You have been provided the following returns data of three assets for your advice in selecting a portfolio of assets: Year Rate of return : : : : Asset A Asset B Asset C 2019 10% 18% 10% 2020 12%…

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(a) Average Return and Standard Deviation for Each Asset

Step 1: Calculate Average Return

The average (mean) return for each asset is calculated as: $$\text{Average Return}=\frac{{\sum }_{i=1}^n{R}_i}{n}$$ where $R_i$ is the return in year $i$ and $n$ is the number of years (5).

For Asset A: $${\text{Average Return}}_A=\frac{10+12+14+16+18}{5}=\frac{70}{5}=14\%$$

For Asset B: $${\text{Average Return}}_B=\frac{18+16+14+12+10}{5}=\frac{70}{5}=14\%$$

For Asset C: $${\text{Average Return}}_C=\frac{10+12+14+16+16}{5}=\frac{68}{5}=13.6\%$$

Step 2: Calculate Standard Deviation

The standard deviation ($\sigma$) is calculated using: $$\sigma =\sqrt{\frac{\sum (R_i-\bar{R}{)}^2}{n}}$$ where $\bar{R}$ is the average return.

For Asset A: $$\begin{array}{rl}(R_i-\bar{R}{)}^2& =(10-14{)}^2+(12-14{)}^2+(14-14{)}^2+(16-14{)}^2+(18-14{)}^2\\ & =16+4+0+4+16=40\\ {\sigma }_A& =\sqrt{\frac{40}{5}}=\sqrt{8}=2.83\%\end{array}$$

For Asset B: $$\begin{array}{rl}(R_i-\bar{R}{)}^2& =(18-14{)}^2+(16-14{)}^2+(14-14{)}^2+(12-14{)}^2+(10-14{)}^2\\ & =16+4+0+4+16=40\\ {\sigma }_B& =\sqrt{\frac{40}{5}}=\sqrt{8}=2.83\%\end{array}$$

For Asset C: $$\begin{array}{rl}(R_i-\bar{R}{)}^2& =(10-13.6{)}^2+(12-13.6{)}^2+(14-13.6{)}^2+(16-13.6{)}^2+(16-13.6{)}^2\\ & =12.96+2.56+0.16+5.76+5.76=27.2\\ {\sigma }_C& =\sqrt{\frac{27.2}{5}}=\sqrt{5.44}=2.33\%\end{array}$$

Final Results:

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(b) Covariance Between Assets A and B, and Assets A and C

Covariance (${\text{Cov}}_{XY}$) is calculated as: $${\text{Cov}}_{XY}=\frac{\sum (R_{X,i}-{\bar{R}}_X)(R_{Y,i}-{\bar{R}}_Y)}{n}$$

Covariance between A and B:

$$\begin{array}{rl}(R_A-{\bar{R}}_A)(R_B-{\bar{R}}_B)& =(10-14)(18-14)+(12-14)(16-14)+(14-14)(14-14)+(16-14)(12-14)+(18-14)(10-14)\\ & =(-4)(4)+(-2)(2)+(0)(0)+(2)(-2)+(4)(-4)\\ & =-16-4+0-4-16=-40\\ {\text{Cov}}_{AB}& =\frac{-40}{5}=-8.00\%\end{array}$$

Covariance between A and C:

$$\begin{array}{rl}(R_A-{\bar{R}}_A)(R_C-{\bar{R}}_C)& =(10-14)(10-13.6)+(12-14)(12-13.6)+(14-14)(14-13.6)+(16-14)(16-13.6)+(18-14)(16-13.6)\\ & =(-4)(-3.6)+(-2)(-1.6)+(0)(0.4)+(2)(2.4)+(4)(2.4)\\ & =14.4+3.2+0+4.8+9.6=32.0\\ {\text{Cov}}_{AC}& =\frac{32.0}{5}=6.40\%\end{array}$$

Final Results:

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(c) Return and Standard Deviation of Portfolios (A+B and A+C)

Portfolio Return

Since the portfolio is equally weighted (50% in each asset), the portfolio return is: $$R_P=0.5\times R_A+0.5\times R_B\text{(for A+B)}$$ $$R_P=0.5\times R_A+0.5\times R_C\text{(for A+C)}$$

For Portfolio A+B: $$R_{AB}=0.5\times 14\%+0.5\times 14\%=14\%$$

For Portfolio A+C: $$R_{AC}=0.5\times 14\%+0.5\times 13.6\%=13.8\%$$

Portfolio Standard Deviation

The portfolio standard deviation (${\sigma }_P$) is calculated using: $${\sigma }_P=\sqrt{w_A^2{\sigma }_A^2+w_B^2{\sigma }_B^2+2w_A{w}_B{\text{Cov}}_{AB}}$$ where $w_A=w_B=0.5$.

For Portfolio A+B: $$\begin{array}{rl}{\sigma }_{AB}& =\sqrt{(0.5{)}^2(2.83{)}^2+(0.5{)}^2(2.83{)}^2+2\times 0.5\times 0.5\times (-8.00)}\\ & =\sqrt{0.25\times 8+0.25\times 8+0.5\times (-8)}\\ & =\sqrt{2+2-4}=\sqrt{0}=0\%\end{array}$$

For Portfolio A+C: $$\begin{array}{rl}{\sigma }_{AC}& =\sqrt{(0.5{)}^2(2.83{)}^2+(0.5{)}^2(2.33{)}^2+2\times 0.5\times 0.5\times 6.40}\\ & =\sqrt{0.25\times 8+0.25\times 5.44+0.5\times 6.40}\\ & =\sqrt{2+1.36+3.2}=\sqrt{6.56}=2.56\%\end{array}$$

Final Results:

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(d) Portfolio Recommendation

Recommendation: The Portfolio A+B is the better choice because:

1. It offers the same return (14%) as holding Asset A alone but with zero risk (standard deviation = 0%).
2. The negative covariance (-8%) between A and B indicates perfect diversification, eliminating all unsystematic risk.
3. Portfolio A+C, while slightly less risky than holding A or C alone, still carries 2.56% risk and a marginally lower return.

Thus, Portfolio A+B is optimal due to its higher return and zero risk.