Lesson 7 of 8 · 14 min

Annualised and continuously compounded returns

Put every return on a per-year basis by compounding it, and use continuously compounded (log) returns when you want returns that add up over time.

In short

With m compounding periods a year, use the periodic rate $R_s/m$ and $mN$ periods: $PV = FV_N(1+R_s/m)^{-mN}$ .
Annualise: $R_{\text{annual}} = (1+R_{\text{period}})^{c} - 1$ , with c = periods per year (365/days, 52/weeks, 12/months; c < 1 for periods longer than a year).
Annualising assumes the period's return could be repeated all year, which is unrealistic for short, lucky periods.
Continuously compounded return: $r = \ln(P_1/P_0) = \ln(1+R)$ , always smaller than the HPR.
Continuously compounded returns add across periods, and $P_T = P_0e^{r_{0,T}}$ .

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