### Problem

Deposits are made into an investment account which pays an annual interest rate of 4%. Assuming each deposit has the same value, and is made at the start of each year, what will the each deposit need to be to accumulate 150,000 after 9 years?

### Solution

The deposits are made at the start of each year for 9 years (n), each deposit is compounded forward at a discount rate of 4% (i), and results in a future value 150,000 (FV).

As the deposits (Pmt) are made at the start of each year the annuity is an annuity due and, as we know the future value, the problem is solved using the future value of an annuity due formula as follows:

FV = 150,000 n = 9 i = 4% FV = Pmt x (1 + i) x ( (1 + i)^{n}- 1 ) / i

Rearranging to solve for Pmt

Pmt = FV / ((1 + i) x ( (1 + i)^{n}- 1 ) / i) Pmt = 150,000 / ((1 + 4%) x ( (1 + 4%)^{9}- 1 ) / 4%) Pmt = 13,628.80

### Explanation

The deposits of 13,628.80 made at the start of each year for the next 9 years are each compounded forward at the rate of 4% to give a future value of 150,000.

### AN6 Annuity Examples

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