Return, Risk, and the Security Market Line Intermediate & Challenge Problems (20-28)

Problem 20

Using CAPM

Question: A stock has a beta of 1.19 and an expected return of 12.4 percent. A risk-free asset currently earns 2.7 percent.

a. What is the expected return on a portfolio that is equally invested in the two assets?

b. If a portfolio of the two assets has a beta of .92, what are the portfolio weights?

c. If a portfolio of the two assets has an expected return of 10 percent, what is its beta?

d. If a portfolio of the two assets has a beta of 2.38, what are the portfolio weights? How do you interpret the weights for the two assets in this case? Explain.

Theoretical Foundation: This problem uses the core principles of portfolio expected return and portfolio beta. The expected return and beta of a portfolio are linear combinations (weighted averages) of the individual assets' returns and betas.

Step-by-step Solution:

• Given: \( \beta_{Stock} = 1.19 \), \( E(R_{Stock}) = 12.4\% \), \( R_f = 2.7\% \).
• a. Equally weighted portfolio:
  • Weights: \( x_{Stock} = 0.50 \), \( x_{RF} = 0.50 \).
  • Expected return: \( E(R_P) = x_{Stock} E(R_{Stock}) + x_{RF} R_f \)
  • \( E(R_P) = 0.50(12.4\%) + 0.50(2.7\%) = 6.2\% + 1.35\% = \textbf{7.55\%} \)
• b. Portfolio with a beta of 0.92:
  • Let \( x_{Stock} \) be the weight in the stock and \( (1-x_{Stock}) \) be the weight in the risk-free asset. The beta of the risk-free asset is 0.
  • Portfolio beta: \( \beta_P = x_{Stock} \beta_{Stock} + (1-x_{Stock})\beta_{RF} \)
  • \( 0.92 = x_{Stock}(1.19) + (1-x_{Stock})(0) \)
  • \( x_{Stock} = 0.92 / 1.19 = \textbf{0.7731}\) or 77.31%
  • Weight in risk-free asset: \( 1 - 0.7731 = \textbf{0.2269}\) or 22.69%
• c. Portfolio with an expected return of 10%:
  • Let \( x_{Stock} \) be the weight in the stock.
  • Expected return: \( E(R_P) = x_{Stock} E(R_{Stock}) + (1-x_{Stock})R_f \)
  • \( 10\% = x_{Stock}(12.4\%) + (1-x_{Stock})(2.7\%) \)
  • \( 0.10 = 0.124x_{Stock} + 0.027 - 0.027x_{Stock} \)
  • \( 0.073 = 0.097x_{Stock} \)
  • \( x_{Stock} = 0.073 / 0.097 = 0.7526 \)
  • Now, find the portfolio beta using this weight: \( \beta_P = x_{Stock} \beta_{Stock} + (1-x_{Stock})\beta_{RF} \)
  • \( \beta_P = 0.7526(1.19) + 0 = \textbf{0.8956} \)
• d. Portfolio with a beta of 2.38:
  • Portfolio beta: \( \beta_P = x_{Stock} \beta_{Stock} + (1-x_{Stock})\beta_{RF} \)
  • \( 2.38 = x_{Stock}(1.19) + (1-x_{Stock})(0) \)
  • \( x_{Stock} = 2.38 / 1.19 = \textbf{2.00}\) or 200%
  • Weight in risk-free asset: \( 1 - 2.00 = \textbf{-1.00}\) or -100%
  • Interpretation: This means the investor has borrowed at the risk-free rate (shorting the risk-free asset) and invested a total of 200% of their own wealth in the risky stock. This is a leveraged position. For every dollar of their own money, they have borrowed an additional dollar to invest in the stock.

Problem 21

Portfolio Returns

Question: Using information from the previous chapter on capital market history, determine the return on a portfolio that is equally invested in large-company stocks and long-term government bonds. What is the return on a portfolio that is equally invested in small-company stocks and Treasury bills?

Theoretical Foundation: This problem requires using historical returns data (from the previous chapter, which is assumed to be available) to calculate the expected returns of two different portfolios. The formula for portfolio expected return is a weighted average of the individual asset returns.

Step-by-step Solution:

• Assume the following historical average returns from the textbook:
  • Large-company stocks: 12.1%
  • Small-company stocks: 16.5%
  • Long-term government bonds: 5.9%
  • Treasury bills: 3.4%
• Portfolio A (Large-company stocks and Long-term government bonds):
  • Weights: \( x_{Large} = 0.50 \), \( x_{Gov} = 0.50 \)
  • Return: \( E(R_A) = 0.50(12.1\%) + 0.50(5.9\%) = 6.05\% + 2.95\% = \textbf{9.0\%} \)
• Portfolio B (Small-company stocks and Treasury bills):
  • Weights: \( x_{Small} = 0.50 \), \( x_{TB} = 0.50 \)
  • Return: \( E(R_B) = 0.50(16.5\%) + 0.50(3.4\%) = 8.25\% + 1.70\% = \textbf{9.95\%} \)

Problem 22

CAPM

Question: Using the CAPM, show that the ratio of the risk premiums on two assets is equal to the ratio of their betas.

Theoretical Foundation: The CAPM states that the expected return of an asset is a function of the risk-free rate, the market risk premium, and the asset's beta. By applying this to two different assets and taking the ratio of their risk premiums, we can demonstrate the relationship.

Step-by-step Solution:

• The CAPM formula for asset \( i \) is: \( E(R_i) = R_f + [E(R_M) - R_f] \times \beta_i \)
• The risk premium for asset \( i \) is: \( E(R_i) - R_f = [E(R_M) - R_f] \times \beta_i \)
• Let's consider two assets, Asset A and Asset B. Their risk premiums are:
  • Risk Premium for A: \( E(R_A) - R_f = [E(R_M) - R_f] \times \beta_A \)
  • Risk Premium for B: \( E(R_B) - R_f = [E(R_M) - R_f] \times \beta_B \)
• Now, form the ratio of these two risk premiums:
• \( \frac{E(R_A) - R_f}{E(R_B) - R_f} = \frac{[E(R_M) - R_f] \times \beta_A}{[E(R_M) - R_f] \times \beta_B} \)
• The market risk premium term, \( [E(R_M) - R_f] \), cancels out.
• \( \frac{E(R_A) - R_f}{E(R_B) - R_f} = \frac{\beta_A}{\beta_B} \)
• This shows that the ratio of the risk premiums on two assets is equal to the ratio of their betas.

Problem 23

Portfolio Returns and Deviations

Question: Consider the following information about three stocks:

a. If your portfolio is invested 40 percent each in A and B and 20 percent in C, what is the portfolio expected return? The variance? The standard deviation?

b. If the expected T-bill rate is 3.70 percent, what is the expected risk premium on the portfolio?

c. If the expected inflation rate is 3.30 percent, what are the approximate and exact expected real returns on the portfolio? What are the approximate and exact expected real risk premiums on the portfolio?

Theoretical Foundation: This problem tests the calculation of portfolio returns, variance, and standard deviation using state-contingent returns. It also introduces the concept of real returns, which adjust for inflation.

Step-by-step Solution:

• a. Portfolio Expected Return, Variance, and Standard Deviation:
  • Portfolio Weights: \( x_A = 0.40 \), \( x_B = 0.40 \), \( x_C = 0.20 \).
  • First, find the portfolio return in each state:
    • \( R_{P,Boom} = 0.40(0.29) + 0.40(0.13) + 0.20(0.60) = 0.116 + 0.052 + 0.120 = 0.288 \)
    • \( R_{P,Normal} = 0.40(0.08) + 0.40(0.11) + 0.20(0.13) = 0.032 + 0.044 + 0.026 = 0.102 \)
    • \( R_{P,Bust} = 0.40(0.02) + 0.40(-0.18) + 0.20(-0.45) = 0.008 - 0.072 - 0.090 = -0.154 \)
  • Expected Return of Portfolio: \( E(R_P) = 0.25(0.288) + 0.60(0.102) + 0.15(-0.154) \)
  • \( E(R_P) = 0.072 + 0.0612 - 0.0231 = \textbf{0.1101}\) or 11.01%
  • Variance: \( \sigma^2_P = 0.25(0.288 - 0.1101)^2 + 0.60(0.102 - 0.1101)^2 + 0.15(-0.154 - 0.1101)^2 \)
  • \( \sigma^2_P = 0.25(0.03164) + 0.60(0.0000656) + 0.15(0.06979) = 0.00791 + 0.000039 + 0.010469 = \textbf{0.018418} \)
  • Standard Deviation: \( \sigma_P = \sqrt{0.018418} = \textbf{0.1357}\) or 13.57%
• b. Expected Risk Premium:
  • Risk Premium = \( E(R_P) - R_f = 11.01\% - 3.70\% = \textbf{7.31\%} \)
• c. Expected Real Returns:
  • Approximate Expected Real Return: \( E(R_{Real, Approx}) = E(R_P) - \text{Inflation} = 11.01\% - 3.30\% = \textbf{7.71\%} \)
  • Exact Expected Real Return: \( E(R_{Real, Exact}) = \frac{1+E(R_P)}{1+\text{Inflation}} - 1 = \frac{1.1101}{1.033} - 1 = 1.0746 - 1 = \textbf{7.46\%} \)
  • Approximate Expected Real Risk Premium: \( \text{Approx. Real RP} = \text{Nominal RP} - \text{Inflation} = 7.31\% \) (no inflation adjustment needed here as nominal and real risk-free rates are usually small).
  • Exact Expected Real Risk Premium: \(\frac{1+E(R_P)}{1+\text{Inflation}} - \frac{1+R_f}{1+\text{Inflation}} = \frac{1.1101-1.037}{1.033} = \frac{0.0731}{1.033} = \textbf{7.08\%} \)

Problem 24

Analyzing a Portfolio

Question: You want to create a portfolio equally as risky as the market, and you have $1,000,000 to invest. Given this information, fill in the rest of the following table:

Theoretical Foundation: A portfolio "equally as risky as the market" has a beta of 1.0. The total investment is $1,000,000. We can use the portfolio beta formula to solve for the missing investment amounts and betas.

Step-by-step Solution:

• The beta of the risk-free asset is 0.
• The total portfolio value is $1,000,000.
• The portfolio beta is \( \beta_P = 1.0 \).
• First, set up the portfolio beta equation: \( \beta_P = x_A \beta_A + x_B \beta_B + x_C \beta_C + x_{RF} \beta_{RF} \)
• The weights are the investment amount divided by the total portfolio value.
  • \( x_A = \$195,000 / \$1,000,000 = 0.195 \)
  • \( x_B = \$365,000 / \$1,000,000 = 0.365 \)
  • \( x_C = \text{Investment}_C / \$1,000,000 \)
  • \( x_{RF} = \text{Investment}_{RF} / \$1,000,000 \)
• Substitute the known values into the portfolio beta equation: \( 1.0 = 0.195(0.80) + 0.365(1.09) + x_C(1.23) + x_{RF}(0) \) \( 1.0 = 0.156 + 0.39785 + 1.23x_C \) \( 1.0 = 0.55385 + 1.23x_C \) \( 0.44615 = 1.23x_C \) \( x_C = 0.44615 / 1.23 = 0.3627 \)
• Now we have the weight for Stock C, so we can find the investment amount. \( \text{Investment}_C = 0.3627 \times \$1,000,000 = \textbf{\$362,700} \)
• The sum of all weights must be 1. We can find the weight of the risk-free asset. \( x_{RF} = 1 - x_A - x_B - x_C \) \( x_{RF} = 1 - 0.195 - 0.365 - 0.3627 = 1 - 0.9227 = 0.0773 \)
• Finally, find the investment amount for the risk-free asset. \( \text{Investment}_{RF} = 0.0773 \times \$1,000,000 = \textbf{\$77,300} \)
• Fill in the table:

Problem 25

Analyzing a Portfolio

Question: You have $100,000 to invest in a portfolio containing Stock X and Stock Y. Your goal is to create a portfolio that has an expected return of 12.1 percent. If Stock X has an expected return of 10.28 percent and a beta of 1.20, and Stock Y has an expected return of 7.52 percent and a beta of .80, how much money will you invest in Stock Y? How do you interpret your answer? What is the beta of your portfolio?

Theoretical Foundation: The expected return of a portfolio is a weighted average of the expected returns of its constituent assets. We can solve for the weights and then determine the investment amounts. The portfolio beta is also a weighted average of the individual betas.

Step-by-step Solution:

• Let \(x_X\) be the weight in Stock X and \(x_Y\) be the weight in Stock Y. We know \(x_X + x_Y = 1\).
• Set up the expected return equation: \( E(R_P) = x_X E(R_X) + (1-x_X) E(R_Y) \)
• \( 12.1\% = x_X(10.28\%) + (1-x_X)(7.52\%) \)
• \( 0.121 = 0.1028x_X + 0.0752 - 0.0752x_X \)
• \( 0.0458 = 0.0276x_X \)
• \( x_X = 0.0458 / 0.0276 = 1.6594 \)
• \( x_Y = 1 - 1.6594 = \textbf{-0.6594} \)
• Investment in Stock Y: \( -0.6594 \times \$100,000 = \textbf{-$65,940} \)
• Interpretation: The negative weight for Stock Y means that the investor has **shorted** this stock, borrowing \$65,940 worth of Stock Y to invest in Stock X, resulting in a leveraged position. The total investment in Stock X is \( (1 + 0.6594) \times \$100,000 = \$165,940 \).
• Portfolio Beta:
  • \( \beta_P = x_X \beta_X + x_Y \beta_Y \)
  • \( \beta_P = 1.6594(1.20) + (-0.6594)(0.80) = 1.9913 - 0.5275 = \textbf{1.4638} \)

Problem 26

Systematic versus Unsystematic Risk

Question: Consider the following information about Stocks I and II:

The market risk premium is 7 percent, and the risk-free rate is 3.5 percent. Which stock has the most systematic risk? Which one has the most unsystematic risk? Which stock is "riskier"? Explain.

Theoretical Foundation: This problem requires distinguishing between systematic and unsystematic risk. Systematic risk is measured by beta, while total risk (which includes unsystematic risk) is measured by standard deviation. The CAPM can be used to determine if a stock is correctly priced relative to its systematic risk. Total risk is the sum of systematic and unsystematic risk.

Step-by-step Solution:

• First, find the expected return and variance for each stock.
  • \( E(R_I) = 0.15(0.05) + 0.70(0.18) + 0.15(0.07) = 0.0075 + 0.126 + 0.0105 = \textbf{0.144} \)
  • \( \sigma^2_I = 0.15(0.05-0.144)^2 + 0.70(0.18-0.144)^2 + 0.15(0.07-0.144)^2 \)
  • \( \sigma^2_I = 0.001270 + 0.000882 + 0.000810 = \textbf{0.002962} \)
  • \( \sigma_I = \sqrt{0.002962} = 0.0544 \) or 5.44%
  • \( E(R_{II}) = 0.15(-0.21) + 0.70(0.10) + 0.15(0.39) = -0.0315 + 0.07 + 0.0585 = \textbf{0.097} \)
  • \( \sigma^2_{II} = 0.15(-0.21-0.097)^2 + 0.70(0.10-0.097)^2 + 0.15(0.39-0.097)^2 \)
  • \( \sigma^2_{II} = 0.013233 + 0.000006 + 0.013054 = \textbf{0.026293} \)
  • \( \sigma_{II} = \sqrt{0.026293} = 0.1621 \) or 16.21%
• Next, use the CAPM to find the beta for each stock, based on their expected returns.
  • \( E(R) = R_f + [E(R_M) - R_f]\beta \Rightarrow \beta = \frac{E(R)-R_f}{E(R_M)-R_f} \)
  • \( \beta_I = \frac{0.144-0.035}{0.07} = \textbf{1.557} \)
  • \( \beta_{II} = \frac{0.097-0.035}{0.07} = \textbf{0.886} \)
• Conclusion:
  • Stock I has the most **systematic risk** because it has a higher beta (\(\beta_I > \beta_{II}\)).
  • Stock II has the most **unsystematic risk**. This is because its total risk (\(\sigma_{II}\) = 16.21%) is much higher than Stock I's total risk (\(\sigma_I\) = 5.44%) even though Stock I has more systematic risk. We can also see this by comparing the expected returns with the total risk; Stock I has a lower standard deviation but a higher expected return.
  • Stock I is arguably **riskier** in the sense that it has more systematic risk, which is the only risk that investors are compensated for. From a total risk perspective, Stock II is much riskier because its total volatility is significantly higher. However, a well-diversified investor would only care about the systematic risk, making Stock I the "riskier" investment from their perspective.

Problem 27

SML

Question: Suppose you observe the following situation:

Assume these securities are correctly priced. Based on the CAPM, what is the expected return on the market? What is the risk-free rate?

Theoretical Foundation: If the securities are correctly priced, they must both lie on the same SML. This means their reward-to-risk ratios must be equal, allowing us to set up two equations with two unknowns (\( E(R_M) \) and \( R_f \)) and solve them simultaneously.

Step-by-step Solution:

• Set up two CAPM equations, one for each security:
  • Pete Corp: \( 0.108 = R_f + [E(R_M) - R_f](1.25) \) (Equation 1)
  • Repete Co: \( 0.082 = R_f + [E(R_M) - R_f](0.87) \) (Equation 2)
• We can use substitution or elimination to solve these equations. Let's solve for the market risk premium, \( E(R_M) - R_f \).
  • Subtract Equation 2 from Equation 1: \( (0.108 - 0.082) = (1.25 - 0.87)[E(R_M) - R_f] \) \( 0.026 = 0.38[E(R_M) - R_f] \)
  • Market Risk Premium: \( E(R_M) - R_f = 0.026 / 0.38 = \textbf{0.0684} \) or 6.84%
• Now, substitute this back into either equation to find \( R_f \). Let's use Equation 2. \( 0.082 = R_f + 0.0684(0.87) \) \( 0.082 = R_f + 0.0595 \) \( R_f = 0.082 - 0.0595 = \textbf{0.0225} \) or 2.25%
• Finally, calculate the expected return on the market. \( E(R_M) = R_f + \text{Market Risk Premium} = 2.25\% + 6.84\% = \textbf{9.09\%} \)

Problem 28

SML

Question: Suppose you observe the following situation:

a. Calculate the expected return on each stock.

b. Assuming the capital asset pricing model holds and Stock A's beta is greater than Stock B's beta by .35, what is the expected market risk premium?

Theoretical Foundation: This problem combines the calculation of expected returns with the CAPM. We can calculate the expected return for each stock, and then use the CAPM relationship and the given information about the difference in their betas to solve for the market risk premium.

Step-by-step Solution:

• a. Calculate Expected Return on each stock:
  • \( E(R_A) = 0.15(-0.08) + 0.60(0.11) + 0.25(0.30) = -0.012 + 0.066 + 0.075 = \textbf{0.129} \) or 12.90%
  • \( E(R_B) = 0.15(-0.10) + 0.60(0.09) + 0.25(0.27) = -0.015 + 0.054 + 0.0675 = \textbf{0.1065} \) or 10.65%
• b. Expected Market Risk Premium:
  • According to the CAPM: \( E(R) = R_f + [E(R_M) - R_f]\beta \)
  • Subtract the two CAPM equations: \( E(R_A) - E(R_B) = ([E(R_M) - R_f]\beta_A) - ([E(R_M) - R_f]\beta_B) \)
  • \( E(R_A) - E(R_B) = [E(R_M) - R_f](\beta_A - \beta_B) \)
  • We are given that \( \beta_A - \beta_B = 0.35 \).
  • Substitute the expected returns from part (a): \( 0.1290 - 0.1065 = [E(R_M) - R_f](0.35) \)
  • \( 0.0225 = [E(R_M) - R_f](0.35) \)
  • Expected Market Risk Premium: \( E(R_M) - R_f = 0.0225 / 0.35 = \textbf{0.0643} \) or 6.43%