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
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 $A$A, with $r=1.125%$r=1.125% and the payment $Q=\$12000$Q=$12000.
Do:
$Q$Q  $=$=  $A\times r$A×r 

$12000$12000  $=$=  $A\times0.01125$A×0.01125 

$\frac{12000}{0.01125}$120000.01125  $=$=  $A$A 
Divide both sides by $0.01125$0.01125. 
$\therefore\ A$∴ A  $=$=  $106666.67$106666.67 
Evaluate to the nearest cent. 
Therefore, he won $\$106666.67$$106666.67.
We can solve problems involving perpetuities using a financial application by setting the present value (PV) and future value (FV) equal to the same amount. However, the present value should be entered as a negative to indicate depositing the money for investment. The payment (PMT) in this case will be positive as this is returned to the investor.
Sarah receives $\$750000$$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:
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.
This is similar to modelling an investment with regular payment with the payment in this case being withdrawn and hence, negative in the form shown below.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.
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:
$V_n=V_{n1}\times(1+r)d$Vn=Vn−1×(1+r)−d, where $V_0=P$V0=P
$V_n=V_{n1}\times(1+r)d$Vn=Vn−1×(1+r)−d, where $V_1=P$V1=P
If the situation is a perpetuity will will have:
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:
$B_n=1.008B_{n1}4000;$Bn=1.008Bn−1−4000; $B_0=500000$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 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.
Farzad invests his workers' compensation payout of $\$2760000$$2760000 in a perpetuity that pays $2.85%$2.85% per annum, compounding quarterly. What is the size of the quarterly payment he will receive?
Hermione invests her superannuation payout of $\$500000$$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.
$\$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 $A_{n+1}=aA_nb$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  $=$=  $\editable{}$ 
with the aid of a financial calculator or computerbased 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