = 7.5% + (11.5% - 7.5%) x 0.765 = 10.56% 6-10A If the expected market return is 12.8 percent and the risk premium is 4.3 percent, the riskless rate of return is 8.5 percent (12.8% - 4.3%) Therefore; Tasaco = 8.5% + (12.8% - 8.5%) x 0.864 = 12.22% LBM = 8.5% + (12.8% - 8.5%) x 0.693 = 11.48% Exxos = 8.5% + (12.8% - 8.5%) x 0.575 = 10.97% 6-11A Asman Time Price $10 12 11 13 Salinas Return 20.00% -8.33 18.18 Price $30 28 32 35 Return -6.67% 14.29 9.38 A holding-period return indicates the rate of return you would earn if you bought a security at the beginning of a time period and sold it at the end of the period, such as the end of the month or year 151 6-12A.a Month kb Sum Zemin (kb - k )2 6.00% 3.00 1.00 -3.00 5.00 0.00 12.00 16.00% 1.00 1.00 25.00 9.00 4.00 56.00 kb Market (kb - k )2 4.00% 2.00 -1.00 -2.00 2.00 2.00 7.00 2.00% 1.17% 24.00% 14.04% 8.03% 0.69 4.69 10.03 0.69 0.69 24.82 (Sum ÷ 6) Variance (Sum  5) 11.20% 3.35% b = 4.97% 2.23% + (Market Return - Risk-Free Rate) X Beta = 8% + [(14% - 8%) X 1.54] = 17.24% c Zemin's historical return of 24 percent exceeds what we would consider a fair return of 17.24 percent, given the stock's systematic risk a The portfolio expected return, k p, equals a weighted average of the individual stock's expected returns 6-13A kp = (0.20)(16%) + (0.30)(14%) + (0.15)(20%) + (0.25)(12%) + (0.10)(24%) = 15.8% 152 b The portfolio beta, ßp, equals a weighted average of the individual stock betas ßp c = (0.20)(1.00) + (0.30)(0.85) + (0.15)(1.20) + (0.25)(0.60) + (0.10)(1.60) = 0.95 Plot the security market line and the individual stocks 25.00 Expected Return 20.00 P M 15.00 10.00 5.00 0.00 0.00 0.50 1.00 1.50 2.00 Beta d A "winner" may be defined as a stock that falls above the security market line, which means these stocks are expected to earn a return exceeding what should be expected given their beta or systematic risk In the above graph, these stocks include 1, 3, and "Losers" would be those stocks falling below the security market line, which are represented by stocks and ever so slightly e Our results are less than certain because we have problems estimating the security market line with certainty For instance, we have difficulty in specifying the market portfolio 153 Reynolds vs Market 0.4 0.3 0.2 Market 0.1 -0.2 -0.1 0.1 0.2 -0.1 -0.2 -0.3 Reynolds Andrews vs Market 0.4 0.3 0.2 Andrews 0.1 -0.2 -0.1 -0.1 -0.2 -0.3 -0.4 -0.5 Marke t 159 0.1 0.2 Reynolds’s returns have a great amount of volatility with some correlation to the market returns The same can be said of Andrews The returns show a great amount of volatility that followed the market returns only part of the time Monthly returns of a portfolio of equal amounts of Reynolds and Andrews 2001 June July August September October November December 2002 January February March April May June July August September October November December 2003 January February March April May Average return Standard deviation 160 Monthly Returns 11.98% -2.32% -16.27% 23.08% 9.74% -0.41% 21.02% 14.70% -9.16% 4.09% 16.20% -8.28% 4.65% -13.81% 8.90% -3.00% 2.84% 2.43% 4.95% 3.66% 7.97% 29.87% -19.80% -0.75% 3.84% 12.29% Reynolds and Andrews 40.00% 50% Reynolds 50% Andrews 30.00% 20.00% 10.00% 0.00% -20.00% -10.00% 0.00% 10.00% 20.00% -10.00% -20.00% -30.00% Market We see in this new graph where both stocks are included as a single portfolio that the relationship of the stocks with the market approximates an average of the relationships taken alone Note the reduction in volatility that occurs when risk is diversified even between just two stocks 161 Monthly holding-period returns for long-term government bonds (ki - k )2 2001 June 5.70% 0.48% 0.000000% July 5.68% 0.47% 0.000001% August 5.54% 0.46% 0.000004% September 5.20% 0.43% 0.000023% October 5.01% 0.42% 0.000041% November 5.25% 0.44% 0.000020% December 5.06% 0.42% 0.000036% 2002 January 5.16% 0.43% 0.000027% February 5.37% 0.45% 0.000012% March 5.58% 0.47% 0.000003% April 5.55% 0.46% 0.000004% May 5.81% 0.48% 0.000000% June 6.04% 0.50% 0.000005% July 5.98% 0.50% 0.000003% August 6.07% 0.51% 0.000006% September 6.07% 0.51% 0.000006% October 6.26% 0.52% 0.000016% November 6.15% 0.51% 0.000009% December 6.35% 0.53% 0.000022% 2003 January 6.63% 0.55% 0.000050% February 6.23% 0.52% 0.000014% March 6.05% 0.50% 0.000005% April 5.85% 0.49% 0.000000% May 6.15% 0.51% 0.000009% Average Monthly Return 0.48% Standard Deviation 0.04% 162 Monthly portfolio returns when portfolio consists of equal amounts invested in Reynolds, Andrews, and long-term government bonds 2001 June July August September October November December 2002 January February March April May June July August September October November December 2003 January February March April May Sum 8.14% -1.39% -10.69% 15.53% 6.63% -0.13% 14.15% 9.94% -5.95% 2.88% 10.95% -5.36% 3.27% -9.04% 6.10% -1.83% 2.07% 1.79% 3.48% 2.63% 5.49% 20.08% -13.04% -0.33% 65.36% Average Monthly Return (ki - k )2 0.0029 0.0017 0.0180 0.0164 0.0015 0.0008 0.0131 0.0052 0.0075 0.0000 0.0068 0.0065 0.0000 0.0138 0.0011 0.0021 0.0000 0.0001 0.0001 0.0000 0.0008 0.0301 0.0248 0.0009 0.1542 2.72% Std Dev 8.19% 163 Comparison of average returns and standard deviations Average Returns 4.44% 3.25% 0.48% 3.84% 2.72% Reynolds Andrews Government security Reynolds & Andrews Reynolds, Andrews, & government security Market 1.25% Standard Deviations 16.93% 18.60% 0.04% 12.29% 8.19% 5.47% From the findings above, we see that higher average returns are associated with higher risk (standard deviations), and that by diversification we can reduce risk, possibly without reducing the average return 10 Based on the standard deviations, Andrews has more risk than Reynolds, 18.60 percent standard deviation versus 16.93 percent standard deviation However, when we only consider systematic risk, Andrews is slightly less risky Reynolds's beta is 1.96 compared to Andrews’ beta of 1.49 (The betas given here for Reynolds and Andrews come from financial services who calculate firms' betas These are not consistent with the graphs above where we see Andrews' returns as being more responsive to the general market We are seeing the problem of using only 24 months of returns as we have done.) 11 = + (Market Return - Risk-Free Rate) X Beta Market Return = 1.25 % Average Monthly Return X 12 Months = 15% (The average returns for the market over a two-year period may be high or low relative to the longer-term past, and as a result should not be considered as “typical” investor expectations For instance, if we used information from Ibbotson & Sinquefield for the years 1926-2002, the market risk premium—market return less risk-free rate—was 8.4 percent, and not the 19 percent that we use below The point: Do not think two years fairly captures what we can expect in the future?) Reynolds: 23.64% = 6% + (15% - 6%) X 1.96 Andrews: 19.41% = 6% + (15% - 6%) X 1.49 And if we used the market premium of 8.4 percent: Reynolds: 22.46% = 6% + 8.4% X 1.96 Andrews: 18.52% = 6% + 8.4% X 1.49 164 Solutions to Problem Set B 6-1B krf = 05 + 07 + (.05 x 07) krf = 1235 or 12.35% = nominal rate of interest 6-2B krf = 03 + 05 + (.03 x 05) krf = 0815 or 8.15% = nominal rate of interest 6-3B (A) Probability P(ki) 15 30 40 15 (B) Return (ki) -3% (A) x (B) Weighted Expected Return Deviation (k k i - k ) P(ki) -0.45% 4.788 0.60 0.127 1.60 0.729 0.90 1.683  = 7.327% k = 2.65%  = 2.707% No, Gautney should not invest in the security The security’s expected rate of return is less than the rate offered on treasury bills 6-4B Security A: (A) Probability P(ki) 0.2 0.5 0.3 (B) Return (ki) - 2% 19 25 (A) x (B) Weighted Expected Return Deviation (k k i - k ) P(ki) -0.4% 69.19% 9.5 2.88 7.5 21.17 16.6% 2 = 93.24% k =  = 9.66% 165 Security B: (A) Probability P(ki) 0.1 0.3 0.4 0.2 (B) Return (ki) 5% 12 14 (A) x (B) Weighted Expected Return Deviation (ki - k )2P(ki) k 0.5% 2.704% 2.1 3.072 4.8 1.296 2.8 2.888  = 9.96% k = 10.2%  = 3.16% Security A k = 16.6%  = 9.66% Security B k = 10.2%  = 3.16% We cannot say which investment is "better." It would depend on the investor's attitude toward the risk-return tradeoff 6-5B Common Stock A: (A) Probability P(ki) 0.2 0.6 0.2 (B) Return (ki) 10% 13 20 (A) x (B) Expected Return k 2.0% 7.8 4.0 k = 13.8% Weighted Deviation (ki - k )2P(ki) 2.89% 0.38 7.69  = 10.96%  = 3.31% Common Stock B (A) Probability P(ki) 0.15 0.30 0.40 0.15 (B) Return (ki) 6% 15 19 (A) x (B) Weighted Expected Return Deviation (ki - k )2P(ki) k 0.9% 5.67% 2.4 5.17 6.0 3.25 2.85 7.04  = 21.13% k = 12.15%  = 4.60% Common Stock A is better It has a higher expected return with less risk 166 6-6B (a) = + Beta = % + 1.5 (16% - 8%) = 20% (b) The 20 percent "fair rate" compensates the investor for the time value of money and for assuming risk However, only nondiversifiable risk is being considered, which is appropriate 6-7B Eye balling the characteristic line for the problem, the rise relative to the run is about 1.75 That is, when the S & P 500 return is four percent Bram's expected return would be about seven percent Thus, the beta is also approximately 1.75 (7 ÷ 4) 6-8B A B C D 6-9B 6.75% 6.75% 6.75% 6.75% + + + + + (12% (12% (12% (12% - 6.75%) 6.75%) 6.75%) 6.75%) x x x x x Beta 1.40 0.75 0.80 1.20 = + (Market Return - Risk-Free Rate) X Beta = 7.5% + (10.5% - 7.5%) x 0.85 = = = = = 14.10% 10.69% 10.95% 13.05% = 10.05% 6-10B If the expected market return is 12.8 percent and the risk premium is 4.3 percent, the riskless rate of return is 8.5 percent (12.8% - 4.3%) Therefore; Dupree = 8.5% + (12.8% - 8.5%) x 0.82 = 12.03% Yofota = 8.5% + (12.8% - 8.5%) x 0.57 = 10.95% MacGrill = 8.5% + (12.8% - 8.5%) x 0.68 = 11.42% 6-11B O'Toole Time Price $22 24 20 25 Return 9.09% -16.67% 25.00% Baltimore Price Return $45 50 11.11% 48 -4.00% 52 8.33% A holding-period return indicates the rate of return you would earn if you bought a security at the beginning of a time period and sold it at the end of the period, such as the end of the month or year, 167 6-12B (a) Sugita kt (kt - k )2 1.80% 0.01% -0.50 5.68 2.00 0.01 -2.00 15.08 5.00 9.71 5.00 9.71 11.30 40.20 Month Sum Market kt (kt - k )2 1.50% 0.06% 1.00 0.06 0.00 1.56 -2.00 10.56 4.00 7.56 3.00 3.06 7.50 22.86 1.88% 1.25% 22.60% 15.00% (Sum ÷ 6) Variance (Sum ÷ 5) 8.04% 4.58% 2.84% 2.14% b = + (Market Return - Risk-Free Rate) X Beta = 8% + [(15% - 8%) X 1.18] = 16.26% c Sugita's historical return of 22.6 percent exceeds what we would consider a fair return of 16.26 percent, given the stock's systematic risk a The portfolio expected return, k p, equals a weighted average of the individual stock's expected returns (0.10)(12%) + (0.25)(11%) + (0.15)(15%) + (0.30)(9%) + kp = (0.20)(14%) 6-13B = 11.7% 168 b The portfolio beta, ßp, equals a weighted average of the individual stock betas ßp c = (0.10)(1.00) + (0.25)(0.75) + (0.15)(1.30) + (0.30)(0.60) + (0.20)(1.20) = 0.90 Plot the security market line and the individual stocks 16.00 Expected Return 14.00 12.00 P M 10.00 8.00 6.00 4.00 2.00 0.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Beta d A "winner" may be defined as a stock that falls above the security market line, which means these stocks are expected to earn a return exceeding what should be expected given their beta or systematic risk In the above graph, these stocks include 1, 2, 3, and "Losers" would be those stocks falling below the security market line, that being stock e Our results are less than certain because we have problems estimating the security market line with certainty For instance, we have difficulty in specifying the market portfolio 169 1.40 6-14B a) Month Jul-02 Aug-02 Sep-02 Oct-02 Nov-02 Dec-02 Jan-03 Feb-03 Mar-03 Apr-03 May-03 Jun-03 Jul-03 Price 1328.72 1320.41 1282.71 1362.93 1388.91 1469.25 1394.46 1366.42 1498.58 1452.43 1420.60 1454.60 1430.83 Sum b) Average Monthly Return Standard deviation Market kt (kt - k )2 -0.63% -2.86% 6.25% 1.91% 5.78% -5.09% -2.01% 9.67% -3.08% -2.19% 2.39% -1.63% 0.0002 0.0013 0.0031 0.0001 0.0026 0.0034 0.0007 0.0080 0.0014 0.0008 0.0003 0.0005 8.52% 0.0225 0.71% 21.00 19.50 17.19 16.88 18.06 24.88 22.75 26.25 33.56 43.31 43.50 43.50 43.63 Hilary’s kt (kt - k )2 -7.14% -11.85% -1.80% 6.99% 37.76% -8.56% 15.38% 27.85% 29.05% 0.44% 0.00% 0.30% 0.0211 0.0369 0.0084 0.0000 0.0924 0.0254 0.0064 0.0419 0.0470 0.0048 0.0054 0.0050 88.42% 0.2948 7.37% 4.52% 170 Price 16.37% c) 50.00% Hilary's 40.00% 30.00% 20.00% 10.00% Market -10.00% 0.00% -5.00% 0.00% 5.00% 10.00% 15.00% -10.00% -20.00% d The Hilary’s returns for the last six months of 2002 and the first six months of 2003 were partially correlated, but with a lot of the variance in the stock’s returns, clearly not explained by the market—as would be expected 171 6-15B Stock A (A) Probability P(ki) 0.10 0.30 0.40 0.20 (B) Return (ki) -4% (A) x (B) Expected Return k -0.40% 13 17 0.60 5.20 3.40 8.80% k= Weighted Deviation (ki - k )2P(ki) 16.384% 2 =  = 13.872 7.056 13.448 50.76% 7.125% Stock B (A) Probability P(ki) 0.13 0.40 0.27 0.20 (B) Return (ki) 4% (A) x (B) Expected Return k 0.52% 10 19 23 k 4.00 5.13 4.60 = 14.25% Weighted Deviation (ki - k )2P(ki) 13.658% 2 =  = 7.225 6.092 15.31 42.285% 6.503% Stock C (A) Probability P(ki) 0.20 0.25 0.45 0.10 (B) Return (ki) -2% (A) x (B) Expected Return k -0.40% 14 25 k = 1.25 6.30 2.50 9.65% Weighted Deviation (ki - k )2P(ki) 27.145% 2 =  = 5.406 8.515 23.562 64.628% 8.039% Stock B has a higher expected rate of return with less risk than Stocks A and C 172 6-16B K G B U + + + + + 5.5% 5.5% 5.5% 5.5% (11% (11% (11% (11% - 5.5%) 5.5%) 5.5%) 5.5%) x x x x x Beta 1.12 1.30 0.75 1.02 = = = = = 6-17B Time Watkins Price $40 45 43 49 Fisher Return 12.50% -4.44 13.95 Price $27 31 35 36 6-18B (a) = + Beta = 4% + 0.95 (7% - 4%) = 6.85% (b) = + Beta = % + 1.25 (7% - 4%) = 7.75% (c) If beta is 0.95: Required rate of return = % + 0.95 (10% - 4%) = 9.7% If beta is 1.25: Required rate of return = % + 1.25 (10% - 4%) = 11.5% 173 Return 14.81% 12.90 2.86 11.66% 12.65% 9.63% 11.11% ... unsystematic risk SOLUTIONS TO END-OF -CHAPTER PROBLEMS Solutions to Problems Set A 6-1A krf = 045 + 073 + (.045 x 073) krf = 1213 or 12.13% = nominal rate of interest 6-2A krf = 064 + 038 + ( .064 x 038)... -14.34% -4.02% 7.15% 18.60% -24.63% 6.17% 32.17% -7 .06% -13.96% 106. 62% 0. 0067 0.0158 0.0153 0.0733 0.0023 0.0134 0.0252 0.1037 0.0592 0.0 006 0.0014 0.0434 0.0009 0.0037 0.0224 0.0353 0.0072... Deviation 0.0007 0.0 006 0.0251 0.0025 0.0046 0.0022 0.0019 0.0008 0.0020 0.0007 0.0 006 0.0014 0.0018 0.0020 0.0004 0.0017 0.0025 0.0000 0.0021 0.0040 0.0011 0.0071 0.0019 0.0012 068 9 Reynolds Computer