# The Present Value Formula

The present value formula gives the value today of a series of future cash flows. When interest rates increase, present values decrease, and vice versa.

Interest rates can be thought of as the price of borrowing to purchase capital (financial resources). It follows, then, that firms should be able to use this price of capital to determine whether a particular project is worth pursuing. The logic is simple: a project is worth pursuing if it provides a higher return (gains on an investment) than simply putting money in the bank. (This is a slight oversimplification but is helpful to consider.) The present value formula provides exactly this information, even when payments and costs are at different points in the future. Present value (PV) is the value from today's perspective of a sequence of future costs and payments.

First, note that if money is saved, it earns interest and is worth more in the future. Therefore, present value (PV, the value today) and future value (FV, the value $t$ periods in the future) are related.
$\text{FV} = \text{PV} (1+\text{i})^{t}$
In this formula, i is the interest rate—in this context, the return on saving. If instead the future value is known and the unknown quantity is how much that future value is worth today, rearrange the formula to solve for the present value.
$\text{PV}=\frac{\text{FV}}{(1+\text{i})^{t}}$
This formula can be used by a saver to answer the following question: "If I want to have $100 next year, how much should I save now?" If the interest rate is six percent, for example, then that value is $\100 / (1 + 0.06)$ , or$94.34. Put another way, the saver would have $100 one year from now if they saved$94.34 at six percent interest. The main present value formula is just an extension of this principle.
$\text{PV}=\frac{\text{FV}_1}{(1+\text{i})}+\frac{\text{FV}_{2}}{(1+\text{i})^2}+\frac{\text{FV}_{3}}{(1+\text{i})^{3}}+\dots$
The FV amounts are the payments, and interest rates one, two, three, and so on are periods in the future. In this formula, from the producer's perspective, future payments on the project are positive and future costs are negative. A project for the producer can then be thought of as worth pursuing if the present value is at least as big as the up-front cost to get the project started.
$\115.76=100(1.05)^3$
A saver who deposits $100 and waits for three years at five percent interest would have$115.76 at the end of the three year period. They would earn $15.76 on top of the$100 they saved.

As the formula shows, present values decrease when interest rates increase, and vice versa. If interest rates are high, a saver would have to put aside less to make the same amount of money as when the interest rates are low. This makes sense because, if the return on saving is higher, then alternative projects look less attractive from today's perspective. The present value formula also explains why the demand for loanable funds slopes downward: at higher interest rates, fewer investment projects look attractive from a present value standpoint, so the quantity of loanable funds demanded is lower.