Introduction to Discounting: Present Value and Future Value

“A hundred today is worth more than a hundred tomorrow” is the axiom of all financial calculation. Converting future money into today’s value is called discounting; the reverse direction—projecting today’s money into the future—is called compounding. Both use the same factor (1+i)^n, one multiplying by it and one dividing. The compound interest calculator and the net present value calculator are two directions of the same mathematical fact, and this guide is the background they share.

Overview

“A hundred today is worth more than a hundred tomorrow” is the axiom of all financial calculation. Converting future money into today’s value is called discounting; the reverse direction—projecting today’s money into the future—is called compounding. Both use the same factor $(1+i)^n$ , one multiplying by it and one dividing. The compound interest calculator and the net present value calculator are two directions of the same mathematical fact, and this guide is the background they share.

One factor, two directions

Let the interest rate per period be $i$ . A present value $PV$ compounded for $n$ periods becomes

FV = PV\,(1+i)^{n}

Solving the same relationship for $PV$ gives discounting:

PV = \frac{FV}{(1+i)^{n}}

$(1+i)^{-n}$ is called the discount factor: it converts “one unit of currency at the end of period $n$ ” into today’s value. In a compounding context $i$ is called the interest rate; in a discounting context it is called the discount rate—the name changes with the direction, the mathematics is identical.

Periods $n$	Discount factor ( $i=8\%$ )	100 at the end of year $n$ , discounted to today
1	0.9259	92.59
2	0.8573	85.73
5	0.6806	68.06
10	0.4632	46.32

The higher the discount rate and the more periods, the less future money is "worth"—this is not inflation (inflation is a separate matter) but the opportunity cost of capital: had the money been in hand today, it could have grown at rate $i$ .

From the discount factor to NPV

A project’s cash flow is “invest today, collect in instalments later”. Discount each period’s cash flow $C_t$ back to today and add the results up, and you have the net present value (NPV):

NPV = \sum_{t=0}^{n} \frac{C_t}{(1+r)^{t}}

$C_0$ is usually negative (the initial outlay; $t=0$ is not discounted). An NPV greater than 0 means the series “earns” when measured at the discount rate $r$ . For example, invest 1,000 and receive 600 in each of the next two years, at $r = 8\%$ :

NPV = -1000 + \frac{600}{1.08} + \frac{600}{1.08^{2}} = -1000 + 555.56 + 514.40 = 69.96

The internal rate of return (IRR) is the discount rate that makes NPV exactly zero—treat NPV as a function of $r$ and IRR is where the curve crosses the horizontal axis. It answers “at what rate does this series of cash flows grow?”.

Nominal and effective rates

The 5% in “5% a year, compounded monthly” is a nominal annual rate (APR): it is merely a convention that multiplies the per-period rate by the number of periods. The true growth over one year is the effective annual rate (EAR):

EAR = \left(1+\frac{r}{m}\right)^{m} - 1

5% compounded monthly corresponds to an EAR of 5.1162%. To compare two plans with different compounding frequencies, compare EARs; to discount cash flows, use the per-period rate that matches the cash-flow interval. Get these two conventions wrong and every number after them is wrong.

Common pitfalls

Confusing discounting with inflation: the discount rate reflects the opportunity cost and risk of capital; inflation is at most one possible component of it. Nominal cash flows go with a nominal discount rate and real cash flows with a real discount rate—never mix the two.
Discounting the $t=0$ outlay: money today is not discounted; $C_0$ enters at face value.
Comparing nominal rates across compounding frequencies: at the same 5%, monthly compounding is actually higher than annual—see the EAR above.
Treating IRR as a universal metric: when cash flows change sign several times the IRR may not be unique; when comparing mutually exclusive projects, NPV is more reliable than IRR.

References and further reading

Brealey, R. A., Myers, S. C., Allen, F., & Edmans, A. Principles of Corporate Finance, 14th ed. McGraw Hill, 2022. Chapter 2, “How to Calculate Present Values”—the standard derivation of present value, annuities and the discount factor.
Wikipedia, Present value（访问日期：2026-09-09）—present value and the discount factor.
Wikipedia, Net present value（访问日期：2026-09-09）—definition of NPV and its relationship to IRR.