# Continuously compounded nominal and real returns

## Nominal return

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

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

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

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

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

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

## Real return

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

${\displaystyle P_{t}^{real}=P_{t}\cdot {\frac {PL_{t-1}}{PL_{t}}}.}$

Then the continuously compounded real rate of return ${\displaystyle RC^{real}}$ is

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

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