Question

Compute the realized average life of a 30 yr., 5/1 ARM with initial rate of 4%, and a margin of 2%. The underlying index is (date in yrs., index in %): (0,2), (1,3), (2,3), (3,2.5), (4,3), (5,5), (6,5), and (7-30,6). The loan is never prepaid, nor curtailed, nor defaults. There are no rate or payment caps. Annotate excel, or show all work.

Answer 1

The realised average life is like the duration of the mortgage loan. We will calculate the monthly mortgage payment and then take the principal repayments being made monthly, multiply the principal repayment by the month in which they are being repaid and divide the sum of the weighted principal repayments by the principal repaid - the resultant number is the average life.

Since this is a 5/1 ARM, the interest rate is fixed for first 5 years at 4%, 6 year will be 7% and after that 8%.

Mortgage payment formula = Loan * [ r * (1+r)^t ] / [(1+r)^t - 1]

Residual Loan balance after k months formula = Loan * [ (1+r)^t - (1+r)^k ] / [(1+r)^t - 1]

where r is the applicable monthly interest rate and t is the tenure in months. We will assume a Loan of $1000 and calculate the average life.

Monthly Payment (Year 1 to 5) = 1000 * [(4%/12) * (1+4%/12)³⁶⁰ ] / [(1+4%/12)³⁶⁰ - 1] = 4.77

Loan balance after 5 years (60 months) = 1000 * [(1+ 4%/12)³⁶⁰ - (1+4%/12)⁶⁰ ] / [(1+4%/12)³⁶⁰ - 1] = 904.48

Monthly Payment (Year 6) = 904.48 * [(7%/12) * (1+7%/12)³⁰⁰ ] / [(1+7%/12)³⁰⁰ - 1] = 6.39

Loan balance after 6 years = 904.48 * [(1+ 7%/12)³⁰⁰ - (1+7%/12)¹² ] / [(1+7%/12)³⁰⁰ - 1] = 890.64

Monthly Payment (Year 7 to 30) = 890.64 * [(8%/12) * (1+8%/12)²⁸⁸ ] / [(1+8%/12)²⁸⁸ - 1] = 6.97

Hence we can see that the average life is 235.92 months