 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
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_{t1} will grow to 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 . 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 RC^{real} is
The continuously compounded real rate of return is just the continuously compounded nominal rate of return minus the continuously compounded inflation rate.
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