Terminal value, exit multiple (`tv.exit_multiple`)¶

Returns the terminal value at the horizon N as a market multiple of a metric of year N, as if the business were sold at N. The metric sets the basis of the value: EV/EBITDA, EV/EBIT and EV/Sales give an enterprise value; P/E and P/BV (price to book value of equity) give an equity value. The two bases are never mixed. Given the rate and the next flow on the same basis, the method also reports the perpetual growth that the multiple implies, which makes a market assumption comparable with a growth assumption.

Formula¶

\[ TV_N = \text{multiple} \times \text{metric}_N, \qquad g_{\text{implied}} = r - \frac{CF_{N+1}}{TV_N} \]

The implied growth is the growth at which tv.gordon on \(CF_{N+1}\) and \(r\) returns the same terminal value. When the implied growth is not below the rate (a next flow of zero or below), the method still returns the value and states in explain.conventions that the multiple is inconsistent with a stable-growth value.

The multiple applies to the metric of year N (a trailing multiple), stated in explain.conventions. The book_value metric is the book value of equity, with the multiple P/BV on the equity basis (maintainer decision of 2026-11-08).

Example¶

from decimal import Decimal

import fairmark as fm

# 8 × EBITDA 125, WACC 10%, FCFF at N + 1 of 60: the multiple implies 4% growth forever
r = fm.tv.exit_multiple(8, "ebitda", 125, rate="0.10", next_flow=60, growth_cap="0.03")
assert r.terminal_value == Decimal("1000")
assert r.basis == "enterprise"
assert r.implied_growth == Decimal("0.04")
assert r.growth_cap["status"] == "exceeded"

# Fed back into the Gordon model, the implied growth returns the same value
g = fm.tv.gordon("0.10", r.implied_growth, next_flow=60)
assert g.terminal_value == r.terminal_value

# P/E: an equity value
pe = fm.tv.exit_multiple(12, "earnings", 50)
assert (pe.terminal_value, pe.basis) == (Decimal("600"), "equity")

# P/BV: 1.4 × book value of equity 750, cost of equity 9%, FCFE 63 at N + 1
pbv = fm.tv.exit_multiple("1.4", "book_value", 750, rate="0.09", next_flow=63)
assert (pbv.terminal_value, pbv.implied_growth) == (Decimal("1050"), Decimal("0.03"))

Inputs¶

Field	Meaning	Default
`multiple`	The multiple, above 0	required
`metric`	`ebitda`, `ebit`, `revenue` (enterprise basis), `earnings` (P/E) or `book_value` (P/BV, book value of equity), both equity basis	required
`metric_value`	The metric at N, in currency units, above 0	required
`rate`	Discount rate on the basis of the metric (WACC or cost of equity), decimal fraction per period; with `next_flow`	none
`next_flow`	Flow at N + 1 on the basis of the metric (FCFF or FCFE), in currency units; with `rate`	none
`growth_cap`	Upper bound for the implied growth, decimal fraction per period; needs `rate` and `next_flow`	none, reported as absent

Outputs¶

Field	Meaning
`terminal_value`	Terminal value at N, not discounted
`basis`	`enterprise` or `equity`
`implied_growth`	\(r - CF_{N+1} / TV_N\); `null` without `rate` and `next_flow`
`growth_cap`	The implied growth against the cap; `null` without a cap

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

Convention	Options	Default
Metric and basis	ebitda, ebit, revenue (enterprise); earnings, book_value (equity)	none: must be chosen
Timing of the metric	metric at N (trailing multiple)	fixed
Implied growth not below the rate	reported in `explain.conventions`, not refused	fixed
Rate and next flow of the implied growth	on the basis of the metric	not computed when absent
Growth cap	value or none	none, reported as absent

Errors¶

Code	When
`FM-E001`	Only one of `rate` and `next_flow` is given, `growth_cap` is given without them, the metric is unknown, or a field is missing or unknown
`FM-E018`	The multiple or the metric is zero or negative
`FM-E008`	Arithmetic overflow (inputs far outside a realistic range)

Accuracy¶

One multiplication for the value, one division and one subtraction for the implied growth: exact up to the 28 significant digits of the decimal type. Fed back into tv.gordon, the implied growth returns the same terminal value to within \(10^{-18}\) (the rounding of one division).

Verified against¶

Golden cases in tests/golden/tv.exit_multiple/, from closed forms and an independent 50-digit computation (scripts/reference_tv.py).
Unit and Python tests feed the implied growth back into tv.gordon on several inputs.

Terminal value, exit multiple (tv.exit_multiple)¶