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.
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 examples
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 $$12000$12000 per quarter without using any of the principal. How much money did Fred win?
Think: $4.5$4.5% p.a. is $1.125$1.125% per quarter. Therefore $1.125$1.125% of the principal = $$12000$12000
Do: $Q=A\times r$Q=A×r
$12000=A\times0.01125$12000=A×0.01125
$A=106666.67$A=106666.67
Therefore he won $\$106666.67$$106666.67
Example 2
Sarah receives $$750000$750000 from an inheritance and wishes to invest the money so that her interest payments cover her monthly living expenses of $$2500$2500 per month.
Ignoring the effects of inflation, solve for the annual interest rate, $r$r, expressed as a percentage, with monthly compounding, that she will need for this investment.
Think:
- Using the financial application of calculator, complete the data for one year.
- Put PV = FV (PV is negative because Sarah is giving this money to the bank and the FV will be positive)
- Payment is positive because from Sarah's point of view the bank is returning the money to her.
- Note we could also use the formula as per example 1.
Do:
| Compound Interest | |
|---|---|
| N | $12$12 |
| I% | ? |
| PV | $-750000$−750000 |
| PMT | $2500$2500 |
| FV | $7500000$7500000 |
| P/Y | $12$12 |
| C/Y | $12$12 |
Using the calculator to solve for $I%$I%, we find that the required rate is $4$4% p.a
Modelling a perpetuity with a recurrence relation
Let's examine the following situation.
Worked example
example 3
Lauren receives a significant inheritance and sets up a perpetuity so that she may live off the earnings. The balance at the end of each month, $B_{n+1}$Bn+1, where the interest and payments are made monthly, is modelled by the recurrence relation:
$B_{n+1}=1.008B_n-4000;$Bn+1=1.008Bn−4000; $B_0=500000$B0=500000
(a) How much did Lauren inherit?
Think: The amount Lauren inherits will be the initial value of the investment
Do: The value for $B_0$B0 is $500000$500000, thus Lauren inherited $\$500000$$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$$4000 each month
(c) What is the annual interest rate for this perpetuity?
Think: Each previous term or previous month's balance is multiplied by $1.008$1.008 which indicates a $0.8%$0.8% interest rate per month.
Do: $0.8\times12=9.6%$0.8×12=9.6% per annum
(d) Show that this investment does in fact represent a perpetuity.
Think: To represent a perpetuity we need to show that the monthly interest accrued is equal to the monthly withdrawal.
Do: Interest = $0.008\times500000=4000$0.008×500000=4000 which is exactly the value of the monthly withdrawal.
Practice questions
Question 1
Jenny receives $\$600000$$600000 from an inheritance and wishes to invest the money so that her interest payments cover her monthly living expenses of $\$1500$$1500 per month.
Ignoring the effects of inflation, solve for the annual interest rate, $r$r, expressed as a percentage, with monthly compounding, that she will need for this investment.
Question 2
Hermione invests her superannuation payout of $\$500000$$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?
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 $I%$I% $\left(\editable{}\right)%$()% $PV$PV $\editable{}$ $PMT$PMT $\editable{}$ $FV$FV $\editable{}$ $P/Y$P/Y $\editable{}$ $C/Y$C/Y $\editable{}$ Hence determine the monthly payment in dollars.
Question 3
$\$16000$$16000 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=aAn−b, $A_0=c$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.
$a$a $=$= $\editable{}$ $b$b $=$= $\editable{}$ $c$c $=$= $\editable{}$