##
9.6 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

$$\begin{array}{c}\text{price}\text{=}\text{}\\ \frac{\text{dividendinyear1}}{\text{1-periodrealinterestfactor}}\text{+}\\ \frac{\text{dividendinyear2}}{\text{2-periodrealinterestfactor}}\text{+}\mathrm{...}\text{+}\frac{\text{dividendinyearT}}{\text{T-periodrealinterestfactor}}\end{array}$$
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 *R*_{1} = (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 *D*_{2} 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 *D*_{2} to the present we divide *D*_{2} by (1 + *r*) × (1 + *r*) = (1 + *r*)^{2}. That is, *R*_{2} = (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 *R*_{1} = (1 + *r*_{1}), then *R*_{2} = (1 + *r*_{1}) × (1 + *r*_{2}), where *r*_{2} is the real interest rate between period 1 and period 2. In the calculation of *R*_{2}, you can think of (1 + *r*_{2}) as discounting the flow from period 2 to period 1 and then (1 + *r*_{1}) as discounting the flow from period 1 to period 0.