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Chapter 6: Problem 53

a) Show that if a loan is amortized with \(n\) level payments of \(R\) b) Verbally interpret the result obtained in \((a)\)

Short Answer

Expert verified
The short answer is: We demonstrated mathematically that the loan can be amortized with n level payments of R by using the present value of an annuity formula, resulting in the equation \( R=\frac{P*i}{1-(1+i)^{-n}} \). Verbally, this means that the borrower must make n equal payments of R so that they will have completely paid off their loan, including both principal and interest, by the end of the loan period. The formula calculates the regular payment amount (R) considering the principal amount (P), annual interest rate (i), and total number of periods (n).

Step by step solution

01

1. Understand the given variables

In this problem, we have a loan with the following variables: - Principal amount (P) - Annual interest rate (i) - Number of periods (n) - Level payments (R)
02

2. Determine the present value of an annuity formula

The present value of an annuity (PV) can be described as the value of a series of cash flows discounted by an interest rate (i). We will use this formula to represent the loan: \( PV=\frac{P}{(1+i)^n} \)
03

3. Apply the formula to the given problem

Now, let's apply the present value of an annuity formula to the loan: \( R\frac{1-(1+i)^{-n}}{i}=P \)
04

4. Solve for R

In order to show that the loan is amortized with n level payments of R, we need to solve the equation above for R: \( R=\frac{P*i}{1-(1+i)^{-n}} \)
05

5. Verbally interpret the result obtained in (a)

The result obtained in part (a) represents the level payments (R) needed to fully amortize the loan. This means that if a borrower makes n equal payments of R, they will have entirely paid off their loan, including both principal and interest, by the end of the loan period. In simpler terms, this formula calculates the regular payment amount (R) that the borrower needs to pay each period, considering the principal amount (P), the annual interest rate (i), and the total number of periods (n), in order to clear the loan at the end of its term.

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Key Concepts

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

Level Payments
Level payments refer to equal periodic payments made over the term of a loan to fully amortize it. In simpler terms, this means that the borrower pays the same amount each period, like monthly or quarterly, until the entire loan amount is cleared.

Using level payments offers predictability and ease since the payment amount remains consistent. This can help the borrower in budgeting and financial planning. In the formula given: \[ R = \frac{P \cdot i}{1 - (1 + i)^{-n}} \], \(R\) represents the level payment amount. This formula helps ascertain that level payments will not only cover interest but also decrease the principal amount over time, leading to the loan's full repayment.
Present Value of Annuity
The present value of an annuity is essentially the total value today of a series of future payments, given a certain interest rate. In the context of a loan, this is important because it helps us understand how much the future series of level payments is worth today.

The formula for the present value of an annuity is instrumental in determining how much a loan's series of repayments would be worth if paid in one lump sum today. This formula is reflected in the expression: \[ PV = \frac{P}{(1+i)^n} \]. It tells us how much is required today to ensure that, after a series of periodic payments, the entire loan (including the interest) is paid off by the end.
Principal Amount
The principal amount is the initial sum of money borrowed in a loan. It's essentially the actual base amount that needs to be repaid, excluding interest.
In the context of loan amortization, the principal plays a crucial role as the base figure from which interest calculations and subsequent repayments are derived.
As represented in the formula, \[ R = \frac{P \cdot i}{1 - (1 + i)^{-n}} \], the variable \(P\) stands for the principal amount.
Each level payment made during the loan duration will consist partly of principal repayment and partly of interest charges.
Annual Interest Rate
The annual interest rate is the percentage of the principal that is paid as interest in a year. In a loan amortization calculation, this rate determines how much extra the borrower will pay on top of the principal amount.

This rate can significantly influence the amount of each level payment, as higher rates lead to higher monthly payments. The formula for calculating level payments \[ R = \frac{P \cdot i}{1 - (1 + i)^{-n}} \] incorporates the annual interest rate \(i\), highlighting its importance in determining overall payment schedules. It's a crucial factor that impacts how quickly the principal amount decreases over the term of the loan.

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Most popular questions from this chapter

A loan of 1 is being amortized over a 10 -year period with continuous payments which vary in such a fashion that the outstanding loan balance is linear. The force of interest is \(10 \%\). Find: \(a\) ) The principal repaid over the first 5 years. b) The interest paid over the first 5 years.

On a loan of \(\$ 10,000\) interest at \(9 \%\) effective must be paid at the end of each year. The borrower also deposits \(\$ X\) at the beginning of each year into a sinking fund earning \(7 \%\) effective. At the end of 10 years the sinking fund is exactly sufficient to pay off the loan. Calculate \(X\)

A loan is being repaid over \(n\) periods with continuous payments at the rate of \(t\) per period at time \(t .\) Find expressions for the outstanding loan balance at time \(k, 0 \leq k \leq n\) a) Prospectively b) Retrospectively.

Nine years ago a family incurred a 20 -year \(\$ 80,000\) mortgage at \(8 \%\) effective on which they were making annual payments. They desire now to make a lump-sum payment of \(\$ 5000\) and to pay off the mortgage in nine more years. Find an expression for the revised annual payment: a) If the lender is satisfied with an \(8 \%\) yield for the past nine years but insists on a \(9 \%\) yield for the next nine years. b) If the lender insists on a \(9 \%\) yield during the entire life of the mortgage.

A borrows \(\$ 2000\) at an effective rate of interest of \(10 \%\) per annum and agrees to repay the loan with payments at the end of each year. The first payment is to be \(\$ 400\) and each payment thereafter is to be \(4 \%\) greater than the preceding payment, with a smaller final payment made one year after the last regular payment. a) Find the outstanding loan balance at the end of three years. b) Find the principal repaid in the third payment.
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