### Problem

Calculate the present value of an annuity due of 300.00 per quarter for 3 years, if interest is compounded monthly at the nominal rate of 9%.

### Solution

In this problem, the payment (effective) period (quarterly) is not the same as the compounding period (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 due formula

#### Calculate the Effective Interest Rate

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

Effective interest rate = (1 + r / m )^{n}- 1 r = annual nominal rate = 9% m = compounding periods in a year = 12 n = number of compounding periods the rate is required for = 3 Effective interest rate = (1 + 9% / 12 )^{3}- 1 Effective interest rate = 2.267% per quarter

#### Calculate the Present Value of the Annuity Due

The present value is calculated using the present value of an annuity due formula. The annuity due is 300.00 per quarter for 3 years (12 quarters) at an effective interest rate of 2.267% per quarter.

PV = Pmt x (1 + i) x (1 - 1 / (1 + i)n) / i Pmt = 300.00 per quarter n = 4 x 3 = 12 quarters i = 2.267% per quarter PV = 300.00 x (1 + 2.267%) x (1 - 1 / (1 + 2.267%)^{12}) / 2.267% PV = 3,191.95

### Explanation

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

### ER1 Effective Interest Rate Examples

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