Present Value

Abstract

The present-value relation says that, under certainty, the value of a capital good or financial asset equals the summed discounted value of the stream of revenues which that asset generates. The discount factor will be that determined by the interest rate over the relevant period. The justification for the present-value relation lies in the fact that (in perfect capital markets) an asset must earn a rate of return exactly equal to the interest rate; otherwise arbitrage opportunities emerge, which is inconsistent with equilibrium. Thus if r _t is the one-period interest rate at t, p _t is the (ex-dividend) price of an asset and d_t is its dividend, it must be true that

$$$$

(1)

since the right-hand side equals the rate of return on the asset. Solving for p _t ,

$$$$

(2)

Replacing t by t + 1, (2) becomes an equation expressing p_t+1 as a function of p_t+2 and p_t+2. Substituting this in (2) and proceeding similarly n times, it follows that

$$$$

(3)