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Arithmetic vs Log Returns

Arithmetic and log returns
R=P1−P0P0r=ln⁡(P1P0)R = \frac{P_1 - P_0}{P_0} \qquad r = \ln\left(\frac{P_1}{P_0}\right)

The same move written two ways. Arithmetic returns add across a portfolio; log returns add across time.

Both are "the return". They differ in which operation they make easy, and choosing wrongly produces real errors.

The defining properties

Log returns add across time.

r0→2=r0→1+r1→2r_{0\to 2} = r_{0\to 1} + r_{1\to 2}

Multi-period returns are sums, which is why every continuous-time model uses them.

Arithmetic returns add across assets.

Rp=∑iwiRiR_p = \sum_i w_i R_i

A portfolio's return is the weighted average of component returns, which is exactly true for arithmetic and only approximately true for log returns.

Key takeaway

Log returns aggregate across time; arithmetic returns aggregate across assets. Neither does both, and this is the whole basis for choosing between them.

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