Perpetuity
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.
The equation for perpetuity is: \text{Withdrawal amount (payment)} = \text{Interest accrued} or Q = A \times r where Q is the amount of interest earned (size of the prize or payment), A is the initial amount invested in dollars, and r is the interest rate for the period as a decimal.
Examples
Example 1
Farzad invests his workers' compensation payout of \$2\,760\,000 in a perpetuity that pays 2.85\% per annum, compounding quarterly. What is the size of the quarterly payment he will receive?
The perpetuity equation:
Perpetuities with the 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.
Examples
Example 2
Hermione invests her superannuation payout of \$500\,000 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\% per annum compounded annually, what monthly payment will Hermione receive?
Fill in the values for each of the following. Put an X next to the variable we wish to solve for.
| N | 1 |
|---|---|
| I\% | ⬚\% |
| PV | ⬚ |
| PMT | ⬚ |
| FV | ⬚ |
| P/Y | ⬚ |
| C/Y | ⬚ |
Hence determine the monthly payment in dollars.
When solving problems involving perpetuities using a financial application, the present value (PV) and future value (FV) equal the same amount, but the present value should be entered as a negative. The payment (PMT) should be entered as a positive.
Perpetuities as recurrence relations
This is similar to modelling an investment with regular payments with the payment in this case being withdrawn and hence, negative in the form shown below.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For a principal investment, P, at the compound interest rate of r per period and a payment d withdrawn per period, the sequence of the value of the investment over time forms a first order linear recurrence.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 sequence which generates the value, V_{n}, of the investment at the end of each instalment period is:
Recursive sequence: V_{n}=V_{n-1} \times \left(1+r \right) - d, where V_{0}=P
The sequence which generates the value, V_{n}, of the investment at the beginning of each instalment period is:
Recursive sequence: V_{n}=V_{n-1} \times \left(1+r \right) - d, where V_{1}=P
If the situation is a perpetuity we will have:
A constant term sequence with each term being the initial investment.
The payment, d, will be equal to the initial investment multiplied by the rate per period. That is, the payment is the interest accrued each period.
Examples
Example 3
\$16\,000 is invested in a perpetuity at 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, A_{0}=c, where A_{n+1} is the amount remaining in the account after n+1 years.
State the values of a, \, b, and c.
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 sequence which generates the value, V_{n}, of the investment at the end of each instalment period is:
Recursive sequence: V_{n}=V_{n-1} \times \left(1+r \right) - d, where V_{0}=P
The sequence which generates the value, V_{n}, of the investment at the beginning of each instalment period is:
Recursive sequence: V_{n}=V_{n-1} \times \left(1+r \right) - d, where V_{1}=P
Where d is equal to the initial investment multiplied by the rate per period.