By Admin on Aug 28, 2009 in Mathematics
hlnov13 asked:
A family has a mortgage on their house with 25yr amortization period. The nominal interest rate is i-5.95%; interest is compounded semiannually. Their monthly payment is p-$3107.77, and their current balance is $237,562.52. Answer the following questions assuming that the interest rate stays the same over the duration of the mortgage.
A family has a mortgage on their house with 25yr amortization period. The nominal interest rate is i-5.95%; interest is compounded semiannually. Their monthly payment is p-$3107.77, and their current balance is $237,562.52. Answer the following questions assuming that the interest rate stays the same over the duration of the mortgage.
a)How many more months will it take until the mortgage is paid off?
b)What was the original amount of the mortgage loan?
Assuming that the interest rate stays constant over the duration of the mortgage:
a) The mortgage will be paid off in exactly eight years; and,
b) The original amount of the mortgage loan was $488,000.
PROOF:
Let “P” be the principal owing on a loan.
Let “M” be the monthly mortgage payment.
Let “T” be the remaining number of months.
Let “R” be the interest rate expressed on a MONTHLY compounded basis.
Let “S” be the interest rate expressed on a SEMIANNUAL compounded basis
The relationship between “P”, “M”, and “R” can be expressed as follows:
Formula 1: P/M = [(1 + R/12)^T - 1] / [R/12 (1 + R/12)^T]
The relationship between “R” and “S” can be expressed as follows:
Formula 2: R/12 = [(1 + S/2)^(1/6)-1]
Substituting the above into my Formula 1, the result becomes:
Formula 3: P/M = [(1 + S/2)^(T/6) - 1] / [((1 + S/2)^(1/6) - 1) x (1 + S/2)^(T/6)]
SOLVING FOR “PART A”
Rewriting my “Formula 3″ in natural logarithmic form, the value of “T” can be expressed as follows:
T = 6 x [ln(1 / (1 + S/2)^(1/6) - 1) - ln((1 / (1 + S/2)^(1/6) - 1) - P/M)] / ln (1 + S/2)
Inserting your known values for “M” ($3,107.77), “S” (5.95%), and “P” ($237,562.52), the value of “T” is found to be 96.0 months, or exactly eight years.
SOLVING FOR “PART B”
“Part B” of the question can be answered by rewriting my “Formula 3″ as follows:
P = M x [(1 + S/2)^(T/6) - 1] / [((1 + S/2)^(1/6) - 1) x (1 + S/2)^(T/6)]
Substituting your known values for “M” ($3,107.77), “S” (5.95%), and “T” (300 months), the value of P is found to be $488,000.
buchoman | Sep 1, 2009 | Reply