# Time value of money (present and future value)

## Key Questions

• The future value of a sum of money is given by 'growing' the current value of that money at the appropriate interest rate (or return rate) over the relevant period of time.

A simple formula is:

$F V = P V {\left(1 + r\right)}^{n}$

What this says is that the future value (FV) is equal to the present value (PV) grown at the rate 'r' over 'n' periods.

An example may make it easier. Consider you have $100 today that you invest at 10% per annum compounded annually. What is the future value of this$100 after 1 year? What about after 3 years?

a) After one year: $F V = 100 {\left(1 + 0.1\right)}^{1} = 110$
b) After three years: $F V = 100 {\left(1 + 0.1\right)}^{3} = 133.$

Note that the present value of a sum of money can be calculated with just a slight manipulation of the same formula:

$P V = \frac{F V}{1 + r} ^ n$

The amount a future sum of money is worth at some period before that.

#### Explanation:

Let's being with a basic rule: an amount of money will be worth different values at different points in time, assuming money has a cost - an interest rate, or rate of return.

Here is a simple example that will help organize our thinking. Let's assume you want to have $10,000 in 5 years so you can celebrate your graduation by trekking the Camino de Santiago. How much will you need to invest today to reach your target? We know the future value is$10,000. And the present value is unknown.

It is easy to calculate the present value using this formula

$P V = \frac{f v}{1 + r} ^ n$

We will assume the rate of return (r) that we can earn on our invested money is 6%. And we need the money in 5 years (n), so

$P V = \frac{10 , 000}{1.06} ^ 5$

$P V = \frac{10 , 000}{1.3382}$

PV = $7,472.58 The answer tells us that the present value of$10,000 in 5 years is $7,472.58 today, if we can invest money at 6%. Your$7,472.58 will grow to \$10,000 in 5 years, if it is invested at 6% annually.