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
$M=A\times i$M=A×i
where $M$M is the regular payment or size of prize
$A$A is the amount invested in dollars
$i$i 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 $\$12\ 000$$12 000 per quarter without using any of the principal. How much money did Fred win?
Think: $4.5%$4.5% p.a. is $1.125%$1.125% per quarter. Hence, we are solving for $A$A, with $i=0.001125$i=0.001125 and the payment $M=\$12\ 000$M=$12 000.
Do:
| $M$M | $=$= | $A\times i$A×i |
|
| $12\ 000$12 000 | $=$= | $A\times0.01125$A×0.01125 |
|
| $\frac{12\ 000}{0.01125}$12 0000.01125 | $=$= | $A$A |
Divide both sides by $0.01125$0.01125. |
| $\therefore\ A$∴ A | $=$= | $1\ 066\ 666.67$1 066 666.67 |
Evaluate to the nearest cent. |
Therefore, he won $\$1\ 066\ 666.67$$1 066 666.67.
Modelling a perpetuity with a recurrence relation
Just as we modelled present value annuities, we can also model a perpetuity as an investment with a regular withdrawal. A perpetuity is a special case of such an investment where the balance stays constant.
For an annuity with initial investment $P$P, regular withdrawals of $d$d per period and earning compound interest at a rate of $i$i per payment period, the recursive sequence which generates the value, $A_n$An, of the investment at the end of each instalment period is:
$A_{n+1}=rA_n-d$An+1=rAn−d, where $V_0=P$V0=P and $r=1+i$r=1+i
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 2
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:
$B_{n+1}=1.008B_n-4000;$Bn+1=1.008Bn−4000; $B_0=500\ 000$B0=500 000
(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 $500\ 000$500 000, thus Lauren inherited $\$500\ 000$$500 000.
(b) How much does she pay herself each month?
Think: Look for the withdrawal amount, that is, the amount subtracted in the recurrence relation.
Do: Lauren withdraws $\$4000$$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\times500\ 000=4000$0.008×500 000=4000 which is indeed the value of the monthly withdrawal.
Practice questions
Question 1
Hourieh invests his workers' compensation payout of $\$2240000$$2240000 in a perpetuity that pays $5.35%$5.35% per annum, compounding quarterly. What is the size of the quarterly payment he will receive?
Question 2
$\$30000$$30000 is invested in a perpetuity at $6%$6% per annum, compounded monthly. A constant amount is withdrawn from the account at the end of each month. This perpetuity can be defined recursively by
$A_{n+1}=a\times A_n-b$An+1=a×An−b, $A_0=c$A0=c
where $A_{n+1}$An+1 is the amount remaining in the account after $n+1$n+1 months.
What is the value of $a$a?
What is the value of $b$b?
What is the value of $c$c?
Perpetuities and using a financial solver
We can solve problems involving perpetuities using an online financial solver 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 3
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, $i$i, expressed as a percentage, with monthly compounding, that she will need for this investment.
Think:
- Using a financial solver, 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 solver to solve for $I%$I%, we find that the required rate is $4%$4% p.a.
Question 3
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.