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 return on the security from t − 1 to t,

$1\; +\; RS\_\{t\}=frac\{P\_\{t\}\}\{P\_\{t-1\}\}.$

Continuously compounded nominal return

The continuously compounded return is the value of RS_t that satisfies

$RS\_\{t\}=lnleft\; (frac\{P\_\{t\}\}\{P\_\{t-1\}\}right\; ).$

Thus,

Real return

Let πt be the purchasing power of a dollar at time t (the number of bundles of consumption that can be purchased for $1). Thus, πt is 1/(PLt), where PLt is the price level at t (the dollar price of a bundle of consumption goods). The simple inflation rate ISt from t₁ to t is

The simple real return rst from t − 1 to t is

$pr\; =\; t\; -\; n\; /\; log\; (rst).$

Continuously compounded real return

The continuously compounded inflation rate is the value of ICt that satisfies. Thus, the continuously compounded real return is the value of rct that satisfies.

Thus, the continuously compounded real return is just the continuously compounded nominal return minus the continuously compounded inflation rate.

Alternatively, the continuously compounded nominal return RCt is the real return rct plus the inflation rate ICt.

Source

• Eugene Fama Notes