# Amortisation and Annuities

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

##### InterestAmortisationAnnuitiesExam-StyleDescriptionHelpMore Finance

This is level 2: Annuities. Give answers which are amounts of money correct to two decimal places.

 Andrew retired at the age of 59. He invested £900000 in an annuity fund which returns 6% p.a. compounded monthly. He wants the money to last for 25 years.1. Calculate how much he can afford to withdraw each month. £ Andrew's sister retired at the age of 55. She invested £300000 in an annuity fund which returns 2.6% p.a. compounded monthly. She withdraws £1500 each month for living expenses and holidays.2. Calculate how long it will take for the money to run out. Round you answer down to a whole number of years. Andrew's brother deposits $690000 in an annuity fund which earns 5.3% p.a. interest compounded quarterly. He wants the money to last for 25 years.3. How much can he afford to withdraw each quarter?$ 4. Find the outstanding balance of Andrew's brother's fund after 10 years. \$ Professor Collins invested £480000 in an annuity fund which returns 2.1% p.a. compounded monthly. He withdraws £3000 each month.5. Calculate how long it will take for the money to run out. Round you answer down to a whole number of years. 6. How many more months would his money last if he only withdrew £2500 each month? Round you answer down to a whole number of months. 7. If £67000 is invested with a 3.5% p.a. interest rate (compounded annually). What is the maximum I can take out of this account at the end of each year if it is to be a perpetuity (a special type of annuity in which the regular payments continue indefinitely)? £ 8. What amount must be initially invested into an annuity account if it is to yield £2000 per month for 20 years at a 2% p.a. rate of interest componded monthly? £ Dr Jane Dose retires at age 63 with €75000 in her savings fund. She rolls this money into an annuity fund earning 3.5% p.a. interest compounded monthly.9. How much will she be able to withdraw each month if her money is to last until her 85th birthday? € 10. How much more will Dr Dose be able to withdraw each month if her money was to only last one decade? €
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This is Amortisation and Annuities level 2. You can also try:
Level 1

## Instructions

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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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## 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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