# Amortisation and Annuities

## Exercises containing problems about gradually paying off loans and calculating pension plans.

##### InterestAmortisationAnnuitiesExam-StyleDescriptionHelpMore Finance

This is level 1: Amortisation. Give answers which are amounts of money correct to two decimal places.

Thomas takes out a loan for £10000 at 6% nominal APR compounded monthly. He makes monthly payments for 6 years.

1. Calculate the amount of the monthly payments.

£

2. Calculate the total amount Thomas will have paid back altogether by the end of the 6 years.

£

A student takes out a loan for £15000 at 7% nominal APR compounded quarterly. The repayments are to be made every three months for 8 years.

3. Calculate the amount of the quarterly payments.

£

4. What is the total amount of interest this student will pay?

£

#### SALE

##### Less than £184 pm
###### Initial deposit of only £5000 then easy monthly repayments over 8 years. Terms and conditions apply.

A car is advertised on sale for £19000. A big orange sign in the car's windscreen advertises a tempting way to pay for the car. In the small print it can be seen that the deal involves a £5000 deposit followed by monthly payments of £183.98 for 8 years.

5. What is the rate of interest (nominal APR) being charged to the customer in this scheme? Give your answer to the nearest whole number.

%

6. What is full cost of the car if purchased using the terms of this scheme?

£

Another scheme to purchase this same car is no deposit but 7 annual payments based on a 5% nominal APR compounded annually.

7. What would the annual payments be?

£

8. What is full cost of the car if purchased using the terms of this second scheme?

£

Thomas's mother takes out a loan for £1100 for emergency house repairs. She will repay the loan with a payment each month at 2.6% nominal APR compounded monthly spread over the course of a year.

9. Calculate the amount of the monthly payments.

£

10. What is the outstanding balance on the loan after 6 months? Give this answer to the nearest pound.

£
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This is Amortisation and Annuities level 1. You can also try:
Level 2

## Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

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## Description of Levels

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Interest - Exercises on compound interest on investments and loans.

Level 1 - Amortisation

Level 2 - Annuities

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More Financial Maths including lesson Starters, visual aids, investigations and self-marking exercises.

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## Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

## Help

These exercises are designed for those who have access to a graphic display calculator (GDC) with a Finance Solver function. It is possible to answer the questions using only a scientific calculator but not easy.

A great way to answer the questions in these exercises without a GDC is by creating a table of interest, payment and balance in a spreeadsheet.

### Amortisation

Amortization is is paying off an amount owed over time by making planned, incremental payments of principal and interest. In accounting, amortisation refers to writing off an asset's cost as an expense over its estimated useful life to reduce a company's taxable income.

The word amortise (which can also be spelled amortize) comes from the latin ad mortem meaning 'to death'

The formula to find the payments for amortisation is:

$$Pmt = PV \times \frac{r(1+r)^n}{(1+r)^n -1}$$
• $$Pmt$$ is the amount of each regular payment being made to pay off the loan.
• $$PV$$ is the present value or the initial amount of the loan.
• $$r$$ is the nominal APR divided by the number of payments per year expressed as a decimal (unlike I used in the Solver function below)
• $$n$$ is the total number of payments that will be made to pay off the loan.

It is recommended to use the Finance Solve on your GDC for this topic. [See TI-Nspire Essentials].

#### menu ⇒ Finance ⇒ Finance Solver

These are the variables used in the Finance Solver function:

• $$N$$ is the total number of payments that will be made to pay off the loan.
• $$I$$ is the rate of compound interest.
• $$PV$$ is the present value (a positive number) or the initial amount of the loan.
• $$Pmt$$ is the amount of each regular payment being made to pay off the loan (a negative value).
• $$FV$$ is the future value. It the loan is to be paid off in full this will be zero.
• $$PpY$$ is the number of payments per year.
• $$CpY$$ is the number of interest calculation periods per year; E.g. this would be 12 if compounded monthly.
• $$PmtAt$$ is usually set to END as this is when the load is completely paid off.
• [Note that N ÷ PpY gives the number of years it will take to pay off the loan]

There is also a function on the GDC that can produce a table showing all the datails of the gradual loan repayment:

#### menu ⇒ Finance ⇒ Amortisation ⇒ Amortisation Table

These are the variables used in the Amortisation Table function:

• $$NPmt$$ is the number of rows you would like the calculator to show as sometimes showing all rows would clutter the calculator screen.
• $$N$$ is the total number of payments that will be made to pay off the loan.
• $$I$$ is the rate of compound interest.
• $$PV$$ is the present value (a positive number) or the initial amount of the loan.
• $$Pmt$$ is the amount of each regular payment being made to pay off the loan.
• $$FV$$ is the future value. It the loan is to be paid off in full this will be zero.
• $$PpY$$ is the number of payments per year.
• $$CpY$$ is the number of interest calculation periods per year; E.g. this would be 12 if compounded monthly.

The values should be typed into the function in this order:

#### amortTbl(NPmt,N,I,PV,Pmt,FV,PpY,CpY)

• Column 1 is the number of the repayment
• Column 2 is the amount of interest paid off at that time.
• Column 3 is the amount of principal paid off at that time.
• Column 4 is the balance of the loan at the end of at that time period.

### Annuity

An annuity is the investment of a lump-sum which provides the fund from which regular withdrawals are made over a fixed time period. The investment earns interest according to the balance of the annuity each time period. The payments may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time.

An annuity which provides for payments for the remainder of a person's lifetime is a life annuity.

It is recommended to use a GDC for your working. See TI-Nspire Essentials for an example of how to use the Finance Solver. Note that PV (present value, the amount of the lump-sum) should be negative and the payments (PMT) should be positive.

The formula for calculating the payments from an annuity is the same as that for an amortisation

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