Question

Ada and Leon purchase a house with price X by mortgage. If they start to pay their first mortgage payment 5 years later, and if they continue to pay annual payment for 20 years, then each annual payment is $10000. Here, assume that that the interest rate of this mortgage is j2 = 6%.

(a) Calculate the price X of this house.

(b) If after the first 10 payments, the interest rate of this mortgage changes to j4 = 6.8 percent, and they still wish to cover the rest of the mortgage with 10 more annual payments, what is the amount of the new annual payment?

(c) Now, assume that Ada and Leon start paying their first annual payment one year after they purchase this house with payment $8000. After this, each annual payment would be $Y higher than pervious one, and there are total 20 annual payments. Determine Y.

(d) Now, assume that they start paying the first annual payment one year after they purchase this house with payment $9000. After this, each annual payment would be $Z higher than the previous one. After the 10th payment, the annual payment stops growing and the rest 10 more payments stay the same as the 10th one. Determine Z.

Answer 1

(a) Price of house = X

Interest rate = 6% pa compounded semi-annually (compounding frequency =2)

Annual interest rate = (1+6%/2)^2 - 1 = 6.09% pa compounded annually

Loan outstanding after 5-years = X*(1+6.09%)^5 = X*1.3439

Annual payments for 20 years = 10000

Present value of annual payments = 10000*(1/6.09%)*(1-1/(1+6.09%)^20) = $113,865.87

X*1.3439 = $113,865.87

X = $84727.93

(b) Value of initial loan after 15 years(10+5) = 84727.93*(1+6.09%)^15 = 205656.92

Future value of the 10 installments paid = 10000*(1/6.09%)*((1+6.09%)^10 - 1) = 132366.38

Loan outstanding after 10 installments = 205656.92 - 132366.38 = 73290.55

Interest rate = 6.8% compounded quarterly

Annual interest rate = (1+6.8%/4)^4 - 1 = 6.975%

Let P be the annual payments made

73290.55 = P*(1/6.975%)*(1-1/(1+6.975%)^10)

P = $10422.87 per annum