Math Problem Statement

Jeremy takes out a 30-year mortgage of 250000 dollars at an annual interest rate of 6.5 percent compounded monthly, with the first payment due in one month. How much does he owe on the loan immediately after the 87th payment?

Solution

Jeremy's mortgage is a standard amortizing loan, where he makes equal payments each month, and each payment is used to cover both interest and principal. We need to calculate how much he still owes after making the 87th payment.

Step 1: Loan Parameters

  • Loan Amount (Principal), P=250,000P = 250,000 dollars.
  • Annual interest rate, r=6.5%r = 6.5\% (or 0.0650.065 as a decimal).
  • Monthly interest rate, i=0.06512=0.00541667i = \frac{0.065}{12} = 0.00541667.
  • Loan term, N=30N = 30 years =30×12=360= 30 \times 12 = 360 months.

Step 2: Monthly Payment Calculation

We first calculate the fixed monthly payment using the mortgage payment formula:

M=Pi(1+i)N(1+i)N1M = P \cdot \frac{i(1 + i)^N}{(1 + i)^N - 1}

Substituting the known values:

M=250,0000.00541667(1+0.00541667)360(1+0.00541667)3601M = 250,000 \cdot \frac{0.00541667(1 + 0.00541667)^{360}}{(1 + 0.00541667)^{360} - 1}

Let’s calculate MM.

Step 3: Remaining Loan Balance After 87 Payments

The remaining balance after tt payments is calculated using the formula:

Bt=P(1+i)N(1+i)t(1+i)N1B_t = P \cdot \frac{(1 + i)^N - (1 + i)^t}{(1 + i)^N - 1}

Where:

  • t=87t = 87 is the number of payments made.
  • N=360N = 360 is the total number of payments.

We'll calculate both the monthly payment and the remaining balance using these formulas.

Let me compute these values.### Results:

  • Monthly payment: $1,580.17
  • Remaining balance after 87 payments: $224,967.47

Thus, Jeremy will still owe $224,967.47 after making the 87th payment.

Would you like a deeper breakdown or have further questions?

Here are five related questions:

  1. How is the monthly interest rate derived from the annual interest rate?
  2. What happens to the loan balance if the interest rate decreases by 1%?
  3. How can extra payments reduce the remaining loan balance?
  4. How is the amortization schedule related to the principal and interest portions of each payment?
  5. What would be the effect on the loan term if the payments are increased by 10%?

Tip: Making extra payments directly towards the principal can significantly reduce both the loan term and the total interest paid.

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Math Problem Analysis

Mathematical Concepts

Amortization
Compound Interest
Loan Payment Calculation

Formulas

M = P * [i(1 + i)^N] / [(1 + i)^N - 1]
B_t = P * [(1 + i)^N - (1 + i)^t] / [(1 + i)^N - 1]

Theorems

Amortization of loans

Suitable Grade Level

Undergraduate Level or Advanced High School (Grades 11-12)