Terminal value, finite horizon (`tv.limited_horizon`)¶

Returns the terminal value at the horizon N of a flow that lasts a given number n of further periods, followed by a residual value at N + n. Italian practice uses this form when the competitive advantage, a concession or a licence has a limited life: instead of a perpetuity, the flow is valued for n explicit periods. The flow is constant, or grows at a given rate; the residual value is zero unless given. The value of the n flows is the flow times the annuity factor of tvm.annuity; only the factor is reused, so no value at N + n is computed.

Formula¶

With the first flow \(CF_{N+1}\) at the end of period N + 1:

\[ TV_N = CF_{N+1}\,\frac{1 - \left(\dfrac{1 + g}{1 + r}\right)^{n}}{r - g} + \frac{RV_{N+n}}{(1 + r)^{n}} \]

and \(CF_{N+1}\,n / (1 + r)\) for the first term when \(g = r\). As n grows the first term tends to the perpetuity \(CF_{N+1} / (r - g)\) when \(g < r\).

Example¶

from decimal import Decimal

import fairmark as fm

# A licence: flow 100 for 10 more periods at 8%, nothing after
r = fm.tv.limited_horizon(100, "0.08", 10)
assert round(r.terminal_value, 10) == Decimal("671.0081398941")

# Growing flow, with a residual value of 500 at N + 5
g = fm.tv.limited_horizon(100, "0.10", 5, growth="0.02", residual_value=500)
assert round(g.residual_value_discounted, 10) == Decimal("310.4606615296")
print(g.conventions[3])  # residual value at N + 5: 500 (given)

# The longest horizon, 10,000 periods, equals the perpetuity 100 / 0.10
long = fm.tv.limited_horizon(100, "0.10", 10000)
assert long.terminal_value == Decimal("1000")

Inputs¶

Field	Meaning	Default
`flow`	First flow after the horizon, at the end of period N + 1, in currency units	required
`rate`	Discount rate of the terminal phase, decimal fraction per period, above −1	required
`periods`	Number of further periods n, from 1 to 10,000	required
`growth`	Growth of the flow per period, decimal fraction, above −1	`0` (constant flow)
`residual_value`	Residual value at the end of period N + n, in currency units	`0`
`growth_cap`	Upper bound for the growth, decimal fraction per period	none, reported as absent

Outputs¶

Field	Meaning
`terminal_value`	Terminal value at N, not discounted
`flows_value`	Value at N of the flows N + 1 to N + n
`annuity_factor`	Value at N of a first flow of 1
`discount_factor`	\((1 + r)^{-n}\)
`residual_value_discounted`	Residual value brought to N
`growth_cap`	The growth (0 when not given) against the cap; `null` without a cap

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

Convention	Options	Default
Flow profile	constant, or growing at `growth`	constant
Number of periods	explicit input	none: must be given
Timing	end of periods N + 1 to N + n; residual value at N + n	fixed
Residual value	given, or 0	0, reported as default

Errors¶

Code	When
`FM-E001`	The growth is −100% or lower, or a field is missing or unknown
`FM-E003`	The rate is −100% or lower
`FM-E018`	`periods` is outside 1 to 10,000 (the same code for both ends of the range)
`FM-E008`	Arithmetic overflow (inputs far outside a realistic range)

Accuracy¶

The value of the flows uses the annuity factor of tvm.annuity times the flow; the residual value uses one discount factor. Measured against a term-by-term sum at 50 digits, the golden cases agree to \(10^{-22}\), including a flow of \(10^9\) over 600 periods. Every n from 1 to 10,000 is accepted: no value at N + n is computed, so long horizons stay within the decimal range. When \(((1 + g)/(1 + r))^n\) falls below the decimal precision (about \(10^{-28}\)), it is taken as 0, its limit, and an explain step says so; the value is then the perpetuity to within that precision.

Verified against¶

Golden cases in tests/golden/tv.limited_horizon/, from an independent 50-digit computation that sums the discounted flows one by one (scripts/reference_tv.py), not the closed form used by the engine.
Unit and Python tests check the convergence to the perpetuity for long horizons.

Terminal value, finite horizon (tv.limited_horizon)¶