Chapter 9, Problem 12CP

A.

Summary Introduction

To determine: The expected return and the alpha for each stock.

Introduction: The Capital Asset Pricing Model explains the relationship among the systematic risk of an asset and the return that are expected.

Answer to Problem 12CP

For each stock X, expected return is 12.2% and the alpha is 1.8%.

For each stock Y, expected return is 18.5% and the alpha is -1.5%.

Explanation of Solution

Given Information:

Forecast returns, standard deviations and the beta values are given.

The capital asset pricing model describes the expected return on beta based security. This model is used for determining the expected return on asset, which is based on systematic risk.

  $Expected\mathrm{Re}turn=RiskFreeRate+Beta[ExpectedMarketRisk-RiskFreeRate]$

  $=r_f+\beta [E(r_M)-r_f]$

  $E_r$ , the expected return

  $r_f$ , the risk free rate of interest

  $\beta$ , the systematic risk

  $r_M$ , return on market portfolio

With substituting the value of beta 0.8, expected rate of return on market $14\%$ and $5\%$ for risk free rate, for stock X

  $\begin{array}{l}{E}_r=r_f+\beta [E(r_M)-r_f]\\ =5\%+0.8[14\%-5\%]\\ =5\%+0.8[9\%]\\ =5\%+7.2\%\\ =12.2\%\end{array}$

For stock X, the expected return is $12.2\%$

With substituting the value of beta $1.5$ , expected rate of return on market $14\%$ and $5\%$ for risk free rate, for stock Y

  $\begin{array}{l}{E}_r=r_f+\beta [E(r_M)-r_f]\\ =5\%+1.5[14\%-5\%]\\ =5\%+1.5[9\%]\\ =5\%+13.5\%\\ =18.5\%\end{array}$

For stock Y, the expected return is $18.5\%$

Calculation of alpha for stock X,

  $\begin{array}{l}{a}_X=Forecasted\mathrm{Re}turn-E(r_x)\\ =14\%-12.2\%=1.8\%\end{array}$

So, the alpha for stock X is $1.8\%.$

Calculation of alpha for stock Y,

  $\begin{array}{l}{a}_X=Forecasted\mathrm{Re}turn-E(r_x)\\ =17\%-18.5\%=-1.5\%\end{array}$

So, the alpha for stock Y is $-1.5\%.$

B.

Summary Introduction

To determine: The most appropriate stock for the investor

Introduction: The Capital Asset Pricing Model explains the relationship among the systematic risk of an asset and the return that are expected.

Answer to Problem 12CP

Stock Y should be chosen

Explanation of Solution

Given Information:

Forecast returns, standard deviations and the beta values are given.

The capital asset pricing model describes the expected return on beta based security. This model is used for determining the expected return on asset, which is based on systematic risk. An investor should invest in single stock, when stock having high expected rate of return and low standard deviation.

  $SharpeRatio=\frac{E(r)-r_f}{\sigma }$

Here, Risk premium is $E(r)-r_f$

Standard deviation is $\sigma$

Expected rate of return is $E(r)$

Risk free return is $r_f$

  $\mathrm{Re}wardToVolatility(SharpeRatio)ForStockX=\frac{E(r)-r_f}{\sigma }$

  $\begin{array}{l}=\frac{(14-5)\%}{36\%}\\ =\frac{9\%}{36\%}=0.25\end{array}$

So, the Sharpe ratio is 25% for stock X

  $\mathrm{Re}wardToVolatility(SharpeRatio)ForStockY=\frac{E(r)-r_f}{\sigma }$

  $\begin{array}{l}=\frac{(17-5)\%}{25\%}\\ =\frac{12\%}{25\%}=0.48\end{array}$

So, the Sharpe ratio is $48\%$ for stock Y

Determination of Sharpe ratio of market index,

  $SharpeRati{o}_{MarketIndex}=\frac{E(r_{MarketIndex})-r_f}{{\sigma }_{MarketIndex}}$

  $\begin{array}{l}=\frac{(14-5)\%}{15\%}\\ =\frac{9\%}{15\%}=0.60\end{array}$

Sharpe ratio of market index is $60\%$

As the Sharpe ratio and the expected return of Sock Y are higher, the risk becomes lesser. So, stock Y is chosen by the investor.