Problem 22 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Suppose you borrow \(\$ 18,000\) at an interest rate of \(8 \%\) compounded annually. You begin paying back money four years from today and make xed payments annually. You pay back the entire debt after six payments. What are your annual payments? Begin by guring out the ballpark gures. Will you pay more than \(\$ 3000\) each year? What is an upper bound for the amount of money you will pay each year?

Step-by-Step Solution

Answer

After calculating the above steps, you will gain the annual payment amount and the upper bound estimate that you are supposed to pay. Compare this with $3000 to understand the comparison.

Step 1: Calculate loan amount including interest

First, as the payments don't start until 4 years into the loan, calculate the new loan amount due to compounding of interest over the initial 4 years. The future value \( FV \) of the initial loan amount \( PV \) can be calculated using the formula: \( FV = PV * (1 + r)^n \), where:\n PV = principle amount = $18000\n r = interest rate = 8% or 0.08,\n n = number of years = 4\n Solution: \( FV = 18000 * (1 + 0.08)^4 \)

Step 2: Determine annual payment

After calculating the compounded loan amount, the annual payments can be calculated using the annuity formula since the debt is paid back in equal annual installments. The formula is: \( P = \frac{FV * r}{(1 - (1 + r)^{-n})} \) where\n r = interest rate = 0.08 and\n n = term of payments = 6. Plug in the future value calculated in step 1 for \( FV \) in this formula.

Step 3: Find the upper bound of payment

To find the upper bound of the annual payment, imagine a scenario where the total repayment was spread evenly with no interest over the six years. This will give an upper limit. Divide the initial loan amount by 6. \( Upperbound = \frac{PV}{6} \) where \( PV = 18000 \) and 6 is the number of repayments.

Step 4: Consider $3000 payment

To check if the annual payment will be more than $3000, compare the calculated annual payment value against $3000. If the value exceeds $3000, then the individual has to pay more than $3000 each year.

Key Concepts

Future Value Calculation

One important concept in finance is the **future value calculation**. This involves determining the value of a present sum of money at a future date, considering a specific interest rate over a set period of time.

To calculate this, the formula used is: \[ FV = PV \times (1 + r)^n \] where:

• \( FV \) stands for future value, which we want to find out.
• \( PV \) represents the present value or the initial amount. In our exercise, it's $18,000.
• \( r \) is the interest rate per period. Here, it's 8% or 0.08 annually.
• \( n \) is the number of periods (years). It's 4 years in this example since payments start four years after borrowing.

This formula helps in knowing how much your loan or investment will grow over time when interest is compounded once every period, which in this case is annually.

Annuity Formula

When it comes time to repay a loan with regular payments, the **annuity formula** becomes crucial. An annuity involves a series of equal payments made at regular intervals.We use the following annuity formula to determine how each payment should be structured: \[ P = \frac{FV \times r}{(1 - (1 + r)^{-n})} \]where:

• \( P \) stands for the payment amount.
• \( FV \) is the future value, which is the compounded value of the loan at the start of repayments. We calculate this using the above future value formula.
• \( r \) is the interest rate per period.
• \( n \) is the total number of payments (in our example, six payments over six years).

This formula allows borrowers to understand what their regular payment amount will be to completely pay off the loan, inclusive of interest, over the specified term.

Loan Repayment

Understanding **loan repayment** is essential for managing debt effectively. Loan repayment involves paying back borrowed money over a specified time, following a contractual agreement. In our situation, repayments begin after a period of delay (in this case, four years) and then occur annually. Here are a few key strategies to consider:

• Consider the total repayable amount, which is impacted by the interest rate and the time for which the loan accrues interest before repayments commence.
• Make calculations to determine if your repayment budget comfortably covers the calculated payment amount. This ensures future financial health and prevents default.
• Estimating upper bounds of payment and considering common payment benchmarks such as $3000 helps in setting realistic financial expectations.

By thoughtfully structuring your repayment plan, you can minimize interest payments and manage your loan more effectively.