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Chapter 4: Problem 19

A sum of $$\$ 100,000$$ is to be repaid over a 10 -yr period through equal installments made at the end of each year. If an interest rate of \(10 \% /\) year is charged on the unpaid balance and interest calculations are made at the end of each year, determine the size of each installment so that the loan (principal plus interest charges) is amortized at the end of \(10 \mathrm{yr}\).

Short Answer

Expert verified
The size of each equal annual installment to fully repay the loan of $100,000 with a 10% annual interest rate in 10 years is approximately $16,274.54.

Step by step solution

01

Understand the problem

We are given a loan amount of $100,000 that has to be paid back over ten years with equal payments at the end of each year. The outstanding balance accrues interest at a rate of 10% per year, so with each payment, we have to cover the interest charged plus a portion of the principal.
02

Use the formula for equal annual payments

The formula to calculate the size of each annual payment is: \[A = P \frac{r(1 + r)^n}{(1 + r)^n - 1}\] Where: - \(A\) is the annual payment, - \(P\) is the loan amount, - \(r\) is the annual interest rate (decimal form), - \(n\) is the number of years. In our case, \(P = 100,000\), \(r = 0.10\) (since 10% = 0.10), and \(n = 10\).
03

Calculate the size of each annual payment

Now, we'll plug the values into the formula: \[A = 100,000 \frac{0.10(1 + 0.10)^{10}}{(1 + 0.10)^{10} - 1}\]
04

Evaluate the expression in the formula

Let's evaluate the formula step by step: 1. Calculate (1 + 0.10) = 1.10. 2. Calculate \(1.10^{10}\) = 2.5937424601. 3. Calculate \(1.10^{10} - 1\) = 2.5937424601 - 1 = 1.5937424601. 4. Calculate \(0.10 \times 1.10^{10}\) = 0.25937424601. 5. Calculate \(\frac{0.25937424601}{1.5937424601}\) = 0.1627453782. Now, we can multiply the result by the loan amount: 6. Calculate \(100,000 \times 0.1627453782\) = $16,274.54 (rounded to the nearest cent).
05

Interpret the result

The size of each installment to fully repay the loan with principal and interest in 10 years is approximately $16,274.54 per year.

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Key Concepts

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

Annual Interest Rate
Understanding the annual interest rate is key when dealing with loans. This rate represents the cost of borrowing money over a period of one year, expressed as a percentage of the principal loan amount. The annual interest rate affects the total amount you will pay over the term of the loan.
It is crucial to convert the percentage into a decimal before using it in calculations. For example, a 10% annual interest rate is converted to 0.10.
This conversion helps in effortlessly applying the interest rate in various financial formulas, like calculating loan installments.
  • Annual Interest Rate: Represents the yearly cost of the loan.
  • Expressed as a percentage of the principal.
  • Conversion to decimals is important for calculations.
Understanding how the interest rate compounds over time is vital, as it influences the total loan repayment significantly.
Loan Installments
Loan installments are the fixed, regular payments made to repay a loan over a set term. They include both principal and interest portions, ensuring the loan is fully amortized by the end of the term.
These payments are typically made monthly or annually, depending on the loan agreement.
For the exercise at hand, it specifically focuses on equal annual installments over a 10-year period. Each installment decreases the principal balance, with subsequent installments having a greater portion going towards principal reduction.
  • Consist of principal and interest.
  • Are usually fixed for the loan duration.
  • Ensure full repayment by the loan's end.
By maintaining consistent payments, you gradually reduce the balance, ultimately paying off the loan by the agreed schedule.
Amortization Formula
The amortization formula is a crucial tool for calculating the size of each installment needed to fully repay a loan. It distributes the loan amount and the accumulated interest across equal payments over the term.
In our example, where a \(100,000 loan is repaid over ten years with a 10% annual interest, the formula is:\[A = P \frac{r(1 + r)^n}{(1 + r)^n - 1}\]
This formula ensures that each payment covers both the interest accrued for the year and a portion of the principal.
Key elements of the formula include:
  • \(A\): The annual payment that keeps consistency in equal installments.
  • \(P\): Principal loan amount, here it's \)100,000.
  • \(r\): Annual interest rate in decimal, which is 0.10.
  • \(n\): Total number of payments, which is 10 years in this case.
Using these components, the formula divides the repayment evenly, making sure the loan is settled by the end of the term.

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Most popular questions from this chapter

Suppose payments will be made for \(6 \frac{1}{2}\) yr at the end of each semiannual period into an ordinary annuity earning interest at the rate of \(7.5 \% /\) year compounded semiannually. If the present value of the annuity is $$\$ 35,000$$, what should be the size of each payment?

The proprietors of The Coachmen Inn secured two loans from Union Bank: one for $$\$ 8000$$ due in 3 yr and one for $$\$ 15,000$$ due in \(6 \mathrm{yr}\), both at an interest rate of \(10 \%\) /year compounded semiannually. The bank has agreed to allow the two loans to be consolidated into one loan payable in 5 yr at the same interest rate. What amount will the proprietors of the inn be required to pay the bank at the end of 5 yr? Hint: Find the present value of the first two loans.

In the last 5 yr, Bendix Mutual Fund grew at the rate of \(10.4 \% /\) year compounded quarterly. Over the same period, Acme Mutual Fund grew at the rate of \(10.6 \% /\) year compounded semiannually. Which mutual fund has a better rate of return?

Fleet Street Savings Bank pays interest at the rate of \(4.25 \%\) /year compounded weekly in a savings account, whereas Washington Bank pays interest at the rate of \(4.125 \%\) /year compounded daily (assume a 365 day year). Which bank offers a better rate of interest?

The parents of a child have just come into a large inheritance and wish to establish a trust fund for her college education. If they estimate that they will need $$\$ 100,000$$ in \(13 \mathrm{yr}\), how much should they set aside in the trust now if they can invest the money at \(8 \frac{1}{2} \% /\) year compounded (a) annually, (b) semiannually, and (c) quarterly?
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