Sara would like to evaluate the performance of her portfolio over the past 10 years. What compound annual rate of return has she achieved if she invested $12,000 10 years ago and now has $25,000?

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BlackTom AI · Accepted Answer

Sara wants the compound annual growth rate (CAGR) over 10 years given a PV of 12,000 and a FV of 25,000.

First, determine the total growth factor: FV / PV = 25,000 / 12,000 = 2.083333...

Next, apply the CAGR formula: CAGR = (FV / PV)^(1/n) − 1, where n = 10 years. So we need the 10th root of 2.083333...

To estimate the 10th root, take natural logarithms for precision: ln(2.083333...) ≈ 0.7340. Divide by 10 to get ≈ 0.07340. Exponentiate to revert from the natural log: e^(0.07340) ≈ 1.0762.

Subtract 1 to convert from growth factor to a rate: 1.0762 − 1 ≈ 0.0762, or about 7.62%.

For a quick sanity check, recall that 7.0% over 10 years would yield roughly a factor of 1.967, and 8.0% over 10 years would yield roughly a factor of 2.159. The observed growth factor of 2.083 lies between these, closer to 7.6–7.7%, which aligns with the calculation above.

Thus, the evaluated annualized return over the 10-year period is approximately 7.6% per year.

Note: If one had used a rough approximation or a calculator, small differences (e.g., 7.4% to 7.8%) can occur due to rounding during intermediate steps, but the result here centers around the mid 7% range.

25_2 FIN401 TVM short answers

Sara would like to evaluate the performance of her portfolio over the past 10 years. What compound annual rate of return has she achieved if she invested $12,000 10 years ago and now has $25,000?

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