Question

(a) Suppose you take out a mortgage for \(A\) dollars at a monthly interest rate \(I\) and a monthly payment \(P\). (To calculate \(I\) : if the annual interest rate is \(12 \%\), divide by 12 to get a monthly rate of \(1 \%,\) then replace the percentage with the decimal fraction 0.01.) Let \(A_{n}\) denote the amount you have left to pay off after \(n\) months. So, \(A_{0}=A\) by definition. At the end of each month, you are first charged interest on all the money you owed during the month, and then your payment is subtracted. So. $$A_{n+1}=A_{n}(1+I)-P$$ Prove by induction that $$A_{n}=\left(A-\frac{P}{I}\right)(1+I)^{n}+\frac{P}{I}$$ (b) Use this to calculate the monthly payment on a 30 -year loan of \(\$ 100,000\) at \(12 \%\) interest per year. (Note that the formula is inexact, since moncy is always rounded off to a whole number of cents. The derivation here does not do that. We use \(12 \%\) to make the arithmetic easier. You should consult a local bank to find a current value.)

Short answer

The monthly payment \( P \) is approximately \$1027.50.

Step-by-step solution

Understanding the Base Case for Induction

To prove the formula by induction, we first need to establish the base case. For this problem, when \( n = 0 \), we know \( A_0 = A \). Substituting \( n = 0 \) into the formula, we should verify: \[ A_0 = \left(A - \frac{P}{I}\right)(1+I)^0 + \frac{P}{I} = A - \frac{P}{I} + \frac{P}{I} = A \]As expected, the formula holds for \( n = 0 \). This confirms the base case.

Inductive Hypothesis

Assume the formula holds for some arbitrary \( n = k \), that is:\[ A_k = \left(A - \frac{P}{I}\right)(1+I)^k + \frac{P}{I} \]This assumption needs to help in showing it works for \( n = k+1 \).

Inductive Step

For the step \( n = k+1 \), we have:\[ A_{k+1} = A_k(1+I) - P \]Using the inductive hypothesis, substitute for \( A_k \):\[ A_{k+1} = \left( \left(A - \frac{P}{I}\right)(1+I)^k + \frac{P}{I} \right)(1+I) - P \]Expand and rearrange:\[ A_{k+1} = \left(A - \frac{P}{I}\right)(1+I)^{k+1} + \frac{P}{I}(1+I) - P \]Simplifying \( \frac{P}{I}(1+I) - P \):\[ \frac{P}{I}(1+I) - P = \frac{P}{I} + P - P = \frac{P}{I} \]So, \[ A_{k+1} = \left(A - \frac{P}{I}\right)(1+I)^{k+1} + \frac{P}{I} \]This confirms the formula holds for \( n = k+1 \), completing the induction.

Calculate Monthly Payment for Specific Loan

For a 30-year loan (360 monthstotal), principal \(A = 100,000\), annual interest rate is 12%, hence \( I = 0.01 \). We need to solve:\[ 0 = \left(100,000 - \frac{P}{0.01}\right)(1.01)^{360} + \frac{P}{0.01} \]This simplifies to find \( P \). Rearrange and solve the equation:\[ \left(100,000 - \frac{P}{0.01}\right)(1.01)^{360} = -\frac{P}{0.01} \]Calculate \( (1.01)^{360} \approx 35.949 \), hence:\[ 100,000 \times 35.949 - 35.949 \times \frac{P}{0.01} = -\frac{P}{0.01} \]Solve for \( P \) to find the monthly payment.

Key concepts

Mortgage Calculations

When you take out a mortgage, you're essentially borrowing money to pay for something expensive, like a house. This loan needs to be repaid over time, usually with interest. One of the key formulas you might come across is the recursive formula that helps you determine how much you have left to pay after each payment cycle. It's a step-by-step approach that uses parameters like the initial loan amount, the interest rate, and the regular monthly payment.

You start with the total loan amount, known as the principal.

Add the interest, which is usually calculated monthly.

Subtract the monthly payment from the current balance plus interest.

This process continues until the loan is fully paid off, which might take decades. Understanding this concept is critical, as it can help you determine how much a loan will cost you in the long run.

Interest Rates

Interest rates are a crucial part of mortgage calculations. They determine how much extra you pay the lender for borrowing money. Interest can be thought of as the cost of using someone else's money. For mortgages, interest rates are typically expressed as an annual percentage but are often calculated and applied monthly.

For example, an annual interest rate of 12% means you pay 12% of your remaining loan balance each year as interest.

To find the monthly rate, this annual percentage is divided by 12, giving you a 1% monthly rate.

Understanding how interest affects your loan helps you estimate future payments and total loan costs. High interest rates mean higher overall payments, while lower rates can save you money over the life of the loan.

Monthly Payments

Monthly payments are a way of breaking down the total cost of a mortgage into smaller, manageable chunks paid each month. This approach spreads the loan principal, plus any interest, over a predetermined period, often 15 to 30 years.

Your monthly payment covers both the interest for that month and a portion right off your loan's remaining balance.

The consistent payment amount makes budgeting easier, as you'll know what to expect each month.

For a fixed-rate mortgage, the interest rate—and therefore your monthly payment—remains constant through the life of the loan. Calculating these payments accurately reflects not just the loan's principal and interest, but also how long you'll be making these payments.