# Present Value of a Lump Sum

…what is it worth today?

The present value of a lump sum receivable in the future is the value the lump sum has today. The present value is arrived at by discounting the future value back to the start of period one.

The concept of present value is used to give a base line to which all lump sums can be discounted, this eliminates the effect of the discount rate and allows the lump sum values to be added together when carrying out time value of money calculations.

To understand the idea of present value let’s look at a simple example.

If a lump sum of money is received today and invested, then it will grow into a larger sum in the future.
For example, if you receive 1,000 and invest it at 10%, in 1 years time it will have earned interest of 1,000 x 10% = 100, and you will have 1,000 + 100 = 1,100.

`Future (1,100) = Present value (1,000) + Interest (100)`

If follows from this that if you receive 1,100 in one years time, you have lost the opportunity to invest it, you have lost the interest of 100.

Rearranging the formula shown above,

`Present value (1,000) = Future value (1,100) - Interest (100)`

So in this example, due to the effect of the discount rate of 10%, receiving 1,100 in one years time is the same as receiving 1,000 today. The present value of the lump sum amount receivable in the future is lower because the opportunity to invest and earn interest has been lost.

## Present Value of a Lump Sum Formula

The formula for the present value of a lump sum follows from the future value formula shown in our future value of a lump sum tutorial.

The future value of a lump sum formula is

`FV = PV x (1 + i)n`

Rearranging this formula gives the formula for the present value of a lump sum

`PV = FV / (1 + i)n`

For example, if 4,000 is receivable at the end of year 3 and the discount rate is 10%, then the present value of the lump sum is given as follows:

```PV = FV / (1 + i)n
PV = 4,000 / (1 + 10%)3
PV = 3,005.26 rounded to 3,005
```

The amount of 4,000 receivable in three years time is only worth 3,005 today at the start of period one, because the opportunity to earn interest for three years has been lost.

## Cash Flow Diagram

Each cash flow stream can be shown in a cash flow diagram. The original 4,000 is receivable at the end of year 3, and has a present value at the start of period 1 of 3,005. The effect of discounting the future value to the present value can be seen by viewing the transaction in the opposite direction starting with the present value of 3,005 and adding compound interest each year to arrive at the future value of 4,000.

This is shown in the table below.

 Period 1 2 3 n PV Opening balance 3,005 3,305 3,636 i Interest @ 10% 300 331 364 FV Closing balance 3,305 3,636 4,000

By receiving the 4,000 at the end of year 3, the opportunity to earn interest of 995 has been lost, and its value today is only 3,005. Putting this another way, receiving the amount of 4,000 in three years time is equivalent to receiving 3,005 now.

The PV of a lump sum is a useful concept, as it allows all future lump sum amounts to be discounted back to a common starting point. The present value eliminates the effect of the discount rate and means that all the lump sums can be added together, an idea which is used when calculating the present value of an annuity or when working out the net present value of a series of lump sums.