Consider the following historical returns of Stock A and B: YearReturn of stock A (%)Return of stock B (%)2021530202210152023150 a. Calculate the average rate of return of stock A and stock B of the…

---

(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%

---

(b) Standard Deviation of Returns

Standard deviation measures the volatility 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%

---

(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:

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

$${\rho }_{AB}=\frac{-50}{(4.08)(12.25)}=\frac{-50}{50.14}\approx -0.997$$

Final Answer:

• Covariance (${\sigma }_{AB}$) = -50
• Correlation Coefficient (${\rho }_{AB}$) ≈ -0.997 (Perfect Negative Correlation)

---

(d) Portfolio Return and Standard Deviation (Equal Investment)

Assume equal investment in Stock A and Stock B (50% each).

Portfolio Return ($R_p$)

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

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

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

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

$${\sigma }_p=\sqrt{(0.5{)}^2(4.08{)}^2+(0.5{)}^2(12.25{)}^2+2\times 0.5\times 0.5\times (-50)}$$

$${\sigma }_p=\sqrt{0.25\times 16.64+0.25\times 150+0.5\times (-50)}$$

$${\sigma }_p=\sqrt{4.16+37.5-25}=\sqrt{16.66}\approx 4.08\%$$

Interpretation:

• The portfolio return (12.5%) is between the individual returns of Stock A (10%) and Stock B (15%).
• The portfolio standard deviation (4.08%) is much lower than either stock individually, indicating diversification benefits.
• The negative covariance (-50) and perfect negative correlation (-0.997) explain why the portfolio risk is significantly reduced despite combining two volatile stocks.

Final Answer:

• Portfolio Return = 12.5%
• Portfolio Standard Deviation ≈ 4.08%