Terminal value, H-model (`tv.h_model`)¶

Returns the terminal value at the horizon N when growth is still above its stable level at N and declines linearly from a short-term growth \(g_s\) to a long-term growth \(g_l\) over 2H periods, then stays at \(g_l\) forever. The H-model (Fuller and Hsia) is a closed-form approximation of that path: the value at stable growth plus the value of the extra growth during the decline. Use it when a transition is already under way at N.

Formula¶

For \(g_l < r\) and \(H \ge 0\):

\[ TV_N = \frac{CF_N\,(1 + g_l)}{r - g_l} + \frac{CF_N\,H\,(g_s - g_l)}{r - g_l} = CF_N\,\frac{(1 + g_l) + H\,(g_s - g_l)}{r - g_l} \]

With \(g_s = g_l\), or \(H = 0\), the second term is zero and the result equals tv.gordon on the last flow. The formula is an approximation, not the exact value of a linearly declining growth path; the method states it in explain.conventions.

Example¶

from decimal import Decimal

import fairmark as fm

# CF_N 100, growth from 10% down to 3% over 2H = 10 periods, rate 9%
r = fm.tv.h_model(100, "0.10", "0.03", 5, "0.09")
assert r.terminal_value == Decimal("2300")
assert round(r.extra_growth_value, 10) == Decimal("583.3333333333")

# Equal growths: the H-model is the Gordon model on the last flow
same = fm.tv.h_model(100, "0.02", "0.02", 4, "0.09")
gordon = fm.tv.gordon("0.09", "0.02", last_flow=100)
assert same.terminal_value == gordon.terminal_value

Inputs¶

Field	Meaning	Default
`last_flow`	Flow of period N, in currency units	required
`short_term_growth`	\(g_s\), growth at the start of the decline, decimal fraction per period, above −1	required
`long_term_growth`	\(g_l\), stable growth, decimal fraction per period, above −1 and below `rate`	required
`half_life`	H, in periods, zero or positive, may be fractional; the decline lasts 2H periods	required
`rate`	Discount rate of the terminal phase, decimal fraction per period	required
`growth_cap`	Upper bound for the long-term growth, decimal fraction per period	none, reported as absent

Outputs¶

Field	Meaning
`terminal_value`	Terminal value at N, not discounted
`stable_growth_value`	\(CF_N (1 + g_l) / (r - g_l)\)
`extra_growth_value`	\(CF_N H (g_s - g_l) / (r - g_l)\)
`growth_cap`	The long-term growth against the cap; `null` without a cap

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

Convention	Options	Default
Growth path	linear decline over 2H periods, then \(g_l\) (approximation)	fixed
Growth checked against the cap	the long-term growth \(g_l\); \(g_s\) is transient	fixed
Rate of the terminal phase	given	none: must be given

Errors¶

Code	When
`FM-E001`	A growth is −100% or lower, or a field is missing or unknown
`FM-E004`	The long-term growth is equal to or greater than the rate
`FM-E018`	The half-life is negative
`FM-E008`	Arithmetic overflow (inputs far outside a realistic range)

Accuracy¶

Rational operations only: exact up to the 28 significant digits of the decimal type. The approximation error of the H-model itself, against an exact declining path, is a property of the model and is not corrected.

Verified against¶

Golden cases in tests/golden/tv.h_model/, from an independent 50-digit computation (scripts/reference_tv.py) and the closed form written in each file.
A unit test checks that equal growths give exactly the result of tv.gordon.
Source of the model: Fuller, R. J. and Hsia, C.-C. (1984), "A Simplified Common Stock Valuation Model", Financial Analysts Journal 40(5).

Terminal value, H-model (tv.h_model)¶