Rate of return is the percentage gain or loss on an investment over a period. It includes both price changes (capital gains/losses) and any cash income paid during the holding period (interest, dividends, rent).
How To Calculate Rate of Return
A common holding-period return (HPR) formula is:
\[ R = \frac{V_1 - V_0 + CF_1}{V_0} \]
where V_0 is the initial value, V_1 is the ending value, and CF_1 is cash income received during the period.
If you just want the price return (ignoring cash income), set CF_1 = 0.
Compounding And Annualization
Returns compound over time. If you earn R_1 in period 1 and R_2 in period 2, the two-period return is:
\[ (1 + R_1)(1 + R_2) - 1 \]
To convert a return over T years into an annualized (compound) rate:
\[ R_{\text{annual}} = (1+R)^{1/T} - 1 \]
Sometimes you also see log (continuously compounded) returns:
\[ r = \ln\left(\frac{V_1}{V_0}\right) \]
Log returns add across time, which can be convenient for some models and statistics.
Nominal vs. Real Rate of Return
A nominal return includes inflation. A real return adjusts for changes in purchasing power:
\[ 1+R_{\text{real}} = \frac{1+R_{\text{nom}}}{1+\pi} \]
For small rates, a useful approximation is R_real \approx R_nom - \pi.
Expected Return, Required Return, And Discounting
Economists and investors distinguish between:
- Realized return: what actually happened.
- Expected return: what you think will happen on average.
- Required rate of return: the minimum return you demand to compensate for time value of money and risk.
In valuation problems, the required return shows up as a discount rate. Holding cash flows fixed, a higher required return implies a lower present value.
Related Terms
- Internal Rate of Return
- Net Present Value
- Discount Rate
- Interest Rate
- Real Interest Rate
- Inflation
- Risk Premium
- Compound Interest