Denotational semantics of the Actor model

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

$1+R{S}_t=\frac{P_t}{P_{t-1}}.$

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

$R{C}_t=\ln \left(\frac{P_t}{P_{t-1}}\right).$

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

$P_t^{real}=P_t\cdot \frac{P{L}_{t-1}}{P{L}_t}.$

Then the continuously compounded real rate of return $R{C}^{real}$ is

$R{C}_t^{real}=\ln \left(\frac{P_t^{real}}{P_{t-1}}\right).$

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

Source

• Eugene Fama Notes