…what is it worth in the future?

The concept of the future value of a lump sum is the starting point for all time value of money calculations.

If a lump sum is invested and earns interest, then over time, the lump sum will grow into a larger sum.

For example, if 3,000 is invested at 10% for a year, then at the end of the year, the interest earned will be 3,000 x 10% = 300, and the lump sum will have grown to 3,000 plus 300, equals 3,300.

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.

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

Each cash flow stream can be represented in a cash flow diagram. 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.

Period | 1 | 2 | 3 |
---|---|---|---|

Cash flow | ↓ -3,000 | 3,993 ↑ |

Each year interest is being earned on the original sum and on the interest from the previous years, this type of interest is referred to as **compound interest**. This is shown in the table below.

n | Period | 1 | 2 | 3 |
---|---|---|---|---|

PV | Opening balance | 3,000 | 3,300 | 3,630 |

i | Interest 10% | 300 | 330 | 363 |

FV | Closing balance | 3,300 | 3,630 | 3,993 |

## Future Value of a Lump Sum Formula

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 of a lump sum 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 of a lump sum 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

*The amount of 3,300 has been replaced with 3,000 x (1 + 10%) ^{1} from the answer for year 1.*

### Future value of a lump sum 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

*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

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

As another example, suppose a lump sum of 4,000 is invested for 19 periods and the interest rate per period is 6%, then at the end of the 19 periods, the value of the lump sum is given by the future value of a lump sum formula as:

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

The future value of a lump sum 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.