Discount factor

The discount factor is the exchange rate between the future and today. You tell it how many years away a payment is and what return you require, and it hands back a number: 0.82, say. That means a thousand euro in five years is worth eight hundred and twenty today.

The formula

$$DF = \frac{1}{(1 + r)^n}$$

• $DF$: the discount factor
• $r$: the required rate of return for one period
• $n$: the number of periods until the payment

The present value then follows from a simple multiplication:

$$PV = FV \cdot DF$$

The factor is always less than one (at a positive rate of return) and falls quickly as $n$ grows. That is the mathematical statement of the fact that the distant future carries little weight today.

A worked example

You require a return of 6% a year. The discount factors for the first five years:

Year	Calculation	Factor
1	1 / 1.06	0.9434
2	1 / 1.06²	0.8900
3	1 / 1.06³	0.8396
4	1 / 1.06⁴	0.7921
5	1 / 1.06⁵	0.7473

If someone promises you €5,000 in four years, today that is worth:

$$5{,}000 \cdot 0.7921 = 3{,}960.50\ \euro$$

Notice how the factor falls ever more slowly in absolute terms, but relentlessly: after twenty years at 6% it is down to 0.31.

Why it pays to work with factors

For a single payment it makes no difference whether you divide or multiply. The advantage shows with several payments: you calculate the factors once and then multiply any amount by them.

That is exactly how a bond or an investment project is valued: every future flow is multiplied by the factor for its year and the results are added up. The sum is the net present value.

How it relates to compounding

The discount factor is the exact opposite of the compounding factor $(1 + r)^n$, which moves money forward. Their product is always one, which is another way of saying that discounting and compound interest are the same operation run in two directions.

Work out your own factors and present values in the discounting calculator.