You buy a house for 200000 and take out a 30year mortgage at

You buy a house for $200000, and take out a 30-year mortgage at 7% interest. For simplicity, assume that interest and payments are done continuously. A) What will be your annual mortgage payment? B) Suppose that regular raises at your job allow you to increase your annual payment by 5% each year. How long will it take to pay off the mortgage?

Solution

assume fixed payments for rest 30 years. calculate the present value of all the payments and equate them to 200000. let fixed payments be c.

it follows that 200000= c/(1+r)+ c/(1+r)² + c/(1+r)³+ .....c/(1+r)³⁰.

for continuous compounding, 200000= c*e^-r+ c*e^-2r + c*e^-3r+ .....c*e^-30r

put r=0.07, apply summation of GP

we get c= 16652 $

for next part, 200000= ce^g-r+ ce^2(g-r)+ ce^3(g-r) + ......+ ce^n(g-r)

where n is unknown, g= .05 , c= 16652

we get n= 13.89 or 14 years.