Terminal value, stable growth (`tv.gordon`)¶

Returns the terminal value at the horizon N of a flow that grows at a constant rate g forever, discounted at the rate of the terminal phase r. This is the Gordon growth model applied at the end of the explicit forecast. The result is a value at N: discounting it to the valuation date is the job of the DCF. The flow of the first period after the horizon is either given (for example a normalised flow) or obtained by growing the last explicit flow by g; the choice is required and reported.

Formula¶

For \(-1 < g < r\):

\[ TV_N = \frac{CF_{N+1}}{r - g}, \qquad CF_{N+1} = CF_N\,(1 + g) \text{ when the last flow is given} \]

When \(g \ge r\) the series does not converge and the method returns an error instead of a value.

Example¶

from decimal import Decimal

import fairmark as fm

# Next flow 50, rate 8%, growth 4%, checked against a 2% cap (the cap is reported, not refused)
r = fm.tv.gordon("0.08", "0.04", next_flow=50, growth_cap="0.02")
assert r.terminal_value == Decimal("1250")
assert r.flow_basis == "next_flow_given"
assert r.growth_cap["status"] == "exceeded"

# Last flow 100 grown by 2.5% to 102.5, rate 9%
last = fm.tv.gordon("0.09", "0.025", last_flow=100)
assert last.next_flow == Decimal("102.5")
print(last.conventions[1])  # flow at N + 1 is the last flow grown by g: CF_N × (1 + g)

Inputs¶

Field	Meaning	Default
`rate`	Discount rate of the terminal phase, decimal fraction per period; may differ from the explicit phase	required
`growth`	Perpetual growth, decimal fraction per period, above −1 and below `rate`	required
`next_flow`	Flow at N + 1, in currency units; give this or `last_flow`	none
`last_flow`	Flow at N, in currency units, grown by `growth`; give this or `next_flow`	none
`growth_cap`	Upper bound for the growth, decimal fraction per period (for example the risk-free rate or the nominal growth of the economy)	none, reported as absent

Outputs¶

Field	Meaning
`terminal_value`	Terminal value at N, not discounted
`next_flow`	Flow at N + 1 used
`flow_basis`	`next_flow_given` or `last_flow_grown`
`growth_cap`	`{growth, cap, status}` with status `respected` (g ≤ cap) or `exceeded`; `null` without a cap

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

Convention	Options	Default
Flow at N + 1	given, or CF_N × (1 + g)	none: one must be chosen
Rate of the terminal phase	given, may differ from the explicit phase	none: must be given
Growth cap	value or none	none, reported as absent
Timing	value at N of flows from N + 1 onwards, end of period	fixed

Errors¶

Code	When
`FM-E001`	Both or neither of `next_flow` and `last_flow` are given, the growth is −100% or lower, or a field is missing or unknown
`FM-E004`	The growth is equal to or greater than the rate
`FM-E008`	Arithmetic overflow (inputs far outside a realistic range)

Accuracy¶

One multiplication and one division: exact up to the 28 significant digits of the decimal type. The golden cases are checked to \(10^{-22}\) against a 50-digit reference.

Verified against¶

tests/golden/tv.gordon/damodaran-fcff-from-roc-table.yaml: a cell of the table "The Big Assumption" in Damodaran's valuation lecture slides (NYU Stern), restated as a Gordon flow, matches at its published precision.
The other golden cases come from an independent 50-digit computation (scripts/reference_tv.py). The formula is the one of tvm.perpetuity for an ordinary growing perpetuity.

Terminal value, stable growth (tv.gordon)¶