A Note on How to Compute Unlevered Betas
for Highly Levered Companies
Abstract
The expected cost of capital is usually obtained by estimating its individual components,
the cost of debt and the cost of equity, and computing a WACC. Unfortunately, in the
presence of risky debt, the standard methodology produces a systematic overestimation.
This bias is increasing in leverage and volatility of assets. In this paper, we propose a novel
methodology to compute the expected return on assets by estimating the unlevered beta
from a time series of assets returns constructed on the basis of Merton (1974)’s model.
Keywords: Cost of Capital, Leverage, Beta, Contingent Pricing, WACC, Merton
JEL Classification: G12, G13, G31, G32
1. Introduction
The cost of capital is a topic of paramount importance for corporate finance, with
applications to capital budgeting, valuation, incentive compensation schemes…
Unfortunately, the literature does not offer adequate or practicable recommendations for
highly levered firms. This note makes a straightforward contribution in the determination of
the cost of capital of such firms.
In computing the weighted average cost of capital (WACC), it is common practice to
use the promised yield of debt contracts, which is far from the expected return (𝐾𝐷 ) of debt
of highly levered firms. The promised yield, usually measured by the internal rate of return,
is the highest of all possible returns. If default is possible, it is not an average of possible
returns. Under the latter assumption, using the promised return of debt in the computation
of WACC is not correct, because WACC should measure the expected return required by
the providers of the capital. This common error has been addressed by Cooper and
Davydenko (2007), who propose a method of estimating 𝐾𝐷 to be used in the computation
of the WACC. They obtain 𝐾𝐷 by making use of Merton (1974)’s model. For ease of
exposition, they assume the absence of taxes, in which case WACC equals the expected
return (𝐾𝐴 ) on assets or, more precisely, the expected return of equity holders were the firm
not levered. We also assume taxes away and use Merton´s model.
A key input in Cooper and Davydenko (2007)’s model is the expected return (𝐾𝐸 ) on
equity (𝐸), which they propose to estimate by a standard application of CAPM, that is,
using the historical beta of equity returns. This amounts to assuming a constant expected
return on equity (𝐾𝐸 ), which is structurally inconsistent with Merton (1974)’s model. In the
latter, 𝐾𝐸 varies constantly with the value of assets (𝐴). Moreover, 𝐾𝐸 tends to infinity as
the present value of debt (𝐷) tends to 𝐴. Therefore, the more needed is the methodology,
the more inaccurate are the results.
An alternative approach proposed to obtain 𝐾𝐴 is to compute it from an estimate of
the beta of assets (𝛽𝐴 ), which equals the weighted average of the beta of equity (𝛽𝐸 ) and the
beta of debt (𝛽𝐷 ). However, the estimation of 𝛽𝐷 is problematic for two reasons: (i) the lack
of liquidity of the most debt instruments, and (ii) the variability of 𝛽𝐷 due to changes in the
𝐴. Typically, 𝛽𝐷 is assumed null, which is grossly inadequate for highly levered companies.
The problem with the previous approaches is that they rely on the common
assumption that 𝛽𝐸 , 𝐾𝐸 , 𝛽𝐷 and 𝐾𝐷 remain constant. Merton’s model contradicts this
assumption as it identifies equity with a long call option on the assets of the company, and
debt with a portfolio comprising those assets and a short call on them. This paper proposes
a novel approach that, instead of computing 𝛽𝐴 as an average of 𝛽𝐸 and 𝛽𝐷 , directly
estimates 𝛽𝐴 from a time series of assets returns, which can be constructed using Merton
(1974)´s model. After 𝛽𝐴 has been estimated, 𝐾𝐴 can be computed using the CAPM.
Briefly, we propose a methodology of estimating 𝐾𝐴 that comprises the following
steps: (i) creating a time series of asset values, (ii) computing the corresponding series of
asset returns, (iii) estimating the beta of assets, and finally (iv) calculating 𝐾𝐴 , the expected
return on assets.
2. Construction of a time series of assets returns
In order to construct the time series of asset values, we compute 𝐴 at each single
observation time using Merton´s model. The steps are as follows:
1. A time series of equity volatility 𝜎𝐸 is obtained. There are different possibilities to
achieve this: a simple exponentially weighted moving average historical volatility,
or a version of GARCH models, or the so-called realized volatility. We would not
discard the use of implied volatility of equity options, although we are aware that
assuming log returns on equity as normal is inconsistent with the following step 3.
2. Debt payments are mapped into a single pair of tenor 𝑇 and payment 𝐷𝑇 .
3. Assuming that assets returns are normally distributed, and given the market value 𝐸
of equity, its volatility 𝜎𝐸 , and the risk free rate 𝑟𝑓 , we can compute both the value 𝐴
of total assets and its volatility 𝜎𝐴 by solving numerically the following twoequation system:
𝐸 = 𝐴 N(𝑑1 ) − 𝐷𝑇 𝑒 −𝑟𝑓 𝑇 N(𝑑2 )
(1)
𝐸 𝜎𝐸 = N(𝑑1 ) 𝐴 𝜎𝐴
(2)
where
𝑑1 =
𝐴
ln 𝐷 + 𝑟𝑓 𝑇
𝑇
𝜎𝐴 √𝑇
+
𝜎𝐴 √𝑇
2
and
𝑑2 = 𝑑1 − 𝜎𝐴 √𝑇.
Equation (1) is Merton (1974)’s formula for the valuation of the equity of a company.
Equation (2) was obtained from (1) by Jones, Mason & Rosenfeld (1984) using Ito’s
formula. The above system of equations is routinely solved by rating agencies to obtain the
risk-neutral default probability 1 − N(𝑑2 ) or N(−𝑑2 ), which is them mapped into a realworld default probability. The rationale of equation (2) can be grasped by considering, on
the one hand, that multiplying the values 𝐸 and 𝐴 of equity and assets by their
corresponding percentage variability, 𝜎𝐸 and 𝜎𝐴 , yields the dollar variability of each, 𝐸𝜎𝐸
𝑑𝐸
and 𝐴𝜎𝐴 , and, on the other hand, that delta N(𝑑1 ) = 𝑑𝐴 is the ratio of dollar changes in 𝐸
and 𝐴.
Although we obtain both 𝐴 and 𝜎𝐴 for each observation time, only the time series of
𝐴 needs to be used to estimate the unlevered beta 𝛽𝐴 , from which the expected return 𝐾𝐴 on
assets can be computed using the CAPM. Moreover, any version of APT can be used to
estimate the expected return on assets from the time series of 𝐴.
3. Possible Extensions
The approach explained in the paper could be complemented in several ways.
First, given the importance of equity volatility 𝜎𝐸 for this model, different estimation
procedures could be explored, discussing strengths and weaknesses of each.
Second, the model could be extended by allowing rollover of debt maturity.
Third, it would be interesting to run an empirical test comparing the time series of
assets values estimated with the proposed methodology against the historical asset values in
those cases in which market value of debt is available. Another relevant experiment would
be the comparison between traditional equity betas unlevered using Modigliani and Miller
(1958)’s framework with those obtained using our approach.
4. Conclusions
In this paper we suggest a novel methodology, based on Merton (1974)’s model, to directly
compute the expected return on assets without estimating the expected returns of debt and
equity. This method aims to solve the typically overseen instability of those components.
References
Cooper, I. A. and S. A. Davydenko (2007). "Estimating the Cost of Risky Debt." Journal of
Applied Corporate Finance 19(3): 90-95.
Jones, E. P., S. P. Mason, and E. Rosenfeld (1984), Contingent Claims Analysis of
Corporate Capital Structure: An Empirical Investigation, Journal of Finance, 39, 61125.
Merton, R. C. (1974), On the Pricing of Corporate Debt: The Risk Structure of Interest
Rates, Journal of Finance, 29, 449-70.
Modigliani, F. and M. H. Miller (1958). "The Cost of Capital, Corporation Finance, and the
Theory of Investment." The American Economic Review XLVIII(3): 261-297.