91Ó°ÊÓ

[In all the installment loans, assume that the term of the loan and the APR remain the same.] (a) Explain why, in the amortization formula, the monthly payment \(M\) is proportional to the principal \(P\). In other words, explain why if the monthly payment on a loan with principal \(P\) is \(M,\) then the monthly payment on a loan with principal \(c P\) (where \(c\) is any positive constant) is \(c M\). (Hint: What happens in the amortization formula when you replace \(P\) by \(c P ?)\) (b) Explain why if the monthly payment on a loan with principal \(P\) is \(M,\) and the monthly payment on a second loan with principal \(Q\) is \(N,\) then the monthly payment on a loan with principal \((P+Q)\) is \((M+N)\).

Step by step solution

01

Understanding the Amortization Formula

The first part of the exercise is dealing with the formula for calculating monthly payments M which is M = P (r(1 + r)^n) / ((1 + r)^n - 1), where P is the principal, r is the APR (interest rate) divided by 12, and n is the number of installments (month). The second part is based on this formula as well.

02

Proportional Relationship between P and M

Let's increase the principal by a factor of constant c, making the new principal cP. Substitute cP in the place of P in the formula, the new monthly payment M' is M' = cP (r(1+ r)^n) / ((1 + r)^n - 1). Divide the new formula by c, we get c M = cP (r(1 + r)^n) / ((1 + r)^n - 1) where we see that monthly payments are proportional to the principal and increase by constant c when the principal does.

03

Sum of Principles

Given two loans with different principals P and Q, and their respective payments are M and N, if we sum the two loans to be a new loan with the principal (P + Q), and substitute this value in monthly payment formula we get (P+Q) (r (1 + r)^n) / ((1 + r)^n - 1). Add M and N together and simplifying it, we get the same expression. Therefore, we can conclude that the monthly payment on a loan with principal (P + Q) is (M + N)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Installment Loans

Installment loans are loans that require the borrower to pay back the money in regular intervals, usually monthly. They are quite common and are used for different purposes like buying a car or a house. The main feature of these loans is that the borrower repays the loan amount along with interest over a set number of payments.

• Each payment is known as an installment and includes both a portion of the principal amount and the interest.
• The term of the loan is predetermined, meaning you know exactly how long it will take to repay the loan.
• Common types of installment loans include mortgages, car loans, and student loans.

Understanding installment loans is essential because it allows you to predict what your monthly financial commitments will be and helps in financial planning.

Principal Proportionality

Principal proportionality refers to the relationship between the principal loan amount and the monthly payments. In the context of installment loans, this relationship is linear, meaning that as the principal amount increases by a certain factor, the monthly payments increase by the same factor. For instance, if the principal of a loan is doubled, the monthly payment will also double. This concept is essential when adjusting loan amounts because:

• It ensures predictability; increasing the loan will proportionally increase the monthly payment.
• Simplifies calculations when dealing with multiple changes in loan terms.

When using the amortization formula, replacing the principal with any constant multiplied by the principal demonstrates this concept, showing clearly why the payments scale proportionately.

Loan Payment Calculation

Loan payment calculation involves determining the amount you need to pay monthly to repay a borrowed amount over a period. The amortization formula is vital here, as it helps to calculate these consistent monthly payments.The formula is:\[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \]where:

• \( M \) is the monthly payment.
• \( P \) is the principal amount.
• \( r \) is the monthly interest rate (annual rate divided by 12).
• \( n \) is the total number of payments.

By plugging in different values for the interest rate or total payments, you can see how these parameters affect the monthly payment amount. Understanding how this calculation works is crucial for managing and planning your finances effectively.

Mathematics of Finance

The mathematics of finance deals with the mathematical models and theories used to evaluate financial scenarios. It involves compound interest, annuities, and different loan calculations. Within this realm, the amortization formula is a practical tool that applies to financial planning and forecasting. It shows how payments are spread over the life of a loan and how each month's payment splits between interest and principal. Key points in this area include:

• The importance of understanding interest rates and how they affect your payments and total wealth over time.
• Application of exponential functions to determine payment schedules and calculate loan affordability.
• The interplay of different financial components like principal, interest rate, and loan term.

A solid grasp on the mathematics of finance can help individuals make smarter choices about loans and investments, ensuring better handling of personal finances.