Question

1. Nina and Rafael Ruiz are married clients have just been approved for a thirty-year, $150,000 mortgage, with an APR of 7 percent. However, they know that they want to make more than the regular payment in order to pay off the loan before thirty years. They have come to you to analyze two options. Option One requires a fixed monthly payment of $1,200. Option Two requires they start with the regular payment but then increase that amount by an effective monthly rate of 0.35 percent.

Provide Nina and Rafael answers to the following questions.

a.    Which payment method will result in a faster payoff?

b.   What is the difference in total interest payments between the two alternative payment methods? Hint: The total of payments for Option Two involves using a formula for the partial sum of a geometric series.

c.    Which repayment method results in higher home equity (the lower loan balance) after fourteen years?

Answer 1

First let's calculate the regular monthly payment. The same can be found out using the PMT function of excel. Inputs are:

Rate = APR of 7% = 7%/12 = 0.5833%; Nper = 12 x 30 = 360 months; PV = - 150,000; FV= 0

Hence, the regular monthly payment = PMT (Rate, Nper, PV, FV) = PMT (0.5833%, 360, -150000, 0) = 997.95

Part (a)

Option 1: PMT = 1,200; We need to find out how long it will take to pay off. The same can be calculated using the Nper function of excel. Number of months to pay off = Nper (Rate, PMT, PV, FV) = Nper (0.5833%, 1200, -150000, 0) = 224.58 months

Option 2: The loan is being repaid using a growing annuity.

PV of a growing annuithy is given by:

(1+9 Pſ - 1 T-9L - 1+r P= First Payment r=rate per period 9 = growth rate n = number of periods

Hence, we need to solve the value of n from the following equation:

150,000 = 997.95 / (0.5833% - 0.35%) x [1 - (1 + 0.35%)n / (1 + 0.5833%)n] = 427,694.46 x (1 - 0.9977n)

Hence, 0.9977n = 1 - 150,000 / 427,694.46 = 0.6493

Hence, n = ln(0.6493) / ln (0.9977) = 185.96

Hence, the payoff will take n = 185.96 months.

Since n of option 2 < Nper of option 1; payment method under option 2 will result in a faster payoff.

Part (b)

Interest paid under option 1 = Total payment - Principal = PMT x Nper - Principal = 1,200 x 224.58 - 150,000 = $ 119,497.62

Interest paid under option 2 = Total payment - Principal = P x [(1 + g)n - 1] / g - 150,000 = 997.95 x [(1 + 0.35%)185.96 - 1] / 0.35% - 150,000 = $ 110,893.83

Hence, the difference in total interest payments between the two alternative payment methods = $ 119,497.62 - $ 110,893.83 = $ 8,603.79

Part (c)

Loan balance under option 1 = PV of all the balance 224.58 - 12 x 14 = 56.58 payments = - PV (rate, nper, pmt, fv) = - PV (0.5833%, 56.58, 1200, 0) = 57,688.08

At the end of 14 years, 12 x 14 = 168 payments would have been made. Under option 2:

P = value of 169th payment = First payment x (1 + g)168 = 997.95 x (1 + 0.35%)168 = $ 1,794.86

And number of payments pending = n = 185.96 - 168 = 17.96

Hence, Loan balance under option 2 = PV of all the balance 192 payments =

(1+9 Pſ - 1 T-9L - 1+r P= First Payment r=rate per period 9 = growth rate n = number of periods

= 1,794.86 / (0.5833% - 0.35%) x [1 - (1 + 0.35%)17.96 / (1 + 0.5833%)17.96] = $ 31,423.16

Hence, repayment method under option 2 results in higher home equity (the lower loan balance) after fourteen years.