Perpetuity: Financial Definition, Formula, and Examples

Perpetuity

Investopedia / Jiaqi Zhou

What Does Perpetuity Mean in Finance?

A perpetuity is a security that pays for an infinite amount of time. In finance, perpetuity is a constant stream of identical cash flows with no end.

The concept of perpetuity is also used in several financial theories, such as in the dividend discount model (DDM).

Key Takeaways

  • A perpetuity, in finance, refers to a security that pays a never-ending cash stream.
  • It is essentially an annuity with no termination date.
  • The present value of a perpetuity is determined by simply dividing the amount of the regular cash flows by the discount rate.
  • A growing perpetuity includes a growth rate that increases the cash flows received each period going forward.
  • Perpetuities today are uncommon financial products, but the concept of a perpetuity is nonetheless important in finance.
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Perpetuity

Understanding Perpetuity

An annuity is a stream of cash flows. A perpetuity is a type of annuity that lasts forever, into perpetuity. The stream of cash flows continues for an infinite amount of time. In finance, a person uses the perpetuity calculation in valuation methodologies to find the present value of a company's cash flows when discounted back at a certain rate.

An example of a financial instrument with perpetual cash flows was the British-issued bonds known as consols, which the Bank of England phased out in 2015. By purchasing a consol from the British government, the bondholder was entitled to receive annual interest payments forever.

Although it may seem a bit illogical, an infinite series of cash flows can have a finite present value. Because of the time value of money, each payment is only a fraction of the last.

Specifically, the perpetuity formula determines the amount of cash flows in the terminal year of operation. In valuation, a company is said to be a going concern, meaning that it goes on forever. For this reason, the terminal year is a perpetuity, and analysts use the perpetuity formula to find its value.

Perpetuity Present Value Formula

The formula to calculate the present value of a perpetuity, or security with perpetual cash flows, is as follows:

PV = C ( 1 + r ) 1 + C ( 1 + r ) 2 + C ( 1 + r ) 3 = C r where: PV = present value C = cash flow r = discount rate \begin{aligned} &\text{PV} = \frac { C }{ ( 1 + r ) ^ 1 } + \frac { C }{ ( 1 + r ) ^ 2 } + \frac { C }{ ( 1 + r ) ^ 3 } \cdots = \frac { C }{ r } \\ &\textbf{where:} \\ &\text{PV} = \text{present value} \\ &C = \text{cash flow} \\ &r = \text{discount rate} \\ \end{aligned} PV=(1+r)1C+(1+r)2C+(1+r)3C=rCwhere:PV=present valueC=cash flowr=discount rate

The basic method used to calculate a perpetuity is to divide cash flows by some discount rate. The formula used to calculate the terminal value in a stream of cash flows for valuation purposes is a bit more complicated. It is the estimate of cash flows in year 10 of the company, multiplied by one plus the company’s long-term growth rate, and then divided by the difference between the cost of capital and the growth rate.

Simply put, the terminal value is some amount of cash flows divided by some discount rate, which is the basic formula for a perpetuity.

Perpetuity Example

For example, if a company is projected to make $100,000 in year 10, and the company’s cost of capital is 8%, with a long-term growth rate of 3%, the value of the perpetuity is as follows:

= Cash Flow Year 10 × ( 1 + g ) r g = $ 100 , 000 × 1.03 0.08 0.03 = $ 103 , 000 0.05 = $ 2.06  million \begin{aligned} &= \frac{ \text{Cash Flow}_\text{Year 10} \times ( 1 + g ) }{ r - g } \\ &= \frac{ \$100,000 \times 1.03 }{ 0.08 - 0.03 } \\ &= \frac{ \$103,000 }{ 0.05 } \\ &= \$2.06 \text{ million} \\ \end{aligned} =rgCash FlowYear 10×(1+g)=0.080.03$100,000×1.03=0.05$103,000=$2.06 million

This means that $100,000 paid into a perpetuity, assuming a 3% rate of growth with an 8% cost of capital, is worth $2.06 million in 10 years. Now, a person must find the value of that $2.06 million today. To do this, analysts use another formula referred to as the present value of a perpetuity.

Growing Perpetuities

The net present value of a perpetuity is not as large as it might seem due to the fact that the time value of money erodes the value of dollars far into the future (e.g., due to inflation). Therefore, the cash flows received by a fixed perpetuity many years from now can become negligible in terms of future buying power.

A growing perpetuity adjusts the amount of perpetual payments each period by the inflation rate, ensuring a constant level of buying power over time. The present value of a growing perpetuity will therefore be greater than a fixed or non-growing perpetuity. The higher the growth rate of future payments per period, the greater the present value.

The formula for a growing perpetuity is nearly identical to the standard formula, but subtracts the rate of inflation (also known as the growth rate, g) from the discount rate, r, in the denominator:

PV = C/(r-g)

Note that the rate of growth in a growing perpetuity remains fixed over its infinite life, making it only able to include a rough estimate of what inflation may be, on average, over the long run.

How Does Perpetuity Work in a Security?

A perpetuity is a financial instrument that offers a stream of cash flows in perpetuity—that is, without end. Before 2015, the U.K. offered a government bond called a “consol” that was structured as a perpetuity, although these instruments have since been discontinued. Unlike other bonds, perpetuities do not have a fixed maturity date, but instead, continue paying interest indefinitely.

How Is a Perpetuity Valued?

At first glance, it may seem as though an instrument that offers an infinite stream of cash flows would be almost infinitely valuable, but this is not the case. Mathematically speaking, the value of a perpetuity is finite, and its value can be determined by discounting its future cash flows to the present using a specified discount rate. This procedure, known as discounted cash flow (DCF) analysis, is also widely used to value other types of securities, such as stocks, bonds, and real estate investments.

What Is the Difference Between a Perpetuity and an Annuity?

A perpetuity and an annuity are similar instruments in that both offer a fixed set of cash flows over time. However, the key difference between them is that annuities have a predetermined end date, known as the “maturity date,” whereas perpetuities are intended to last forever. Importantly, both annuities and perpetuities can be valued using DCF analysis.

How Long Does a Perpetuity Last?

Forever.

The Bottom Line

Perpetuities are investments that make payments indefinitely, with no maturity or expiration date. They are essentially never-ending annuities. Perpetuities as financial products are quite rare today, but the abstract concept of a perpetuity and the calculation of its present value (by dividing the cash flow amount by the discount rate) remains a key concept in finance.

Article Sources
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  1. Federal Reserve Bank of St. Louis. "Consols: The Never-Ending Bonds."

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