Chapter 07

7-1Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

Chapter 7

Annuities

Introductory Mathematics & Statistics


Learning Objectives

• Understand and apply annuities

• Distinguish between future and present values of annuities

• Solve problems involving the future value of an annuity

• Calculate the present value of an annuity

• Calculate the periodic payment of a present value annuity (amortisation)

• Calculate the periodic payment of a future value annuity (sinking fund)


7.1 Introduction

• An annuity annuity is a series of equal payments, often made under contract, paid at equal intervals (e.g. quarterly or monthly)

Definitions1. The rent period (or payment period) is the interval of

time between the payments of an annuity

2. The term of an annuity is the time from the beginning of the first payment period to the end of the last payment

period


7.1 Introduction (cont…)

• The two main types of annuities are simple and general

• A simple annuity is where interest is compounded at the same times as the annuity payments

• A general annuity is where interest is compounded at times that are either greater or smaller than when the annuity payments are made


7.1 Introduction (cont…) Simple annuities can be classified into four types

1. An ordinary annuity is one where the payments are made at the end of each period

2. An annuity due is one where the payments are made at the beginning of each period

3. A deferred annuity is one where the payments do not commence until a period of time has elapsed

4. A perpetuity is an annuity in which the payments continue indefinitely


7.1 Introduction (cont…)

• Ordinary annuities have the following aspects

1. The future value of an annuity: this is the value of the annuity at the end of its term

2. The present value of an annuity: this is the value of the annuity at its beginning

3. Amortisation: this is the making of periodic payments to repay a debt (including the principal and interest)

4. Sinking fund: this is a fund into which the periodic payments necessary to realise a given sum of money in the future are made


7.2 Future value of an annuity• The future value (or accumulated value) of an annuity is the amount

due at the end of the term• That is, it is the sum of all the periodic payments made and interest

accrued up to and including the final payment period

Where: S = future value of the annuity R = amount of the annuity payment made per periodi = interest rate per payment periodn = total number of payments

i

1i1RSn


7.2 Future value of an annuity (cont…)

The formula can also be written as

Where

The value of sn i (or sn at i %) may be thought of as the future value at an interest rate of i% of $1 paid at the end of each period for n periods.

insRS

i

1i1sn

in



ExampleA customer deposits $250 every 3 months into a building society account that pays interest at a rate of 8% per annum convertible quarterly. How much money will be in the account at the end of 10 years?

Solution40104n,02.0

408.0i,250$R

40198318.6002.0

102.01s40

02.040



Solution (cont…)

Hence, the customer will have $15 100.50 in the account at the end of 10 years.

50.15100$40198318.60250$

sRS in


7.3 Present value of an annuity

• The present value (or discounted value) of an annuity is its value at the beginning of the initial rent period.

• That is, it is the sum of the compound present values of all the payments one period before the initial payment.

Where: A = present value R = annuity payment per period i = interest rate per period n = number of payments

i

i11RAn


7.3 Present value of an annuity (cont…)

• Another way of writing the formula

• Where

• The value of a n i (or an at i %) may be thought of as the present value at an interest rate of i % per period of $1 paid at the end of each period for n periods

inaRA

iia

n

in

11


7.3 Present value of an annuity (cont…)Example

Melinda took out a loan from a credit union in order to purchase a home computer. She was to repay the loan in monthly instalments of $120 for 5 years. Calculate the present value of these repayments if the interest rate was 9%, convertible monthly

Solution

From Table 3:

60512n,00785.01209.0i,120$R

17337352.48a 0075.060


7.3 Present value of an annuity (cont…)

Solution (cont…)

That is, the repayments are worth $5780.80 at the beginning of the loan

80.5780$17337352.48120$

aRA in


7.4 Amortisation

• When an individual or business pays a debt (including interest) by making periodic payments at regular intervals, the debt is said to be amortised.

• An amortisation problem involves finding the periodic payment that will discharge a debt.

• In particular, as each periodic payment is made, this amount covers part of the principal and interest on the balance of the principal.

• In turn, there is a reduction in the remaining principal such that at the end of the term of the annuity the debt is extinguished.


7.4 Amortisation (cont…)Example

A family undertakes a mortgage of $40 000 from a bank in order to buy its new home. The bank charges interest at a rate of 12% per annum, compounded quarterly over 20 years. What quarterly payments will the family have to make on this loan?

Solution

From Table 3:

80204n,03.0412.0i,40000$A

20076345.30a 03.080


7.4 Amortisation (cont…)Solution (cont…)

Hence, the family must repay the loan at the rate of $1324.47 per quarter.

Note that the family will, in fact, be paying back to the bank a total of $1324.47 × 80 = $105 957.60, considerably more than the original $40 000 loan!

47.1324$20076345.3040000$R

aR40000

aRA

03.080

in


7.5 Sinking funds

• A sinking fund is a fund into which periodic payments are made so as to accumulate a nominated amount of money at the end of a specified period

• It is assumed that each deposit earns compound interest until the end of the period

Where: S = future value R = annuity payment per period i = interest rate per period n = number of payments

i

1i1SR n

ins

SR or


7.5 Sinking funds (cont…)

Example

Connie is planning to spend her Christmas holidays in 2 years time in the United States. She estimates that she will need an amount of $6000 to pay for her airfares, accommodation and other expenses. She is going to save her money in an account that attracts 10% interest per annum, compounded quarterly. How much will Connie have to deposit into her account at the end of each quarter, to have the desired amount at the end of 2 years?


7.5 Sinking funds (cont…)Solution

So

If Connie deposits $686.80 every 3 months over the 2 years, she will have the required $6000 at the end of that time

842n,025.0410.0i,6000$S

73611590.8025.0

1025.01s8

025.08

80.686$76311590.8

6000$R


Summary

• We looked at understanding and applying annuities

• We found the difference between future and present value of annuities

• We solved problems involving the future value of an annuity

• We calculated the present value of an annuity

• We calculated the periodic payment of a present value annuity (amortisation)

• We calculated the periodic payment of a future value annuity (sinking fund)

Documents

Chapter 07