## 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.

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}