10.6 Appendix: A General Formulation of Discounted Present Value

This section presents a more general way of thinking about discounted present value. The economic idea is the same as the one we encountered when discussing the pricing of orange trees. Here the idea is to isolate the central ideas of discounted present value. We then use this more general formulation to talk about the pricing of stocks in an asset market.

We begin by defining the t-period real interest factor between the present date and some future date t years from now. The t-period real interest factor is simply the amount by which you must discount when calculating a discounted present value of a flow benefit (already adjusted for inflation) that will be received t years from now.

Suppose we have an asset that will provide real dividend payments every year for t years. Suppose that D_t is the real dividend in period t, and R_t is the real interest factor from the current period to period t. Then the price of the asset is given by

$$\begin{array}{c}\text{price}\text{ =}\\ \frac{\text{dividend in year 1}}{\text{ 1-period real interest factor}}\text{ +}\\ \frac{\text{dividend in year 2}}{\text{2-period real interest factor}}\text{ + }\mathrm{...}\text{ +}\frac{\text{dividend in year T}}{\text{ T-period real interest factor}}\end{array}$$

or

$$q=\frac{D_1}{R_1}+\frac{D_2}{R_2}\mathrm{...}+\frac{D_T}{R_T}\text{.}$$

All we did was to divide the dividends (D) due in period t by the interest factor R_t and then add them together.

If interest rates are constant over time, then the interest factors are easy to determine. Suppose that the annual real interest rate for one year is r. Then R₁ = (1 + r) because this is the factor we would use to discount from next year to the present. What about discounting dividends two periods from now? To discount D₂ to period 1, we would divide by (1 + r). To discount that back again to the current period we would again divide by (1 + r). So to discount D₂ to the present we divide D₂ by (1 + r) × (1 + r) = (1 + r)². That is, R₂ = (1 + r)². In general, R_t = (1 + r)^t when interest rates are constant.

If real interest rates are not constant over time, the calculation of R_t is more tedious. If R₁ = (1 + r₁), then R₂ = (1 + r₁) × (1 + r₂), where r₂ is the real interest rate between period 1 and period 2. In the calculation of R₂, you can think of (1 + r₂) as discounting the flow from period 2 to period 1 and then (1 + r₁) as discounting the flow from period 1 to period 0.