Cost of equity (`coc.cost_of_equity`)¶

Returns the cost of equity by the capital asset pricing model (CAPM), with the premia that practice adds for private companies: country risk, size and company-specific risk. Every premium is an explicit input, and every premium, given or defaulted, is reported in the audit trail. The levered beta is either given or relevered inside the method from an unlevered beta, as in coc.beta_relever.

Formula¶

\[ k_e = r_f + \beta_L \cdot ERP + CRP \cdot \lambda + SP + SRP \]

where \(r_f\) is the risk-free rate, \(\beta_L\) the levered beta, \(ERP\) the equity risk premium of the market, \(CRP\) the country risk premium, \(\lambda\) the exposure of the company to the country risk, \(SP\) the size premium and \(SRP\) the company-specific risk premium. All rates are annual effective rates, decimal fractions.

When relever is given, the levered beta is

\[ \beta_L = \beta_U + (\beta_U - \beta_D)\, k, \qquad k = (1 - t)\,\frac{D}{E}\ \text{(Hamada)} \quad\text{or}\quad k = \frac{D}{E}\ \text{(Harris-Pringle)} \]

When to use each premium¶

Country risk premium and exposure \(\lambda\): when the equity risk premium refers to a mature market and the company earns its cash flows in a riskier one. \(\lambda\) scales the premium to the share of the business exposed (for example the share of revenue earned in that country); the default of 1 means full exposure.
Size premium: when the analyst holds that small companies earn returns above those explained by beta, typically taken from a published size study.
Specific risk premium: for risks of the individual company not captured elsewhere, such as dependence on a few customers or on a key person. Because it is a judgement, a non-zero value requires a written justification, which is echoed in explain.conventions.

Each premium appears in the conventions as given or as the default 0, so a reader of the result can see which premia were applied and which were left out.

Example¶

Risk-free rate 3.5%, beta 1.1, equity risk premium 5.5%, country premium 2% with exposure 0.5, size premium 2% and a justified specific premium of 1%; then an unlevered beta of 0.8 relevered by Hamada at D/E 0.5 and tax rate 24%.

from decimal import Decimal

import fairmark as fm

r = fm.coc.cost_of_equity(
    "0.035",
    "0.055",
    levered_beta="1.1",
    country_risk_premium="0.02",
    country_exposure="0.5",
    size_premium="0.02",
    specific_risk_premium="0.01",
    specific_risk_justification="Dependence on two customers for 60% of revenue",
)
# 0.035 + 1.1 × 0.055 + 0.02 × 0.5 + 0.02 + 0.01
assert r.cost_of_equity == Decimal("0.1355")
print(r.conventions[-1])  # specific risk premium justification: Dependence on two ...

relevered = fm.coc.cost_of_equity(
    "0.03",
    "0.05",
    relever={
        "unlevered_beta": "0.8",
        "debt_to_equity": "0.5",
        "tax_rate": "0.24",
        "formula": "hamada",
    },
)
# beta 0.8 × (1 + 0.76 × 0.5) = 1.104; ke 0.03 + 1.104 × 0.05 = 0.0852
assert relevered.levered_beta == Decimal("1.104")
assert relevered.cost_of_equity == Decimal("0.0852")

Inputs¶

Field	Meaning	Default
`risk_free_rate`	Risk-free rate, annual decimal fraction	required
`equity_risk_premium`	Equity risk premium of the market, annual decimal fraction	required
`levered_beta`	Levered beta, dimensionless; give exactly one of `levered_beta` and `relever`	none
`relever`	`{unlevered_beta, debt_to_equity, tax_rate, formula, debt_beta}` to relever an unlevered beta; `formula` is `hamada` or `harris_pringle`, `debt_beta` defaults to 0	none
`country_risk_premium`	Country risk premium, annual decimal fraction, not negative	`0`
`country_exposure`	Exposure to the country risk (\(\lambda\)), dimensionless, not negative	`1`
`size_premium`	Size premium, annual decimal fraction	`0`
`specific_risk_premium`	Company-specific risk premium, annual decimal fraction	`0`
`specific_risk_justification`	Written reason for the specific premium; required when the premium is not zero, recorded in the conventions when given with a zero premium	none

Outputs¶

Field	Meaning
`cost_of_equity`	Cost of equity, annual decimal fraction
`levered_beta`	Levered beta used, given or relevered

The steps show the relevered beta (when computed), the market risk component \(\beta_L \cdot ERP\), the country risk component \(CRP \cdot \lambda\) and the sum.

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

Convention	Options	Default
Rates	annual effective, decimal fractions	fixed
Beta	given, or relevered by Hamada or Harris-Pringle	none: one must be given
Debt beta (when relevering)	any value	0, risk-free debt
Country risk premium, exposure	any value	0 and 1
Size premium	any value	0
Specific risk premium	any value, with a written justification when not zero	0

Errors¶

Code	When
`FM-E001`	Both or neither of `levered_beta` and `relever` are given; a non-zero specific premium has no justification; the country risk premium or the country exposure is negative; in `relever`, D/E is negative or the tax rate is outside 0 to below 1; a field is missing or unknown
`FM-E008`	Arithmetic overflow (inputs far outside a realistic range)

Accuracy¶

The formula, relevering included, uses only additions and multiplications: results are exact up to the 28 significant digits of the decimal type.

Verified against¶

Golden cases in tests/golden/coc.cost_of_equity/, whose expected values are closed-form computations written out in each file and checkable by hand.
CAPM with additive premia and the relevering formulas in their commonly published form.

Cost of equity (coc.cost_of_equity)¶