Question

Use the table to answer the question that follows.

ROR Portfolio 1 Portfolio 2 Portfolio 3
3.9% $1,250 $950 $900
1.7% $575 $2,025 $2,350
10.6% $895 $1,185 $310
−3.2% $800 $445 $1,600
8.1% $1,775 $625 $2,780

Calculate the weighted mean of the RORs for each portfolio. Based on the results, which list shows a comparison of the overall performance of the portfolios, from best to worst?

Portfolio 1, Portfolio 3, Portfolio 2
Portfolio 2, Portfolio 3, Portfolio 1
Portfolio 1, Portfolio 2, Portfolio 3
Portfolio 3, Portfolio 2, Portfolio 1

Answer 1

C) Portfolio 1, Portfolio 2, Portfolio 3

Explanation:

`\boxed{\begin{minipage}{5 cm}\underline{Weighted Mean Formula}\\\\$ \overline{x}=(\displaystyle\sum^(n)_(i=1)x_iw_i)/(\displaystyle\sum^(n)_(i=1)w_i)$\\\\\\where:\\\phantom{ww} $\bullet$ $x_i$ is the data value\\\phantom{ww} $\bullet$ $w_i$ is the weight\\\end{minipage}}`

To calculate the weighted mean of the RoR for each portfolio:

1. Multiply the RoR (xi) in decimal form by the corresponding amount invested (wi).
2. Sum the values calculated in step 1.
3. Divide the sum from step 2 by the sum of the amounts invested (in that portfolio).

Portfolio 1

`\implies \overline{x}=(1250 \cdot 0.039+575 \cdot 0.017 + 895 \cdot .0106 + 800 \cdot -0.032 + 1775 \cdot 0.081)/(1250+575+895+800+1775)`

`\implies \overline{x}=(271.57)/(5295)`

`\implies \overline{x}=0.051288075...`

`\implies \overline{x}=5.13\%\;\; \sf (2 \;d.p.)`

Portfolio 2

`\implies \overline{x}=(950\cdot 0.039+2025\cdot 0.017 + 1185\cdot .0106 + 445\cdot -0.032 + 625\cdot 0.081)/(950+2025+1185+445+625)`

`\implies \overline{x}=(233.47)/(5230)`

`\implies \overline{x}=0.0446405353...`

`\implies \overline{x}=4.46\%\;\; \sf (2 \;d.p.)`

Portfolio 3

`\implies \overline{x}=(900\cdot 0.039+2350\cdot 0.017 + 310\cdot .0106 + 1600\cdot -0.032 + 2780\cdot 0.081)/(900+2350+310+1600+2780)`

`\implies \overline{x}=(281.89)/(7940)`

`\implies \overline{x}=0.0355025188...`

`\implies \overline{x}=3.55\%\;\; \sf (2 \;d.p.)`

The RoR (rate of return) is the net gain (or loss) of an investment over a specified time period, expressed as a percentage of the investment's initial cost.

Therefore, based on the weighted means of the RoRs for each portfolio, the best to worst portfolios are:

• Portfolio 1, Portfolio 2, Portfolio 3

(as Portfolio 1 has the highest RoR and Portfolio 3 has the lowest RoR).