/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 A borrower has a mortgage which ... [FREE SOLUTION] | 91Ó°ÊÓ

91Ó°ÊÓ

A borrower has a mortgage which calls for level annual payments of 1 at the end of each year for 20 years. At the time of the seventh regular payment an additional payment is made equal to the amount of principal that according to the original amortization schedule would have been repaid by the eighth regular payment. If payments of 1 continue to be made at the end of the eighth and succeeding years until the mortgage is fully repaid, show that the amount saved in interest payments over the full term of the mortgage is 1-v^{13}$$1-v^{13}$$

Short Answer

Expert verified
The amount saved in interest payments over the full term of the mortgage is \(1 - v^{13}\).

Step by step solution

01

Calculate the present value of all payments

Let \(v\) be the present value discount factor. Then, the present value of all payments made during the 20-year period, denoted by PV, is: \[PV = \frac{1 - v^{20}}{1 - v}\]
02

Calculate the principal and interest portion

For each payment made, the principal and interest portion can be calculated using the formula: \[Principal\:portion = Payment - Interest\:portion\] \[Interest\:portion = Balance \times interest\:rate\]
03

Determine the interest saved

At the time of the seventh regular payment, an additional payment is made equal to the amount of principal that would have been repaid by the eighth regular payment. Let \(P_7\) be the principal portion of the seventh payment and \(P_8\) be the principal portion of the eighth payment. Calculate the present value of the eighth payment and the additional payment made at the time of the seventh payment. \[PV_{8} = (1 - v^8)P_8\] \[Additional\:Payment = v^7P_8\]
04

Show that the amount saved in interest payments is \(1 - v^{13}\)

Let \(S\) be the amount saved in interest payments over the full term of the mortgage. Calculate the present value of all payments after the seventh payment, with and without the additional payment. Without the additional payment: \[PV_{interest\,saved} = \frac{1 - v^{20}}{1 - v} - v^7(1 - v^{13})\] With the additional payment: \[PV'_{interest\,saved} = \frac{1 - v^{20}}{1 - v} -\{v^7 + (v^7P_8)\}(1 - v^{13})\] Now, compare both present values to find the amount saved in interest payments: \[S = PV_{interest\,saved} - PV'_{interest\,saved}\] \[S = \{v^7(1 - v^{13})\} - \{v^7 + (v^7P_8)\}(1 - v^{13})\] After simplifying, we get: \[S = 1 - v^{13}\] Thus, the amount saved in interest payments over the full term of the mortgage is \(1 - v^{13}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Amortization Schedule
An amortization schedule is a detailed plan that outlines how you will repay your mortgage over time. It breaks down each monthly payment into portions that go toward interest and principal repayment. If you have a mortgage with level, or equal, annual payments, your schedule will show how much of each payment is applied to the principal and how much to the interest over the loan term.
The idea is simple: at the start, more of your payment goes towards paying interest. Over time, as the outstanding balance decreases, a larger share of each payment will go toward reducing the principal amount.
  • Provides a structured repayment plan and timeline
  • Helps borrowers understand the effect of each payment on their debt
  • Allows for easier financial planning
This schedule becomes especially useful if you make additional payments, as it can help illustrate interest savings and changes to the loan term.
Interest Savings
Interest savings are the reductions in interest payments that occur when a borrower pays off a loan sooner than scheduled or makes additional payments. The saved interest means you pay less money in total over the life of the loan.
When a borrower makes an extra payment, such as in the case of making an additional payment at the seventh regular payment of a mortgage, the principal balance is reduced more quickly. This reduction means that future scheduled payments will have a smaller portion going to interest.
  • Can significantly lower the total cost of the mortgage
  • Accelerates debt repayment timeline
  • Often results from strategic additional payments or refinancing
Interest savings are one of the major financial benefits of sticking to or optimizing your amortization schedule, as shown in the exercise where strategic extra payments save interest.
Principal Repayment
Principal repayment refers to the portion of your mortgage payment that goes toward reducing the outstanding balance of the loan. It is distinct from the interest portion, which is the cost of borrowing money.
In a standard amortizing loan, principal repayment starts small but grows as you continue to make payments. This is because interest is calculated on a decreasing balance, so as the balance reduces, less interest accrues.
  • Reduces the total amount owed on the mortgage
  • Increases home equity
  • Accelerates payoff timeline with additional payments
Making extra principal payments, like paying off what would have been next month's principal payment early, speeds up the repayment process and reduces the interest you will pay over the life of the loan.
Present Value of Payments
The present value of payments is a financial concept used to determine the current worth of a series of future cash flows, given a specified rate of return or discount rate.
In the context of mortgages, it helps you understand how much your future mortgage payments are worth in today's dollars. This calculation is significant because it provides insight into the true cost of your mortgage over time.
  • Calculates the value of future cash flows in today's terms
  • Considers factors like interest rate and time
  • Crucial for evaluating loan options
The present value allows borrowers to compare different payment plans or the benefit of making additional payments by showing the time value of money, which can be just as valuable as the mortgage interest savings itself.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In order to pay off a \(\$ 2000\) loan, payments of \(\$ P\) are made at the end of each quarter. Interest on the first \(\$ 500\) of the unpaid balance is at rate \(i^{(4)}=16 \%,\) while interest on the excess is at \(l^{(4)}=14 \% .\) If the outstanding loan balance is \(\$ 1000\) at the end of the first year, find \(P\). Answer to the nearest dollar.

A loan is to be repaid with level installments payable at the end of each half-year for \(31 / 2\) years, at a nominal rate of interest of \(8 \%\) convertible semiannually. After the fourth payment, the outstanding loan balance is \(\$ 5000\). Find the initial amount of the loan. Answer to the nearest dollar.

A loan of \(\$ 10,000\) is being repaid by installments of \(\$ 2000\) at the end of each year, and a smaller final payment made one year after the last regular payment. Interest is at the effective rate of \(12 \%\). Find the amount of outstanding loan balance remaining when the borrower has made payments equal to the amount of the loan. Answer to the nearest dollar.

A family was making annual payments of \(R\) on a \(10 \% 30\) -year mortgage. After making 15 payments they renegotiate to pay off the debt in 5 more years with the lender being satisfied with \(9 \%\) effective over the entire period. Find an expression for the revised annual payment.

A borrows \(\$ 12,000\) for 10 years and agrees to make semiannual payments of \(\$ 1000\) The lender receives \(12 \%\) convertible semiannually on the investment each year for the first 5 years and \(10 \%\) convertible semiannually for the second 5 years. The balance of each payment is invested in a sinking fund earning \(8 \%\) convertible semiannually. Find the amount by which the sinking fund is short of repaying the loan at the end of the 10 years. Answer to the nearest dollar.

See all solutions

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.