Perpetuity Formula: Meaning, Calculation, and Examples
Learn how to calculate the perpetuity formula.
What is a Perpetuity?
A perpetuity is a type of investment that pays a set amount of cash flows indefinitely, with no end date. Perpetuities are seen as a theoretical concept in corporate finance as nothing can really last forever.
Although perpetuities do not currently exist, the financial theory of perpetuity is used in valuation models such as the dividend discount model. Using the dividend discount model, an investor can assume a fair stock price for a company based on the sum of the present value of all future dividend payments. The formula for the dividend discount model assumes dividends will continue indefinitely in the future because the company is assumed to be a going concern. That means the company is expected to be able to meet its upcoming financial obligations as they come due and continue in its usual course of business for the foreseeable future.
Perpetuities are rare and there are hardly any examples of perpetuities in current times. Previously, the British government had issued a type of perpetuity– a bond known as a consol. It was first introduced in 1751 and had been in circulation for centuries. In 2015, the British government redeemed all consols in circulation.
What Is The Perpetuity Formula?
The formula for perpetuity is:
PV = CF/ R
Where:
- PV = Present value
- CF = Cash flow payments
- R = Interest rate, discount rate, or yield
How Do I Calculate The Present Value of a Perpetuity?
In finance, the present value of a financial instrument is the current value of the future cash flows discounted back to the present based on a given rate of return. You can calculate the present value of a perpetuity by dividing the cash flows by the stated interest rate (also known as the discount rate or yield).
Example of Present Value of The Perpetuity Formula
Assume you are to receive an annual payment of $10,000 indefinitely. The stated interest rate on the financial instrument is 4%. The present value of the perpetuity would be:
$10,000 / .04 = $250,000
- $10,000 = Cash flow payment
- 4% = .04 = Interest rate, discount rate, or yield
- $250,000 = Present Value ofPerpetuity
Growing Perpetuity
To understand a growing perpetuity, we first need to define the time value of money. The time value of money is a financial principle that states that the value of a dollar today is worth more than the value of the same dollar in the future. This is due to inflation reducing the value of the dollar and being able to grow that dollar over time in savings accounts earning interest.
If you have a perpetuity with a fixed payment, the cash flows will be worth less in the future due to the time value of money. A growing perpetuity counteracts the effects of inflation by adjusting the payment amount each period to the current inflation rate in order to maintain the same buying power over time.
The formula for the present value of a growing perpetuity is:
PV of a Growing Perpetuity = CF1 / (R - G)
Where:
- CF1 = Cash flow from period 1(dividend or coupon payment)
- R = Interest rate, discount rate, or yield
- G = Growth rate of the growing perpetuity
Example of Present Value ofGrowing Perpetuity Formula
Assume you are to receive an annual cash flow that will continue indefinitely. The amount you will receive in the first year is $10,000. The cash flow is expected to grow at 5% per year, and the required interest rate is 10%. The present value of the growing perpetuity would, therefore, be:
$10,000 / (.10 - .05) = $200,000
- $10,000 = Cash Flow in Period 1
- 10% = .10 = Interest rate, discount rate, or yield
- 5% = .05 = Growth rate
- $200,000 = Present Value of the Growing Perpetuity
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Real Company Example: British Government Consols
The British government issued consols, a type of perpetual bond, in 1751. The consols had an initial interest rate of 3.5%, but the amount fluctuated in later years. The bonds had no end date and fixed coupon payments were made to bondholders until the British government redeemed all consols in circulation in 2015.
How are Perpetuity and Annuity Different?
A perpetuity does not have an end date, while an annuity has a pre-defined end date.
Annuity
An annuity is a financial instrument that pays a set amount of cash flow payments over a predetermined time period. Annuities offer a guaranteed income stream, and it can be a useful way for retirees to ensure they do not outlive their savings. An investor can contract with an insurance company or financial institution to invest in an annuity and determine an appropriate payout schedule based on their planned standard of living in the future.
There are two phases in an annuity’s life cycle:
- The accumulation phase
- The annuitization phase
The accumulation phase occurs when an annuity is being funded by the investor, either in a lump sum or through periodic payments. The annuitization phase occurs once payouts begin.
Types of annuities:
- Immediate annuity
- Deferred annuity
- Fixed annuity
- Variable annuity
- Ordinary annuity
- Annuity due
An immediate annuity can be purchased at any age. It allows the investor to exchange a lump sum of money for specific cash flow payouts over a period of time. People who win the lottery or a large lawsuit settlement may opt for this kind of annuity to ensure constant payments over time. A deferred annuity allows the investor’s money to grow tax-deferred until payouts begin in the annuitization phase. It is commonly used by people who want guaranteed income in their retirement years.
A fixed annuity pays specific, guaranteed periodic payments to the investor. It is more common for retirement investors. A variable annuity fluctuates based on the performance of the investment portfolio. It is less stable than a fixed annuity, but it allows the investor to capitalize on investments with strong returns.
An ordinary annuity– or annuity in arrears– is one in which the payments are made at the end of each period. When you pay your mortgage, you pay at the end of the month for the prior month.
An annuity due is one in which the payments are made at the beginning of each period. An example of an annuity due would be typical rental payments, which are made at the beginning of a period for the following month.
Annuity End Dates:
- Life Only
- Life With Refund
- Life With Period Certain
- Period Certain Only
Annuities have end dates that are determined upfront in the contract. With a life-only annuity, payments continue until your death. A joint life annuity allows payments to continue until the death of a second person, such as your spouse. Life with a refund guarantees that your beneficiary will receive the amount you paid into the annuity if you happen to die prior to that payout.
Life with period certain guarantees that you will receive payments until your death, or for a minimum number of years even if you die prior. In that case, your beneficiary will receive the remaining payouts. Period certain only guarantees payments for a specific number of years. If you outlive the time period, your payments stop. If you die during the period, it continues to pay out to your beneficiaries.
Perpetuity
Unlike an annuity, a perpetuity does not have an end date. Perpetuities continue indefinitely. Since an annuity has a set time period, it uses compound interest to determine its present value. However, a perpetuity uses a stated interest rate to determine its present value.
How long does a perpetuity last?
A perpetuity lasts forever. It does not have an end date.
Additional Resources
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