finance Finance refers to monetary resources and to the study and Academic discipline, discipline of money, currency, assets and Liability (financial accounting), liabilities. As a subject of study, is a field of Business administration, Business Admin ...

, a perpetuity is an

annuity In investment, an annuity is a series of payments made at equal intervals based on a contract with a lump sum of money. Insurance companies are common annuity providers and are used by clients for things like retirement or death benefits. Examples ...

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 Northwestern Europe, off the coast of European mainland, the continental mainland. It comprises England, Scotlan ...

(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 bond, government debt issues in the form of perpetual bonds, redeemable at the option of the government. The first British co ...

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

investment Investment is traditionally defined as the "commitment of resources into something expected to gain value over time". If an investment involves money, then it can be defined as a "commitment of money to receive more money later". From a broade ...

s that effect 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 refers to the fact that there is normally a 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 time ...

methods for valuing

financial asset A financial asset is a non-physical asset whose value is derived from a contractual claim, such as deposit (finance), bank deposits, bond (finance), bonds, and participations in companies' share capital. Financial assets are usually more market li ...

s. Perpetuities can be structured as a

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 pri ...

and 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

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 (PDV), 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 ha ...

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, :

PV \ = \

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, ...

. 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, or the price-sensitivity to a small change in the interest rate r, of a perpetuity is given by the following formula: :

D \ = \

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 Accountancy, 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 (a ...

or the

net cash flow Cash flow, in general, refers to payments made into or out of a business, project, or financial product. It can also refer more specifically to a real or virtual movement of money. *Cash flow, in its narrow sense, is a payment (in a currency), es ...

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 average price of goods and services in terms of money. This increase is measured using a price index, typically a consumer price index (CPI). When the general price level rises, each unit of curre ...

. 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

) 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 ...

, 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 financial economics, the dividend discount model (DDM) is a method of valuing the price of a company's capital stock or business value based on the assertion that intrinsic value is determined by the sum of future cash flows from dividend pay ...

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.

Detailed description

Real-life examples

References

See also