To solve this problem, we need to calculate the weighted mean of the Rate of Return (ROR) for each portfolio. The weighted mean ROR for a portfolio is computed by multiplying each ROR by its corresponding investment amount, summing these products, and then dividing by the total investment in the portfolio.
Here are the RORs and investments:
[tex]\[
\begin{array}{|l|l|l|l|}
\hline
& \text{ROR} & \text{Portfolio 1} & \text{Portfolio 2} & \text{Portfolio 3} \\
\hline
& 12.6\% & \$1,250 & \$950 & \$900 \\
\hline
& 2.8\% & \$575 & \$2,025 & \$2,350 \\
\hline
& 10.4\% & \$895 & \$1,185 & \$310 \\
\hline
& 1.8\% & \$800 & \$445 & \$1,600 \\
\hline
& -5.6\% & \$1,775 & \$625 & \$2,780 \\
\hline
\end{array}
\][/tex]
First, let's calculate the total investment for each portfolio:
[tex]\[
\text{Total Investment for Portfolio 1} = 1250 + 575 + 895 + 800 + 1775 = 5295
\][/tex]
[tex]\[
\text{Total Investment for Portfolio 2} = 950 + 2025 + 1185 + 445 + 625 = 5230
\][/tex]
[tex]\[
\text{Total Investment for Portfolio 3} = 900 + 2350 + 310 + 1600 + 2780 = 7940
\][/tex]
Next, let's calculate the weighted mean ROR for each portfolio:
### Portfolio 1:
[tex]\[
\text{Weighted ROR for Portfolio 1} = \frac{(12.6\% \times 1250) + (2.8\% \times 575) + (10.4\% \times 895) + (1.8\% \times 800) + (-5.6\% \times 1775)}{5295}
\][/tex]
[tex]\[
\approx \frac{15750 + 1610 + 9308 + 1440 - 9940}{5295} \approx \frac{ 7823}{5295} \approx 3.43\%
\][/tex]
### Portfolio 2:
[tex]\[
\text{Weighted ROR for Portfolio 2} = \frac{(12.6\% \times 950) + (2.8\% \times 2025) + (10.4\% \times 1185) + (1.8\% \times 445) + (-5.6\% \times 625)}{5230}
\][/tex]
[tex]\[
\approx \frac{11970 + 5670 + 12324 + 801 - 3500}{5230} \approx \frac{27181}{5230} \approx 5.21\%
\][/tex]
### Portfolio 3:
[tex]\[
\text{Weighted ROR for Portfolio 3} = \frac{(12.6\% \times 900) + (2.8\% \times 2350) + (10.4\% \times 310) + (1.8\% \times 1600) + (-5.6\% \times 2780)}{7940}
\][/tex]
[tex]\[
\approx \frac{11340 + 6580 + 3224 + 2880 - 15568}{7940} \approx \frac{1456}{7940} \approx 1.06\%
\][/tex]
Now that we have calculated the weighted mean RORs:
[tex]\[
\text{Portfolio 1:} \approx 3.43\%
\][/tex]
[tex]\[
\text{Portfolio 2:} \approx 5.21\%
\][/tex]
[tex]\[
\text{Portfolio 3:} \approx 1.06\%
\][/tex]
Based on these results, the overall performance of the portfolios from best to worst is:
Portfolio 2, Portfolio 1, Portfolio 3
So, the correct answer is:
Portfolio 2, Portfolio 1, Portfolio 3
