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A fixed-term, fixed-interest-rate loan has four important parameters: the principal \(P\) (the amount borrowed), the interest rate \(r\), the repayment interval \(\tau\), and the number of payments \(n\). The loan is repaid in \(n\) equal payments over the loan term \(n \tau\). Each payment consists of a principal and an interest portion. The interest portion is the interest accumulated on the principal outstanding during the term; the principal portion reduces the outstanding principal. The dimensionless quantity determining the type of loan is \(\beta \equiv n \tau r . .\) a. Estimate the payment (the amount per term) in the easy case \(\beta=0\), in terms of \(P, n\), and \(\tau\). (The term \(n \tau\) and the repayment interval \(\tau\) don't vary that much \(-\tau\) is usually 1 month and \(n \tau\) is somewhere between 3 to 30 years - so \(\beta \ll 1\) is usually reached by lowering the interest rate \(r\).) b. Estimate the payment in the slightly harder case where \(\beta \ll 1\) (which includes the \(\beta=0\) case). In this regime, the loan is called an installment loan. c. Estimate the payment in the easy case \(\beta \gg 1 .\) In this regime, the loan is called an annuity. (This regime is usually reached by increasing \(n\).)

Step by step solution

01

Case a: \(\beta = 0\)

To find the payment when \(\beta = 0\), we can simply divide the total principal by the number of payments. Payment \( = \frac{P}{n} \)

02

Case b: \(\beta \ll 1\)

When \(\beta \ll 1\), the loan is called an installment loan. To estimate the payment, we can use the formula: Payment \( = P \cdot \frac{r(1+r)^n}{(1+r)^n-1}\) Since \(\beta = n\tau r\) and \(r\) is small in this case, the exponent \((1+r)^n \approx 1 + nr\) can be considered. So the payment becomes: Payment \( = P \cdot \frac{r(1+nr)}{(1+nr)-1} = P \cdot \frac{r(1+nr)}{nr} \) Here, the term \(1+nr\) considers the interest rate accumulation over the loan term.

03

Case c: \(\beta \gg 1\)

When \(\beta \gg 1\), the loan is called an annuity. In this case, we can use the formula: Payment \( = P \cdot \frac{r}{1-(1+r)^{-n}}\) Since \(n\) is large, the term \((1+r)^{-n}\) becomes close to zero. So the payment becomes: Payment \( = P \cdot \frac{r}{1-0} = P \cdot r\) In this case, the payment is just the principal times the interest rate, as the interest accumulated is considered to be negligible compared to the principal amount for each payment.

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Key Concepts

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

Fixed-Term Loan

A fixed-term loan is a type of loan where the repayment schedule is set for a specific period. It typically comes with a fixed interest rate, meaning that the rate does not change throughout the duration of the loan. This setup is particularly beneficial for budgeting because you know exactly how much you'll pay each week, month, or year until the end of the loan.
Here are the essential aspects of a fixed-term loan:

• Principal (P): The initial amount borrowed from the lender.
• Interest Rate (r): The percentage charged on the borrowed principal amount.
• Repayment Interval (Ï„): The frequency of the payments, which is usually monthly.
• Number of Payments (n): Total installments required to repay the loan.

Understanding these components is crucial for proper financial planning and selecting the right loan for your needs.

Interest Rate Calculation

Calculating the interest rate for a loan involves understanding how much extra you'll pay on top of the borrowed amount. The interest is usually expressed as a percentage of the principal.
For a fixed-term loan, interest rate calculations determine how much interest you'll pay over the life of the loan and, consequently, how much you will owe overall.
Interest payments are calculated using:

• The principal amount
• The interest rate
• The length of time the loan is outstanding

The formula for the interest portion of a payment on a loan with periodic interest compounding is:\[Interest = Principal \times Interest\ Rate \times Time\]This calculation helps to determine how much money will be added to the original loan amount due to interest, allowing for clear financial forecasting.

Installment Loan

An installment loan involves borrowing a set amount of money and paying it back over a pre-determined number of payments. With each installment, you pay both a portion of the principal and the interest accrued.
This type of loan is particularly popular because:

• It offers predictable monthly payments, making it easier to budget.
• The terms often suit major purchases like homes and vehicles.
• You gradually reduce the principal with each payment.

When dealing with installment loans, the payments need to be calculated carefully, especially with a small interest rate and the number of payments in mind. The general formula is:\[Payment = P \cdot \frac{r(1+r)^n}{(1+r)^n-1}\]Learning how this formula works helps in understanding how much of each payment goes toward interest versus reducing the loan's principal.

Annuity Loan

An annuity loan is where repayments are structured typically for long terms, often leading to large numbers of payments (large value of \(n\)). It involves making fixed payments at regular intervals, with the loan fully paid off by the end of its term.
Here’s why annuity loans are noteworthy:

• The payment gets divided into principal and interest.
• The interest payment decreases over time as more of the principal gets paid down.
• It is beneficial for long-term financial strategies.

The payment formula for an annuity loan, especially when \(\beta \gg 1\), simplifies when the interest is assumed to be mainly serviced. The formula used is:\[Payment = P \cdot \frac{r}{1-(1+r)^{-n}}\]This structure lets you systematically reduce the owed amount over time with clear visibility on the financial commitment.

Principal Repayment

Principal repayment refers to the portion of the loan payment that goes toward reducing the original loan amount. In every installment, apart from paying the interest, part of the repayment decreases the outstanding debt.
Understanding principal repayment is essential because:

• It determines how much of the loan is still owed after each payment.
• Lenders generally require the full principal to be returned over the loan term.
• Effective financial planning can shorten the overall loan duration and reduce total interest paid.

For most loans, the principal is returned through either equal installments or increasing installments, where the principal payment grows larger over time, completing the repayment structure. Knowing how much goes towards principal repayment helps borrowers manage debt more efficiently and plan financially sound futures.