Consider the following probability distribution and returns for stock Alpha and Beta State of EconomyProbabilityReturn of Alpha (%)Return of Beta (%)Rapid economic growth0.22530Normal economic…

a. Expected Returns of Stocks Alpha and Beta

The expected return of a stock is calculated using the formula:

$$E(R)=\sum (P_i\times R_i)$$

where:

• $P_i$ = Probability of each state of the economy
• $R_i$ = Return of the stock in each state

Stock Alpha

$$E(R_{\text{Alpha}})=(0.2\times 25)+(0.6\times 15)+(0.2\times (-5))$$ $$=5+9+(-1)$$ $$=\ast \ast 13\%\ast \ast$$

Stock Beta

$$E(R_{\text{Beta}})=(0.2\times 30)+(0.6\times 20)+(0.2\times (-10))$$ $$=6+12+(-2)$$ $$=\ast \ast 16\%\ast \ast$$

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b. Expected Coefficient of Variation (CV) and Risk Comparison

The coefficient of variation (CV) is calculated as:

$$CV=\frac{\sigma }{\mu }$$

where:

• $\sigma$ = Standard deviation of returns
• $\mu$ = Expected return

Step 1: Calculate Variance and Standard Deviation

Stock Alpha

$$E(R_{\text{Alpha}})=13\%$$

$$\text{Variance}=\sum (P_i\times (R_i-\mu {)}^2)$$ $$=(0.2\times 144)+(0.6\times 4)+(0.2\times 324)$$ $$=28.8+2.4+64.8=96$$

$${\sigma }_{\text{Alpha}}=\sqrt{96}=9.7979\approx 9.80\%$$

Stock Beta

$$E(R_{\text{Beta}})=16\%$$

$$\text{Variance}=\sum (P_i\times (R_i-\mu {)}^2)$$ $$=(0.2\times 196)+(0.6\times 16)+(0.2\times 676)$$ $$=39.2+9.6+135.2=184$$

$${\sigma }_{\text{Beta}}=\sqrt{184}=13.5647\approx 13.56\%$$

Step 2: Calculate Coefficient of Variation (CV)

$$C{V}_{\text{Alpha}}=\frac{9.80}{13}=0.7538\approx 0.75$$

$$C{V}_{\text{Beta}}=\frac{13.56}{16}=0.8475\approx 0.85$$

Comparison of Risk

The coefficient of variation (CV) measures risk-adjusted returns. A lower CV indicates less risk per unit of return.

Since Stock Alpha has a lower CV (0.75) compared to Stock Beta (0.85), Stock Alpha is less risky when considering risk per unit of return. This means that despite having a lower expected return, Stock Alpha provides better risk-adjusted performance.