Perpetuity (`tvm.perpetuity`)¶

Returns the present value of a payment that continues indefinitely, either level or growing at a constant rate below the discount rate. The growing perpetuity is the Gordon growth model, used for terminal values and for the value of stable dividend streams. Payments start at the end of the first period (ordinary) or immediately (due).

Formula¶

The payment of period \(k\) is \(P (1 + g)^{k - 1}\). For \(-1 < g < r\):

\[ PV_{\text{ordinary}} = \frac{P}{r - g} \qquad PV_{\text{due}} = \frac{P}{r - g}\,(1 + r) \]

With \(g = 0\) the ordinary formula is \(P / r\). When \(g \ge r\) the series does not converge and the method returns an error instead of a value. A growth of −100% or lower makes the payments vanish or alternate in sign, so it is rejected as in tvm.annuity.

Example¶

A first payment of 100, growing at 2%, discounted at 8% (Gordon), and a level perpetuity due of 100 at 8%:

from decimal import Decimal

import fairmark as fm

gordon = fm.tvm.perpetuity(100, "0.08", timing="ordinary", growth="0.02")
assert round(gordon.present_value, 20) == Decimal("1666.66666666666666666667")

due = fm.tvm.perpetuity(100, "0.08", timing="due")
assert due.present_value == Decimal("1350")
print(due.conventions[-1])  # payment grows by 0 per period (default: level payments)

try:
    fm.tvm.perpetuity(100, "0.05", timing="ordinary", growth="0.05")
except fm.FairmarkError as err:
    assert err.code == "FM-E004"  # growth must be below the rate
    print(err.message)

Inputs¶

Field	Meaning	Default
`payment`	First payment	required
`rate`	Discount rate per period, decimal fraction; must exceed `growth`	required
`growth`	Growth of the payment per period, decimal fraction, greater than −1	`0` (level perpetuity)
`timing`	`ordinary` (first payment at \(t = 1\)) or `due` (first payment at \(t = 0\))	required

Outputs¶

Field	Meaning
`present_value`	Present value at \(t = 0\)

Conventions (explicit, reported in `explain.conventions`)¶

Convention	Options	Default
Timing	ordinary, due	none: must be chosen
Growth	above −100% and below the rate	0, reported as default

The rate and the growth are per period and must use the same period and compounding.

Errors¶

Code	When
`FM-E001`	`growth` is −100% or lower, a field is missing or unknown, or `timing` is not `ordinary` or `due`
`FM-E003`	The rate is −100% or lower
`FM-E004`	The growth is equal to or greater than the rate

Accuracy¶

The formula uses one division and, for the due timing, one multiplication. Measured against an independent 60-digit computation, it agrees to about \(10^{-24}\) relative or better. The golden cases are checked to \(10^{-19}\).

Verified against¶

Golden cases in tests/golden/tvm.perpetuity/, from an independent 50-digit computation (scripts/reference_time.py).
The closed form is the Gordon growth model, the same formula used for the terminal value in dcf.fcff.

Perpetuity (tvm.perpetuity)¶