Consider the following historical returns of Stock A and B: Year Return of stock A (%) Return of stock B (%) 2021 5 30 2022 10 15 2023 15 0 a. Calculate the average rate of return of stock A and…

a. Average Rate of Return

The average rate of return (mean return) is calculated as the arithmetic mean of the annual returns over the given period.

Stock A

$${\text{Average Return}}_A=\frac{5+10+15}{3}=\frac{30}{3}=10\%$$

Stock B

$${\text{Average Return}}_B=\frac{30+15+0}{3}=\frac{45}{3}=15\%$$

Final Answer:

• Average return of Stock A = 10%
• Average return of Stock B = 15%

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b. Standard Deviation of Returns

Standard deviation measures the volatility (risk) of returns. It is calculated using the formula:

$$\sigma =\sqrt{\frac{\sum (R_i-\bar{R}{)}^2}{N}}$$

where:

• $R_i$ = individual return,
• $\bar{R}$ = average return,
• $N$ = number of observations.

Stock A

$${\sigma }_A=\sqrt{\frac{25+0+25}{3}}=\sqrt{\frac{50}{3}}=\sqrt{16.67}\approx 4.08\%$$

Stock B

$${\sigma }_B=\sqrt{\frac{225+0+225}{3}}=\sqrt{\frac{450}{3}}=\sqrt{150}\approx 12.25\%$$

Final Answer:

• Standard deviation of Stock A ≈ 4.08%
• Standard deviation of Stock B ≈ 12.25%

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c. Covariance and Correlation Coefficient

Covariance (${\sigma }_{AB}$)

Covariance measures how returns of two stocks move together. The formula is:

$${\sigma }_{AB}=\frac{\sum (R_{A_i}-{\bar{R}}_A)(R_{B_i}-{\bar{R}}_B)}{N}$$

$${\sigma }_{AB}=\frac{-75+0-75}{3}=\frac{-150}{3}=-50$$

Correlation Coefficient (${\rho }_{AB}$)

The correlation coefficient is calculated as:

$${\rho }_{AB}=\frac{{\sigma }_{AB}}{{\sigma }_A{\sigma }_B}$$

$${\rho }_{AB}=\frac{-50}{(4.08)(12.25)}=\frac{-50}{50.04}\approx -0.999$$

Interpretation:

• Covariance = -50 (negative, indicating an inverse relationship).
• Correlation ≈ -1 (perfect negative correlation; when Stock A rises, Stock B falls, and vice versa).

Final Answer:

• Covariance = -50
• Correlation coefficient ≈ -1

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d. Portfolio Return and Standard Deviation (Equal Investment)

Assume an investor allocates 50% in Stock A and 50% in Stock B.

Portfolio Return ($R_p$)

$$R_p=w_A{R}_A+w_B{R}_B$$ where $w_A=w_B=0.5$.

For each year:

Average Portfolio Return: $${\bar{R}}_p=\frac{17.5+12.5+7.5}{3}=\frac{37.5}{3}=12.5\%$$

Portfolio Standard Deviation (${\sigma }_p$)

The formula for portfolio variance is:

$${\sigma }_p^2=w_A^2{\sigma }_A^2+w_B^2{\sigma }_B^2+2w_A{w}_B{\sigma }_A{\sigma }_B{\rho }_{AB}$$

Substituting values: $${\sigma }_p^2=(0.5{)}^2(4.08{)}^2+(0.5{)}^2(12.25{)}^2+2(0.5)(0.5)(4.08)(12.25)(-1)$$ $${\sigma }_p^2=0.25(16.65)+0.25(150.06)+0.5(4.08)(12.25)(-1)$$ $${\sigma }_p^2=4.16+37.52-25=16.68$$ $${\sigma }_p=\sqrt{16.68}\approx 4.08\%$$

Interpretation:

• The portfolio return (12.5%) is between the returns of Stock A (10%) and Stock B (15%).
• The portfolio standard deviation (4.08%) is lower than both individual stocks, indicating diversification benefits due to the perfect negative correlation between the two stocks. This reduces risk significantly.

Final Answer:

• Portfolio return = 12.5%
• Portfolio standard deviation ≈ 4.08%