Solutions ManualCHAPTER 22 ESTIMATING RISK AND RETURN ON ASSETS SUGGESTED ANSWERS TO THE REVIEW QUESTIONS AND PROBLEMS I. Questions 1. Risk is the variability of an asset’s future returns. When only one return is possible, there is no risk. When more than one return is possible, the asset is risky. 2. An objective probability distribution is generally based on past outcomes of similar events while a subjective probability distribution is based on opinions or “educated guesses” about the likelihood that an event will have a particular future outcome. 3. A discrete probability distribution is an arrangement of the probabilities associated with the values of a variable that can assume a limited or finite number of values (outcomes) while a continuous probability distribution is an arrangement of probabilities associated with the values of a variable that can assume an infinite number of possible values (outcomes). 4. The accuracy of forecasted returns generally decreases as the length of the project being forecast increases. This increases the variability of an asset’s returns and therefore risk. 5. The expected value of return of a single asset is the weighted average of the returns, with the weights being the probabilities of each return. 6. The risk of a single asset is measured by its standard deviation or coefficient of variation. The standard deviation measures the variability of outcomes around the expected value and is an absolute measure of risk. The coefficient of variation is the ratio of the standard deviation to the expected value and is a relative measure of risk. 7. The characteristic of a normal curve is a bell-shaped distribution that is dependent upon the mean and the standard deviation of the population under investigation. Since the normal distribution is a continuous rather than a discrete distribution, it is not possible to speak of the probability 22-1 Chapter 22 Estimating Risk and Return on Assets of a point but only of the probability of falling within some specified range of values. Thus, the area under the curve between any two points must then also depend upon the values of the mean and standard deviation. However, it is possible to standardize any normal distribution so that it has a mean of zero and a standard deviation of one. 8. Decision makers may be classified into three categories according to their risk preferences: risk-averse, risk-neutral and risk-taker. Financial theory assumes that decision makers are risk-averse. 9. Portfolio risk is measured by the portfolio standard deviation. Portfolio risk is influenced by diversification. Risk reduction is achieved through diversification whenever the returns of the assets combined in a portfolio are not perfectly positively correlated. Correlation measures the tendency of two variables to move together. 10. No to both questions. The portfolio expected return is a weighted average of the asset returns, so it must be less than the largest asset return and greater than the smallest asset return. 11. False. The variance of the individual assets is a measure of the total risk. The variance on a well-diversified portfolio is a function of systematic risk only. 12. Yes, the standard deviation can be less than that of every asset in the portfolio. However, p cannot be less than the smallest beta because p is a weighted average of the individual asset betas. II. Multiple Choice Questions 1. B 3. D 5. B 7. A 2. C 4. A 6. B III. Problems Problem 1 (a) The bar charts for Stock A and Stock B are shown in the next page. (b) Stock A’s probability distribution is skewed to the left and Stock B’s probability distribution is symmetrical. (c) Stock A’s range of returns is 24 percentage points (25 – 1) and Stock B’s range of returns is 20 points (30 – 10). 22-2 Estimating Risk and Return on Assets Chapter 22 (d) Stock A is riskier than Stock B because Stock A has a wider range of returns and a flatter probability distribution. Stock A 0.60 0.50 0.40 Pro bili ba ty 0.30 0.20 0.10 - 40 0 5 10 15 20 25 30 35 40 10 -5 Return (%) Stock B 0.60 0.50 0.40 bab ility Pro 0.30 0.20 0.10 - -5 0 5 10 15 20 25 30 35 40 10 -5 Return 22-3 (%) Chapter 22 Estimating Risk and Return on Assets Problem 2 The expected value of the returns for each stock is: Stock A ȓA = (0.05) (0.01) + (0.20) (0.05) + (0.25) (0.10) + (0.35) (0.15) + (0.15) (0.25) = 0.1255 or 12.55% Stock B ȓB = (0.10) (0.10) + (0.20) (0.15) + (0.40) (0.20) + (0.20) (0.25) + (0.10) (0.30) = 0.20 or 20% Problem 3 (a) The calculation of the expected value can be set up in tabular form. i pi ri (%) pi ri (%) 1 0.1 0 0 2 0.2 10 2.0 3 0.4 20 8.0 4 0.2 30 6.0 5 0.1 40 4.0 ȓ = 20.0% (b) The calculation of the standard deviation can also be set up in tabular form. The square root of the variance, σ 2, of 120 percent is 10.95 percent (rounded). i ri (%) ȓ (%) ri (%) - ȓ (%) (ri - ȓ) 2 (%) pi pi (ri - ȓ) 2 (%) 1 0 20 - 20 400 0.1 40.0 2 10 20 - 10 100 0.2 20.0 3 20 20 0 0 0.4 0 4 30 20 10 100 0.2 20.0 5 40 20 20 400 0.1 40.0 σ 2 = 120.0 22-4 Estimating Risk and Return on Assets Chapter 22 √ 120.0 = 10.95% (c) The coefficient variation is: 10.95 CV = 20.00 = 0.55 A coefficient of variation of 0.55 means that there is 0.55 percent risk for every 1 percent of return. Problem 4 (a) Project Y is riskier than Project X when ranked by their standard deviations. However, the two projects are equally risky when ranked by their coefficients of variation. (b) In this situation, the coefficient variation is the more appropriate risk measure because the projects have different net investments and expected values. Thus, a relative measure of risk (coefficient of variation) is needed rather than an absolute measure (standard deviation). Problem 5 (a) The ranges for Project X are shown below: Standard Expected Deviation Value (20,000) Range P100,000 ±1 P80,000 − P120,000 P100,000 ±2 P60,000 − P140,000 (b) Approximately 68 percent of the returns should lie between ± 1 standard deviation of the expected value and about 95 percent within ± 2 standard deviations. Problem 6 (a) Calculating the probability that the return will exceed P130,000 requires three steps: First, compute the z value as follows: 22-5 Chapter 22 Estimating Risk and Return on Assets P130,000 – P100,000 z = P20,000 = 1.50 Next, find Pr (0 < z < 1.5) which is 0.4332 or 43.32 percent. This probability is the chance of getting a return between the expected return of P100,000 and a return of P130,000. Finally, the probability of getting a return greater than P130,000 must be calculated. Remember that in a normal distribution, 50 percent of the outcomes lies on each side of the expected value. The probability of receiving a return of more than P130,000 is 0.0668 (0.5000 − 0.4332) or 6.68 percent. (b) Determining the probability of a return falling between P50,000 and P130,000 requires several additional steps. First, determine the z value for a return between P50,000 and P100,000. P50,000 – P100,000 z = P20,000 = − 2.50 Next, find Pr (- 2.50 < z < 0) which is 0.4938 or 49.38 percent. Finally, the probability of a return between P50,000 and P130,000 is obtained by adding the probability of a return between P50,000 and P100,000, or 0.4938, and the probability of a return between P100,000 and P130,000, or 0.4332. This produces a probability of 0.9270 (0.4938 + 0.4332) or 92.70 percent chance of obtaining a return between P50,000 and P130,000. Problem 7 (a) The expected portfolio return for each plan is: Plan A Plan B ȓp = (0.6) (0.24) + (0.4) (0.08) ȓp = (0.2) (0.24) + (0.8) (0.08) = 0.176 or 17.6% = 0.112 or 11.2% The portfolio standard deviation for each plan is: Plan A σp = √ (0.6) 2 (0.16) 2 + (0.4) 2 (0.02) 2 + (2) (0.6) (0.4) (0.5) (0.16) (0.02) = √ 0.00922 + 0.00006 + 0.00077 22-6 Estimating Risk and Return on Assets Chapter 22 = √ 0.01005 = 0.10025 or 10.025% Plan B σp = √ (0.2) 2 (0.16) 2 + (0.8) 2 (0.02) 2 + (2) (0.2) (0.8) (0.5) (0.16) (0.02) = √ 0.00102 + 0.00026 + 0.00051 = √ 0.00179 = 0.04231 or 4.231% (b) As shown in Plan B, both the expected portfolio return and portfolio standard deviation decrease as a greater proportion of the portfolio is invested in Mega Value Food Stores. Thus, the influence of Gigabyte Computer’s higher expected risk and return are replaced in the portfolio by Mega Value Food Stores’ lower expected risk and return. Problem 8 The expected returns are just the possible returns multiplied by the associated probabilities: E (RA) = (.20 x -.15) + (.50 x .20) + (.30 x .60) = 25% E (RB) = (.20 x.20) + (.50 x .30) + (.30 x .40) = 31% The variances are given by the sums of the squared deviations from the expected returns multiplied by their probabilities: σ2A = .20 x (-.15 − .25) 2 + .50 x (.20 − .25) 2 + .30 x (.60 − .25) 2 = (.20 x -.402) + (.50 x -.052) + (.30 x .352) = (.20 x .16) + (.50 x .0025) + (.30 x .1225) = .0700 σ2B = .20 x (.20 − .31) 2 + .50 x (.30 − .31) 2 + .30 x (.40 − .31) 2 = (.20 x -.112) + (.50 x -.012) + (.30 x .092) = (.20 x .0121) + (.50 x .0001) + (.30 x .0081) 22-7 Chapter 22 Estimating Risk and Return on Assets = .0749 The standard deviations are thus: σA = √ .0700 σB = √ .0049 = 26.46% = 7% Problem 9 The portfolio weights are P15,000/20,000 = .75 and P5,000/20,000 = .25. The expected return is thus: E (RP) = .75 x E (RA) + .25 x E (RB) = (.75 x 25%) + (.25 x 31%) = 26.5% Alternatively, we could calculate the portfolio’s return in each of the states: Probability Weighted State of of State of Portfolio Return if Returns Economy Economy State Occurs (%) Recession .20 (.75 x −.15) + (.25 x .20) = −.0625 − 1.25% Normal .50 (.75 x .20) + (.25 x .30) = .2250 11.25% Boom .30 (.75 x .60) + (.25 x .40) = .5500 16.5% 26.5% Problem 10 The portfolio weight of an asset is total investment in that asset divided by the total portfolio value. First, we will find the portfolio value, which is: Total value = 180 (P45) + 140 (P27) = P11,880 The portfolio weight for each stock is: WeightA = 180 (P45) / P11,880 = .6818 22-8 Estimating Risk and Return on Assets Chapter 22 WeightB = 140 (P27) / P11,880 = .3182 Problem 11 The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset. The total value of the portfolio is: Total value = P2,950 + 3,700 = P6,650 So, the expected return of this portfolio is: E (Rp) = (P2,950/P6,650)(0.11) + (P3,700/P6,650)(0.15) = .1323 or 13.23% Problem 12 The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset. So, the expected return of the portfolio is: E (Rp) = .60 (.09) + .25 (.17) + .15 (.13) = .1160 or 11.60% Problem 13 Here, we are given the expected return of the portfolio and the expected return of each asset in the portfolio, and are asked to find the weight of each asset. We can use the equation for the expected return of a portfolio to solve this problem. Since the total weight of a portfolio must equal 1 (100%), the weight of Stock Y must be one minus the weight of Stock X. Mathematically speaking, this means: E (Rp) = .124 = .14wX + .105(1 – wX) We can now solve this equation for the weight of Stock X as: .124 = .14wX + .105 – .105wX .019 = .035wX wX = 0.542857 So, the amount invested in Stock X is the weight of Stock X times the total portfolio value, or: Investment in X = 0.542857 (P10,000) = P5,428.57 22-9 Chapter 22 Estimating Risk and Return on Assets And the amount invested in Stock Y is: Investment in Y = (1 – 0.542857) (P10,000) = P4,574.43 Problem 14 The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of the asset is: E (R) = .25 (–.08) + .75 (.21) = .1375 or 13.75% Problem 15 The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of the asset is: E (R) = .20 (–.05) + .50 (.12) + .30 (.25) = .1250 or 12.50% Problem 16 The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of each stock asset is: E (RA) = .15 (.05) + .65 (.08) + .20 (.13) = .0855 or 8.55% E (RB) = .15 (–.17) + .65 (.12) + .20 (.29) = .1105 or 11.05% To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, then add all of these up. The result is the variance. So, the variance and standard deviation of each stock is: A2 = .15(.05 – .0855)2 + .65(.08 – .0855)2 + .20(.13 – .0855)2 = .00060 A = (.00060)1/2 = .0246 or 2.46% B2 = .15(–.17 – .1105)2 + .65(.12 – .1105)2 + .20(.29 – .1105)2 = .01830 22-10 Estimating Risk and Return on Assets Chapter 22 B = (.01830)1/2 = .1353 or 13.53% Problem 17 The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset. So, the expected return of the portfolio is: E (Rp) = .25(.08) + .55(.15) + .20(.24) = .1505 or 15.05% If we own this portfolio, we would expect to get a return of 15.05 percent. Problem 18 (a) To find the expected return of the portfolio, we need to find the return of the portfolio in each state of the economy. This portfolio is a special case since all three assets have the same weight. To find the expected return in an equally weighted portfolio, we can sum the returns of each asset and divide by the number of assets, so the expected return of the portfolio in each state of the economy is: Boom: E (Rp) = (.07 + .15 + .33)/3 = .1833 or 18.33% Recession: E (Rp) = (.13 + .03 .06)/3 = .0333 or 3.33% To find the expected return of the portfolio, we multiply the return in each state of the economy by the probability of that state occurring, and then sum. Doing this, we find: E (Rp) = .35(.1833) + .65(.0333) = .0858 or 8.58% (b) This portfolio does not have an equal weight in each asset. We still need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: Boom: E (Rp) = .20(.07) +.20(.15) + .60(.33) =.2420 or 24.20% Recession: E (Rp) = .20(.13) +.20(.03) + .60(.06) = –.0040 or –0.40% 22-11 Chapter 22 Estimating Risk and Return on Assets And the expected return of the portfolio is: E (Rp) = .35(.2420) + .65(.004) = .0821 or 8.21% To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, than add all of these up. The result is the variance. So, the variance and standard deviation of the portfolio is: p2 = .35(.2420 – .0821)2 + .65(.0040 – .0821)2 = √.013767 = 11.73% Problem 19 (a) This portfolio does not have an equal weight in each asset. We first need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: Boom: E (Rp) = .30(.3) + .40(.45) + .30(.33) = .3690 or 36.90% Good: E (Rp) = .30(.12) + .40(.10) + .30(.15) = .1210 or 12.10% Poor: E (Rp) = .30(.01) + .40(–.15) + .30(–.05) = –.0720 or –7.20% Recession: E (Rp) = .30(–.06) + .40(–.30) + .30(–.09) = –.1650 or –16.50% And the expected return of the portfolio is: E (Rp) = .15(.3690) + .45(.1210) + .35(–.0720) + .05(–.1650) = .0764 or 7.64% (b) To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, than add all of these up. The result is the variance. So, the variance and standard deviation of the portfolio is: p2 = .15(.3690 – .0764)2 + .45(.1210 – .0764) 2 + .35(–.0720 – .0764)2 + .05(–.1650 – .0764)2 22-12 Estimating Risk and Return on Assets Chapter 22 p2 = .02436 p = √.02436 = .1561 or 15.61% 22-13