A perpetuity in finance is a stream of payments or cash flows that is presumed to extend indefinitely into the future. Learn the importance of perpetuities, with the help of examples of investments.
A perpetuity is an extension of the concept of an annuity. In finance, an annuity is a stream of equal payments for a set period of time. Examples of annuities are bonds and fixed-rate mortgages. Both have a fixed payment and a maturity date.
The value of an annuity stream is determined by the amount of the expected cash flows, the number of compounding periods (timing of payments), the maturity value (which would be par value for a bond or zero for a fully amortizing mortgage), and an interest rate assumption. Knowing these variables, one can determine the present value (see "net present value") of an annuity at any point in time. Determining the value of an annuity in this manner could help investors looking to buy or sell bonds, or a bank looking to purchase a mortgage from another financial institution.
A perpetuity comes into play when an annuity has no maturity date. A preferred stock, for example, can continue to pay dividends for as long as the issuing corporation decides to do so. The preferred shares thus provide a stream of payments indefinitely into the future. The intrinsic value of such an investment cannot be determined with an annuity formula, since that would require a finite time period.
Because payments are fixed and regular, the intrinsic value of a perpetuity can be determined with a standardized formula and calculated for any point in time.
A perpetuity is therefore an annuity that has no set period of time and is instead assumed to continue, theoretically at least, forever. This assumption, along with an interest rate (also referred to as a discount rate), allows the value of perpetuities to be represented by a rather simple equation.
A perpetuity generates payments or cash flows indefinitely and a perpetuity calculation can be used to determine either a present value for an investment or its projected future value, based solely on its perpetual income stream and the expected discount rate. That means it will be most applicable to investments for which income is the only way that investors receive benefit, as it does not consider any other factors that might lead to an investment appreciating or returning capital. That makes it applicable to investments like real estate or preferred stocks. It can also be used as part of discounted cash flow (DCF) analysis on common stock.
The formula that is used to describe a simple perpetuity is:
PV = present value,
R = the interest or discount rate.
(Both the cash flow and the interest rate need to be expressed for the same time period, which is normally years.)
Present value describes the value today for the cash flow and interest rate shown. The present value can be taken as the value someone should be willing to pay for the perpetuity indicated.
Tip: Calculated present values of annuities and perpetuities are theoretical calculations and can be used to determine a “fair present value”. The actual trading value of an annuity or perpetuity may be different, depending on supply and demand or a different interest rate assumption.
The formula above is for the simplest form of perpetuity – one that assumes a constant interest rate and a constant cash flow. More complex calculations are required when either the interest rate or cash flow is expected to change over time. One scenario, however, which requires only a slight modification of the formula is where cash flow is expected to grow over time at a steady annual growth rate (G).
The formula for a growing perpetuity is:
The growth factor here reduces the denominator of the formula, resulting in a higher PV than if expected growth was 0. It is expected that any growth factor is a positive number, as opposed to negative.
Preferred stock with a $2.25 annual dividend.
As an example, let’s assume the ABC Widget Company has a preferred stock that pays a dividend of $2.25 per year, where there is no maturity date. What might the fair value of those preferred shares be worth if current interest rates for comparably risky investments were at 4% annually?
Using the perpetuity formula, we would have:
PV = CF/R
PV = 2.25/.04 = $56.25
The investor should be willing to pay $56.25 to achieve a 4% return.
If the current interest rate level were 7%, the Present Value of this perpetuity would naturally decrease. We could calculate it as:
PV = 2.25/.07 = $32.14
Let's assume the same preferred shares, with a 4% discount rate, but add an expected growth rate in the dividend of 2% per year.
PV = CF/(R - G)
PV = 2.25/(.04 -.02) = 2.25/.02 = $112.50
A modest growth rate of only 2% per year in the dividend payment would assume that next year’s dividend would be $2.25 X 1.02 = $2.30 and so on. Because perpetuities extend indefinitely, it is easy to see how even a small increase each year can boost the value of the investment considerably. In this particular example, the present value is doubled.
Any investment that pays income and does not have a maturity date can be viewed, at least in part, as a perpetuity. As mentioned above, preferred stocks are often modeled as perpetuities, as are income-producing real estate investments, such as REITs. Funds or ETFs composed of these instruments may also be considered perpetuities.
In addition, any common stock can be modeled as a perpetuity using the company’s total cash flows, rather than its dividends. Discounted cash flow analysis is a commonly used method of determining the intrinsic value of a stock, even when there are no dividends paid by the company at all.
Tip: Investments that are essentially perpetuities can be attractive to investors, especially if the cash flows are expected to increase over time and the issuer has strong financials to support the cash flow. Perpetuity valuations, however, are highly sensitive to interest rates and can decline in value when interest rates rise.
Income-producing investments that have fixed, regular payments and no maturity date can be modeled as perpetuities and valued by a simple formula. Investors can utilize the perpetuity formula to help determine the fair value of an investment as well as the projected future value they might expect at a later date.
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