Ideas
Perpetuity
A perpetuity is similar to an annuity, in that interest is generated on an account and an amount is withdrawn, but in the case of a perpetuity the withdrawal is exactly equal to the interest generated. In this way, the balance of the perpetuity never increases or decreases - it remains perpetually constant, and lasts indefinitely. As such, the relationship between a perpetuity and an annuity is the same as the relationship between an interest only loan and a reducing balance loan.
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
Othman wants to start a scholarship where the top student each year receives a \$4000 prize. If the interest on the initial investment averages 4\% per annum, compounded annually, how much should his initial investment be?
The perpetuity equation:
Perpetuity with an amortisation table
A perpetuity has a much simpler looking amortisation table than an annuity. In fact, every value remains constant between each time period, and so each row is identical (other than the period number).
For example, consider a scholarship fund of \$50\,000 which is invested into a perpetuity at a rate of 12\% per annum, where the interest and payments are calculated monthly.
| Period | Payment | Interest earned | Reduction in principal | Balance |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 50\,000 |
| 1 | 500 | 500 | 0 | 50\,000 |
| 2 | 500 | 500 | 0 | 50\,000 |
| \ldots | \ldots | \ldots | \ldots | \ldots |
| n | 500 | 500 | 0 | 50\,000 |
As long as the withdrawal payment always matches the amount of interest earned, then the investment will last indefinitely.
In perpetuity, the investment will last indefinitely as long as the withdrawal payment always equals the amount of interest received.
Perpetuity with a recurrence relation
The type of recurrence relation used to model a perpetuity is the same as that of an annuity, since there is interest generated and a payment/withdrawal taken out at each time period.
Interest on a perpetuity can be modelled using the following recurrence relation: A_{n+1}=RA_n-d,\,A_0=P where A_{n+1} is the value of the investment after n+1 time periods, R equals 1+\dfrac{r\%}{100}, usually expressed as a decimal, where r\% is the interest rate, d is the amount of the payment, which is equal to the amount of interest earned, and P is the initial value of the investment or loan (the principal value).
Examples
Example 2
\$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.
Interest on a perpetuity can be modelled using the following recurrence relation:
Perpetuity with the financial application
A perpetuity has a similar setup as that of an annuity investment, but with two important differences.
The number of periods, N, is irrelevant since a perpetuity lasts forever. So it is simplest to set this value to be 1, and consider what happens in just one time period.
PV and FV have the same value (since the overall balance doesn't change) just with opposing signs. (PV will be negative, as initially money has been invested, while FV will be positive as the final value is returned to the investor.)
Examples
Example 3
A community club has \$100\,000 and decides to invest in a perpetuity which earns 5.2\% per annum, compounding annually. Each year the club grants the \$5200 payout from the perpetuity to a local sporting club.
How much extra would the annual grant be if the interest were compounded monthly instead?
A perpetuity has a similar setup as that of an annuity investment, but with two important differences.
The number of periods, N, is irrelevant since a perpetuity lasts forever. So it is simplest to set this value to be 1, and consider what happens in just one time period.
PV and FV have the same value (since the overall balance doesn't change) just with opposing signs. (PV will be negative, as initially money has been invested, while FV will be positive as the final value is returned to the investor.)