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Stock \(X\) has an expected return of 10 percent, a beta coefficient of \(0.9,\) and a standard deviation of expected returns of 35 percent. Stock \(Y\) has an expected return of 12.5 percent, a beta coefficient of \(1.2,\) and a standard deviation of expected returns of 25 percent. The risk-free rate is 6 percent, and the market risk premium is 5 percent. a. Calculate each stock's coefficient of variation. b. Which stock is riskier for diversified investors? c. Calculate each stock's required rate of return. d. On the basis of the two stocks' expected and required returns, which stock would be most attractive to a diversified investor? e. Calculate the required return of a portfolio that has \(\$ 7,500\) invested in Stock \(X\) and \(\$ 2,500\) invested in Stock \(Y\) f. If the market risk premium increased to 6 percent, which of the two stocks would have the largest increase in their required return?

Step by step solution

01

Calculate Coefficient of Variation for Stock X and Y

The coefficient of variation (CV) is calculated by dividing the standard deviation by the expected return. For Stock X: \[ CV_X = \frac{35\%}{10\%} = 3.5 \] For Stock Y: \[ CV_Y = \frac{25\%}{12.5\%} = 2.0 \] The coefficient of variation measures relative risk per unit of return.

02

Analyze Risk for Diversified Investors

For diversified investors, risk is typically measured by beta rather than standard deviation, as beta represents the stock's sensitivity to market movements. Stock X has a beta of 0.9 while Stock Y has a beta of 1.2, suggesting Stock Y is riskier.

03

Calculate Required Return Using CAPM

The required return can be calculated using the Capital Asset Pricing Model (CAPM): \[ r_i = r_f + \beta_i (r_m - r_f) \] where \( r_f = 6\% \) and \( r_m - r_f = 5\% \). For Stock X, the required return is: \[ r_X = 6\% + 0.9 \times 5\% = 10.5\% \] For Stock Y, the required return is: \[ r_Y = 6\% + 1.2 \times 5\% = 12\% \].

04

Compare Expected and Required Returns

Stock X has an expected return of 10% and a required return of 10.5%, yielding an expected return less than required. Stock Y has an expected return of 12.5% and a required return of 12%, yielding an expected return more than required. Therefore, Stock Y is more attractive.

05

Calculate Required Return of Portfolio

The portfolio's required return is the weighted average of the required returns of the individual stocks: \[ r_p = \frac{7500}{10000} \times 10.5\% + \frac{2500}{10000} \times 12\% = 10.875\% \].

06

Effect of Increased Market Risk Premium

With an increase in the market risk premium to 6%, the new required return for each stock is calculated. For Stock X: \[ r_X = 6\% + 0.9 \times 6\% = 11.4\% \].For Stock Y: \[ r_Y = 6\% + 1.2 \times 6\% = 13.2\% \].The increase in required return is 0.9% for Stock X and 1.2% for Stock Y. Thus, Stock Y has the largest increase.

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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 in relation to the overall market. In simpler terms, it tells us how much a stock's price might change when the market moves. A beta greater than 1 indicates that the stock is more volatile than the market, meaning it will likely experience larger swings in price. If beta is less than 1, the stock is less volatile than the market and generally has smaller price fluctuations.
A beta coefficient of 1 means the stock's price moves exactly with the market. If the market goes up 10%, the stock will likely do the same. In our exercise, Stock X has a beta of 0.9, suggesting it is less volatile than the market. Stock Y, with a beta of 1.2, is more volatile, thus considered riskier by diversified investors who rely on beta to gauge risk because it captures how stocks react to market conditions.

Required Rate of Return

The required rate of return is the minimum annual percentage return needed by an investor to consider an investment worthwhile. It accounts for the risk-free rate combined with the risk of the investment as captured by its beta. This rate helps investors evaluate if the potential gains on an investment are worth its risks compared to other opportunities.
Using the Capital Asset Pricing Model (CAPM), this can be calculated with the formula: \[ r_i = r_f + \beta_i (r_m - r_f) \]Where \( r_f \) is the risk-free rate, \( \beta_i \) is the beta of the stock, and \( (r_m - r_f) \) is the market risk premium. Investors use this to decide whether a stock's potential rewards outweigh its risks. For example, if a stock's expected return is lower than its required return, it may be considered less appealing.

Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM) is a fundamental financial theory used to determine the expected return on an asset, considering its risk compared to the market. This model serves as a cornerstone for investors, providing a formula to evaluate whether a stock offers a reasonable return for its associated risk.
The CAPM formula is expressed as:\[ r_i = r_f + \beta_i (r_m - r_f) \]Here, \( r_f \) is the risk-free rate, such as government bond yields, and \( (r_m - r_f) \) represents the market risk premium. The value \( \beta_i \) quantifies the asset's market risk. The model holds that the expected return should compensate investors for both the time value of money (the risk-free rate) and the risk taken above a safe investment. It helps align expectations of returns with inherent risks across different assets, aiding diversified investors in portfolio decisions.

Market Risk Premium

Market risk premium (MRP) is a key element of the CAPM. It reflects the extra return that investors require for choosing a risky investment over a risk-free one. This premium is essential because it helps to quantify the risk-reward trade-off of investing in the market.

• If the MRP is higher, it suggests that investors see the market as riskier and demand higher returns for taking on that risk.
• Conversely, a lower MRP implies the market is perceived as less risky.

The MRP plays a significant role in calculating the required rate of return through CAPM by influencing how much return above the risk-free rate an investor expects. In the exercise, a change in MRP directly affects the required returns for Stocks X and Y, highlighting its impact on investor decision-making.