What is the present value of an annuity if payments of $434.00 are made at the end of every month for 15 years at an interest rate of 7.5% per year computed monthly?

Solution:

1. Given:

- Monthly payment: USD 434.00

- Number of years: 15

- Annual interest rate: 7.5%

- Number of compounding periods per year: 12

2. Convert the annual interest rate to a monthly interest rate:

- Monthly interest rate r = \frac{7.5\%}{12} = \frac{0.075}{12}

- Monthly interest rate r = 0.00625

3. Calculate the total number of payments:

- Total number of payments n = 15 \, \text{years} \times 12 \, \text{payments/year}

- Total number of payments n = 180

4. Use the present value of an annuity formula:

- Present Value (PV) formula: PV = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)

- Where:

* C = 434 (monthly payment)

* r = 0.00625 (monthly interest rate)

* n = 180 (total number of payments)

5. Substitute the values into the formula:

PV = 434 \times \left( \frac{1 - (1 + 0.00625)^{-180}}{0.00625} \right)

6. Calculate:

PV\approx46817.07

The present value of the annuity is approximately USD 46817.07.