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Chapter 11: Problem 33

Beta and CAPM Suppose the risk-free rate is 4.8 percent and the market portfolio has an expected return of 11.4 percent. The market portfolio has a variance of .0429. Portfolio \(Z\) has a correlation coefficient with the market of .39 and a variance of .1783. According to the capital asset pricing model, what is the expected return on portfolio \(Z\) ?

Short Answer

Expert verified
The expected return on Portfolio \(Z\) according to the CAPM is \(E[R_Z] = 0.048 + \beta_Z (0.114 - 0.048)\), where \(\beta_Z = \frac{0.39 * \sqrt{0.1783} * \sqrt{0.0429}}{0.0429}\). Calculate the beta and substitute it into the formula to find the expected return, which will help investors to balance their portfolios based on their risk tolerance and expected returns.

Step by step solution

01

Calculate the Covariance between Portfolio Z and the Market

To calculate the beta of Portfolio \(Z\), we need to find the covariance between Portfolio \(Z\) and the market. We can use the following formula to calculate the covariance: \( Cov(Z, M) = \rho(Z, M) * \sigma_Z * \sigma_M \) Where: - \(Cov(Z, M)\) is the covariance between Portfolio \(Z\) and the market - \(\rho(Z, M)\) is the correlation coefficient between Portfolio \(Z\) and the market, 0.39 in our case - \(\sigma_Z\) is the standard deviation of Portfolio \(Z\) - \(\sigma_M\) is the standard deviation of the market First, we need to calculate the standard deviations of Portfolio \(Z\) and the market by taking the square root of their variances. \( \sigma_Z = \sqrt{0.1783} \) and \( \sigma_M = \sqrt{0.0429} \) Now we can calculate the covariance between Portfolio \(Z\) and the market using the formula: \( Cov(Z, M) = 0.39 * \sqrt{0.1783} * \sqrt{0.0429} \)
02

Calculate the Beta of Portfolio Z

Next, we calculate the beta of Portfolio \(Z\) using the covariance and the variance of the market portfolio. The formula for beta is: \( \beta_Z = \frac{Cov(Z, M)}{Var(M)} \) We already calculated the covariance between Portfolio \(Z\) and the market in the previous step. Substitute the values into the formula: \( \beta_Z = \frac{0.39 * \sqrt{0.1783} * \sqrt{0.0429}}{0.0429} \)
03

Calculate the Expected Return using CAPM

Now we can use the CAPM formula to calculate the expected return on Portfolio \(Z\). The CAPM formula is: \( E[R_Z] = R_f + \beta_Z (E[R_M] - R_f) \) Where: - \(E[R_Z]\) is the expected return on Portfolio \(Z\) - \(R_f\) is the risk-free rate, 4.8% (or 0.048) in our case - \(\beta_Z\) is the beta of Portfolio \(Z\), calculated in previous step - \(E[R_M]\) is the expected return on the market portfolio, 11.4% (or 0.114) in our case Substituting the values into the formula: \( E[R_Z] = 0.048 + \beta_Z (0.114 - 0.048) \) Calculate the expected return on Portfolio \(Z\).
04

Interpret the Result

Once you have calculated the expected return on Portfolio \(Z\), interpret the result in the context of the CAPM. A higher return indicates a higher risk associated with the portfolio, while a lower return indicates a lower risk. This helps investors to balance their portfolios and make investment decisions based on their risk tolerance and expected returns.

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Key Concepts

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

Capital Asset Pricing Model
The Capital Asset Pricing Model (CAPM) is a foundational concept in finance used to determine the expected return on an investment given its risk in relation to the market as a whole. It offers a formula that predicts the return of a security or portfolio by linking its risk profile to the expected market return, after accounting for the risk-free rate. The key insight of CAPM is that not all risks should affect the pricing of an asset; rather, only the systemic risk, represented by the asset's beta in relation to the market, should influence its expected return.
Beta Coefficient
The beta coefficient is a measure of an investment's volatility in comparison to the market as a whole. It's a crucial component in the CAPM formula and has significant implications for investors. A beta of 1 implies that the investment's price will move with the market. A beta greater than 1 denotes higher volatility and, therefore, higher risk and potential return; conversely, a beta of less than 1 indicates lower volatility and risk.
Risk-free Rate
The risk-free rate is the return on an investment with zero risk of financial loss, which in practice is typically represented by government bonds. In the CAPM formula, the risk-free rate serves as a baseline for an investment's expected return. The idea is that investors should, at a minimum, receive the risk-free rate. It acts as a starting point for calculating the additional return required for taking on additional risk.
Market Portfolio
When discussing CAPM, the market portfolio represents the totality of all possible investments in the global financial markets, assuming they are held in proportion to their market values. In practice, the market portfolio is often approximated by a broad-based market index, such as the S&P 500. It offers a benchmark against which the performance and risk of other investments or portfolios can be gauged.
Expected Market Return
Expected market return is the overall anticipated return of the market portfolio and is a critical factor in determining an asset's expected return via CAPM. This encompasses all dividends, capital gains, and other income components that investors expect to receive over a certain time period. The expected market return reflects the collective risk the market is willing to bear and is often derived from historical market performance.
Portfolio Variance
Portfolio variance is a measurement of the dispersion of returns in a portfolio. It quantifies the volatility of the portfolio, which is a key component of risk. The lower the variance, the more stable the returns are expected to be. In assessing individual investments through CAPM, the variance of the market portfolio is particularly important as it helps to determine an asset's beta.
Covariance
Covariance is a measure of how two stocks move together. A positive covariance means that the stocks tend to move in the same direction, while a negative covariance indicates that they move in opposite directions. In the context of CAPM, the covariance between an individual portfolio and the market portfolio is used to calculate the beta of the portfolio, thus linking an individual portfolio鈥檚 movements to the broader market.
Correlation Coefficient
The correlation coefficient measures the strength and direction of a linear relationship between two investment returns. It can range between -1 and 1, where 1 indicates that the returns move perfectly in sync, -1 indicates perfect opposition, and zero signifies no linear correlation. In the context of the CAPM, the correlation coefficient, together with the standard deviations of the portfolio and the market, is used to determine the covariance, hence playing a part in calculating beta.

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Most popular questions from this chapter

Calculating Expected Returns A portfolio is invested 15 percent in stock \(G\), 65 percent in stock \(J\), and 20 percent in stock \(K\). The expected returns on these stocks are 8 percent, 15 percent, and 24 percent, respectively. What is the portfolio's expected return? How do you interpret your answer?

Calculating Portfolio Betas You own a stock portfolio invested 25 percent in stock \(Q, 20\) percent in stock \(R, 15\) percent in stock \(S\), and 40 percent in stock \(T\). The betas for these four stocks are \(.75,1.90,1.38\), and 1.16 , respectively. What is the portfolio beta?

Standard Deviation and Beta There are two stocks in the market, stock \(\boldsymbol{A}\) and stock \(B\). The price of stock \(A\) today is \(\$ 75\). The price of stock \(A\) next year will be \(\$ 63\) if the economy is in a recession, \(\$ 83\) if the economy is normal, and \(\$ 96\) if the economy is expanding. The probabilities of recession, normal times, and expansion are \(.2, .6\), and .2, respectively. Stock \(A\) pays no dividends and has a correlation of .8 with the market portfolio. Stock \(B\) has an expected return of 13 percent, a standard deviation of 34 percent, a correlation with the market portfolio of .25, and a correlation with stock \(A\) of .48. The market portfolio has a standard deviation of 18 percent. Assume the CAPM holds. 1\. If you are a typical, risk-averse investor with a well-diversified portfolio, which stock would you prefer? Why? 2\. What are the expected return and standard deviation of a portfolio consisting of 70 percent of stock \(A\) and 30 percent of stock \(B\) ? 3\. What is the beta of the portfolio in part (b)?

Portfolio Standard Deviation Suppose the expected returns and standard deviations of stocks \(\boldsymbol{A}\) and \(\boldsymbol{B}\) are \(\mathrm{E}\left(\boldsymbol{R}_A\right)=.13, \mathrm{E}\left(\boldsymbol{R}_{\boldsymbol{B}}\right)=.19, \sigma_A=.38\), and \(\sigma_B=.62\), respectively. 1\. Calculate the expected return and standard deviation of a portfolio that is composed of \(\mathbf{4 5}\) percent \(A\) and 55 percent \(B\) when the correlation between the returns on \(A\) and \(B\) is .5. 2\. Calculate the standard deviation of a portfolio that is composed of 40 percent \(A\) and 60 percent \(B\) when the correlation coefficient between the returns on \(A\) and \(B\) is -.5 . 3\. How does the correlation between the returns on \(\boldsymbol{A}\) and \(\boldsymbol{B}\) affect the standard deviation of the portfolio?

Portfolio Expected Return You own a portfolio that has \(\$ 1,900\) invested in stock \(\boldsymbol{A}\) and \(\$ 2,300\) invested in stock \(B\). If the expected returns on these stocks are 10 percent and 15 percent, respectively, what is the expected return on the portfolio?
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