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

We start by identifying the key data: the initial investment (P0) is $12,000, the ending value (P1) is $25,000, and the time period is 10 years. The compound annual growth rate (CAGR) is defined as CAGR = (P1 / P0)^(1 / n) − 1, where n is the number of years.

First, compute the total growth factor: P1 / P0 = 25,000 / 12,000 = 2.083333...

Next, take the 10th root of the growth factor to find the annual rate: (2.083333...)^(1/10).

A practical way to evaluate this is to use natural logarithms: ln(2.083333...) ≈ 0.733969. Divide by 10 years: 0.733969 / 10 ≈ 0.0733969.

Exponentiate to convert back from the natural log: e^(0.0733969) ≈ 1.07615.

Subtract 1 to obtain the annual growth rate: 1.07615 − 1 ≈ 0.07615, which is about 7.615%, typically rounded to 7.62%.

Sanity check: If CAGR is around 7.62%, over 10 years a $12,000 investment would grow by a factor of roughly (1 + 0.0762)^10 ≈ 2.083, which matches the observed ending value ratio of 25,000 / 12,000.

Potential pitfalls to avoid: using a simple average return (arithmetic mean) instead of CAGR can misstate the true annualized performance, and forgetting to take the 10th root would yield an incorrect rate. Ensure consistency of units and the time horizon, and verify the final result by back-calculating forward.

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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