You discover an antique in your attic that you purchased at an estate sale 10 years ago for $400. You auction it on eBay and receive $8,000 for your item. What annual rate of return did you earn?

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BlackTom AI · Accepted Answer

We start by identifying the time horizon and the money involved. You bought an antique for $400 and sold it for $8,000, so your total return is 8,000 / 400 = 20 times your initial investment. The question asks for the annual rate of return over 10 years, so we need the compound annual growth rate (CAGR).

The CAGR formula is: CAGR = (Ending value / Beginning value)^(1 / number of years) − 1.
Plugging in the numbers: CAGR = (8,000 / 400)^(1/10) − 1 = 20^(0.1) − 1.

Compute 20^(0.1). A natural way to estimate is to use natural logs: ln(20) ≈ 2.9957; divide by 10 gives ≈ 0.29957; exponentiate to get e^(0.29957) ≈ 1.3499. Subtract 1 to obtain ≈ 0.3499, or about 34.99%.

Now compare this to each option:
- Option a: 20.00%. This is far below the calculated CAGR of about 35%. It would imply only a modest growth over 10 years, which contradicts a 20x multiplier.
- Option b: 19.00%. Similar to option a, this underestimates dramatically and does not reflect the 20x total return over a decade.
- Option c: 30.47%. This is closer to the correct value but still noticeably lower than the precise CAGR of approximately 34.99%, so it underestimates the annual growth rate required to turn $400 into $8,000 over 10 years.
- Option d: 34.93%. This is the closest to the computed CAGR of about 34.99%, matching the result after applying the exact CAGR formula and rounding, and it aligns with the logic of compounding over 10 years for a 20x return.

In summary, the correct annualized return is the rate that, when compounded for 10 years, grows $400 to $8,000. The calculation yields roughly 34.99%, making the option that lists a value around 34.9% the closest representation of the true CAGR. The other options fall short by a substantial margin or are only modest approximations, failing to match the 20x growth over a decade.

MCD2170 - T1 - 2025 Key Concept 2 Video quiz

You discover an antique in your attic that you purchased at an estate sale 10 years ago for $400. You auction it on eBay and receive $8,000 for your item. What annual rate of return did you earn?

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