A perpetuity is an
annuity
In investment, an annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings account, mo ...
that has no end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence. For example, the
United Kingdom
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a country in Europe, off the north-western coast of the continental mainland. It comprises England, Scotland, Wales and North ...
(UK) government issued them in the past; these were known as
consols
Consols (originally short for consolidated annuities, but subsequently taken to mean consolidated stock) were government debt issues in the form of perpetual bonds, redeemable at the option of the government. They were issued by the Bank of Englan ...
and were all finally redeemed in 2015. Real estate and
preferred stock
Preferred stock (also called preferred shares, preference shares, or simply preferreds) is a component of share capital that may have any combination of features not possessed by common stock, including properties of both an equity and a debt ins ...
are among some types of investments that affect the results of a perpetuity, and prices can be established using techniques for valuing a perpetuity. Perpetuities are but one of the
time value of money
The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference.
The t ...
methods for valuing financial assets. Perpetuities are a form of ordinary annuities.
The concept is closely linked to
terminal value and terminal growth rate in
valuation.
Detailed description
A perpetuity is an
annuity
In investment, an annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings account, mo ...
in which the periodic payments begin on a fixed date and continue indefinitely. It is sometimes referred to as a perpetual annuity. Fixed coupon payments on permanently invested (irredeemable) sums of money are prime examples of perpetuities. Scholarships paid perpetually from an endowment fit the definition of perpetuity.
The value of the perpetuity is finite because receipts that are anticipated far in the future have extremely low present value (
present value
In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has inte ...
of the future cash flows). Unlike a typical bond, because the
principal is never repaid, there is no present value for the principal. Assuming that payments begin at the ''end'' of the current period, the price of a perpetuity is simply the coupon amount over the appropriate
discount rate or yield; that is,
:
where PV = present value of the perpetuity, A = the amount of the periodic payment, and r = yield,
discount rate or
interest rate
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
.
To give a numerical example, a 3% UK government war loan will trade at 50 pence per pound in a yield environment of 6%, while at 3% yield it is trading at par. That is, if the face value of the loan is £100 and the annual payment £3, the value of the loan is £50 when market interest rates are 6%, and £100 when they are 3%.
The
duration
Duration may refer to:
* The amount of time elapsed between two events
* Duration (music) – an amount of time or a particular time interval, often cited as one of the fundamental aspects of music
* Duration (philosophy) – a theory of time and ...
, or the price-sensitivity to a small change in the interest rate r, of a perpetuity is given by the following formula:
:
This of course follows the fact that for bigger changes the new price must be calculated with the present-value formula given that for changes greater than a few basis-points the calculated duration is not reflective of the true-change in price.
Real-life examples
Valuing real estate with a
capitalization rate
Capitalization rate (or "cap rate") is a real estate valuation measure used to compare different real estate investments. Although there are many variations, the cap rate is generally calculated as the ratio between the annual rental income produ ...
or cap rate (the convention used in
real estate finance) is a more current example . Using a cap rate, the value of a particular real estate asset is either the
net income
In business and accounting, net income (also total comprehensive income, net earnings, net profit, bottom line, sales profit, or credit sales) is an entity's income minus cost of goods sold, expenses, depreciation and amortization, interest, a ...
or the
net cash flow
A cash flow is a real or virtual movement of money:
*a cash flow in its narrow sense is a payment (in a currency), especially from one central bank account to another; the term 'cash flow' is mostly used to describe payments that are expected ...
of the property, divided by the cap rate. Effectively, the use of a cap rate to value a piece of real estate assumes that the current income from the property continues in perpetuity. Underlying this valuation is the assumption that rents will rise at the same rate as
inflation
In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reductio ...
. Although the property may be sold in future (or even the very near future), the assumption is that other investors will apply the same valuation approach to the property.
UK government perpetuities (called
consols
Consols (originally short for consolidated annuities, but subsequently taken to mean consolidated stock) were government debt issues in the form of perpetual bonds, redeemable at the option of the government. They were issued by the Bank of Englan ...
) were undated as well as irredeemable except by act of Parliament. As with
war bonds
War bonds (sometimes referred to as Victory bonds, particularly in propaganda) are debt securities issued by a government to finance military operations and other expenditure in times of war without raising taxes to an unpopular level. They are a ...
, they paid fixed coupons (interest payments), and traded actively in the bond market until the government redeemed them in 2015. Very long dated bonds have financial characteristics that can appeal to some investors and in some circumstances: ''e.g.'' long-dated bonds have prices that change rapidly (either up or down) when yields change (fall or rise) in the financial markets.
The constant growth
dividend discount model In finance and investing, the dividend discount model (DDM) is a method of valuing the price of a company's stock based on the fact that its stock is worth the sum of all of its future dividend payments, discounted back to their present value. In ot ...
for the valuation of the common stock of a corporation is another example. This model assumes that the market price per share is equal to the discounted stream of all future dividends, which is assumed to be perpetual. If the discount rate for stocks (shares) with this level of
systematic risk
In finance and economics, systematic risk (in economics often called aggregate risk or undiversifiable risk) is vulnerability to events which affect aggregate outcomes such as broad market returns, total economy-wide resource holdings, or aggrega ...
is 12.50%, then a constant perpetuity of dividend income per dollar is eight dollars. However, if the future dividends represent a perpetuity increasing at 5.00% per year, then the dividend discount model, in effect, subtracts 5.00% off the discount rate of 12.50% for 7.50% implying that the price per dollar of income is $13.33.
References
See also
{{EB1911 poster, Perpetuity
*
Geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For e ...
*
Perpetual bond
A perpetual bond, also known colloquially as a perpetual or perp, is a bond with no maturity date, therefore allowing it to be treated as equity, not as debt. Issuers pay coupons on perpetual bonds forever, and they do not have to redeem the prin ...
*
Jacob Lund Fisker
Jacob Lund Fisker (born 1975) is a Danish astrophysicist and writer. He is known as the author of a philosophy of extreme early retirement that has inspired a lifestyle movement. Fisker's book ''Early Retirement Extreme'' discusses how to becom ...
Mathematical finance
Annuities