Lesson 6 of 10 · All lessons in this module
  1. Financial assets, yields and expected versus actual returns
  2. What an interest rate is made of
  3. Required return, risk premia and excess returns
  4. Holding period return
  5. Arithmetic and geometric mean returns
  6. Annualised and continuously compounded returnsVideo lesson
  7. Gross, net, after-tax, real and leveraged returnsVideo lesson
  8. Taxing price and income returns
  9. Leverage through call options
  10. Integrated cases: from total return to what the investor keeps

This lesson is part of the 2027 curriculum. You are following the 2026 curriculum, where it is not taught in this form. Switch if you are sitting the exam under the 2027 curriculum.

Lesson 6 of 10 · 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 Rs/mR_s/m and mNmN periods: PV=FVN(1+Rs/m)−mNPV = FV_N(1+R_s/m)^{-mN}.
  • Annualise: Rannual=(1+Rperiod)c−1R_{\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⁡(P1/P0)=ln⁡(1+R)r = \ln(P_1/P_0) = \ln(1+R), always smaller than the HPR.
  • Continuously compounded returns add across periods, and PT=P0er0,TP_T = P_0e^{r_{0,T}}.

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