Perpetuity: Financial Definition, Formula, and Examples
What is a Perpetuity?
A perpetuity is an infinite, identical series of cash flows. It can also be thought of as a particular kind of annuity. In this case, periodic payments to you will commence on a specific date and will be paid to you for the remainder of your life.
To comprehend perpetuity one must be familiar with the concept of the time value of money (TVM). A dollar received today has more value (in terms of being able to invest it) than it will at any point in the future, and thus a dollar you might receive 100 years from now is nearly worthless today. Because of this, while a perpetuity pays an infinite amount of cash over an infinite period of time, it still only has a present value of zero.
Although the term 'time value of money' (TVM) refers to concepts relating primarily to time limited periods (a 30-year mortgage, a 10-year bond or a 5-year car loan), one of the exceptions is a financial instrument known as a perpetuity. A perpetuity is a theoretical financial instrument that represents a series of cash flows that will last forever.
Key Features:
- Infinite Life : There is no specified time until maturity for a perpetuity, unlike a regular annuity.
- Fixed Payment Schedule : The cash flow will always occur at a specified interval (usually once per year).
- Decreasing Present Value : As time progresses, the present value of a perpetuity's cash flows becomes very small.
The Mathematics of Perpetuity
When you calculate the value of a perpetuity, the method you use is to capitalize the cash flows at a discount rate. There are two basic types of perpetuities; Constant Perpetuity and Growing Perpetuity.
A. Constant Perpetuity
This form is the easiest type of perpetuity to calculate, as it is basically a constant payment (C), which will be made indefinitely.The Formula:
Pv = 𝑪/𝒓
Where :
Pv = Present Value of the Perpetuity
C = Cash flow amount per period
r = Discount Rate (or required rate of return)
Example Calculation:
A wealthy philanthropist wants to set up a scholarship fund that pays Rs.10,000 to a student every single year, forever. The university's endowment fund earns a consistent 5% annual return. How much must the philanthropist donate today to fund this?
- Using the Formula : PV = 10000/0.05 = Rs. 2,00,000
A one-time donation of Rs.200,000 invested at 5% will generate exactly Rs.10,000 in interest every year (200,000 × 0.05). The principal remains untouched, allowing the payments to continue indefinitely.
B. Growing Perpetuity
To fit the real world, you have to consider that inflation reduces your purchasing power eventually. The present value of Rs. 10,000 today will mean less than it will mean 50 years into the future. To combat this problem, we create a Continually Growing Cash Flow Model, where all cash flows keep increasing at the same percentage rate (g)each year indefinitely.The Formula:
Pv = 𝑪/𝒓-g
Where :
g = The constant growth rate of the cash flows.
Constraint: The discount rate (r) must be higher than the growth rate (g), otherwise the value becomes infinite.
Example Calculation:
Assume the same philanthropist wants the scholarship to keep up with inflation, which is expected to be 2% per year. The endowment still earns 5%.
- PV = 10000/0.05 − 0.02 = 10000/0.03 = Rs. 3,33,333.
Notice the significant jump in required capital. To ensure the payment grows by 2% forever, the donor must put up an additional Rs.133,333 today.
Perpetuity vs. Annuity
Annuities and perpetuities are commonly misunderstood. The most significant difference between the two is related to their periods and the calculation method used to calculate them.
| Feature | Annuity | Perpetuity |
|---|---|---|
| Duration | Finite (e.g., 20 years) | Infinite (Forever) |
| Principal | Eventually depleted | Never depleted (Principal remains intact) |
| Formula | Complex (involves time factor n) | Simple (PV = c/r) |
| Example | Mortgage payments, Pension | Preferred Stock, Consols |
Unlike an annuity, a perpetuity is can be treated as an annuity if the number of times the annuity is compounded is equal to infinity.
Examples of Badly Managed Treaties.
Although Shelter Insurance and Meridian Life are the only two treaties for which it is easy to obtain real-world experience testifying to the fact that these types of treaties can and will be renegotiated, they are not the only examples of this principle being violated. Below are a few examples of other "badly managed" treaties that may be useful as case studies for comparative analysis of global trade agreements and why so many fail to produce any positive results.
- A. The Agreements between British and American
Merchants
One of the worst examples of a badly managed treaty is the 1794 Agreement between the British and American governments regarding trade between their respective countries and the establishment of a system of "inland" customs on both sides of the Atlantic. This agreement had been signed as an exclusive agreement between the American and British governments and was never put into place.
- B. The Treaty of Ghent, 1814
Another prime example of a badly managed treaty is the Treaty of Ghent, which attempted to resolve the unresolved issues between the United States and Great Britain after the conclusion of the War of 1812. Although the Treaty of Ghent ultimately produced a resolution of all outstanding issues between the United States and Great Britain, it was still an unmitigated failure and one of the most wasteful peace agreements ever concluded.
Example : If Coca-Cola issues a preferred stock paying a Rs.2.00 annual dividend and the required return for such risk is 4%, the stock should trade at roughly Rs.50.
- C. Real Estate Valuation (Cap Rates)
The Cap Rate (Capitalization Rate) is a concept that is used by real estate investors in order to determine the value of a property. The Cap Rate essentially represents the discount rate (r) in the perpetuity formula.
Property Value = Net Operating Income (NOI)/Cap Rate
By utilizing a Cap Rate to value a property, it assumes continual generation of Net Operating Income (NOI) by the property indefinitely, as long as the property continues to be maintained and that rent is collected.
Financial Modeling Applications
Perpetuity's ability to provide continuous cash flow is incredibly beneficial and is not limited to the purchase of perpetual bonds. Instead, the most potent use of perpetuity is when valuing a company.
Dividend Discount Model (DDM) - Gordon Growth Model
According to the DDM, the value of a share is calculated by assuming it pays dividends that grow at a fixed annual rate indefinitely.
P0 = D1/r-g
The Growing Perpetuity formula is mathematically identical to DDM's Gordon Growth Model.
The value of the company is equivalent to the present value of all dividend payments.
Terminal Value Methodology (DCF)
The DCF method provides a framework through which investment banks evaluate private companies based on projected cash flows for 5-10 years.
However, after the 5-10 year projection period ends, a private company will still exist (Year 11 → Infinity). A Terminal Value must be estimated to value all business operating profits that occur in all years after the end of the 5-10 year projection period.
This Terminal Value is almost always calculated using the Growing Perpetuity method:
Terminal Value = Final Projected Cash Flow × (1 + g)/WACC-G
Here , WACC(Weighted Average Cost of Capital) is used here as a discount rate, which makes up a significant portion 60-80% of total valuation using DCF, and underlines the importance of appreciating the concept of perpetuity.
Limitations and Risks
While these computations are mathematically sound, its application is fraught with uncertainty because of interest rate risk. Rising rates will cause a huge drop in the value of a perpetual annuity; furthermore, since that annuity has a duration of forever, its value will change dramatically in reaction to interest rate fluctuations.
Therefore, non-inflation-adjusted perpetuities will be considerably devalued by inflation with time. What is worth Rs100 today will have virtually no purchasing power 40 years from now. Additionally, no company or nation can realistically be considered "forever." The assumption of a perpetual cash flow from an asset might lend itself to plausible, perpetual growth (assumes a continued existence of the asset indefinitely).
Conclusion
Perpetuity is a useful concept to consider when determining an investment's value. It provides a means for valuing a preferred stock, considering the value of endowment scholarships or valuing the tech giants using DCF models. The present value of an asset is equivalent to the sum of the cash flows of that asset, discounted to the present. Understanding and applying perpetuity provides investors with an opportunity to see beyond their nearest horizon and realise the long-term growth potential of a capital investment.