Returns — simple, log and annualised

Simple returns

The simple (arithmetic) return from $t-1$ to $t$ is

$$R_t = \frac{P_t - P_{t-1}}{P_{t-1}} = \frac{P_t}{P_{t-1}} - 1$$

Simple returns aggregate across assets: a portfolio’s return is the weighted sum of its holdings’ simple returns, $R_p = \sum_i w_i R_i$. That makes them the right choice for portfolio arithmetic.

Log returns

The log (continuously compounded) return is

$$r_t = \ln\frac{P_t}{P_{t-1}} = \ln(1 + R_t)$$

Log returns aggregate across time: the log return over many periods is just the sum of the single-period log returns,

$$r_{0 \to T} = \sum_{t=1}^{T} r_t$$

That makes them convenient for statistics over time (means, regressions, the central limit theorem). For small moves the two are close, $r \approx R - R^2/2$, but they diverge quickly for big ones: a $-50\%$ simple return is a $-69\%$ log return, and a $+100\%$ simple return is only $+69\%$ in log terms.

Arithmetic vs geometric mean

The average of simple returns overstates what you actually earn. Gain 50%, then lose 50%: the arithmetic mean is 0%, but you finish with 75% of your money. The compound (geometric) growth rate is

$$g = \Big(\prod_t (1 + R_t)\Big)^{1/T} - 1$$

and, approximately,

$$g \approx \bar{R} - \frac{\sigma^2}{2}$$

The $\sigma^2/2$ term is volatility drag. Two strategies with the same average return do not grow at the same rate; the more volatile one compounds slower.

Annualising

If returns are independent from period to period, the mean scales with time and the volatility with its square root. With $N$ periods per year (252 trading days for equities, 365 for crypto, 8,760 hours for hourly crypto):

$$\mu_{\text{annual}} \approx N\,\bar{r}, \qquad \sigma_{\text{annual}} \approx \sqrt{N}\,\sigma$$

So a daily volatility of 1% is roughly 16% annualised in equities ($\sqrt{252} \approx 15.9$), and a daily 3% crypto vol is about 57% a year ($\sqrt{365} \approx 19.1$).