Using technology, calculate the difference between the arithmetic average ROR and the weighted average ROR. Round to the nearest tenth of a percent.
A. $0.5 \%$
B. $1.1 \%$
C. $2.3 \%$
D. $3.5 \%
Answer (2)
Calculate the arithmetic average ROR: 4 2.1 + 4.4 − 7.8 + 10.5 = 2.3% .
Calculate the weighted average ROR: 3200 + 1750 + 1235 + 2300 ( 3200 ) ( 0.021 ) + ( 1750 ) ( 0.044 ) + ( 1235 ) ( − 0.078 ) + ( 2300 ) ( 0.105 ) ≈ 3.41% .
Find the absolute difference: ∣2.3 − 3.41∣ = 1.11% .
Round to the nearest tenth of a percent: 1.1% .
Explanation
Understanding the Problem We are given a table of investments with their amounts and rates of return (ROR). Our goal is to find the difference between the arithmetic average ROR and the weighted average ROR, rounded to the nearest tenth of a percent.
Calculating Arithmetic Average ROR First, let's calculate the arithmetic average ROR. This is simply the sum of the RORs divided by the number of investments, which is 4. A r i t hm e t i c A v er a g e ROR = 4 2.1 + 4.4 + ( − 7.8 ) + 10.5 = 4 9.2 = 2.3%
Calculating Weighted Average ROR Next, we calculate the weighted average ROR. This involves multiplying each investment amount by its ROR, summing these products, and then dividing by the total investment amount. Total investment = $3200 + $1750 + $1235 + $2300 = 8485 W e i g h t e d a v er a g e ROR = \frac{(3200)(0.021) + (1750)(0.044) + (1235)(-0.078) + (2300)(0.105)}{8485} = \frac{67.2 + 77 - 96.33 + 241.5}{8485} = \frac{289.37}{8485} \approx 0.0341 = 3.41 %$
Finding the Difference Now, we find the difference between the arithmetic average ROR and the weighted average ROR. Difference = |Arithmetic average ROR - Weighted average ROR| = |2.3 - 3.41| = |-1.11| = 1.11 %
Rounding the Difference Finally, we round the difference to the nearest tenth of a percent. Rounded difference = 1.1 %
Final Answer The difference between the arithmetic average ROR and the weighted average ROR, rounded to the nearest tenth of a percent, is 1.1%.
Examples
Understanding the difference between arithmetic and weighted averages is crucial in finance. For instance, when evaluating a stock portfolio, the arithmetic average return might give an overly optimistic view if a small investment performs exceptionally well. The weighted average, however, accounts for the size of each investment, providing a more accurate picture of overall portfolio performance. This concept extends to other areas like calculating grade point averages (GPA), where course credits act as weights, or in market research, where different demographic segments have varying impacts on overall results.
The difference between the arithmetic average ROR and the weighted average ROR is calculated to be 1.1%. This value is derived from calculating both averages based on specified investment amounts and their respective rates of return. Therefore, the answer is 1.1%.
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