In finance, a perpetuity is an annuity with payments that continue indefinitely. 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.
Types and assumptions Perpetuity valuation depends on the timing of payments and on whether payments are level or grow over time. The discount rate, r {\displaystyle r} is stated per payment period (for example, per year if payments are annual).
Ordinary perpetuity Payments are made at the end of each period, with the first payment one period from now.
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.
Level perpetuity The payment amount is constant each period. Standard formulas assume a constant discount rate and r > 0 {\displaystyle r>0} , so the discounted sum converges to a finite value.
Growing perpetuity Payments increase at a constant rate g {\displaystyle g} per period. The usual closed-form valuation applies only when r > g {\displaystyle r>g} . If r ≤ g {\displaystyle r\leq g} , the discounted sum does not converge to a finite value.
Valuation Valuation follows from discounting each payment and summing the resulting infinite series. Let r {\displaystyle r} 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 A {\displaystyle A} each period has present value:
P V = ∑ t = 1 ∞ A ( 1 + r ) t PV=\sum _{t=1}^{\infty }{\frac {A}{(1+r)^{t}}}
Expanding the first terms makes the constant ratio between successive terms clear:
P V = A 1 + r + A ( 1 + r ) 2 + A ( 1 + r ) 3 + ⋯ PV={\frac {A}{1+r}}+{\frac {A}{(1+r)^{2}}}+{\frac {A}{(1+r)^{3}}}+\cdots
Factor out the first term:
P V = A 1 + r ( 1 + 1 1 + r + 1 ( 1 + r ) 2 + ⋯ ) {\displaystyle PV={\frac {A}{1+r}}\left(1+{\frac {1}{1+r}}+{\frac {1}{(1+r)^{2}}}+\cdots \right)} . The bracketed sum is a geometric series with ratio q = 1 1 + r {\displaystyle q={\frac {1}{1+r}}} , which converges when | q | < 1 {\displaystyle |q|<1} . For r > 0 {\displaystyle r>0} , this condition holds, so the sum can be evaluated:
P V = A 1 + r ⋅ 1 1 − 1 1 + r = A r PV={\frac {A}{1+r}}\cdot {\frac {1}{1-{\frac {1}{1+r}}}}={\frac {A}{r}}
This result means a level perpetuity's value rises in direct proportion to the payment A {\displaystyle A} and falls as the discount rate r {\displaystyle r} increases (future payments are discounted more heavily).
Perpetuity due If the first payment is made immediately (a perpetuity due), the present value is larger by one undiscounted payment:
P V due = A + A r PV_{\text{due}}=A+{\frac {A}{r}}
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 t = 1 {\displaystyle t=1} .
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