Estimate 1.015 to the 12th power

Question

A fund returns 1.5\% each month. Compounded over twelve months, its growth factor is (1.015)^{12}. Estimate (1.015)^{12} to three decimal places. Aim: under 20 seconds.

Accepted Answer

Binomial expansion of (1 + x)^n with x = 0.015, n = 12: (1.015)^{12} \approx 1 + 12(0.015) + \binom{12}{2}(0.015)^2 = 1 + 0.18 + 66 	imes 0.000225 = 1.195. The two big terms are the naive +18\% and the compounding bonus 66 	imes 0.000225 = 0.01485. The cubic term adds \binom{12}{3}(0.015)^3 = 220 	imes 0.0000034 \approx 0.001, giving (1.015)^{12} \approx 1.196. True value: 1.19562\ldots, so the two-term-plus-nudge estimate is essentially exact. Takeaway. For compounding estimates, "simple interest plus half the pairwise cross terms": (1+x)^n \approx 1 + nx + \frac{n(n-1)}{2}x^2. Here that …

Estimate 1.015 to the 12th power

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