Australian Curriculum 12 General Mathematics - 2020 Edition 6.06 Perpetuities
Lesson

A perpetuity is a type of investment in which regular withdrawals are made. However, the balance remains stable as the withdrawal amount exactly equals the interest accrued for each time period. A good way to remember this is to think of a 'perpetual trophy' which is a trophy that continues to be awarded each year. A perpetuity fund continues forever.

Perpetuity

Withdrawal amount (payment) = Interest accrued

$Q=A\times r$Q=A×r

where $Q$Q is the amount of interest earned (size of the prize or payment)

$A$A is the initial amount invested in dollars

$r$r is the interest rate for the period as a decimal

#### Worked example

##### Example 1

Fred won Lotto and invested the money into a perpetuity which pays $4.5%$4.5% p.a. compounded quarterly. He is able to pay himself $\$12000$$12000 per quarter without using any of the principal. How much money did Fred win? Think: 4.5%4.5% p.a. is 1.125%1.125% per quarter. Hence, we are solving for AA, with r=1.125%r=1.125% and the payment Q=\12000Q=12000. Do:  QQ == A\times rA×r 1200012000 == A\times0.01125A×0.01125 \frac{12000}{0.01125}120000.01125​ == AA Divide both sides by 0.011250.01125. \therefore\ A∴ A == 106666.67106666.67 Evaluate to the nearest cent. Therefore, he won \106666.67$$106666.67.

### Perpetuities and using a financial application

We can solve problems involving perpetuities using a financial application by setting the present value (PV) and future value (FV) equal to the same amount. However, the present value should be entered as a negative to indicate depositing the money for investment. The payment (PMT) in this case will be positive as this is returned to the investor.

#### Worked example

##### Example 2

Sarah receives $\$750000$$750000 from an inheritance and wishes to invest the money so that her interest payments cover her monthly living expenses of \2500$$2500 per month.

Ignoring the effects of inflation, solve for the annual interest rate, $r$r, expressed as a percentage, with monthly compounding, that she will need for this investment.

Think:

• Using the financial application of calculator, complete the data for one year.
• Enter the same amount for PV and FV. With PV negative as Sarah is giving this money to the bank and the FV positive.
• The payment (PMT) is positive because from Sarah's point of view the bank is returning the money to her.
• Note we could also use the formula as per example 1.

Do:

Compound Interest
N $12$12
I% ?
PV $-750000$750000
PMT $2500$2500
FV $7500000$7500000
P/Y $12$12
C/Y $12$12

Using the calculator to solve for  $I%$I%, we find that the required rate is $4%$4% p.a.

## Modelling a perpetuity with a recurrence relation

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. A perpetuity is a special case of such an investment where the balance stays constant.

Sequence - Investment with regular withdrawal

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

The sequence which generates the value, $V_n$Vn, of the investment at the end of each instalment period is:

• Recursive sequence:

$V_n=V_{n-1}\times(1+r)-d$Vn=Vn1×(1+r)d, where $V_0=P$V0=P

The sequence which generates the value, $V_n$Vn, of the investment at the beginning of each instalment period is:

• Recursive sequence:

$V_n=V_{n-1}\times(1+r)-d$Vn=Vn1×(1+r)d, where $V_1=P$V1=P

If the situation is a perpetuity will will have:

• A constant term sequence with each term being the initial investment.
• The payment, $d$d,  will be equal to the initial investment multiplied by the rate per period. That is, the payment is the interest accrued each period.

#### Worked example

##### example 3

Lauren receives a significant inheritance and sets up a perpetuity so that she may live off the earnings. The balance at the end of each month, $B_n$Bn, where the interest and payments are made monthly, is modelled by the recurrence relation:

$B_n=1.008B_{n-1}-4000;$Bn=1.008Bn14000;  $B_0=500000$B0=500000

(a)  How much did Lauren inherit?

Think: The amount Lauren inherits will be the initial value of the investment

Do: The value for $B_0$B0 is $500000$500000, thus Lauren inherited $\$500000$$500000 (b) How much does she pay herself each month? Think: Look for the withdrawal amount, that is, the amount subtracted in the recurrence relation. Do: Lauren withdraws \4000$$4000 each month

(c)  What is the nominal annual interest rate for this perpetuity?

Think: Each previous term or previous month's balance is multiplied by $1.008$1.008 which indicates a $0.8%$0.8% interest rate per month.

Do: $0.8\times12=9.6%$0.8×12=9.6% per annum compounded monthly

(d)  Show that this investment does in fact represent a perpetuity.

Think: To represent a perpetuity we need to show that the monthly interest accrued is equal to the monthly withdrawal.

Do: Interest = $0.008\times500000=4000$0.008×500000=4000 which is indeed the value of the monthly withdrawal.

#### Practice questions

Farzad invests his workers' compensation payout of $\$2760000$$2760000 in a perpetuity that pays 2.85%2.85% per annum, compounding quarterly. What is the size of the quarterly payment he will receive? ##### Question 2 Hermione invests her superannuation payout of \500000$$500000 into a perpetuity that will provide a monthly income without using any of the initial investment. If the interest rate of the perpetuity is $9%$9% per annum compounded annually, what monthly payment will Hermione receive?

1. Fill in the values for each of the following. Type an $X$X next to the variable we wish to solve for.

$N$N $1$1 $\left(\editable{}\right)%$()% $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Hence determine the monthly payment in dollars.

##### Question 3

$\$1600016000 is invested in a perpetuity at $3%$3% per annum, compounded annually. A constant amount is withdrawn from the account at the end of each year.

This perpetuity can be defined recursively by $A_{n+1}=aA_n-b$An+1=aAnb, $A_0=c$A0=c, where $A_{n+1}$An+1 is the amount remaining in the account after $n+1$n+1 years.

State the values of $a$a, $b$b and $c$c.

1.  $a$a $=$= $\editable{}$ $b$b $=$= $\editable{}$ $c$c $=$= $\editable{}$

### Outcomes

#### 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