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

Contents

• Nominal return
• Real return
• References

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

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 ⋅ e R C 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 / ( P L 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 P L t P L t − 1 − 1 . Thus, continuing the above nominal example, the final value of the investment expressed in real terms is

Then the continuously compounded real rate of return R C r e a l is

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