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Chapter 6: Problem 13

Stock \(R\) has a beta of \(1.5,\) Stock \(S\) has a beta of \(0.75,\) the expected rate of return on an average stock is 13 percent, and the risk-free rate of return is 7 percent. By how much does the required return on the riskier stock exceed the required return on the less risky stock?

Short Answer

Expert verified
The required return on the riskier stock exceeds the other by 4.5%.

Step by step solution

01

Identify the formulas needed

To solve this problem, we'll use the Capital Asset Pricing Model (CAPM) formula, which is given by: \[ R_i = R_f + \beta_i \times (R_m - R_f) \]where \( R_i \) is the required return on stock \( i \), \( R_f \) is the risk-free rate, \( \beta_i \) is the beta of stock \( i \), and \( R_m \) is the expected return on the market.
02

Calculate the required return for Stock R

Using the CAPM formula, substitute the given values for Stock \( R \):\[ R_R = 7\% + 1.5 \times (13\% - 7\%) \]Calculate the market risk premium: \[ 13\% - 7\% = 6\% \]Thus, the required return for Stock \( R \) becomes:\[ R_R = 7\% + 1.5 \times 6\% = 7\% + 9\% = 16\% \]
03

Calculate the required return for Stock S

Substitute the given values for Stock \( S \) into the CAPM formula:\[ R_S = 7\% + 0.75 \times (13\% - 7\%) \]Using the market risk premium calculated earlier, the required return for Stock \( S \) is:\[ R_S = 7\% + 0.75 \times 6\% = 7\% + 4.5\% = 11.5\% \]
04

Calculate the difference in required returns

Find the difference between the required returns of the riskier stock, \( R \), and the less risky stock, \( S \):\[ 16\% - 11.5\% = 4.5\% \]Thus, the required return on the riskier stock exceeds the required return on the less risky stock by 4.5%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Beta Coefficient
The beta coefficient is a measure of a stock's volatility relative to the broader market. It plays a crucial role in the Capital Asset Pricing Model (CAPM) by helping to determine the risk level of a stock when compared with the market. A beta greater than 1 means the stock is more volatile, while a beta less than 1 means it is less volatile. In our example, Stock R has a beta of 1.5, indicating it is riskier than the market, while Stock S has a beta of 0.75, indicating it is less risky.
Risk-Free Rate
The risk-free rate represents the return on an investment with no risk of financial loss. It usually corresponds to government bonds or treasury bills. In calculations, it provides a baseline upon which additional risk is assessed. For Stock R and Stock S, the risk-free rate is 7 percent, serving as the foundational level of return under CAPM.
Market Risk Premium
The market risk premium is the excess return expected from the market above the risk-free rate. It reflects the added risk investors take on when investing in the market compared to a risk-free asset. In our problem, the market risk premium is calculated by subtracting the risk-free rate (7%) from the expected market return (13%), resulting in a 6% market risk premium.
Required Return
The required return is the minimum expected return an investor demands to invest in a particular stock. It takes into account both the risk-free rate and the stock's beta coefficient using CAPM. For Stock R, the required return is calculated to be 16% using its higher beta. Conversely, Stock S, with a lower beta, has a required return of 11.5%. These calculations incorporate both the risk-free rate and the market risk premium.
Stock Risk Comparison
Stock risk comparison involves analyzing different stocks based on their beta coefficients and required returns. Stock R, being the riskier stock with a beta coefficient of 1.5, exhibits greater expected volatility and requires a higher return for the risk undertaken. Stock S, with a beta of 0.75, is seen as less volatile and demands a lower required return. The difference in required returns, calculated as 4.5%, highlights the premium investors demand for taking on greater risk in Stock R.

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Most popular questions from this chapter

The market and Stock J have the following probability distributions: $$\begin{array}{ccc} \text { PROBABILITY } & \mathbf{k}_{M} & \mathbf{k}_{\mathbf{J}} \\ \hline 0.3 & 15 \% & 20 \% \\ 0.4 & 9 & 5 \\ 0.3 & 18 & 12 \end{array}$$ a. Calculate the expected rates of return for the market and Stock J. b. Calculate the standard deviations for the market and Stock J. c. Calculate the coefficients of variation for the market and Stock J.

The tendency of a stock's price to move up and down with the market is reflected in its beta coefficient. Therefore, beta is a measure of an investment's market risk and is a key element of the CAPM. In this exercise you will find betas using Yahoo!Finance, located at http:// finance.yahoo.com. To find a company's beta, enter the desired stock symbol and request a basic quote. Once you have the basic quote, select the "Profile" option in the "More Info" section of the basic quote screen. Scroll down this page to find the stock's beta. a. According to Yahoo!Finance, what is the beta for a company called ELXSI, whose stock symbol is ELXS? b. From Yahoo!Finance obtain a report on MBNA America Bank's holding company, \(\mathrm{KRB},\) whose stock symbol is KRB. What is KRB's beta? c. Obtain and view a report for Exxon Mobil Corporation and identify its beta. Use Yahoo!Finance's look-up feature to obtain Exxon Mobil's trading symbol. To do this, click on symbol lookup, type part of the company name, say Exxon, and then click on Lookup. (Hint: You should find that the company's stock symbol is XOM.) d. Obtain and view a report on Ford Motor Company, and identify its beta. Use Yahoo!Finance's look-up feature to obtain Ford's trading symbol. e. If you made an equal dollar investment in each of the four stocks above, ELXSI, KRB, Exxon Mobil, and Ford Motor Company, what would be your portfolio's beta? a. According to Yahoo!Finance, what is the beta for a company called ELXSI, whose stock symbol is ELXS? b. From Yahoo!Finance obtain a report on MBNA America Bank's holding company, \(\mathrm{KRB},\) whose stock symbol is KRB. What is KRB's beta? c. Obtain and view a report for Exxon Mobil Corporation and identify its beta. Use Yahoo!Finance's look-up feature to obtain Exxon Mobil's trading symbol. To do this, click on symbol lookup, type part of the company name, say Exxon, and then click on Lookup. (Hint: You should find that the company's stock symbol is XOM.) d. Obtain and view a report on Ford Motor Company, and identify its beta. Use Yahoo!Finance's look-up feature to obtain Ford's trading symbol. e. If you made an equal dollar investment in each of the four stocks above, ELXSI, KRB, Exxon Mobil, and Ford Motor Company, what would be your portfolio's beta?

A stock has a required return of 11 percent. The risk-free rate is 7 percent, and the market risk premium is 4 percent. a. What is the stock's beta? b. If the market risk premium increases to 6 percent, what will happen to the stock's required rate of return? Assume the risk-free rate and the stock's beta remain unchanged.

Assume that the risk-free rate is 5 percent and the market risk premium is 6 percent. What is the expected return for the overall stock market? What is the required rate of return on a stock that has a beta of \(1.2 ?\)

Stocks \(X\) and \(Y\) have the following probability distributions of expected future returns: $$\begin{array}{ccc} \text { PROBABILITY } & \mathbf{X} & \mathbf{Y} \\ \hline 0.1 & (10 \%) & (35 \%) \\ 0.2 & 2 & 0 \\ 0.4 & 12 & 20 \\ 0.2 & 20 & 25 \\ 0.1 & 38 & 45 \end{array}$$ a. Calculate the expected rate of return, \(\hat{\mathrm{k}},\) for Stock \(\mathrm{Y}\). ( \(\hat{\mathrm{k}}_{\mathrm{X}}=12 \%\).) b. Calculate the standard deviation of expected returns for Stock X. (That for Stock \(\mathrm{Y}\) is 20.35 percent.) Now calculate the coefficient of variation for Stock \(\mathrm{Y}\). Is it possible that most investors might regard Stock \(Y\) as being less risky than Stock \(X\) ? Explain.
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