Editing Perpetuity

{{short description|Payments made at equal intervals forever}}
{{for|the sculpture|Perpetuity (sculpture)}}

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In [[finance]], a '''perpetuity''' is an [[Annuity (finance theory)|annuity]] 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]] (UK) government issued them in the past; these were known as [[consols]] and were all finally redeemed in 2015. 

[[Real estate]] and [[preferred stock]] are among some types of [[investment]]s that effect the results of a perpetuity, and prices can be established using techniques for valuing a perpetuity.<ref>{{cite web|title=Perpetuity|url=http://www.investopedia.com/terms/p/perpetuity.asp?layout=infini&v=5D&orig=1&adtest=5D|website=Investopedia|access-date=26 May 2016|date=24 November 2003}}</ref> Perpetuities are but one of the [[time value of money]] methods for valuing [[financial asset]]s. 

Perpetuities can be structured as a [[perpetual bond]] and are a form of ordinary annuities. The concept is closely linked to [[Terminal value (finance)|terminal value]] and terminal growth rate in [[Valuation (finance)|valuation]].

==Detailed description==
A '''perpetuity''' is an [[Annuity (finance theory)|annuity]] 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]] of the future cash flows).  Unlike a typical bond, because the [[:wikt:principal|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 [[Discounting|discount]] rate or yield; that is,

:<math> PV \ = \ {A \over r}  </math>

where '''PV''' = present value of the perpetuity, '''A''' = the amount of the periodic payment, and '''r''' = yield, [[discount window|discount rate]] or [[interest rate]].<ref>{{Cite book |title =Investment Science |page =45 |last =Luenberger |first =David |author-link =David Luenberger |publisher =[[Oxford University Press]] |year =2009 |place =[[New York City]] }}</ref>

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 [[Bond duration|duration]], or the price-sensitivity to a small change in the interest rate '''r''', of a perpetuity is given by the following formula:<ref>{{Cite web|title=fixed income – Duration of perpetual bond|url=https://quant.stackexchange.com/questions/22288/duration-of-perpetual-bond|access-date=2021-05-13|website=Quantitative Finance Stack Exchange}}</ref>

: <math> D \ = \ {1 \over r}  </math>

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]] 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]] or the [[net cash flow]] 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]]. 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]]) were '''undated''' as well as '''irredeemable'''  except by act of Parliament. As with [[war bonds]], 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]] 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]] 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==
{{reflist}}

==See also==
{{EB1911 poster|Perpetuity}}
*[[Geometric progression]]
*[[Perpetual bond]]

[[Category:Mathematical finance]]
[[Category:Annuities]]