Continuously compounded nominal and real returns
# Continuously compounded nominal and real returns

Return rate is a corporate finance and accounting tool which calculates the gain and loss of investment over a certain period of time.

## 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 $\pi _{t}$ be the purchasing power of a dollar at time t (the number of bundles of consumption that can be purchased for $1). Then $\pi _{t}=1/(PL_{t})$, 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.

