Continuously compounded nominal and real returns

Nominal return

Let P_t be the price of a security at time t, including any cash dividends or interest, and let P_t − 1 be its price at t − 1. Let RS_t be the simple rate of return on the security from t − 1 to t. Then

$1 + RS_{t}=\frac{P_{t}}{P_{t-1}}.$

The continuously compounded rate of return or instantaneous rate of return RC_t obtained during that period is

$RC_{t}=\ln\left (\frac{P_{t}}{P_{t-1}}\right ).$

If this instantaneous return is received continuously for one period, then the initial value P_t-1 will grow to $P_t = P_{t-1} \cdot e^{RC_t}$ during that period. See also continuous compounding.

Since this analysis did not adjust for the effects of inflation on the purchasing power of P_t, RS and RC are referred to as nominal rates of return.

Real return

Let π_t be the purchasing power of a dollar at time t (the number of bundles of consumption that can be purchased for $1). Then π_t = 1 / (PL_t), where PL_t is the price level at t (the dollar price of a bundle of consumption goods). The simple inflation rate IS_t from t –1 to t is $\tfrac {PL_t}{PL_{t-1}} - 1$. Thus, continuing the above nominal example, the final value of the investment expressed in real terms is

$P_t^{real} = P_t \cdot \frac{PL_{t-1}}{PL_t}.$

Then the continuously compounded real rate of return RC^real is

$RC_{t}^{real}=\ln\left (\frac{P_{t}^{real}}{P_{t-1}}\right ).$

The continuously compounded real rate of return is just the continuously compounded nominal rate of return minus the continuously compounded inflation rate.

Source

• Eugene Fama Notes