# Time value of money

The time value of money is the concept that allows somebody to compare the value of money at two different points in time. It relies on the principle that money that is available now could be invested to generate interest and that receiving money at a later point in time has implied opportunity costs.

## Present Value and Future Value

The most basic application of the time value of money is the conversion between the Present Value (PV) and the Future Value (FV) of money. The important assumption is that, at the beginning of each period (often, this is one year), all the available money is invested at an interest rate of r per period.

After Period 1:

$FV = PV \cdot (1+r)$

After Period 2:

$FV = (PV \cdot (1+r)) \cdot (1+r) = PV \cdot (1+r)^2$

After Period n:

$FV = PV \cdot (1+r)^n$

Likewise, the Present Value of an amount of money we get after n periods would be:

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

## Present and Future Value - an example

A shop has an offer: When you buy a certain product, you can choose between paying $950 now or$1000 in one year. We assume that our bank would give us 5% interest for a 1-year investment.

The Future Value of $950 after 1 year (1 period) would be: $FV = \950 \cdot (1+0.05) = \997.5$ So the Future Value of the$950 is less than what we would have to pay for the product in the future. This means that we wouldn't be able to pay the $1000 for the product using only the the$950 we invest today. So it's better to pay right away.

We also could have calculated the Present Value of "$1000 in one year": $PV = \frac{\1000}{(1+0.05)} = \952.38$ The conclusion is (of course) the same: The Present Value of "$1000 in one year" is higher than the \$950 we would have to pay today, so we gain nothing from waiting (on the contrary, even).