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R.C. Higgins

Fall 2004

LEVERING AND UNLEVERING BETAS

Purpose

. To show the relation between a company's asset, or unlevered, beta and its equity beta, and to demonstrate why one might want to know this relation.

Derivation. By definition, the market value of a levered firm equals the market value of its debt plus the

market value of equity.

Modigliani and Miller tell us that the value of a levered firm can be written as the value of the firm

unlevered plus the present value of the tax shields due to debt financing. (This relationship assumes

perpetual debt and ignores any bankruptcy or distress costs accompanying debt financing.)

Equating these two expressions, tDVED

u where D is interest-bearing debt, E is the market value of equity, V u is the value of the firm without any debt, and t is the marginal tax rate.

An important property of beta is that the beta of a portfolio is the weighted-average of the betas of the

individual assets comprising the portfolio. Applying this insight to both sides of the equation above, ITS

uu uu ED tDVtD tDVV EDE EDD where D is the beta of debt, e is the beta of equity, u is the beta of the unlevered firm, or equivalently, the firm's asset beta, and ITS is the beta of the firm's interest tax shields. Now two assumptions: 1) The firm's debt is for practical purposes riskless, so that D equals 0, and 2) the risk of interest tax shields equals the risk of the firm's unlevered asset cash flows, so that ITS u 1 Given these assumptions, the above equation simplifies to Eu EDE In words, a firm's asset beta equals its equity-to-value ratio times its equity beta. An Example. To illustrate the use of this equation, suppose Zenos Manufacturing is contemplating

investing in a new business. An existing company in the target industry has an equity beta of 0.90 but is

very conservatively financed with a market value, equity-to-value ratio 95%. Zenos intends to use an

equity-to-value ratio of 30% and wonders what beta to use in estimating a cost of capital in the new business. ITS 1

A popular alternative assumption is that

D = 0, which yields the following relation between asset beta and equity beta, Eu EDt 111
ITS

This alternative assumption may be plausible for existing debt, but if firm indebtedness varies over time

with firm value, it is much less plausible for new debt. For new debt I think the more reasonable assumption is that = u as above. I prefer this assumption because it is at least as plausible as the alternative, it yields a simpler expression, and it is consistent with traditional WACC valuation.

R.C. Higgins

Fall 2004

Zenos needs to unlever and then re-lever the target firm's equity beta. Unlevering the target's equity beta

yields an estimated asset beta for the new industry. Thus, u = .855.090.095.0 Relevering this

asset beta to reflect Zenos target capital structure yields the appropriate equity beta for Zenos to use in

estimating a cost of capital in the new business. .85.2855.030.01 uE E EDquotesdbs_dbs15.pdfusesText_21