# What is the present worth of a series of equal end of month payments of $3,500 if the series...

## Question:

## Effective Annually Rate:

Interest rate on loans are often quoted in terms of annual percentage rate, which is the effective periodic rate times the number of periods in a year. This way of quoting interest rates often understates the true annual cost of borrowing because the annual percentage rate does not account for the compounding of interest.

## Answer and Explanation: 1

We can use the following formula to compute the present value of the series of monthly payments:

- {eq}\frac{M*(1 - (1 + r)^{-T})}{r}
{/eq}, where
*M*is the monthly payment,*r*is the effective monthly interest rate, and*T*is the number of monthly payments.

In this question, we know monthly payment is 3,500, and there 12*8 = 96 payments in 8 years. To find the value of the payments, we need to determine the effective monthly interest rate.

a) Given quarterly compounding, the effective monthly interest rate is {eq}e^{ln(1 + 7.5\%/4)/3} - 1 = 0.62\%{/eq}. Applying the formula, the present value of the payments is:

- {eq}\frac{3500*(1 - (1 + 0.62\%)^{-96})}{0.62\%} = 252,515.27 {/eq}

b) Given continuously compounding, the effective monthly interest rate is {eq}e^{7.5\%/12} - 1 = 0.63\%{/eq}. Applying the formula, the present value of the payments is:

- {eq}\frac{3500*(1 - (1 + 0.63\%)^{-96})}{0.63\%} = 251,876.73 {/eq}

b) Given semi-annually compounding, the effective monthly interest rate is {eq}e^{ln(1 + 7.5\%/2) / 6} - 1 = 0.615\%{/eq}. Applying the formula, the present value of the payments is:

- {eq}\frac{3500*(1 - (1 + 0.615\%)^{-96})}{0.615\%} =253,140.48 {/eq}

#### Learn more about this topic:

from

Chapter 8 / Lesson 3Learn how to find present value of annuity using the formula and see its derivation. Study its examples and see a difference between Ordinary Annuity and Annuity Due.