topic badge

6.07 Annuities

Lesson

Introduction

An annuity is a style of investment from which individuals usually withdraw a regular amount of funds. The withdrawal in many cases is repeated until the funds run out, that is the value of the annuity reaches \$0. A special case of annuities are perpetuities where rather than the funds dwindling the investment value remains constant.

We can solve annuity problems using the finance application of our calculators, or by modelling the problem with a recurrence relation.

Model annuities with the financial application

We can solve problems involving annuities using a financial application by setting the present value (PV) as the initial value invested and future value (FV) as the remaining funds at the end of the investment (often \$0). The present value should be entered as a negative to indicate depositing the money for investment and the payment (PMT) will be positive as this is returned to the investor.

Examples

Example 1

Victoria invests \$190\,000 at a rate of 12\% per annum compounded monthly.

Victoria wants to determine what her equal monthly withdrawal should be if she wants the investment to last 20 years.

a

Fill in the value for each of the following. Put an X next to the variable we wish to solve for.

VariableValue
N
I(\%)⬚\%
PV
PMT
FV
P/Y
C/Y
Worked Solution
Create a strategy

Use the given information to find each value, keeping in mind that the PV and PMT should be opposite in sign to FV.

Apply the idea

N is the number of instalment periods which is 20\times 12=240 months.

I\% is our annual interest rate which is 12\%.

PV is the present value of her investment which is initially \$190\,000. Since she is investing her money, she no longer has it, so we can think of it as she is losing it. So we should make it negative: -\$190\,000.

FV is the future value of our investment, and we're looking to solve for when the money runs outso it should be zero.

P/Y is the number of contributions per year which is 12 since she makes a payment every month.

C/Y is the number of compounding periods each year which is also 12.

Value
N240
I\%12\%
PV-190\,000
PMTX
FV0
P/Y12
C/Y12
b

Hence determine the amount of the monthly withdrawal. Give your answer to the nearest cent.

Worked Solution
Create a strategy

Use the financial application on your calculator using the values from part (a).

Apply the idea

By pressing enter next to PMT in the financial application, we should get:

\displaystyle PMT\displaystyle =\displaystyle -2092.06

The monthly withdrawal is \$2092.06.

Idea summary

When using a financial application for an annuity:

N is the number of instalment periods, or how long the annuity lasts.

I\% is our annual interest rate.

PV is the present value of our investment. This should be represented by a negative value, as the investment represents money has set aside.

PMT is the amount withdrawal each time period.

FV is the future value of our investment which should be 0.

P/Y is the number of payments or withdrawals made each year.

C/Y is the number of compounding periods each year.

Model annuities with recurrence relations

This is similar to modelling an  investment with regular payment  with the payment in this case being withdrawn and hence, negative in the form shown below.

For a principal investment, P, at the compound interest rate of r per period and a payment d withdrawn per period, the sequence of the value of the investment over time forms a first order linear recurrence.

The recursive sequence which generates the value, V_{n}, of the investment at the end of each instalment period is: V_{n}=V_{n-1}\times (1+r)-d, \, \text{where }V_{0}=P

The recursive sequence which generates the value, V_{n}, of the investment at the beginning of each instalment period is: V_{n}=V_{n-1}\times (1+r)-d, \, \text{where }V_{1}=P

If we want to calculate annuities without technology, we will need to find an annuity formula that perfectly describes what is happening to the annuity at every step.

We know that an annuity starts with a principal investment P which gains interest each compounding period according to some rate r, and after this we then withdraw a payment of d. Then, we repeat this until the annuity's value becomes zero.

Let's start by representing this information in an equation for the first few compounding periods:

\displaystyle V_{0}\displaystyle =\displaystyle P
\displaystyle V_{1}\displaystyle =\displaystyle P\times (1+r)-d
\displaystyle V_{2}\displaystyle =\displaystyle (P\times (1+r)-d)\times(1+r)-d
\displaystyle =\displaystyle P\times (1+r)^{2}-d\times((1+r)+1)
\displaystyle V_{3}\displaystyle =\displaystyle (P\times (1+r)^{2}-d)\times((1+r)+1))\times (1+r)-d
\displaystyle =\displaystyle P\times (1+r)^{3}-d\times((1+r)^{2}+(1+r)+1)

Because the sequence is recursive, we begin to see a pattern emerge after expanding and simplifying the result of each compounding period. In fact, we can generalise these equations to get:

V_{n}=P\times (1+r)^{n}-d\times ((1+r)^{n-1}+(1+r)^{n-2}+\ldots+(1+r)^{2}+(1+r)+1)

We can simplify this using the equivalence:x^{n-1}+x^{n-2}+\ldots+x^{2}+x+1=\dfrac{x^{n}-1}{x-1} by letting x=(1+r):(1+r)^{n-1}+(1+r)^{n-2}+\ldots+(1+r)^{2}+(1+r)+1=\dfrac{(1+r)^{n}-1}{(1+r)-1}

Now we can rewrite the general equation as: V_{n}=P\times (1+r)^{n}-d\times \dfrac{(1+r)^{n}-1}{1+r-1}which simplifies to the formula for annuities: V_{n}=P\times (1+r)^{n}-d\times\dfrac{(1+r)^{n}-1}{r}

where V_{n} is the annuity value after n compounding periods, P is the principal investment, r is the interest rate per compounding period, and d is the amount being withdrawn at the end of each period.

Examples

Example 2

Lachlan received an inheritance of \$100\,000. He invests the money at 8\% per annum with interest compounded annually at the end of the year. After the interest is paid, Lachlan withdraws \$9000 and the amount remaining in the account is invested for another year.

a

How much is in the account at the end of the first year?

Worked Solution
Create a strategy

Increase the inheritance by the interest rate for one year and then subtract the withdrawal made.

Apply the idea

To increase the inheritance of \$100\,000 by 8\% we can multiply it by 1.08. Then we need to subtract the withdrawal of \$9000.

\displaystyle \text{Balance}\displaystyle =\displaystyle 100\,000 \times 1.08 -9000Increase by the rate and subtract the withdrawal
\displaystyle =\displaystyle \$99\,000Evaluate
b

Write a recursive rule for A_n in terms of A_{n - 1} that gives the value of the account after n years and an initial condition A_{0}.

Worked Solution
Create a strategy

To use the recursive rule V_{n}=V_{n-1}\times (1+r)-d, where V_{0} = P, we need to find r,\,d,\, and P.

Apply the idea

The balance is increased by r=8\% =0.08 each year. Lachlan withdraws the amount of d=\$9000. The original investment was P=\$100\,000.

\displaystyle A_{n}\displaystyle =\displaystyle (1+r)\times A_{n-1} - d,\,A_{0}=PWrite the recursive rule
\displaystyle =\displaystyle (1+0.08)\times A_{n-1} - 9000,\,A_{0}=100\,000Substitute r, \,d, \,P
\displaystyle A_{n}\displaystyle =\displaystyle 1.08\times A_{n-1} - 9000,\,A_{0}=100\,000Simplify
c

What is the value of the investment at the end of year 10?

Worked Solution
Create a strategy

Enter the recursive rule and initial value we found in part (b) into your calculator and look for the value of the investment at N=10.

Apply the idea

Using your calculator you should get that the value for the 10th year is \$85\,513.44.

\text{Balance}=\$85\,513.44

Reflect and check

We could also have done this using the explicit formula with r=0.08, \, d=9000, \, P=100\,000, and n=10:

\displaystyle A_{n}\displaystyle =\displaystyle P (1+r)^{n}-d\times\dfrac{(1+r)^{n}-1}{r}Write the formula
\displaystyle A_{10}\displaystyle =\displaystyle 100\,000 (1.08)^{10}-9000\times\dfrac{(1.08)^{10}-1}{0.08}Substitute r, \, d, \, P, \, n
\displaystyle =\displaystyle \$85\,513.44Evaluate and round
d

By the end of which year will the annuity have run out?

Worked Solution
Create a strategy

Enter the recursive rule and initial value we found in part (b) into your calculator and look for a negative balance.

Apply the idea

Using your calculator you should get that the 29th value is when the balance starts to be negative.

The annuity will run out by the end of the 29th year.

Idea summary

For a principal investment, P, at the compound interest rate of r per period and a payment d withdrawn per period, the sequence of the value of the investment over time forms a first order linear recurrence.

The recursive sequence which generates the value, V_{n}, of the investment at the end of each instalment period is:V_{n}=V_{n-1}\times (1+r)-d, \, \text{where }V_{0}=P

The recursive sequence which generates the value, V_{n}, of the investment at the beginning of each instalment period is:V_{n}=V_{n-1}\times (1+r)-d, \, \text{where }V_{1}=P

Explicit formula for annuities:

\displaystyle V_{n}=P (1+r)^{n}-d\times\dfrac{(1+r)^{n}-1}{r}
\bm{V_{n}}
is the annuity value after n compounding periods
\bm{P}
is the principal investment
\bm{r}
is the interest rate per compounding period
\bm{d}
is the amount being withdrawn at the end of each period

Outcomes

ACMGM099

use a recurrence relation to model an annuity, and investigate (numerically or graphically) the effect of the amount invested, the interest rate, and the payment amount on the duration of the annuity

ACMGM100

with the aid of a financial calculator or computer-based financial software, solve problems involving annuities (including perpetuities as a special case); for example, determining the amount to be invested in an annuity to provide a regular monthly income of a certain amount

What is Mathspace

About Mathspace