Alright, let's tackle this problem step by step.
First, we'll define the problem clearly:
- We are given the average rates of return (AOR) for different portfolios over a certain period.
- Our task is to determine the weighted mean of the rates of return (ROR) for each portfolio.
Here is the table presented:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline AOR & Portfolio 1 & Portfolio 2 & Portfolio 3 \\
\hline 7.3 & 81,150 & 8,500 & 1,100 \\
\hline 100 & 51,800 & 52,500 & 8,525 \\
\hline -6.7 & 81,405 & 250 & 825 \\
\hline 20.4 & 81,065 & 81,200 & 5,400 \\
\hline 270 & 81,450 & 511,800 & 12,225 \\
\hline
\end{tabular}
\][/tex]
### Step-by-Step Solution:
1. Calculate the total value for each portfolio:
- Sum of the values in Portfolio 1:
[tex]\[
81,150 + 51,800 + 81,405 + 81,065 + 81,450 = 376,870
\][/tex]
- Sum of the values in Portfolio 2:
[tex]\[
8,500 + 52,500 + 250 + 81,200 + 511,800 = 654,250
\][/tex]
- Sum of the values in Portfolio 3:
[tex]\[
1,100 + 8,525 + 825 + 5,400 + 12,225 = 28,075
\][/tex]
2. Calculate the weighted mean for each portfolio:
- For Portfolio 1:
[tex]\[
\text{Weighted Mean Portfolio 1} = \frac{(7.3 \times 81,150) + (100 \times 51,800) + (-6.7 \times 81,405) + (20.4 \times 81,065) + (270 \times 81,450)}{376,870}
\][/tex]
This results in:
[tex]\[
76.6105221959827
\][/tex]
- For Portfolio 2:
[tex]\[
\text{Weighted Mean Portfolio 2} = \frac{(7.3 \times 8,500) + (100 \times 52,500) + (-6.7 \times 250) + (20.4 \times 81,200) + (270 \times 511,800)}{654,250}
\][/tex]
This results in:
[tex]\[
221.86145204432557
\][/tex]
- For Portfolio 3:
[tex]\[
\text{Weighted Mean Portfolio 3} = \frac{(7.3 \times 1,100) + (100 \times 8,525) + (-6.7 \times 825) + (20.4 \times 5,400) + (270 \times 12,225)}{28,075}
\][/tex]
This results in:
[tex]\[
151.94701691896705
\][/tex]
3. Determine and compare the overall performance of the portfolios:
- The weighted means we have computed are as follows:
- Portfolio 1: 76.6105221959827
- Portfolio 2: 221.86145204432557
- Portfolio 3: 151.94701691896705
4. Sort the portfolios from worst to best based on their weighted means:
- Lower weighted mean indicates poorer performance, while a higher weighted mean indicates better performance.
- Sorting the portfolios in ascending order of their weighted means:
[tex]\[
\text{Portfolio 1 (76.61)}, \text{Portfolio 3 (151.95)}, \text{Portfolio 2 (221.86)}
\][/tex]
Thus, the correct order from worst to best is:
[tex]\[ \text{Portfolio 1, Portfolio 3, Portfolio 2} \][/tex]
So, the correct list that shows a comparison of the overall performance of the portfolios, from worst to best, is:
[tex]\[ \text{Portfolio 1, Portfolio 3, Portfolio 2} \][/tex]
