Beta relevering (`coc.beta_relever`)¶

Converts a beta between its unlevered (asset) form and its levered (equity) form at a given capital structure. Use it to relever an industry or peer beta to the target debt-to-equity ratio of the company being valued, or to unlever an observed beta.

Formula¶

\[ \beta_L = \beta_U + (\beta_U - \beta_D)\, k \qquad \beta_U = \frac{\beta_L + \beta_D\, k}{1 + k} \]

\[ k = (1 - t)\,\frac{D}{E}\ \text{(Hamada)} \qquad k = \frac{D}{E}\ \text{(Harris-Pringle)} \]

where \(\beta_L\) is the levered beta, \(\beta_U\) the unlevered beta, \(\beta_D\) the debt beta, \(D/E\) the debt-to-equity ratio and \(t\) the tax rate of the tax shield. With \(\beta_D = 0\) these are the familiar \(\beta_L = \beta_U [1 + (1 - t) D/E]\) and \(\beta_L = \beta_U (1 + D/E)\).

Choosing the formula¶

Hamada assumes debt fixed in amount, so the tax shield is as risky as the debt and reduces the leverage effect by \((1 - t)\). Harris-Pringle assumes debt rebalanced to a target ratio, so the tax shield carries the risk of the operating assets and the tax rate drops out. A positive debt beta recognises that the debt of a leveraged company bears part of the business risk; it lowers the levered beta for a given unlevered beta, and the default of 0 treats debt as risk-free.

Example¶

from decimal import Decimal

import fairmark as fm

# Hamada, risk-free debt: 0.8 × (1 + (1 − 0.24) × 0.5)
r = fm.coc.beta_relever(
    "0.8", direction="relever", debt_to_equity="0.5", tax_rate="0.24", formula="hamada"
)
assert r.beta == Decimal("1.104")
print(r.conventions[1])  # debt beta 0 (default: risk-free debt)

# Harris-Pringle with a debt beta of 0.2: 0.8 + (0.8 − 0.2) × 0.5
hp = fm.coc.beta_relever(
    "0.8",
    direction="relever",
    debt_to_equity="0.5",
    tax_rate="0.24",
    formula="harris_pringle",
    debt_beta="0.2",
)
assert hp.beta == Decimal("1.1")

# Hamada unlevering with a debt beta of 0.1: 1.245 / 1.45
u = fm.coc.beta_relever(
    "1.2",
    direction="unlever",
    debt_to_equity="0.6",
    tax_rate="0.25",
    formula="hamada",
    debt_beta="0.1",
)
assert u.beta == Decimal("0.8586206896551724137931034483")

Inputs¶

Field	Meaning	Default
`beta`	Beta to convert: unlevered for `relever`, levered for `unlever`; dimensionless	required
`direction`	`relever` or `unlever`	required
`debt_to_equity`	Debt-to-equity ratio D/E, decimal (0.5 = debt equal to half of equity), not negative	required
`tax_rate`	Tax rate of the tax shield, decimal fraction from 0 to below 1	required
`formula`	`hamada` or `harris_pringle`	required
`debt_beta`	Beta of debt, dimensionless	`0` (risk-free debt)

The tax rate is the one the analyst considers deductible for the tax shield; the engine applies no tax-specific logic (for Italian companies, whether IRAP is included is decided in the rate passed). Harris-Pringle does not use it, but it is still validated and reported.

Outputs¶

Field	Meaning
`beta`	The converted beta: levered for `relever`, unlevered for `unlever`

The steps show the leverage factor \(k\) and the converted beta.

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

Convention	Options	Default
Formula	Hamada, Harris-Pringle	none: must be chosen
Debt beta	any value	0, risk-free debt

Errors¶

Code	When
`FM-E001`	D/E is negative, the tax rate is outside 0 to below 1, the formula or direction is not one of the listed options, or a field is missing or unknown
`FM-E008`	Arithmetic overflow (inputs far outside a realistic range)

Accuracy¶

The formulas are rational: results are exact up to the 28 significant digits of the decimal type. Unlevering and then relevering at the same structure returns the original beta to within \(10^{-26}\) (property test, both formulas, with and without debt beta).

Verified against¶

Published worked examples (Damodaran, public NYU Stern lecture slides): Tata Motors, unlevered beta 0.8601 relevered at D/E 41.41% and tax 32.45% gives 1.1007; a median beta of 1.24 unlevered at gross D/E 27.06% and tax 40% gives 1.0668. Both are golden cases. The same slides state the version with a debt beta, algebraically identical to the one used here.
Golden cases in tests/golden/coc.beta_relever/, whose expected values are closed-form computations written out in each file.
The Hamada and Harris-Pringle formulas, with and without debt beta, in their commonly published form.

Beta relevering (coc.beta_relever)¶