A book returns +6%, -4%, +3% and -5% over four quarters. The four moves sum to zero. What is the cumulative return to four decimal places, and what accounts for the gap?

A book returns +6%, -4%, +3% and -5% over four quarters. The four moves sum to zero. What is the cumulative return to four decimal places, and what accounts for the gap?

Approach: Multiply the four growth factors, then reproduce the answer in log space to show where the shortfall comes from and how large it should be from the squared terms alone.

-0.4278%. Percentage changes compound multiplicatively, so multiply the factors: 1.06 * 0.96 = 1.0176, then 1.0176 * 1.03 = 1.048128, then 1.048128 * 0.95 = 0.9957216, a loss of 0.4278%. The naive sum of the four moves is exactly zero, and the whole gap comes from the cross terms. Log returns size it directly: ln(1.06) + ln(0.96) + ln(1.03) + ln(0.95) = 0.05827 - 0.04082 + 0.02956 - 0.05129 = -0.00429, and exponentiating that returns 0.99572. Each log return sits below its simple return by roughly r^2/2, so the second order correction is the sum of 0.0018, 0.0008, 0.00045 and 0.00125 with a minus sign, which is -0.0043 and reproduces the answer without touching the products at all.

Follow-up: What ordering of the same four moves gives the highest cumulative return, and by how much does order change the answer?

Key concepts: multiplicative compounding, cross terms, log returns, second order correction.