Perpetuity

{{short description|Annuity with payments made at equal intervals indefinitely}} {{for|the sculpture|Perpetuity (sculpture)}} {{for|the legal doctrine|Rule against perpetuities}} In finance, a '''perpetuity''' is an annuity with payments that continue indefinitely.<ref name="Brealey2011" />

Perpetuity formulas are used in time value of money and discounted cash flow valuation. Common applications include valuing shares under the dividend discount model, estimating terminal value in company valuation, and capitalising stabilised income in real estate appraisal.<ref name="DamodaranApp3" /><ref name="BrueggemanFisher2010" />

==Types and assumptions== Perpetuity valuation depends on the timing of payments and on whether payments are level or grow over time. The discount rate, <math>r</math> is stated per payment period (for example, per year if payments are annual).<ref name="DamodaranApp3" /><ref name="OpenStaxPerpetuities" />

=== Ordinary perpetuity === Payments are made at the end of each period, with the first payment one period from now.<ref name="DamodaranApp3" /><ref name="OpenStaxPerpetuities" />

=== Perpetuity due === Payments are made at the start of each period, with the first payment immediately. Because each payment is received one period earlier, a perpetuity due has a higher present value than an otherwise identical ordinary perpetuity.<ref name="DamodaranApp3" /><ref name="OpenStaxPerpetuities" />

=== Level perpetuity === The payment amount is constant each period. Standard formulas assume a constant discount rate and <math>r>0</math>, so the discounted sum converges to a finite value.<ref name="Brealey2011" /><ref name="OpenStaxPerpetuities" />

=== Growing perpetuity === Payments increase at a constant rate <math>g</math> per period. The usual closed-form valuation applies only when <math>r>g</math>. If <math>r\leq g</math>, the discounted sum does not converge to a finite value.<ref name="Brealey2011" /><ref name="DamodaranApp3" /><ref name="OpenStaxPerpetuities" />

=={{Anchor|Detailed description}} Valuation== Valuation follows from discounting each payment and summing the resulting infinite series. Let <math>r</math> be the discount rate per period.

===Level perpetuity (ordinary perpetuity)=== Under the ordinary-perpetuity convention (first payment one period from now), a level perpetuity paying <math>A</math> each period has present value:<ref name="Brealey2011" /><ref name="DamodaranApp3" /><ref name="OpenStaxPerpetuities" />

<math display>PV=\sum_{t=1}^{\infty}\frac{A}{(1+r)^t}</math>

Expanding the first terms makes the constant ratio between successive terms clear:<ref name="Brealey2011" /><ref name="OpenStaxPerpetuities" />

<math display>PV=\frac{A}{1+r}+\frac{A}{(1+r)^2}+\frac{A}{(1+r)^3}+\cdots</math>

Factor out the first term: <math>PV=\frac{A}{1+r}\left(1+\frac{1}{1+r}+\frac{1}{(1+r)^2}+\cdots\right)</math>. The bracketed sum is a geometric series with ratio <math>q=\frac{1}{1+r}</math>, which converges when <math>|q|<1</math>. For <math>r>0</math>, this condition holds, so the sum can be evaluated:<ref name="Brealey2011" /><ref name="OpenStaxPerpetuities" />

This result means a level perpetuity's value rises in direct proportion to the payment <math>A</math> and falls as the discount rate <math>r</math> increases (future payments are discounted more heavily).<ref name="Brealey2011" /><ref name="OpenStaxPerpetuities" />

===Perpetuity due=== If the first payment is made immediately (a perpetuity due), the present value is larger by one undiscounted payment:<ref name="DamodaranApp3" /><ref name="OpenStaxPerpetuities" />

This result means receiving each payment one period earlier increases value by exactly one extra payment today, because the ordinary-perpetuity value starts discounting from <math>t=1</math>.<ref name="DamodaranApp3" /><ref name="OpenStaxPerpetuities" />

===Growing perpetuity=== A growing perpetuity is derived in the same way. Let <math>CF_1</math> be the expected payment next period, and let payments grow at constant rate <math>g</math>, so <math>CF_t=CF_1(1+g)^{t-1}</math>. Discounting and summing gives:<ref name="Brealey2011" /><ref name="DamodaranApp3" /><ref name="OpenStaxPerpetuities" />

<math display>PV=\sum_{t=1}^{\infty}\frac{CF_1(1+g)^{t-1}}{(1+r)^t} =\frac{CF_1}{1+r}\sum_{t=1}^{\infty}\left(\frac{1+g}{1+r}\right)^{t-1}</math>

The ratio is <math>\frac{1+g}{1+r}</math>, so the sum converges when <math>\frac{1+g}{1+r}<1</math>, which is equivalent to <math>r>g</math>. Evaluating the series yields:<ref name="Brealey2011" /><ref name="DamodaranApp3" /><ref name="OpenStaxPerpetuities" />

This result means the value increases with the next-period cash flow <math>CF_1</math> and increases as growth <math>g</math> rises, but only while <math>r>g</math>. As <math>g</math> approaches <math>r</math>, the denominator shrinks and the valuation becomes very sensitive to small changes in <math>r</math> or <math>g</math>.<ref name="Brealey2011" /><ref name="DamodaranApp3" /><ref name="OpenStaxPerpetuities" />

==Applications== Perpetuities are used as simplified models for valuing assets that are expected to generate cash flows for a very long time, or for representing the value of cash flows beyond an explicit forecast horizon.<ref name="Brealey2011" /><ref name="DamodaranChap12" />

In real estate appraisal, direct capitalisation converts an estimate of a single year's stabilised income into an indication of value by dividing by a market-derived capitalisation rate (cap rate). Under the standard stabilised-income assumption, this has the same algebraic form as valuing a level perpetuity, with the cap rate playing the role of a required return for that income stream.<ref name="AppraisalInstitute2020" /><ref name="BrueggemanFisher2010" />

In corporate finance and equity valuation, the constant-growth dividend discount model treats dividends as a growing perpetuity and expresses the share price in terms of next-period dividends, the discount rate and the growth rate. Related growing-perpetuity expressions are also used to estimate terminal value in discounted cash flow analysis once a firm is assumed to have reached stable growth.<ref name="Brealey2011" /><ref name="DamodaranChap12" />

Some financial instruments can be approximated as perpetuities. For example, shares of preferred stock with fixed dividends and no maturity date are commonly valued by discounting the dividend stream as a level perpetuity, using a required return appropriate to the risk of the dividends.<ref name="DamodaranChap12" /><ref name="OpenStaxPerpetuities" />

Perpetuities appear in instructional examples for endowments designed to preserve principal while distributing a regular amount indefinitely (for example, to fund a scholarship).<ref name="OpenStaxPerpetuities" />

=={{Anchor|Real-life examples}} Examples== Consols are government securities with no scheduled maturity date that pay a fixed coupon until they are redeemed by the issuer.<ref name="Britannica-Consol" /> British consols originated in the 18th century and were issued in several forms, including 4% consols in the late 1920s and early 1930s.<ref name="Britannica-Consol" />

In the 2010s, the UK government moved to eliminate the remaining undated gilts. In October 2014 it announced the redemption of ''4% Consolidated Loan'' (redeemed 1 February 2015), noting that the issue dated from 1927 and that cumulative interest payments since then were estimated at £1.26 billion.<ref name="UKGOV-ConsolLoan-2014" /><ref name="UKDMOConsolLoan2014" /> In December 2014 it announced the redemption of ''{{frac|3|1|2}}% War Loan'' at par on 9 March 2015.<ref name="UKGOV-WarLoan-2014" /><ref name="UKDMOWarLoan2014" /> Further operational notices in 2015 covered additional undated and "rump" gilts redeemed later that year, and legislation provided a framework for redeeming the remaining undated government stocks.<ref name="UKDMO-RumpGilts-2015" /><ref name="UKDMO-UndatedGilts-2015" /><ref name="FinanceAct2015-s124" /><ref name="FinanceAct2015-s124-EN" /><ref name="DMOAnnualReview201415" />

Modern examples of "perpetual" cash-flow instruments are common in bank regulatory capital. Under the Basel III capital rules, an instrument eligible as Additional Tier 1 capital must be perpetual (no maturity date), although it may include issuer call options (subject to conditions such as a minimum period before the first call and supervisory approval).<ref name="BIS-BCBS-d417" /> These instruments are often structured to absorb losses in stress (for example via conversion or write-down), which makes them riskier than conventional bonds despite their bond-like coupons.<ref name="BIS-BCBS-d417" />

Perpetuities are used to illustrate long-lived funding arrangements. Many university endowments are defined as funds whose principal is invested in perpetuity, with spending limited to investment returns under an institutional spending policy.<ref name="Stanford-EndowmentPayout" /><ref name="Harvard-EndowmentDefinitions" />

==References== <references> <ref name="Brealey2011">{{cite book |last1=Brealey |first1=Richard A. |last2=Myers |first2=Stewart C. |last3=Allen |first3=Franklin |title=Principles of Corporate Finance |edition=10th |publisher=McGraw-Hill/Irwin |location=New York |year=2011 |isbn=978-0-07-338238-8}}</ref>

<ref name="DamodaranApp3">{{cite report |last=Damodaran |first=Aswath |title=Appendix 3: Time Value of Money |publisher=Stern School of Business, New York University |url=https://people.stern.nyu.edu/adamodar/pdfiles/acf2E/App3.pdf |access-date=10 January 2026}}</ref>

<ref name="BrueggemanFisher2010">{{cite book |last1=Brueggeman |first1=William B. |last2=Fisher |first2=Jeffrey D. |title=Real Estate Finance & Investments |edition=14th |publisher=McGraw-Hill |year=2010 |isbn=978-0-07-714456-2}}</ref>

<ref name="OpenStaxPerpetuities">{{cite web |title=8.1 Perpetuities |website=Principles of Finance |publisher=OpenStax |date=24 March 2022 |url=https://openstax.org/books/principles-finance/pages/8-1-perpetuities |access-date=10 January 2026}}</ref>

<ref name="DamodaranChap12">{{cite report |last=Damodaran |first=Aswath |title=Chapter 12: Valuation: Principles and Practice |publisher=Stern School of Business, New York University |url=https://pages.stern.nyu.edu/~adamodar/pdfiles/acf2E/Chap12.pdf |access-date=10 January 2026}}</ref>

<ref name="AppraisalInstitute2020">{{cite book |title=The Appraisal of Real Estate |edition=15th |publisher=Appraisal Institute |location=Chicago |year=2020 |isbn=978-1-935328-78-0}}</ref>

<ref name="Britannica-Consol">{{cite web |title=Consol |website=Encyclopaedia Britannica |url=https://www.britannica.com/money/consol-economics |access-date=11 January 2026}}</ref>

<ref name="UKGOV-ConsolLoan-2014">{{cite web |title=Chancellor Osborne to repay part of our First World War debt |website=GOV.UK |publisher=HM Treasury |date=31 October 2014 |url=https://www.gov.uk/government/news/chancellor-osborne-to-repay-part-of-our-first-world-war-debt |access-date=11 January 2026}}</ref>

<ref name="UKGOV-WarLoan-2014">{{cite web |title=Chancellor to repay the nation's First World War debt |website=GOV.UK |publisher=HM Treasury |date=3 December 2014 |url=https://www.gov.uk/government/news/chancellor-to-repay-the-nations-first-world-war-debt |access-date=11 January 2026}}</ref>

<ref name="UKDMO-RumpGilts-2015">{{cite press release |title=Redemption of 3% Treasury Stock |publisher=UK Debt Management Office |date=6 February 2015 |url=https://www.dmo.gov.uk/media/a0olvmsx/pr060215.pdf |access-date=11 January 2026}}</ref>

<ref name="UKDMO-UndatedGilts-2015">{{cite press release |title=Redemption of 2{{citefrac|3|4}}% Annuities, 2{{citefrac|1|2}}% Annuities, 2{{citefrac|1|2}}% Consolidated Stock and 2{{citefrac|1|2}}% Treasury Stock |publisher=UK Debt Management Office |date=27 March 2015 |url=https://www.dmo.gov.uk/media/i2sfsfj4/pr270315.pdf |access-date=11 January 2026}}</ref>

<ref name="FinanceAct2015-s124">{{cite web |title=Finance Act 2015, section 124: Redemption of undated government stocks |website=legislation.gov.uk |url=https://www.legislation.gov.uk/ukpga/2015/11/section/124 |access-date=11 January 2026}}</ref>

<ref name="FinanceAct2015-s124-EN">{{cite web |title=Finance Act 2015: Explanatory Notes for section 124 (Redemption of Undated Government Stocks) |website=legislation.gov.uk |url=https://www.legislation.gov.uk/ukpga/2015/11/notes/division/1/124 |access-date=11 January 2026}}</ref>

<ref name="DMOAnnualReview201415">{{cite report |title=Debt Management Office Annual Review 2014–15 |publisher=UK Debt Management Office |date=1 August 2015 |url=https://www.dmo.gov.uk/media/llxjzqo2/gar1415.pdf |access-date=11 January 2026}}</ref>

<ref name="BIS-BCBS-d417">{{cite report |title=Basel III definition of capital: Frequently asked questions |publisher=Bank for International Settlements (Basel Committee on Banking Supervision) |date=December 2017 |url=https://www.bis.org/bcbs/publ/d417.pdf |access-date=11 January 2026}}</ref>

<ref name="Stanford-EndowmentPayout">{{cite web |title=Endowment payout process |website=Fingate |publisher=Stanford University |url=https://fingate.stanford.edu/managing-funds/endowment-payout-process |access-date=11 January 2026}}</ref>

<ref name="Harvard-EndowmentDefinitions">{{cite web |title=Gifts and endowments: definitions |website=Office of Finance |publisher=Harvard University Faculty of Arts & Sciences |url=https://finance.fas.harvard.edu/pages/gifts-and-endowments-definitions |access-date=11 January 2026}}</ref>

<ref name="UKDMOConsolLoan2014">{{cite press release |title=Redemption of 4% Consolidated Loan |publisher=UK Debt Management Office |date=31 October 2014 |url=https://www.dmo.gov.uk/media/0yfp3tr2/pr311014.pdf |access-date=10 January 2026}}</ref>

<ref name="UKDMOWarLoan2014">{{cite press release |title=Redemption of 3{{citefrac|1|2}}% War Loan |publisher=UK Debt Management Office |date=3 December 2014 |url=https://www.dmo.gov.uk/media/kmwf5vfk/pr031214.pdf |access-date=10 January 2026}}</ref>

</references>

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

Category:Mathematical finance Category:Annuities