This is “Discounted Present Value”, section 16.3 from the book Theory and Applications of Macroeconomics (v. 1.0).

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**Discounted present value** is a technique used to add dollar amounts over time. We need this technique because a dollar today has a different value from a dollar in the future.

The discounted present value this year of $1.00 that you will receive next year is as follows:

$$\frac{\$1}{\text{nominalinterestfactor}}=\frac{\$1}{1+\text{nominalinterestrate}}\text{.}$$If the **nominal interest rate** is 10 percent, then the **nominal interest factor** is 1.1, so $1 next year is worth $1/1.1 = $0.91 this year. As the interest rate increases, the discounted present value decreases.

More generally, we can compute the value of an asset this year from the following formula:

$$\text{valueofassetthisyear}=\frac{\text{flowbenefitfromassetthisyear}+\text{priceofassetnextyear}}{\text{nominalinterestfactor}}\text{.}$$The flow benefit depends on the asset. For a bond, the flow benefit is a coupon payment. For a stock, the flow benefit is a dividend payment. For a fruit tree, the flow benefit is the yield of a crop.

If an asset (such as a bond) yields a payment next year of $10 and has a price next year of $90, then the “flow benefit from asset + price of the asset next year” is $100. The value of the asset this year is then $\frac{\$100}{\text{nominalinterestfactor}}\text{.}$ If the nominal interest rate is 20 percent, then the value of the asset is $100/1.2 = 83.33.

We discount nominal flows using a nominal interest factor. We discount real flows (that is, flows already corrected for inflation) using a **real interest factor**, which is equal to (1 + **real interest rate**).

- If the interest rate is positive, then the discounted present value is less than the direct sum of flows.
- If the interest rate increases, the discounted present value will decrease.

Denote the dividend on an asset in period *t* as *D*_{t}. Define *R*_{t} as the *cumulative* effect of interest rates up to period *t*. For example, *R*_{2} = (1 + *r*_{1})(1 + *r*_{2}). Then the value of an asset that yields *D*_{t} dollars in every year up to year *T* is given by

If the interest rate is constant (equal to *r*), then the one period interest factor is *R* = 1 + *r*, and *R*_{t} = *R*^{t}.

The discounted present value tool is illustrated in Table 16.1 "Discounted Present Value with Different Interest Rates". The number of years (*T*) is set equal to 5. The table gives the value of the dividends in each year and computes the discounted present values for two different interest rates. For this example, the annual interest rates are constant over time.

Table 16.1 Discounted Present Value with Different Interest Rates

Year | Dividend ($) | Discounted Present Value with R = 1.05 ($) |
Discounted Present Value with R = 1.10 ($) |
---|---|---|---|

1 | 100 | 100 | 100 |

2 | 100 | 95.24 | 90.91 |

3 | 90 | 81.63 | 74.38 |

4 | 120 | 103.66 | 90.16 |

5 | 400 | 329.08 | 273.20 |

Discounted present value | 709.61 | 628.65 |