Math Problem Statement

What are the arithmetic and geometric average returns for a stock with annual returns of 4 percent, 9 percent, −5 percent, and 17 percent?

Multiple Choice

6.25%; 5.95%

8.63%; 5.95%

5.95%; 6.25%

6.25%; 8.63%

8.63%; 6.25%

Solution

To solve for the arithmetic and geometric average returns, let’s go through each one:

Arithmetic Average Return

The arithmetic average return is simply the sum of the returns divided by the number of periods.

Given returns: 4%,9%,5%,17%4\%, 9\%, -5\%, 17\%

Arithmetic Average=4+95+174=254=6.25%\text{Arithmetic Average} = \frac{4 + 9 - 5 + 17}{4} = \frac{25}{4} = 6.25\%

Geometric Average Return

The geometric average return is calculated as the nn-th root of the product of (1+ri)(1 + r_i) values, minus 1, where rir_i is each annual return and nn is the number of periods.

Geometric Average=((1+0.04)(1+0.09)(10.05)(1+0.17))141\text{Geometric Average} = \left( (1 + 0.04)(1 + 0.09)(1 - 0.05)(1 + 0.17) \right)^{\frac{1}{4}} - 1

Calculating this step-by-step:

  1. 1+0.04=1.041 + 0.04 = 1.04
  2. 1+0.09=1.091 + 0.09 = 1.09
  3. 10.05=0.951 - 0.05 = 0.95
  4. 1+0.17=1.171 + 0.17 = 1.17

Now, multiplying them:

1.04×1.09×0.95×1.171.23871.04 \times 1.09 \times 0.95 \times 1.17 \approx 1.2387

Taking the fourth root:

1.2387141.05951.2387^{\frac{1}{4}} \approx 1.0595

Subtracting 1:

Geometric Average0.0595=5.95%\text{Geometric Average} \approx 0.0595 = 5.95\%

Answer

The correct answer is:

6.25%; 5.95%


Would you like further details or have any questions?

Related Questions:

  1. How do you interpret the difference between arithmetic and geometric average returns?
  2. Why might an investor prefer the geometric mean over the arithmetic mean?
  3. How would the averages change if there was a negative return greater than 17%?
  4. What would the effect of compounding be on these averages over more years?
  5. How do arithmetic and geometric returns differ in interpreting stock volatility?

Tip: Arithmetic returns are useful for estimating average short-term performance, while geometric returns are better for understanding compounded growth over time.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Average
Geometric Average
Stock Returns

Formulas

Arithmetic Average = (Sum of returns) / (Number of periods)
Geometric Average = (Product of (1 + each return))^(1/Number of periods) - 1

Theorems

Arithmetic Mean
Geometric Mean

Suitable Grade Level

Undergraduate Finance or Business Studies