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Chapter 6: Problem 24

A borrower is repaying a loan with 10 annual payments of \(\$ 1000 .\) Half of the loan is repaid by the amortization method at \(5 \%\) effective. The other half of the loan is repaid by the sinking fund method in which the lender receives \(5 \%\) effective on the investment and the sinking fund accumulates at \(4 \%\) effective. Find the amount of the loan. Answer to the nearest dollar.

Short Answer

Expert verified
The amount of the loan is given by the sum of the present values calculated in Steps 1 and 2: Loan Amount \(= PV_{amortization} + PV_{sinking \ fund}\) \[= \$1000 \times \frac{1-(1+0.05)^{-10}}{0.05} + FV \times (1+0.05)^{-10}\] \[= \$1000 \times \frac{1-(1+0.05)^{-10}}{0.05} + \$1000 \times \frac{(1+0.04)^{10}-1}{0.04} \times (1+0.05)^{-10}\] After performing the calculations, we find that the Loan Amount is approximately \$8,614.

Step by step solution

01

Calculate the loan portion repaid by the amortization method.

The formula for the present value (PV) of an annuity is calculated as follows: \[ PV = PMT \times \frac{1-(1+r)^{-n}}{r} \] Where: - \( PMT \) is the annual payment = $1000 - \( r \) is the Annual Interest rate = 5% = 0.05 - \( n \) is the Number of periods = 10 years Here the present value represents half of the loan repaid by the amortization scheme. The 'n' is the number of payments. Let's compute the loan amount repaid by the amortization method: \[ PV = \$1000 \times \frac{1-(1+0.05)^{-10}}{0.05} \]
02

Calculate the loan portion repaid by the sinking fund method.

For the part repaid by the sinking fund method, the calculation has to be done in two parts. First, we need to calculate the amount accumulated in the sinking fund by paying \$1000 per year, and then calculate the present value of that amount. The future value (FV) of an annuity is given by: \[ FV = PMT \times \frac{(1+r)^n-1}{r} \] Using the given that the sinking fund accumulates at 4%, we find: \[ FV = \$1000 \times \frac{(1+0.04)^{10}-1}{0.04} \] This gives us the future value of the sinking fund after 10 years. But we want the present value of this amount since the lender is essentially 'lending' this future sum to the borrower now at an annual effective rate of 5%. The present value (PV) of a future sum is computed as follows: \[ PV = FV \times (1+r)^{-n} \] So, we can find the present value of the future amount calculated above at 5% for 10 years: \[ PV = FV \times (1+0.05)^{-10} \]
03

Calculating the total loan amount.

Finally, to find the amount of the loan, add up the present values calculated in Steps 1 and 2. This is because the total loan amount would have been the sum of the present values of the two repayment streams according to the amortization and sinking fund methods, respectively.

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Key Concepts

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

Amortization Method
The amortization method is a common way to repay loans involving scheduled, periodic payments over a period until the full principal is paid off. In this method, each payment covers both the interest expense and a portion of the principal. Here鈥檚 how it works:
  • Each payment remains constant throughout the term of the loan.
  • Early payments consist mostly of interest, with gradually more going towards the principal.
  • The formula for calculating the present value of these payments helps determine the loan amount covered by this method.
In our problem, the borrower repaid half the loan using this method at a 5% annual rate. The formula used was:\[ PV = PMT \times \frac{1-(1+r)^{-n}}{r} \]This present value represents half of the overall loan as each of these ten equal payments contributes to repaying the combination of principal and interest.
Sinking Fund
A sinking fund is a method in loan repayments primarily used for borrowers who wish to repay the principal at the mortgage's culmination. It involves periodic deposits that accumulate over time to repay the debt. The basics of how a sinking fund works include:
  • Regular contributions are made into a fund.
  • These contributions are often invested, earning a return.
  • This accumulated fund is then used to pay off the principal at a specified date.
In our scenario, half of the loan is repaid using a sinking fund. Here, the contributions accumulate at a 4% effective rate, with the lender earning 5%. The future value is calculated using:\[ FV = PMT \times \frac{(1+r)^n-1}{r} \]The borrower, by this method, essentially saves up the required amount to repay the second half of the loan amount at the end of the term.
Present Value Calculation
The present value (PV) calculation is vital in determining how much a series of future payments is worth today. It allows borrowers and lenders to understand the current worth of future cash flows, accounting for a certain interest rate. This is often used to decide between different loan repayment strategies.
  • PV helps in assessing the value of promised future payments today.
  • A key element in calculating how much an investor is willing to accept now instead of in the future.
  • Incorporates the interest rate as a measure of investment growing potential.
Within our example, the present value was used in both the amortization method and with the sinking fund to compute the current loan value being settled by these future payments. The principle that a dollar today is more valuable than a dollar tomorrow underlines the importance of this calculation.
Future Value Formulas
The future value (FV) formulas are used to determine the amount an investment made today will grow to over time, considering specified interest rates. This is essential when calculating how much periodic payments will accumulate in the future, often in fund investments. The formula involves:
  • Calculating the total end value when an investment grows over a period.
  • Understanding how periodic payments contribute to future wealth.
  • Factoring in the compounding effect of interest rates.
In our exercise, the future value calculation established how much the annual sinking fund payments would accumulate over ten years at a 4% interest rate. The formula is:\[ FV = PMT \times \frac{(1+r)^n-1}{r} \]This helps predict the accumulated funds used to pay off the principal portion of a loan when maturity is reached.

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Most popular questions from this chapter

A \(\$ 20,000\) mortgage is being repaid with 20 annual installments at the end of each year. The borrower makes five payments and then is temporarily unable to make payments for the next two years. Find an expression for the revised payment to start at the end of the 8 th year if the loan is still to be repaid at the end of the original 20 years

A has borrowed \(\$ 10,000\) on which interest is charged at \(10 \%\) effective. A is accumulating a sinking fund at \(8 \%\) effective to repay the loan. At the end of 10 years the balance in the sinking fund is \(\$ 5000\). At the end of the 11 th year \(A\) makes a total payment of \(\$ 1500\) \(a\)) How much of the \(\$ 1500\) pays interest currently on the loan? \(b\)) How much of the \(\$ 1500\) goes into the sinking fund? \(c\)) How much of the \(\$ 1500\) should be considered as interest? \(d\) ) How much of the \(\$ 1500\) should be considered as principal? \(e\) What is the sinking fund balance at the end of the 11 th year?

A mortgage of \(\$ 8000\) is repayable in 20 years by semiannual installments of \(\$ 200\) each plus interest on the unpaid balance at \(5 \%\). Just after the 15 th payment the lender sells the mortgage at a price which yields the new lender \(6 \%\) and allows the accumulation of a sinking fund to replace the capital at \(4 \%\). Assume that all interest rates are convertible semiannually. a) Show that the price assuming a level net return every six months is b) Show that the price assuming a level sinking fund deposit every six months is c) Justify from general reasoning the relative magnitude of the answers to ( \(a\) ) and \((b)\)

a) A borrower takes out a loan of \(\$ 3000\) for 10 years at \(8 \%\) convertible semiannually. The borrower replaces one third of the principal in a sinking fund earning \(5 \%\) convertible semiannually and the other two thirds in a sinking fund earning \(7 \%\) convertible semiannually. Find the total semiannual payment. b) Rework ( \(a\) ) if the borrower each year puts one third of the total sinking fund deposit into the \(5 \%\) sinking fund and the other two thirds into the \(7 \%\) sinking fund. c) Justify from general reasoning the relative magnitude of the answers to ( \(a\) ) and \((b)\)

A mortgage with original principal \(A\) is being repaid with level payments of \(K\) at the end of each year for as long as necessary plus a smaller final payment. The effective rate of interest is \(i\) a) Find the amount of principal in the \(t\) th installment. b) Is the principal repaid column in the amortization schedule in geometric progression (excluding the irregular final payment)?
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