Net Present Value

Discounts future money back to what it is worth today, to judge whether an investment pays. The net present value is Σ(cash in year t ÷ (1 + rate)^t) − the initial outlay. A positive figure means the investment is worth more than it costs.

1000000 today and 1000000 a year from now are not worth the same. Money in hand can be invested. At 5% a year, 1000000 today becomes 1050000 in a year, which means 1000000 received a year from now is worth only 952381 today. That figure is its present value.

Net present value takes every payment an investment will bring in, converts each to its present value, adds them up and subtracts what was paid out at the start.

NPV = \sum_{t=1}^{n} \dfrac{C_t}{(1 + r)^{t}} - I

$NPV$ is the net present value
$C_t$ is the cash arriving in year $t$
$r$ is the discount rate, the annual percentage divided by 100
$I$ is the initial investment
$n$ is the number of years cash arrives

Cash in year $t$ is divided by $(1 + r)$ exactly $t$ times. The further out it lands, the more divisions it takes, and the less it is worth today.

Example

Take an initial investment of 1000000, a discount rate of 5%, and income of 300000 a year for five years.

Added up without discounting, the income is 1500000, which looks like 500000 more than the 1000000 invested. But the 300000 arriving in a year is worth 285714 today, and the 300000 arriving in five years only 235058. All five years together come to about 1298843.

Subtracting the 1000000 investment leaves a net present value of about 298843. It is positive, so the investment is worth making.

The profitability index is the present value divided by the initial investment, about 1.2988 here. Each unit invested generates 1.3 units of value, which is what makes it useful for ranking projects of different sizes against each other.

Choosing the discount rate

The discount rate reflects what the money would earn elsewhere, or what it costs to borrow. Companies commonly use their cost of capital.

Raising the rate discounts distant income more heavily and pushes the net present value down. In this example 15% leaves about 5647, barely above zero, and 16% turns it to about −17712. In other words, if better than 15% a year is available elsewhere, this investment is not worth making.

Points to watch

Income is counted from year 1 onwards. Anything received at the moment of investing belongs in the initial investment figure instead.

The calculation assumes you know the future income. In practice forecasts are wrong. Run it at several discount rates to see how much the case can absorb before it stops working.