Question

Real estate prices are so high in Tokyo that some mortgages are written for 99 years in an attempt to keep down the monthly payments. A) What is the monthly payment on a $500,000 mortgage at 8% for 99 years? B) How much does this save the borrower each month compared to a 30-year mortgage at 8% for the same amount? C) Which term results in the higher total payment?

Answer 1

(A)

Loan Amount P = $500000

Interest Rate = 8% or 0.08/12 monthly

Number of payment periods = n = 99*12 = 1188 months

Let monthly payments made be X

Hence, the sum of present value of monthly payments must be equal to the value of the loan amount

=> X/(1+r) + X/(1+r)² +....+ X/(1+r)^N = P

=> X[1- (1+r)^-N]/r = P

=> X = rP(1+r)^N/[(1+r)^N-1]

Hence, Monthly Payments =  rP(1+r)^N/[(1+r)^N-1]

= 500000*( 0.08/12)*(1+ 0.08/12)¹¹⁸⁸/((1+ 0.08/12)¹¹⁸⁸-1) = $3334.58

Total Payment = Monthly Payments * Number of months = 3334.58*1188 = $3961481.04

(B)

Loan Amount P = $500000

Interest Rate = 8% or 0.08/12 monthly

Number of payment periods = n = 30*12 = 360 months

Let monthly payments made be X

Hence, the sum of present value of monthly payments must be equal to the value of the loan amount

=> X/(1+r) + X/(1+r)² +....+ X/(1+r)^N = P

=> X[1- (1+r)^-N]/r = P

=> X = rP(1+r)^N/[(1+r)^N-1]

Hence, Monthly Payments =  rP(1+r)^N/[(1+r)^N-1]

= 500000*( 0.08/12)*(1+ 0.08/12)³⁶⁰/((1+ 0.08/12)³⁶⁰-1) = $3668.82

Total Payment = Monthly Payments * Number of months = 3668.82*360 = $1320775.2

(C)

Total payment is higher for 99 year year mortgage as calculated above