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

[math]\displaystyle{ 1 + RS_{t}=\frac{P_{t}}{P_{t-1}}. }[/math]

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

[math]\displaystyle{ RC_{t}=\ln\left (\frac{P_{t}}{P_{t-1}}\right ). }[/math]

If this instantaneous return is received continuously for one period, then the initial value P_t-1 will grow to [math]\displaystyle{ P_t = P_{t-1} \cdot e^{RC_t} }[/math] 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 [math]\displaystyle{ \pi _ t }[/math] be the purchasing power of a dollar at time t (the number of bundles of consumption that can be purchased for $1). Then [math]\displaystyle{ \pi_t = 1/(PL_t) }[/math], 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 [math]\displaystyle{ \tfrac {PL_t}{PL_{t-1}} - 1 }[/math]. Thus, continuing the above nominal example, the final value of the investment expressed in real terms is

[math]\displaystyle{ P_t^{real} = P_t \cdot \frac{PL_{t-1}}{PL_t}. }[/math]

Then the continuously compounded real rate of return [math]\displaystyle{ RC^{real} }[/math] is

[math]\displaystyle{ RC_{t}^{real}=\ln\left (\frac{P_{t}^{real}}{P_{t-1}}\right ). }[/math]

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