### Problem

An investment will pay 6,000 the first year, 10,000 the second year, 18,000 the third year, (all payments are at the end of each year). What it the value of the investment today if the annual discount rate is 6.00%?

### Solution

The value of the investment today is the present value of each of the payments added together. As the payments are not the same in amount an annuity formula cannot be used.

This problem is solved using the present value of a lump sum formula on each of the amounts as follows.

###### Payment 1

Payment 1 is for 6,000 (FV), made at the end of year 1 (n), at a discount rate of 6% (i)

FV = 6,000 n = 1 i = 6% PV = FV /(1 + i)^{n}PV = 6,000/(1 + 6%)^{1}PV = 5,660.38

###### Payment 2

Payment 2 is for 10,000 (FV), made at the end of year 2 (n), at a discount rate of 6% (i)

FV = 10,000 n = 2 i = 6% PV = FV /(1 + i)^{n}PV = 10,000/(1 + 6%)^{2}PV = 8,899.96

###### Payment 3

Payment 3 is for 18,000 (FV), made at the end of year 3 (n), at a discount rate of 6% (i)

FV = 18,000 n = 3 i = 6% PV = FV /(1 + i)^{n}PV = 18,000/(1 + 6%)^{3}PV = 15,113.15

The total present value of all three payments and the value of the investment is

PV = 5,660.38 + 8,899.96 + 15,113.15 PV = 29,673.49

### Explanation

At a discount rate of 6%, receiving payments of 6,000 in one year, 10,000 in two years, and 18,000 in three years is equivalent to receiving 29,673.49 today, which is the value of the investment.

### PV4 Present Value Example

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