An investor who was not as astute as he believed invested $281,000 into an account 6 years ago. Today, that account is worth $217,200. What was the annual rate of return on this account?

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BlackTom AI · Accepted Answer

We begin by identifying the key values and the time period: an initial investment (present value) of 281,000 is grown to a final value of 217,200 over 6 years. The problem asks for the annual rate of return, r, assuming compounding annually, so the formula is FV = PV × (1 + r)^t.

Option-by-option analysis:
- Option A: −3.68%
  To test, compute (1 + r)^6 with r = −0.0368. This is (0.9632)^6. Since 0.9632 is less than 1 but not dramatically, the sixth power would yield a factor somewhat less than 0.80, but the observed growth factor is 0.773. Without precise calculation, this value is plausible but would need checking; however, small deviations in r produce noticeable changes after six years. 
- Option B: 5.12%
  Here r = 0.0512, so (1 + r)^6 = 1.0512^6. This would produce a growth factor well above 1.3, not a decrease to 0.773. Therefore this positive return cannot match the observed decline from 281,000 to 217,200. It is inconsistent with the data.
- Option C: −4.20%
  With r = −0.042, we get (1 − 0.042)^6 = 0.958^6. A sixth power of a number just under 1 results in a sizable reduction, and 0.958^6 closely approximates 0.773, matching the observed ratio FV/PV. This aligns well with the given numbers.
- Option D: 4.39%
  r = 0.0439 gives (1.0439)^6, which would yield a growth factor greater than 1, again contradicting the fact that the final value is smaller than the initial investment. This cannot be correct for the observed outcome.
- Option E: −3.92%
  r = −0.0392 leads to (0.9608)^6. While this is a negative return, the exact sixth-power value would need calculation, but since −4.2% provides a closer match to the required 0.773 factor, this option is unlikely and would not fit as precisely as the −4.20% candidate.

An investor who was not as astute as he believed invested $281,000 into an account 6 years ago. Today, that account is worth $217,200. What was the annual rate of return on this account?

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