Final answer:
To answer this question, we need to use the formula for the present value of an annuity. We can use this formula to find the number of payments, the size of the last payment, and the difference between the total amount required to amortize the mortgage and the contractual monthly payments.
Step-by-step explanation:
To answer this question, we need to use the formula for the present value of an annuity. The formula is:
PV = PMT * ((1 - (1 + r/n)^(n*t)) / (r/n))
Where:
- PV is the present value or mortgage balance
- PMT is the monthly payment
- r is the annual interest rate (in decimal form)
- n is the number of times the interest is compounded per year
- t is the number of years
(a) To find the number of payments, we can rearrange the formula to solve for t. Plugging in the given values, we have:
t = log((PMT * (1 - (1+r/n)^(n*t))) / (PV * (r/n))) / (n * log(1 + r/n))
Using a financial calculator or Excel, we can find that t is approximately 29.83. Therefore, the mortgagor will have to make 30 payments.
(b) The size of the last payment can be found by subtracting the sum of the previous payments from the mortgage balance. In this case, the sum of the previous payments is $29,000 * 29 = $841,000. Therefore, the size of the last payment is $1,000,000 - $841,000 = $159,000.
(c) To determine the difference between the total amount required to amortize the mortgage with the contractual monthly payments rounded to the nearest cent, we can calculate the total amount paid using the given monthly payment and the number of payments found in part (a). The total amount paid is $29,000 * 30 = $870,000. The difference between the total amount paid and the mortgage balance is $1,000,000 - $870,000 = $130,000.