Unlevering and Relevering Beta

You are valuing a private packaging business. It has no share price, so it has no beta, so CAPM has nothing to chew on. The obvious move is to borrow a beta from listed packaging companies. The obvious move is also wrong as stated, because the beta you can look up on a screen is not measuring what you want.

Think about what a levered equity actually is. Two firms own identical factories making identical boxes. One is debt-free; the other funded half its assets with a loan. When box demand drops 10%, both firms' assets lose the same value. But the levered firm's lenders are owed a fixed amount regardless, so the entire loss lands on a much smaller equity base. Same business, twice the swing in the share price. A regression of that share price against the index will report a higher beta — and none of the difference is about boxes.

So an observed levered (equity) beta is a mixture: the risk of the business, amplified by however much debt happens to sit on that particular company's balance sheet. Unlevering removes the amplifier. Relevering re-applies it at the leverage you have in mind.

The formula

The standard relationship is the Hamada equation:

$$\beta_L = \beta_U \left[ 1 + (1 - t)\frac{D}{E} \right].$$

$\beta_L$ is the levered beta you observe, $\beta_U$ the unlevered or asset beta — the beta the firm would have with no debt — $t$ the marginal tax rate, and $D/E$ the ratio of debt to equity at market values. In words: leverage multiplies business risk by a factor bigger than one, and the tax shield takes a little of the sting out because the government absorbs $t$ of every interest payment. Run it backwards to unlever:

$$\beta_U = \frac{\beta_L}{1 + (1 - t)\,D/E}.$$

0.94 0.70 0.46

0 0.35 1.0 market D/E levered beta

Worked example: building a beta from comparables

Three listed packaging peers, with betas from a five-year monthly regression against a broad index, debt and equity at market value, and a 25% marginal tax rate.

Comp	Levered β	Market D/E	Unlevered β
A	1.30	0.50	1.30 / (1 + 0.75 × 0.50) = 0.945
B	1.05	0.20	1.05 / (1 + 0.75 × 0.20) = 0.913
C	1.62	0.90	1.62 / (1 + 0.75 × 0.90) = 0.967

The raw betas span 1.05 to 1.62 — a 54% range that would swing a valuation wildly. Unlevered, they collapse to 0.913–0.967. That tightening is the whole point: it says these three really are the same business, and the spread was funding, not fundamentals.

Take the mean, $\beta_U = 0.942$, and relever to the target structure for the private firm, $D/E = 0.35$:

$$\beta_L = 0.942 \times (1 + 0.75 \times 0.35) = 0.942 \times 1.2625 = 1.189.$$

With a risk-free rate of 4.2% and an equity risk premium of 5.0%, the cost of equity is $4.2 + 1.189 \times 5.0 = 10.15\%$. Finish the job: $D/E = 0.35$ means $D/V = 0.35/1.35 = 25.9\%$, and at a pre-tax cost of debt of 5.5%,

$$\text{WACC} = 0.741 \times 10.15\% + 0.259 \times 5.5\% \times 0.75 = 8.59\%.$$

Refinements you will be asked about

Debt beta. Hamada assumes debt is riskless, $\beta_D = 0$. Fine for an investment-grade borrower; not fine for Comp C at 0.9× leverage. The general form is $\beta_U = [\beta_L + \beta_D(1-t)D/E] \,/\, [1 + (1-t)D/E]$. Give Comp C a debt beta of 0.10 and its unlevered beta rises from 0.967 to 1.008 — you had been giving away risk that lenders were actually bearing.

Which leverage policy. The $(1-t)$ term embeds Myers' assumption of fixed debt. If instead the company rebalances to a constant debt ratio, the tax shield carries business risk and the Harris–Pringle version drops the tax term entirely: $\beta_U = \beta_L / (1 + D/E)$. Sponsors modelling an APV with a fixed amortisation schedule use the first; a corporate with a published target ratio fits the second.

Regression hygiene. Conventions differ and the answer moves with them: five years of monthly returns against a local index (Damodaran, Value Line) versus two years of weekly (Bloomberg's default). Monthly data reduces the non-synchronous-trading bias that drags illiquid small caps' betas down; weekly data reacts faster to a genuine change in the business. Raw betas are noisy enough that most providers apply a Blume adjustment toward 1.0 before you ever see them — check whether your screen is showing you raw or adjusted, because unlevering an already-shrunk beta double-counts the correction.