Continuously compounded nominal and real returns

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Continuously Compounded Nominal and Real Returns

Contents 1 Nominal Return 1.1 Continuously Compounded Nominal Return 2 Real Return 2.1 Continuously Compounded Real Return 3 Source

[edit] 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,

[edit] Continuously Compounded Nominal Return

The continuously compounded return is the value of RS_t that satisfies

Thus,

[edit] Real Return

Let πt be the purchasing power of a dollar at 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 from t 1 to t, ISt, is,

The simple real return from t-1 to t, rst, is, pr = t-n / log rst

[edit] 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

[edit] Source

• Eugene Fama Notes