Annualised Return

Non-annual Compounding

In general, with more than one compounding period in a year, we can express the formula for present value as follows:

where

m = number of compounding periods per year,

$R_s$ = quoted annual interest rate, and

N = number of years.

Annualising Returns

A general equation to annualise returns is given, where c is the number of periods in a year.

$R_{annual} = (1 + R_{period})^c -1$

c is equal to the number of holding periods in a quarter or in a week.

Continuously Compounded Returns

The continuously compounded return associated with a holding period return is the natural logarithm of one plus that holding period return, or equivalently, the natural logarithm of the ending price over the beginning price (the price relative).

We use r to refer specifically to continuously compounded returns.

The continuously compounded return from t to t+1 is

$r_{t,t+1} = \ln{\frac{P_{t+1}}{P_t}} = \ln(1 + R_{t,t+1})$

If our investment horizon extends from t = 0 to t = T, then the continuously compounded return to T is

$r_{0,T} = \ln(\frac{P_T}{P_0})$

Applying the exponential function to both sides of the equation,

$P_T = P_0\exp(r_{0,T})$

We can also express $\frac{P_T}{P_0}$ as the product of price relatives:

$\frac{P_T}{P_0} = (\frac{P_T}{P_{T-1}})(\frac{P_{T-1}}{P_{T-2}})\dots(\frac{P_1}{P_0})$

Taking logs of both sides of this equation, we find that the continuously compounded return to time T is the sum of the one-period continuously compounded returns:

$r_{0,T} = r_{T-1,T} + r_{T-2,T-1} + \dots + r_{0,1}$