Nominal return

Let P_t be the price of a security at time t, including any cash dividends or interest

Interest

Interest is a fee paid by a borrower of assets to the owner as a form of compensation for the use of the assets. It is most commonly the price paid for the use of borrowed money, or money earned by deposited funds....

, 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_t-1 will grow to during that period. See also continuous compounding

Compound interest

Compound interest arises when interest is added to the principal, so that from that moment on, the interest that has been added also itself earns interest. This addition of interest to the principal is called compounding...

.

Since this analysis did not adjust for the effects of inflation

Inflation

In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...

 on the purchasing power of P_t, RS and RC are referred to as nominal rates of return.

Real return

Let be the purchasing power of a dollar at time t (the number of bundles of consumption that can be purchased for $1). Then , 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 is

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

Source

• Eugene Fama Notes