Roddy Richards invested $12014.88 in Atron Enterprises five years ago. The investment had yearly arithmetic returns of -9.7%, -8.1%, 15%, 7.2%, and 15.4%.

What is the geometric average return of Roddy's Richard's investment?

Question

Roddy Richards invested $12014.88 in Atron Enterprises five years ago. The investment had yearly arithmetic returns of -9.7%, -8.1%, 15%, 7.2%, and 15.4%.

What is the geometric average return of Roddy's Richard's investment?

BlackTom AI · Accepted Answer

We start by restating the essential parts of the problem to set the stage for calculation. The question asks for the geometric average return of an investment with annual arithmetic returns of -9.7%, -8.1%, 15%, 7.2%, and 15.4%. The available option is 3.38%.

To compute the geometric average return, we convert each yearly arithmetic return into a gross return factor, multiply all factors together, take the fifth root (since there are five years), and then subtract 1.

Step 1: Convert each return to a growth factor.
- Year 1: 1 + (-9.7%) = 0.903
- Year 2: 1 + (-8.1%) = 0.919
- Year 3: 1 + 15% = 1.15
- Year 4: 1 + 7.2% = 1.072
- Year 5: 1 + 15.4% = 1.154

Step 2: Multiply the five growth factors.
0.903 × 0.919 × 1.15 × 1.072 × 1.154 ≈ 1.17936

Step 3: Take the fifth root to obtain the geometric mean factor.
Geometric factor = (1.17936)^(1/5) ≈ 1.0337

Step 4: Subtract 1 to get the geometric average annual return.
Geometric average return ≈ 1.0337 − 1 = 0.0337, or about 3.37% to 3.38%.

Now evaluating the given option:
- Option: 3.38% — this aligns with the computed geometric average return (approximately 3.37% to 3.38%), which is consistent with the math above.

Explanation of why this value makes sense in context:
- The geometric mean accounts for compounding across years, unlike the simple arithmetic mean.
- The presence of negative returns in the sequence lowers the product, but the sizable positive returns in later years pull the geometric mean up above the simple average of the percentages.

If one wonders about the numeric sensitivity: small changes in the rounding of the intermediate product (1.17936) can shift the fifth-root result slightly, but the final answer remains around 3.38% rather than, say, a much larger or smaller value. Thus the option 3.38% is consistent with the standard method for computing a geometric average return over multiple periods.

In short, the steps above show that the geometric mean of the five annual factors yields a result near 3.38%, which matches the provided option.

Roddy Richards invested $12014.88 in Atron Enterprises five years ago. The investment had yearly arithmetic returns of -9.7%, -8.1%, 15%, 7.2%, and 15.4%. What is the geometric average return of Roddy's Richard's investment? 单项选择题

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