An investment portfolio is shown below.
| Investment | Amount Invested | ROR |
| ---------------------- | --------------- | -------- |
| Money Market Account | $3,200 | 2.1% |
| Government Bond | $1,750 | 4.4% |
| Preferred Stock | $1,235 | -7.8% |
| Common Stock | $2,300 | 10.5% |
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 (1)
Calculate the arithmetic average ROR: 4 2.1 + 4.4 − 7.8 + 10.5 = 2.3
Calculate the total investment: 3200 + 1750 + 1235 + 2300 = 8485
Calculate the weighted average ROR: 8485 3200 × 2.1 + 8485 1750 × 4.4 + 8485 1235 × ( − 7.8 ) + 8485 2300 × 10.5 ≈ 3.41
Find the difference and round: ∣2.3 − 3.41∣ ≈ 1.1%
Explanation
Understanding the Problem We are given an investment portfolio with four different investments: Money Market Account, Government Bond, Preferred Stock, and Common Stock. We are given the amount invested in each and the Rate of Return (ROR) for each. 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 the Arithmetic Average ROR First, let's calculate the arithmetic average ROR. This is simply the average of the four RORs: 4 2.1 + 4.4 + ( − 7.8 ) + 10.5 = 4 9.2 = 2.3
Calculating the Total Investment Next, we need to calculate the weighted average ROR. To do this, we first need to find the total investment: 3200 + 1750 + 1235 + 2300 = 8485
Calculating the Weighted Average ROR Now we can calculate the weight of each investment by dividing the amount invested in that investment by the total investment. Then, we multiply each ROR by its corresponding weight and sum these values to find the weighted average ROR:
Weighted Average ROR = 8485 3200 × 2.1 + 8485 1750 × 4.4 + 8485 1235 × ( − 7.8 ) + 8485 2300 × 10.5
Weighted Average ROR = 0.3771 × 2.1 + 0.2062 × 4.4 + 0.1456 × ( − 7.8 ) + 0.2711 × 10.5
Weighted Average ROR = 0.7920 + 0.9073 − 1.1357 + 2.8466 = 3.4102 (approximately)
Finding the Difference and Rounding Finally, we find the difference between the arithmetic average ROR and the weighted average ROR: ∣2.3 − 3.4102∣ = ∣ − 1.1102∣ = 1.1102 Rounding to the nearest tenth of a percent, we get 1.1%
Examples
Understanding the difference between arithmetic and weighted averages is crucial in finance. For instance, when evaluating investment performance, the weighted average ROR provides a more accurate picture of how the overall portfolio is performing, as it takes into account the size of each investment. This concept extends to other areas like calculating grade point averages (GPA), where course credits act as weights, or determining the average salary in a company, weighted by the number of employees at each salary level. By using weighted averages, we gain a more realistic understanding of the data, reflecting the true impact of each component.
