### Problem

Calculate the present value of an annuity of 600.00 per quarter for 2 years, if interest is compounded every 6 months at the nominal rate of 9%.

### Solution

In this problem, the payment (effective) period (quarterly) is not the same as the compounding period (6 monthly).

This problem is solved in two steps.

- Calculate the effective interest rate for the payment period.
- Use the calculated effective rate in the present value of an annuity formula

#### Calculate the Effective Interest Rate

The effective interest rate for the payment period is calculated using the effective interest rate formula.

The compounding period is 6 monthly, so there are 2 compounding periods in a year (m). The payment period is quarterly, so there is 1/2 a compounding period in a payment period (n).

Effective interest rate = (1 + r / m )^{n}- 1 r = annual nominal rate = 9% m = compounding periods in a year = 2 n = number of compounding periods the rate is required for = 1/2 Effective interest rate = (1 + 9% / 2 )^{1/2}- 1 Effective interest rate = 2.225% per quarter

#### Calculate the Present Value of the Annuity

The present value is calculated using the present value of an annuity formula. The annuity is 600.00 per quarter for 2 years (8 quarters) at an effective interest rate of 2.225% per quarter.

PV = Pmt x (1 - 1 / (1 + i)^{n}) / i Pmt = 600.00 per quarter n = 4 x 2 = 8 quarters i = 2.225% per quarter PV = 600 x (1 - 1 / (1 + 2.225%)^{8}) / 2.225% PV = 4,352.97

### Explanation

The payments on this annuity are made quarterly (effective period) and do not coincide with the compounding periods which are six monthly.

To solve the problem it is necessary to first calculate the effective rate of interest for each payment period, and then use this effective rate in the present value of an annuity formula.

### ER2 Effective Interest Rate Examples

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