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Chapter 8: Problem 9

Stock \(\mathrm{R}\) has a beta of \(1.5,\) Stock \(\mathrm{S}\) 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 less risky stock by 4.5%.

Step by step solution

01

Identify Key Information

Given values: \( \beta_R = 1.5 \) for Stock R and \( \beta_S = 0.75 \) for Stock S. The expected market return \( r_m = 13\% \), and the risk-free rate \( r_f = 7\% \).
02

Calculate Market Risk Premium

The market risk premium \( MRP \) is calculated as \( r_m - r_f = 13\% - 7\% = 6\% \).
03

Calculate Required Return for Stock R

Using the Capital Asset Pricing Model (CAPM), we find the required return for Stock R: \( r_R = r_f + \beta_R \times MRP = 7\% + 1.5 \times 6\% = 16\% \).
04

Calculate Required Return for Stock S

Using the CAPM for Stock S: \( r_S = r_f + \beta_S \times MRP = 7\% + 0.75 \times 6\% = 11.5\% \).
05

Find the Difference in Required Returns

Subtract the required return of Stock S from that of Stock R: \( 16\% - 11.5\% = 4.5\% \).

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Key Concepts

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

Required Return
The required return is a fundamental concept in investing that represents the minimum expected profit an investor should receive for taking on the risk of an investment. In simple terms, it is the return needed to justify the risk. The Capital Asset Pricing Model (CAPM) provides a way to calculate a stock's required return. It uses the formula: \( r = r_f + \beta \times (r_m - r_f) \).Let's break this down:
  • \( r \) is the required return you want to calculate.
  • \( r_f \) is the risk-free rate, which is the return on an investment with zero risk.
  • \( \beta \) is the beta coefficient, which measures the stock's volatility compared to the market.
  • \( (r_m - r_f) \) is the market risk premium, which indicates the additional return expected from the market over the risk-free rate.
Using the CAPM helps investors to determine whether a stock's return is worth the risk. With this method, investors can compare different stocks to decide where to allocate their money.
Market Risk Premium
The market risk premium is an integral part of the CAPM formula. It represents the extra return that investors expect from holding a risky market portfolio instead of risk-free assets. Essentially, it gauges how much additional return you should expect for investing in a stock market.Here's what you need to know:
  • The market risk premium is calculated as the difference between the expected return on the market (\( r_m \)) and the risk-free rate (\( r_f \)).
  • The formula is: \( MRP = r_m - r_f \).
In the given problem, the expected market return is 13%, and the risk-free rate is 7%. Therefore, the market risk premium is \( 13\% - 7\% = 6\% \).Investors use the market risk premium to assess the expected returns for taking on additional risk. A higher market risk premium suggests that investors require a larger compensation for higher risk levels. Understanding this concept can help you evaluate if potential investments meet your return expectations.
Beta Coefficient
The beta coefficient is a measure of a stock's volatility in relation to the overall market. It is a critical component of the CAPM as it impacts the required return. Here's how beta works:
  • A beta of 1 indicates that the stock's price moves with the market.
  • A beta greater than 1 suggests that the stock is more volatile than the market, meaning it's riskier.
  • A beta less than 1 means the stock is less volatile or risky compared to the market.
In our exercise, Stock R has a beta of 1.5, indicating it's more volatile than the market, whereas Stock S has a beta of 0.75, suggesting it is less volatile and thus, less risky. The required return adjusts according to these beta values, ensuring that investors are compensated for taking on more or less risk. By understanding a stock's beta coefficient, investors can make informed decisions on how a stock's volatility aligns with their risk appetite and overall portfolio strategy.

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Most popular questions from this chapter

Bartman Industries' and Reynolds Inc.'s stock prices and dividends, along with the Winslow 5000 Index, are shown here for the period \(2000-2005\) The Winslow 5000 data are adjusted to include dividends. a. Use the data to calculate annual rates of return for Bartman, Reynolds, and the Winslow 5000 Index, and then calculate each entity's average return over the 5 -year period. (Hint: Remember, returns are calculated by subtracting the beginning price from the ending price to get the capital gain or loss, adding the dividend to the capital gain or loss, and dividing the result by the beginning price. Assume that dividends are already included in the index. Also, you cannot calculate the rate of return for 2000 because you do not have 1999 data. b. Calculate the standard deviations of the returns for Bartman, Reynolds, and the Winslow 5000 . (Hint: Use the sample standard deviation formula, 8 - \(3 a\), to this chapter, which corresponds to the STDEV function in Excel.) c. \(\quad\) Now calculate the coefficients of variation for Bartman, Reynolds, and the Winslow 5000 d. Construct a scatter diagram that shows Bartman's and Reynolds's returns on the vertical axis and the Winslow Index's returns on the horizontal axis. e. Estimate Bartman's and Reynolds's betas by running regressions of their returns against the index's returns. Are these betas consistent with your graph? f. Assume that the risk-free rate on long-term Treasury bonds is 6.04 percent. Assume also that the average annual return on the Winslow 5000 is not a good estimate of the market's required return- \(-\) it is too high, so use 11 percent as the expected return on the market. Now use the SML equation to calculate the two companies' required returns. g. If you formed a portfolio that consisted of 50 percent Bartman and 50 percent Reynolds, what would the beta and the required return be? h. Suppose an investor wants to include Bartman Industries' stock in his or her portfolio. Stocks \(A, B,\) and \(C\) are currently in the portfolio, and their betas are 0.769 \(0.985,\) and \(1.423,\) respectively. Calculate the new portfolio's required return if it consists of 25 percent of Bartman, 15 percent of Stock \(A, 40\) percent of Stock \(B,\) and 20 percent of Stock \(C\)

A mutual fund manager has a \(\$ 20,000,000\) portfolio with a beta of \(1.5 .\) The risk-free rate is 4.5 percent and the market risk premium is 5.5 percent. The manager expects to receive an additional \(\$ 5,000,000,\) which she plans to invest in a number of stocks. After investing the additional funds, she wants the fund's required return to be 13 percent. What should be the average beta of the new stocks added to the portfolio?

Security Market Line You plan to invest in the Kish Hedge Fund, which has total capital of \(\$ 500\) million invested in five stocks: Kish's beta coefficient can be found as a weighted average of its stocks' betas. The riskfree rate is 6 percent, and you believe the following probability distribution for future market returns is realistic: $$\begin{array}{cc} \text { Probability } & \text { Market Return } \\ \hline 0.1 & 7 \% \\ 0.2 & 9 \\ 0.4 & 11 \\ 0.2 & 13 \\ 0.1 & 15 \end{array}$$ a. What is the equation for the Security Market Line (SML)? (Hint: First determine the expected market return.) b. Calculate Kish's required rate of return. c. Suppose Rick Kish, the president, receives a proposal from a company seeking new capital. The amount needed to take a position in the stock is \(\$ 50\) million, it has an expected return of 15 percent, and its estimated beta is \(2.0 .\) Should Kish invest in the new company? At what expected rate of return should Kish be indifferent to purchasing the stock?

Portfolio beta An individual has \(\$ 35,000\) invested in a stock with a beta of 0.8 and another \(\$ 40,000\) invested in a stock with a beta of \(1.4 .\) If these are the only two investments in her portfolio, what is her portfolio's beta?

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