Question

question 22 a home buyer is considering a 20 - year, 8% mortgage loan of $72,500. the loan agent is offering an alternative. the buyer can pay $2,500 today to lower the loan interest rate to 7.5%. the mortgage loan requires annual payments. should the buyer pay $2,500 to lower the loan interest rate? (assume the buyer will not refinance the mortgage or pay it off earlier) a yes b no c indifferent (i.e., the two options are equivalent) d it depends. e none of the choices

Explanation:

Step1: Calculate the present - value of the savings from the lower interest rate.

We use the formula for the present - value of an ordinary annuity $PV = A\times\frac{1-(1 + r)^{-n}}{r}$, where $A$ is the annual payment, $r$ is the interest rate per period, and $n$ is the number of periods.
First, find the annual payment for the 8% loan. Let the loan amount $L=\$72500$. For an 8% interest rate $r_1 = 0.08$ and $n = 20$ years. Using the loan - payment formula $A_1=\frac{r_1L}{1-(1 + r_1)^{-n}}=\frac{0.08\times72500}{1-(1 + 0.08)^{-20}}\approx\frac{5800}{1 - 0.2145}\approx\frac{5800}{0.7855}\approx7384$.
For a 7.5% interest rate $r_2=0.075$ and $n = 20$ years, $A_2=\frac{r_2L}{1-(1 + r_2)^{-n}}=\frac{0.075\times72500}{1-(1 + 0.075)^{-20}}\approx\frac{5437.5}{1 - 0.2354}\approx\frac{5437.5}{0.7646}\approx7112$.
The annual savings in payment is $\Delta A=A_1 - A_2=7384 - 7112 = 272$.
The present - value of these annual savings over 20 years at a 7.5% discount rate (since we are comparing to the 7.5% loan) is $PV=\Delta A\times\frac{1-(1 + r_2)^{-n}}{r_2}=272\times\frac{1-(1 + 0.075)^{-20}}{0.075}=272\times\frac{1 - 0.2354}{0.075}=272\times\frac{0.7646}{0.075}\approx272\times10.195\approx2773$.

Step2: Compare the present - value of savings with the up - front cost.

The up - front cost is $\$2500$. Since the present - value of the savings ($\$2773$) is greater than the up - front cost ($\$2500$), it is beneficial to pay the $\$2500$ to lower the interest rate.

Answer:

A. Yes