# Exercise-3: Computation of present value of an annuity

**Learning objective:**

This exercise illustrates the computation of present value of an annuity assuming different interest rates.

Sophia will need an amount of $2,000 to go on vacations with her husband at the end of each year for 10 years. For this purpose she wants to invest some money in a saving bank but does not know the exact amount of money to invest.

**Require:** If the interest is compounded annually, what amount does she need to invest now to receive an income of $2,000 at the end of each year for 10 years assuming:

- an interest rate of 15% compounded annually.
- an interest rate is 18% compounded annually.

## Solution:

Because Sophia needs equal amounts at the end of each year, it is an annuity. She needs to invest an amount that is equal to the present value of this annuity at given interest rate. Let’s calculate the present value of annuity for Sophia assuming the given interest rates of 15% and 18%.

### (1) If the interest rate is 15%:

= $2,000 × [((1 + 15%)^{10} – 1)/(15%(1 + i)^{10})]

= $2,000 × 5.019*

= $10,038

*Value of [(1 + 15%)^{10} – 1/15%(1 + i)^{10}] from present value of an annuity of $1 in arrears table.

At 15% interest rate (compounded annually), Sophia needs to invest $10,038 now to receive an annuity of $2,000 for 10 years.

### (2) If the interest rate is 18%:

= $2,000 × [(1 + 18%)^{10} – 1/18%(1 + i)^{10}]

= $2,000 × 4.494*

= $8,988

*Value of [(1 + 18%)^{10} – 1/18%(1 + i)^{10}] from present value of an annuity of $1 in arrears table.

At 18% interest rate (compounded annually), Sophia needs to invest $8,988 now to receive an annuity of $2,000 for 10 years.

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