Return, Risk, and the Security Market Line Basic Problems (1-19)

Problem 1

Determining Portfolio Weights

Question: What are the portfolio weights for a portfolio that has 145 shares of Stock A that sell for $47 per share and 200 shares of Stock B that sell for $21 per share?

Theoretical Foundation: Portfolio weights are the percentage of the total portfolio value invested in each asset. The total value of the portfolio is the sum of the market values of all individual assets within it.

Step-by-step Solution:

• First, calculate the market value of each stock.
• Value of Stock A: \(145 \text{ shares} \times \$47/\text{share} = \$6,815\)
• Value of Stock B: \(200 \text{ shares} \times \$21/\text{share} = \$4,200\)
• Next, calculate the total value of the portfolio.
• Total Portfolio Value: \(\$6,815 + \$4,200 = \$11,015\)
• Finally, calculate the portfolio weight for each stock.
• Portfolio Weight of Stock A: \(\$6,815 / \$11,015 = \textbf{0.6187}\) or 61.87%
• Portfolio Weight of Stock B: \(\$4,200 / \$11,015 = \textbf{0.3813}\) or 38.13%

Problem 2

Portfolio Expected Return

Question: You own a portfolio that has $4,450 invested in Stock A and $9,680 invested in Stock B. If the expected returns on these stocks are 8 percent and 11 percent, respectively, what is the expected return on the portfolio?

Theoretical Foundation: The expected return of a portfolio is the weighted average of the expected returns of the individual assets. The weights are determined by the proportion of the total investment in each asset.

Step-by-step Solution:

• First, calculate the total value of the portfolio.
• Total Portfolio Value: \(\$4,450 + \$9,680 = \$14,130\)
• Next, calculate the portfolio weights for each stock.
• Portfolio Weight of Stock A: \(\$4,450 / \$14,130 = 0.3149\)
• Portfolio Weight of Stock B: \(\$9,680 / \$14,130 = 0.6851\)
• Finally, calculate the portfolio's expected return using the weighted average formula: \(E(R_P) = x_A \times E(R_A) + x_B \times E(R_B)\)
• Expected Return on Portfolio: \((0.3149 \times 8\%) + (0.6851 \times 11\%) = 2.5192\% + 7.5361\% = \textbf{10.06\%}\)

Problem 3

Portfolio Expected Return

Question: You own a portfolio that is invested 15 percent in Stock X, 35 percent in Stock Y, and 50 percent in Stock Z. The expected returns on these three stocks are 9 percent, 15 percent, and 12 percent, respectively. What is the expected return on the portfolio?

Theoretical Foundation: The expected return of a portfolio is the weighted average of the expected returns of the individual assets. Since the portfolio weights are already given, you can directly apply the formula.

Step-by-step Solution:

• Apply the portfolio expected return formula: \(E(R_P) = x_X E(R_X) + x_Y E(R_Y) + x_Z E(R_Z)\)
• Expected Return on Portfolio: \((0.15 \times 9\%) + (0.35 \times 15\%) + (0.50 \times 12\%) \)
• \( = 1.35\% + 5.25\% + 6.00\% = \textbf{12.60\%} \)

Problem 4

Portfolio Expected Return

Question: You have $10,000 to invest in a stock portfolio. Your choices are Stock X with an expected return of 12.4 percent and Stock Y with an expected return of 10.1 percent. If your goal is to create a portfolio with an expected return of 10.85 percent, how much money will you invest in Stock X? In Stock Y?

Theoretical Foundation: The portfolio's expected return is a weighted average of the individual stock returns. We can set up an equation with the portfolio weights as the unknown variables and solve for them. The sum of the weights must equal 1.

Step-by-step Solution:

• Let \(x_X\) be the portfolio weight of Stock X and \(x_Y\) be the portfolio weight of Stock Y. We know \(x_X + x_Y = 1\), so \(x_Y = 1 - x_X\).
• Set up the equation for the portfolio's expected return: \(E(R_P) = x_X E(R_X) + (1-x_X) E(R_Y)\)
• Substitute the given values: \(10.85\% = x_X(12.4\%) + (1-x_X)(10.1\%)\)
• Solve for \(x_X\):
• \(0.1085 = 0.124x_X + 0.101 - 0.101x_X\)
• \(0.0075 = 0.023x_X\)
• \(x_X = 0.0075 / 0.023 = \textbf{0.3261}\) or 32.61%
• Now, find the portfolio weight for Stock Y:
• \(x_Y = 1 - 0.3261 = \textbf{0.6739}\) or 67.39%
• Finally, calculate the amount of money to invest in each stock.
• Investment in Stock X: \(0.3261 \times \$10,000 = \textbf{\$3,261}\)
• Investment in Stock Y: \(0.6739 \times \$10,000 = \textbf{\$6,739}\)

Problem 5

Calculating Expected Return

Question: Based on the following information, calculate the expected return:

Theoretical Foundation: The expected return is the sum of each possible return multiplied by its probability.

Step-by-step Solution:

• Expected Return: \(E(R) = (0.20 \times -0.13) + (0.80 \times 0.19)\)
• \( = -0.026 + 0.152 = \textbf{0.126}\) or 12.60%

Problem 6

Calculating Expected Return

Question: Based on the following information, calculate the expected return:

Theoretical Foundation: The expected return is the sum of each possible return multiplied by its probability.

Step-by-step Solution:

• Expected Return: \(E(R) = (0.10 \times -0.17) + (0.60 \times 0.08) + (0.30 \times 0.27)\)
• \( = -0.017 + 0.048 + 0.081 = \textbf{0.112}\) or 11.20%

Problem 7

Calculating Returns and Standard Deviations

Question: Based on the following information, calculate the expected return and standard deviation for Stock A and Stock B:

Theoretical Foundation: The expected return is the weighted average of possible returns. The variance is the weighted average of the squared deviations from the expected return. The standard deviation is the square root of the variance.

Step-by-step Solution:

• Stock A:
• Expected Return: \(E(R_A) = (0.15 \times 0.04) + (0.55 \times 0.09) + (0.30 \times 0.17)\)
• \( = 0.006 + 0.0495 + 0.051 = \textbf{0.1065}\) or 10.65%
• Variance: \(\sigma^2_A = 0.15(0.04 - 0.1065)^2 + 0.55(0.09 - 0.1065)^2 + 0.30(0.17 - 0.1065)^2\)
• \( = 0.15(0.004422) + 0.55(0.000272) + 0.30(0.004032) \)
• \( = 0.000663 + 0.000150 + 0.001210 = \textbf{0.002023} \)
• Standard Deviation: \(\sigma_A = \sqrt{0.002023} = \textbf{0.0450}\) or 4.50%

• Stock B:
• Expected Return: \(E(R_B) = (0.15 \times -0.17) + (0.55 \times 0.12) + (0.30 \times 0.27)\)
• \( = -0.0255 + 0.066 + 0.081 = \textbf{0.1215}\) or 12.15%
• Variance: \(\sigma^2_B = 0.15(-0.17 - 0.1215)^2 + 0.55(0.12 - 0.1215)^2 + 0.30(0.27 - 0.1215)^2\)
• \( = 0.15(0.084932) + 0.55(0.000002) + 0.30(0.022052) \)
• \( = 0.012740 + 0.000001 + 0.006616 = \textbf{0.019357} \)
• Standard Deviation: \(\sigma_B = \sqrt{0.019357} = \textbf{0.1391}\) or 13.91%

Problem 8

Calculating Expected Returns

Question: A portfolio is invested 45 percent in Stock G, 40 percent in Stock J, and 15 percent in Stock K. The expected returns on these stocks are 11 percent, 9 percent, and 15 percent, respectively. What is the portfolio's expected return? How do you interpret your answer?

Theoretical Foundation: The expected return of a portfolio is the weighted average of the expected returns of the individual assets. The weights are the percentages of the total investment in each asset.

Step-by-step Solution:

• Apply the portfolio expected return formula: \(E(R_P) = x_G E(R_G) + x_J E(R_J) + x_K E(R_K)\)
• Expected Return on Portfolio: \((0.45 \times 11\%) + (0.40 \times 9\%) + (0.15 \times 15\%)\)
• \( = 4.95\% + 3.60\% + 2.25\% = \textbf{10.80\%} \)

Interpretation: The portfolio's expected return of 10.80% means that, over a long period, an investor would expect to earn an average annual return of 10.80% on this portfolio. It is a value that lies between the highest and lowest expected returns of the individual stocks, weighted by their respective proportions in the portfolio.

Problem 9

Returns and Variances

Question: Consider the following information:

a. What is the expected return on an equally weighted portfolio of these three stocks?

b. What is the variance of a portfolio invested 20 percent each in A and B and 60 percent in C?

Theoretical Foundation: The expected return of a portfolio is a weighted average. The variance must be calculated by first finding the portfolio return in each state of the economy and then using the weighted average of squared deviations from the portfolio's expected return.

Step-by-step Solution:

• First, calculate the expected return for each individual stock:
• \(E(R_A) = (0.75 \times 0.07) + (0.25 \times 0.12) = 0.0525 + 0.03 = \textbf{0.0825}\)
• \(E(R_B) = (0.75 \times 0.18) + (0.25 \times -0.08) = 0.135 - 0.02 = \textbf{0.1150}\)
• \(E(R_C) = (0.75 \times 0.27) + (0.25 \times -0.21) = 0.2025 - 0.0525 = \textbf{0.1500}\)
• a. Equally Weighted Portfolio:
• The portfolio weights are \(1/3\) for each stock.
• \(E(R_P) = (1/3)(0.0825) + (1/3)(0.1150) + (1/3)(0.1500) = \textbf{0.1158}\) or 11.58%
• b. Variance of the specified portfolio:
• The portfolio weights are \(x_A = 0.20\), \(x_B = 0.20\), and \(x_C = 0.60\).
• First, find the portfolio return in each state of the economy:
• Boom: \(R_{P,Boom} = (0.20 \times 0.07) + (0.20 \times 0.18) + (0.60 \times 0.27) = 0.014 + 0.036 + 0.162 = \textbf{0.212}\)
• Bust: \(R_{P,Bust} = (0.20 \times 0.12) + (0.20 \times -0.08) + (0.60 \times -0.21) = 0.024 - 0.016 - 0.126 = \textbf{-0.118}\)
• Next, find the portfolio's expected return:
• \(E(R_P) = (0.75 \times 0.212) + (0.25 \times -0.118) = 0.159 - 0.0295 = \textbf{0.1295}\)
• Finally, calculate the variance:
• \(\sigma^2_P = 0.75(0.212 - 0.1295)^2 + 0.25(-0.118 - 0.1295)^2\)
• \( = 0.75(0.00680625) + 0.25(0.06125625) \)
• \( = 0.0051047 + 0.0153141 = \textbf{0.0204}\)

Problem 10

Returns and Standard Deviations

Question: Consider the following information:

a. Your portfolio is invested 30 percent each in A and C, and 40 percent in B. What is the expected return of the portfolio?

b. What is the variance of this portfolio? The standard deviation?

Theoretical Foundation: The expected return and variance of a portfolio are calculated based on the weighted average of the returns in each state of the economy. The standard deviation is the square root of the variance.

Step-by-step Solution:

• a. Expected Return of the Portfolio:
• Portfolio Weights: \(x_A = 0.30\), \(x_B = 0.40\), \(x_C = 0.30\).
• First, find the portfolio return in each state of the economy:
• Boom: \(R_{P,Boom} = 0.30(0.35) + 0.40(0.40) + 0.30(0.28) = 0.105 + 0.160 + 0.084 = 0.349\)
• Good: \(R_{P,Good} = 0.30(0.16) + 0.40(0.17) + 0.30(0.09) = 0.048 + 0.068 + 0.027 = 0.143\)
• Poor: \(R_{P,Poor} = 0.30(-0.01) + 0.40(-0.03) + 0.30(0.01) = -0.003 - 0.012 + 0.003 = -0.012\)
• Bust: \(R_{P,Bust} = 0.30(-0.10) + 0.40(-0.12) + 0.30(-0.09) = -0.030 - 0.048 - 0.027 = -0.105\)
• Expected Return of Portfolio: \(E(R_P) = 0.15(0.349) + 0.45(0.143) + 0.30(-0.012) + 0.10(-0.105)\)
• \( = 0.05235 + 0.06435 - 0.0036 - 0.0105 = \textbf{0.1026}\) or 10.26%
• b. Variance and Standard Deviation:
• Variance: \(\sigma^2_P = 0.15(0.349 - 0.1026)^2 + 0.45(0.143 - 0.1026)^2 + 0.30(-0.012 - 0.1026)^2 + 0.10(-0.105 - 0.1026)^2\)
• \( = 0.15(0.060714) + 0.45(0.001632) + 0.30(0.013133) + 0.10(0.043105) \)
• \( = 0.009107 + 0.000734 + 0.003940 + 0.004311 = \textbf{0.018092}\)
• Standard Deviation: \(\sigma_P = \sqrt{0.018092} = \textbf{0.1345}\) or 13.45%

Problem 11

Calculating Portfolio Betas

Question: You own a stock portfolio invested 15 percent in Stock Q, 20 percent in Stock R, 30 percent in Stock S, and 35 percent in Stock T. The betas for these four stocks are .79, 1.23, 1.13, and 1.36, respectively. What is the portfolio beta?

Theoretical Foundation: The beta of a portfolio is the weighted average of the betas of the individual assets in the portfolio. The weights are the percentages of the total investment in each asset.

Step-by-step Solution:

• Apply the portfolio beta formula: \(\beta_P = x_Q \beta_Q + x_R \beta_R + x_S \beta_S + x_T \beta_T\)
• Portfolio Beta: \(\beta_P = (0.15 \times 0.79) + (0.20 \times 1.23) + (0.30 \times 1.13) + (0.35 \times 1.36)\)
• \( = 0.1185 + 0.2460 + 0.3390 + 0.4760 = \textbf{1.1795} \)

Problem 12

Calculating Portfolio Betas

Question: You own a portfolio equally invested in a risk-free asset and two stocks. If one of the stocks has a beta of 1.34 and the total portfolio is equally as risky as the market, what must the beta be for the other stock in your portfolio?

Theoretical Foundation: The beta of a portfolio is the weighted average of the betas of the individual assets. A portfolio that is "equally as risky as the market" has a portfolio beta of 1.0.

Step-by-step Solution:

• The portfolio is equally invested in three assets, so each asset has a weight of \(1/3\).
• The beta of the risk-free asset is 0.
• The beta of the total portfolio is 1.0.
• Set up the portfolio beta equation: \(\beta_P = x_{RF}\beta_{RF} + x_1\beta_1 + x_2\beta_2\)
• \(1.0 = (1/3)(0) + (1/3)(1.34) + (1/3)\beta_2\)
• \(1.0 = 0.4467 + 0.3333\beta_2\)
• \(0.5533 = 0.3333\beta_2\)
• \(\beta_2 = 0.5533 / 0.3333 = \textbf{1.66}\)

Problem 13

Using CAPM

Question: A stock has a beta of 1.15, the expected return on the market is 11.3 percent, and the risk-free rate is 3.6 percent. What must the expected return on this stock be?

Theoretical Foundation: The Capital Asset Pricing Model (CAPM) describes the relationship between systematic risk and expected return. The formula is: \(E(R_i) = R_f + [E(R_M) - R_f] \times \beta_i\).

Step-by-step Solution:

• Identify the given variables: \(\beta_i = 1.15\), \(E(R_M) = 11.3\%\), and \(R_f = 3.6\%\).
• Calculate the market risk premium: \(E(R_M) - R_f = 11.3\% - 3.6\% = 7.7\%\)
• Apply the CAPM formula: \(E(R_i) = 3.6\% + (7.7\% \times 1.15)\)
• \( = 3.6\% + 8.855\% = \textbf{12.455\%} \)

Problem 14

Using CAPM

Question: A stock has an expected return of 11.4 percent, the risk-free rate is 3.9 percent, and the market risk premium is 6.8 percent. What must the beta of this stock be?

Theoretical Foundation: The CAPM describes the relationship between systematic risk and expected return. We can rearrange the formula to solve for the beta of the stock.

Step-by-step Solution:

• Identify the given variables: \(E(R_i) = 11.4\%\), \(R_f = 3.9\%\), and \(E(R_M) - R_f = 6.8\%\).
• Apply the CAPM formula and solve for \(\beta_i\): \(E(R_i) = R_f + [E(R_M) - R_f] \times \beta_i\)
• \(11.4\% = 3.9\% + (6.8\% \times \beta_i)\)
• \(7.5\% = 6.8\% \times \beta_i\)
• \(\beta_i = 7.5\% / 6.8\% = \textbf{1.10} \)

Problem 15

Using CAPM

Question: A stock has an expected return of 11.85 percent, its beta is 1.08, and the risk-free rate is 3.9 percent. What must the expected return on the market be?

Theoretical Foundation: The CAPM describes the relationship between systematic risk and expected return. We can rearrange the formula to solve for the expected return on the market.

Step-by-step Solution:

• Identify the given variables: \(E(R_i) = 11.85\%\), \(\beta_i = 1.08\), and \(R_f = 3.9\%\).
• Apply the CAPM formula and solve for \(E(R_M)\): \(E(R_i) = R_f + [E(R_M) - R_f] \times \beta_i\)
• \(11.85\% = 3.9\% + [E(R_M) - 3.9\%] \times 1.08\)
• \(7.95\% = [E(R_M) - 3.9\%] \times 1.08\)
• \(7.361\% = E(R_M) - 3.9\%\)
• \(E(R_M) = 7.361\% + 3.9\% = \textbf{11.261\%} \)

Problem 16

Using CAPM

Question: A stock has an expected return of 10.45 percent, its beta is .85, and the expected return on the market is 11.8 percent. What must the risk-free rate be?

Theoretical Foundation: The CAPM describes the relationship between systematic risk and expected return. We can rearrange the formula to solve for the risk-free rate.

Step-by-step Solution:

• Identify the given variables: \(E(R_i) = 10.45\%\), \(\beta_i = 0.85\), and \(E(R_M) = 11.8\%\).
• Apply the CAPM formula and solve for \(R_f\): \(E(R_i) = R_f + [E(R_M) - R_f] \times \beta_i\)
• \(10.45\% = R_f + (11.8\% - R_f) \times 0.85\)
• \(10.45\% = R_f + 10.03\% - 0.85R_f\)
• \(0.42\% = 0.15R_f\)
• \(R_f = 0.42\% / 0.15 = \textbf{2.80\%}\)

Problem 17

Using the SML

Question: Asset W has an expected return of 8.8 percent and a beta of .90. If the risk-free rate is 2.6 percent, complete the following table for portfolios of Asset W and a risk-free asset. Illustrate the relationship between portfolio expected return and portfolio beta by plotting the expected returns against the betas. What is the slope of the line that results?

Theoretical Foundation: The expected return and beta of a portfolio are linear combinations (weighted averages) of the returns and betas of the individual assets. The slope of the Security Market Line is the reward-to-risk ratio.

Step-by-step Solution:

• The portfolio's expected return is \(E(R_P) = x_W E(R_W) + (1-x_W)R_f\).
• The portfolio's beta is \(\beta_P = x_W \beta_W + (1-x_W)\beta_{RF}\). Since \(\beta_{RF} = 0\), this simplifies to \(\beta_P = x_W \beta_W\).
• Given \(E(R_W) = 8.8\%\), \(\beta_W = 0.90\), and \(R_f = 2.6\%\).
• Calculate the values for each row:

The slope of the line is the reward-to-risk ratio: \(\frac{E(R_W) - R_f}{\beta_W} = \frac{8.8\% - 2.6\%}{0.90} = \frac{6.2\%}{0.90} = \textbf{6.89\%}\).

Problem 18

Reward-to-Risk Ratios

Question: Stock Y has a beta of 1.2 and an expected return of 11.5 percent. Stock Z has a beta of .80 and an expected return of 8.5 percent. If the risk-free rate is 3.2 percent and the market risk premium is 6.8 percent, are these stocks correctly priced?

Theoretical Foundation: In a correctly priced market, all assets must plot on the Security Market Line (SML) and have the same reward-to-risk ratio. We can use the CAPM to find the expected returns for each stock and compare them to the given expected returns. Alternatively, we can check if their reward-to-risk ratios are equal to the market's reward-to-risk ratio.

Step-by-step Solution:

• First, find the expected return for each stock using the CAPM.
• Expected Return on Market: \(E(R_M) = R_f + \text{Market Risk Premium} = 3.2\% + 6.8\% = 10.0\%\)
• SML equation: \(E(R) = 3.2\% + \beta(6.8\%)\)
• Expected Return for Stock Y: \(E(R_Y) = 3.2\% + 1.2(6.8\%) = 3.2\% + 8.16\% = \textbf{11.36\%}\)
• Expected Return for Stock Z: \(E(R_Z) = 3.2\% + 0.8(6.8\%) = 3.2\% + 5.44\% = \textbf{8.64\%}\)
• Compare these to the given expected returns:
• Stock Y's given expected return (11.5%) is higher than its CAPM-predicted return (11.36%). This means it is **undervalued**.
• Stock Z's given expected return (8.5%) is lower than its CAPM-predicted return (8.64%). This means it is **overvalued**.

Problem 19

Reward-to-Risk Ratios

Question: In the previous problem, what would the risk-free rate have to be for the two stocks to be correctly priced?

Theoretical Foundation: If two stocks are correctly priced, their reward-to-risk ratios must be equal. We can set up an equation where the reward-to-risk ratios of the two stocks are equal and solve for the unknown risk-free rate.

Step-by-step Solution:

• Set the reward-to-risk ratios equal to each other: \(\frac{E(R_Y) - R_f}{\beta_Y} = \frac{E(R_Z) - R_f}{\beta_Z}\)
• Substitute the given values: \(\frac{11.5\% - R_f}{1.2} = \frac{8.5\% - R_f}{0.80}\)
• Solve for \(R_f\):
• \(0.80(11.5\% - R_f) = 1.2(8.5\% - R_f)\)
• \(9.2\% - 0.80R_f = 10.2\% - 1.2R_f\)
• \(0.40R_f = 1.0\%\)
• \(R_f = 1.0\% / 0.40 = \textbf{2.5\%}\)