A $1 investment made 10%, 20%, 30%, 10%, and 15% p.a returns in the first five years. What is the constant annual return for the five-year period that would have resulted in the same outcome at the end of Year 5 with compounded interest

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BlackTom AI · Accepted Answer

We start by restating the problem in our own words and identifying what is being asked: find the constant annual return r over five years that would yield the same final investment value as a sequence of annual returns of 10%, 20%, 30%, 10%, and 15%.

First, compute the overall growth factor from the given five-year sequence by multiplying the year-by-year factors: (1 + 0.10) × (1 + 0.20) × (1 + 0.30) × (1 + 0.10) × (1 + 0.15).
- After Year 1: 1.10
- After Year 2: 1.10 × 1.20 = 1.32
- After Year 3: 1.32 × 1.30 = 1.716
- After Year 4: 1.716 × 1.10 = 1.8876
- After Year 5: 1.8876 × 1.15 = 2.17174
Thus, over five years the investment grows by a factor of 2.17174.

Next, to find the constant annual return r that would achieve the same end value with compounding, we solve (1 + r)^5 = 2.17174.
Take the fifth root of both sides: 1 + r = (2.17174)^(1/5).
Then r = (2.17174)^(1/5) − 1.

Estimating (2.17174)^(1/5) can be done by trying nearby powers or using natural logarithms: ln(2.17174) ≈ 0.775, divide by 5 gives ≈ 0.155, and exp(0.155) ≈ 1.168. Subtracting 1 yields r ≈ 0.168, or about 16.8%.

A more precise calculation using iterative refinement shows the fifth root to be around 1.1677, giving r ≈ 0.1677, i.e., approximately 16.77%.

Key ideas at play:
- Compound growth factors multiply across periods, so the total growth is the product of (1 + each year's rate).
- The equivalent constant annual rate is the fifth root of that total growth factor minus 1.

Common pitfalls to avoid:
- Adding annual rates instead of multiplying growth factors would misstate the total growth.
- Trying to match a single year’s rate or ignoring the compounding effect would lead to an incorrect, typically much smaller or larger rate.
- Assuming the same rate across all years without solving the root equation would not reflect the actual equivalent constant rate.

In summary, solving (1 + r)^5 = 2.17174 gives r ≈ 0.1677, i.e., about 16.77% per year over five years would produce the same final value as the given sequence of yearly returns.

FINANCE 261 Quiz 1

A $1 investment made 10%, 20%, 30%, 10%, and 15% p.a returns in the first five years. What is the constant annual return for the five-year period that would have resulted in the same outcome at the end of Year 5 with compounded interest

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