Beta from comparable companies (`coc.beta_peers`)¶

Estimates the beta of a private company from listed comparables: unlevers the observed beta of each comparable at its own capital structure and tax rate, takes the mean or the median of the unlevered betas, and relevers the selected statistic to the target structure of the company being valued. The relevering formulas are those of coc.beta_relever.

Formula¶

\[ \beta_{U,i} = \frac{\beta_{L,i} + \beta_D\, k_i}{1 + k_i}, \qquad \bar\beta_U = \operatorname{mean}\ \text{or}\ \operatorname{median}(\beta_{U,i}), \qquad \beta_L = \bar\beta_U + (\bar\beta_U - \beta_D)\, k^{*} \]

where \(k_i = (1 - t_i)\,(D/E)_i\) (Hamada) or \((D/E)_i\) (Harris-Pringle) for comparable \(i\), and \(k^{*}\) is the same factor at the target D/E and tax rate. For an even number of comparables the median is the mean of the two central values.

Example¶

Five comparables unlevered by Hamada; the median is relevered to D/E 0.5 at a 24% tax rate.

from decimal import Decimal

import fairmark as fm

peers = [
    {"name": "Peer A", "levered_beta": "1.15", "debt_to_equity": "0.40", "tax_rate": "0.24"},
    {"name": "Peer B", "levered_beta": "0.95", "debt_to_equity": "0.20", "tax_rate": "0.25"},
    {"name": "Peer C", "levered_beta": "1.30", "debt_to_equity": "0.75", "tax_rate": "0.21"},
    {"name": "Peer D", "levered_beta": "1.05", "debt_to_equity": "0.35", "tax_rate": "0.24"},
    {"name": "Peer E", "levered_beta": "0.88", "debt_to_equity": "0.10", "tax_rate": "0.28"},
]
r = fm.coc.beta_peers(
    peers,
    formula="hamada",
    statistic="median",
    target_debt_to_equity="0.5",
    target_tax_rate="0.24",
)
# Median is Peer B: 0.95 / (1 + 0.75 × 0.2) = 0.95 / 1.15; relevered × (1 + 0.76 × 0.5)
assert r.median_unlevered_beta == Decimal("0.8260869565217391304347826087")
assert r.mean_unlevered_beta == Decimal("0.8349189472537530893029774557")
assert r.relevered_beta == Decimal("1.14")
print(r.conventions[-1])  # selected statistic: median (mean of the two central values ...)

Inputs¶

Field	Meaning	Default
`peers`	Comparables, at least one: `{name, levered_beta, debt_to_equity, tax_rate}`; D/E not negative, tax rate from 0 to below 1; `name` is used in the audit trail only	required
`formula`	`hamada` or `harris_pringle`, used both to unlever and to relever	required
`statistic`	`mean` or `median` of the unlevered betas	required
`target_debt_to_equity`	Target D/E of the company being valued, decimal, not negative	required
`target_tax_rate`	Tax rate of the company being valued, decimal fraction from 0 to below 1	required
`debt_beta`	Beta of debt, dimensionless, applied to comparables and target	`0` (risk-free debt)

Outputs¶

Field	Meaning
`unlevered_betas`	Unlevered beta of each comparable, in input order
`mean_unlevered_beta`	Arithmetic mean of the unlevered betas
`median_unlevered_beta`	Median of the unlevered betas
`selected_unlevered_beta`	The statistic chosen by `statistic`
`relevered_beta`	Selected unlevered beta relevered to the target structure

Both statistics are always returned, so the effect of the choice is visible. The steps show each comparable by name, both statistics and the relevered beta.

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

Convention	Options	Default
Formula	Hamada, Harris-Pringle	none: must be chosen
Statistic	mean, median (mean of the two central values for an even count)	none: must be chosen
Debt beta	any value	0, risk-free debt

Errors¶

Code	When
`FM-E001`	`peers` is empty, a D/E is negative, a tax rate is outside 0 to below 1, an option is not one of the listed ones, 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. The golden case is checked to \(3 \times 10^{-28}\).

Verified against¶

Golden case in tests/golden/coc.beta_peers/: closed form per comparable written out in the file, confirmed by an independent 50-digit computation (scripts/reference_coc.py).
The Hamada and Harris-Pringle formulas in their commonly published form.

Beta from comparable companies (coc.beta_peers)¶