Borrowing a listed company's beta for a privately held, more highly geared asset, step by step.
To unlever a comparable's beta, strip out its debt: asset beta = equity beta × E/V + debt beta × D/V. To relever, add your own asset's gearing back: equity beta = asset beta + (asset beta − debt beta) × D/E. Three illustrative listed comparables with equity betas around 0.75 unlever to an asset beta of 0.409, which relevers to 1.131 at 70 per cent gearing, a cost of equity of 10.22 per cent rather than the 8.12 per cent their raw betas suggest.
Worked in full in The Infrastructure Investment Analyst by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
An infrastructure asset almost never has a quoted share price, so its beta is borrowed from listed companies in the same business: utilities, toll road operators, airport groups. Those companies are geared at perhaps 45 to 55 per cent; a regulated network or a concession held privately is typically geared much higher. A listed utility's equity beta measures the risk of equity sitting on top of its own debt. Applied to an asset carrying more debt, it understates the risk of the thinner equity slice. Unlevering removes the comparable's capital structure; relevering puts on the target's.
Unlever: βA = βE × E/V + βD × D/V
Relever: βE = βA + (βA − βD) × D/E
Cost of equity: rE = risk-free rate + βE × equity risk premium
Excel: =BetaE*E_V+BetaD*D_V, then =BetaA+(BetaA-BetaD)*D_E
This is the form regulators of water and energy networks commonly use: no tax term, and a small positive debt beta because highly geared debt carries some market risk. Use net debt, not gross, and market values of equity, not book.
| Comparable | Equity beta | Net debt / equity | D/V |
|---|---|---|---|
| Listed utility A | 0.75 | 1.10 | 52.4% |
| Listed utility B | 0.68 | 0.90 | 47.4% |
| Listed toll road C | 0.82 | 1.30 | 56.5% |
Debt beta 0.10, risk-free rate 4.0 per cent and equity risk premium 5.5 per cent, all illustrative. The target is a privately held asset at 70 per cent net debt to enterprise value, a D/E of 2.333.
Step 1, unlever each comparable. For utility A, E/V is 1 / 2.10 = 0.476 and D/V is 0.524. The equity part is 0.75 × 0.476 = 0.357; the debt part is 0.10 × 0.524 = 0.052. Asset beta: 0.410.
| Comparable | Equity beta | Asset beta |
|---|---|---|
| Listed utility A | 0.75 | 0.410 |
| Listed utility B | 0.68 | 0.405 |
| Listed toll road C | 0.82 | 0.413 |
| Average | 0.750 | 0.409 |
The equity betas spread from 0.68 to 0.82; the asset betas sit within 0.405 to 0.413. Most of the difference between the listed companies was gearing, not business risk, which is the point of the exercise.
Step 2, relever to the target. 0.409 + (0.409 − 0.10) × 2.333 = 0.409 + 0.722 = 1.131.
Step 3, cost of equity. 4.0% + 1.131 × 5.5% = 4.0% + 6.22% = 10.22 per cent.
The asset's equity carries a beta of 1.131, half as much again as the comparables' 0.750, and a cost of equity of 10.22 per cent against 8.12 per cent if their betas were used as they are. The 2.10 point gap is not a premium for anything the asset does; it is the arithmetic of the extra debt. Because the target's gearing will fall as the debt amortises, the cost of equity should be relevered period by period in a discounted cash flow, a point taken further in does a constant WACC overstate value when the debt amortises.
| Net debt / EV | D/E | Equity beta | Cost of equity |
|---|---|---|---|
| 50% | 1.00 | 0.719 | 7.95% |
| 60% | 1.50 | 0.873 | 8.80% |
| 70% | 2.33 | 1.131 | 10.22% |
| 80% | 4.00 | 1.646 | 13.06% |
The relationship is not linear: D/E explodes as gearing approaches 100 per cent, so the last ten points of debt add 2.84 points to the cost of equity, more than three times the 0.85 point step from 50 to 60. The debt beta matters less than people fear but is not nothing. With a debt beta of zero the asset beta falls to 0.357, the relevered beta rises to 1.191 and the cost of equity to 10.55 per cent; at 0.15 it is 10.06 per cent.
A corporate-finance alternative, the Hamada formula, includes a tax term and assumes a zero debt beta: βA = βE / (1 + (1 − t) × D/E). At 25 per cent tax it gives 1.129 here, almost the same, because the tax term and the debt beta happen to offset at these gearings. They do not offset in general. Pick one formula and use it both ways. The corporate version, with Hamada inside a full WACC, is worked in how to calculate WACC step by step.
A regulated asset valued both ways, with the cost of equity relevered every year on net debt, is in the free workbook for this book.
Regulators of networks have typically used small positive values for investment-grade debt, and a figure around 0.10 is a common illustrative choice. It matters at high gearing: in the worked case a debt beta of zero gives a cost of equity of 10.55 per cent at 70 per cent gearing, 0.10 gives 10.22 and 0.15 gives 10.06. Whatever you choose, use it in both the unlevering and the relevering.
Either, consistently. Hamada includes a tax term and assumes riskless debt; the debt-beta form drops the tax term, as many regulators do. In the illustrative case they give 1.129 and 1.131 at 70 per cent gearing, but only because the two effects offset. The costly error is mixing assumptions: unlevering with a 0.10 debt beta and relevering with none and no tax term gives 1.364, a cost of equity 1.28 points too high.
Because the relevering multiplies by debt over equity, which grows without limit as gearing approaches 100 per cent. With an asset beta of 0.409, moving from 70 to 80 per cent net debt to EV lifts D/E from 2.33 to 4.00 and the cost of equity from 10.22 to 13.06 per cent, a bigger step than from 50 to 70 per cent.
This article is one calculation from The Infrastructure Investment Analyst. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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