# Future Value of a Lump Sum

…what is it worth in the future?

The future value of a lump sum is the amount of money that a sum of money today will grow to at a future date, assuming a certain rate of return.

To illustrate, suppose 3,000 is invested at 10% for a year. In this case at the end of the year the interest earned will be 3,000 x 10% = 300. Furthermore the lump sum will have grown to 3,000 plus 300, equals 3,300.

Generally the original lump sum (3,000) is referred to as the present value (PV) and is the value today, at the start of period 1. The final larger lump sum (3,300) is referred to as the future value (FV), and in this case, is the value of the lump sum at the end of year 1.

To illustrate, at the end of year 1 the future value is given by.

```FV = PV + Interest
FV = 3,000 + 3,000 x 10%
FV = 3,000 + 300
FV = 3,300```

If the amount of 3,300 remains invested for another year, then at the end of year 2, the future value is given by.

```FV = 3,300 + 3,300 x 10%
FV = 3,300 + 330
FV = 3,630```

If the amount of 3,630 remains invested for another year, then at the end of year 3, the future value is given by.

```FV = 3,630 + 3,630 x 10%
FV = 3,630 + 363
FV = 3,993```

## Future Value of a Lump Sum Cash Flow Diagram

Additionally each cash flow stream can be represented in a cash flow diagram. As can be seen the original 3,000 is invested (cash out) at the start of period 1, and is returned (cash in) with interest as the larger lump sum 3,993, at the end of period 3.

It is important to realize that each year interest is being earned on the original sum and on the interest from the previous years. Consequently this type of interest is referred to as compound interest. To illustrate the compounding is shown in the table below.

Future value of a lump sum

## Future Value of a Lump Sum Formula

To summarize it would be useful to write the future value of a lump sum answers above in terms of the original lump sum (3,000).

### Future value at the end of year 1:

```FV = PV + Interest
FV = 3,000 + 3,000 x 10%
FV = 3,000 x (1 + 10%)
FV = 3,000 x (1 + 10%)1
FV = 3,300```

### Future value at the end of year 2:

```FV = 3,300 + 3,300 x 10%
FV = 3,300 x (1 + 10%)
FV = 3,000 x (1 + 10%)1 x (1 + 10%)
FV = 3,000 x (1 + 10%)2
FV = 3,630```

As can be seen above the amount of 3,300 has been replaced with 3,000 x (1 + 10%)1 from the answer for year 1.

### Future value at the end of year 3:

```FV = 3,630 + 3,630 x 10%
FV = 3,630 x (1 + 10%)
FV = 3,000 x (1 + 10%)2 x (1 + 10%)
FV = 3,000 x (1 + 10%)3
FV = 3,993```

In similar fashion the amount of 3,630 has been replaced with 3,000 x (1 + 10%)2 from the answer for year 2.

To summarize:

```End of year 1 FV = 3,000 x (1 + 10%)1
End of year 2 FV = 3,000 x (1 + 10%)2
End of year 3 FV = 3,000 x (1 + 10%)3```

This can be continued indefinitely so for example at the end of year 20

```FV = 3,000 x (1 + 10%)20
FV = 20,182.50```

Furthermore if we replace our original lump sum of 3,000 with the more general term PV, the discount rate of 10% with i, and the number of periods 20 with n, we arrive at the future value of a lump sum formula as follows:

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

Variables used in the future value of a lump sum formula
PV = Present value, the value today at the start of period 1
FV = Future value, the value at the end of period n
i = Discount rate, the rate per period
n = Number of periods

## Future Value of a Lump Sum Example

To illustrate the use of the formula suppose a lump sum of 4,000 is invested for 19 periods. Additionally assume and the interest rate per period is 6%. In this case at the end of the 19 periods, the value of the lump sum is given by the formula as:

```FV = PV x (1 + i)n
FV = 4,000 x (1 + 6%)19
FV = 12,102.40```

To summarize, the future value concept forms the basis of many time value of money calculations. The formula allows any lump sum (PV) to be compounded forward a number of periods (n) at a discount rate of i per period.