# Levered and unlevered Beta

**ABSTRACT** We prove that in a world without leverage cost the relationship between the levered beta ( L) and the unlevered beta ( u) is the No-costs-of-leverage formula: L = u + ( u - d) D (1 - T) / E. We also analyze 6 alternative valuation theories proposed in the literature to estimate the relationship between the levered beta and the unlevered beta (Harris and Pringle (1985), Modigliani and Miller (1963), Damodaran (1994), Myers (1974), Miles and Ezzell (1980), and practitioners) and prove that all provide inconsistent results.

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**ABSTRACT:**This study analyzes the impacts of tax policy on market risk for the listed firms in the Viet Nam commercial electric industry as it becomes necessary. First, by using quantitative and analytical methods to estimate asset and equity beta of total 18 listed companies in Viet Nam commercial electric industry with a proper traditional model, we found out that the beta values, in general, for many companies are acceptable. Second, under 3 different scenarios of changing tax rates (20%, 25% and 28%), we recognized that there is not large disperse in equity beta values, estimated at 0,625, 0,626 and 0,627.These values are lower than those of the listed VN construction firms. Third, by changing tax rates in 3 scenarios (25%, 20% and 28%), we recognized equity /asset beta mean increase if tax rate increases from 20% to 25%, then goes up to 28%. Finally, this paper provides some outcomes that could provide companies and government more evidence in establishing their policies in governance.

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Pablo Fernández. IESE Business SchoolBeta levered and beta unlevered

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Beta levered and beta unlevered

Pablo Fernández*

PricewaterhouseCoopers Professor of Corporate Finance

IESE Business School

Camino del Cerro del Aguila 3. 28023 Madrid, Spain.

Telephone 34-91-357 08 09 Fax 34-91-357 29 13 e-mail: fernandezpa@iese.edu

ABSTRACT

We prove that in a world without leverage cost the relationship between the levered beta (βL) and the

unlevered beta (βu) is the No-cost-of-leverage formula: βL = βu + (βu – βd) D (1 – T) / E.

We also prove that the value of tax shields in a world with no leverage cost is the present value of the

debt (D) times the tax rate (T) times the required return to the unlevered equity (Ku), discounted at the

unlevered cost of equity (Ku): D T Ku / (Ku – g).

We then analyze 7 valuation theories proposed in the literature to estimate the relationship between

the levered beta and the unlevered beta: Modigliani-Miller (1963), Myers (1974), Miles-Ezzell (1980),

Harris-Pringle (1985), Damodaran (1994), No-cost-of-leverage and Practitioners. ? Without leverage costs

the relationship between the betas is the No-cost-of-leverage formula. Only Damodaran provides consistent

valuations once leverage costs are allowed for, but he introduces leverage costs in an ad hoc way.

JEL Classification: G12, G31, M21

Keywords: unlevered beta, levered beta, asset beta, value of tax shields, required return to equity, leverage cost,

March 4, 2002

* I would like to thank my colleagues José Manuel Campa and Miguel Angel Ariño, and an anonymous reviewer for very

helpful comments, and to Charlie Porter for his wonderful help revising previous manuscripts of this paper.

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1. Introduction

This paper provides clear, theoretically sound, guidelines to evaluate the appropriateness of various

relationships between the levered beta and the unlevered beta.

The relationship between the levered beta (βL) and the unlevered beta (βu) is

[10] βL = βu + (βu – βd) D (1 – T) / E.

For constant growth companies, we prove that the value of tax shields in a world with no leverage cost

is the present value of the debt (D) times the tax rate (T) times the required return to the unlevered equity

(Ku), discounted at the unlevered cost of equity (Ku):

[32] VTS = D T Ku / (Ku – g)

Please note that it does not mean that the appropriate discount for tax shields is the unlevered cost of

equity. We discount D T Ku, which is higher than the tax shield. As shown in Fernandez (2001) equation

[32] is the difference of two present values.

The paper is organized as follows.

In Section 2, we derive the relationship between the levered beta and the unlevered beta for

perpetuities in a world without leverage costs. This relationship is equation [10].

[10] βL = βu + (βu – βd) D (1 – T) / E

In Section 3, we revise the financial literature about the relationship between the levered beta and the

unlevered beta. We prove that the relationship between the levered beta and the unlevered beta for growing

perpetuities in a world without leverage costs is:

[36] βL = βu + (βu – βd) D (1 – T) / E

Note that [36] is equal to [10]. So, we may conclude that formula [10] is not only applicable for

perpetuities, but also for growing perpetuities and for the general case in a world without leverage cost.

In Section 4 we analyze the 7 theories for perpetuities. We prove that five of the seven theories

provide inconsistent results: Harris-Pringle (1985), Miles-Ezzell (1980) Modigliani-Miller (1963), Myers

(1974), and Practitioners. No-cost-of-leverage is the method to use in a world without leverage costs.

Damodaran provides us acceptable results in a world with leverage costs (although he introduces leverage

costs in an ad hoc way)

Our conclusions are in Section 5.

Appendix 1 contains the dictionary of the initials used in the paper, and Appendix 2 the main

valuation formulas according to the seven valuation theories that we analyze.

2. Relationship between the levered beta and the unlevered beta for perpetuities in a world

without leverage costs

The value of tax shields for perpetuities in a world without leverage cost is DT.

[1] VTS = Value of tax shields = DT

Many authors report this result. For example, Brealey and Myers (2000), Modigliani and Miller

(1963), Taggart (1991), Copeland et al. (2000), and Fernandez (2001). Fernandez (2001) proves [1]

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computing the difference between the present value of the taxes paid in the unlevered company and the

present value of the taxes paid in the levered company (GL).

But one problem of equation [1] is that DT can be understood as = D α T / α. At first glance, α can

be anything, related or unrelated to the company that we are valuing. In Section 3 it will be seen that

Modigliani and Miller (1963) assume that α is risk-free rate (RF). Myers (1974) assumes that α is the cost

of debt (Kd). Fernandez (2001) shows that the value of tax shields is the difference between the present

values of the taxes paid by the unlevered company and the taxes paid by the levered company and proves

that the correct α is the required return to unlevered equity (Ku). Fernandez (2001) also proves that for

growing companies both Modigliani and Miller (1963) and Myers (1974) provide inconsistent results.

The formula for the adjusted present value [2] indicates that the value of the debt today (D) plus that

of the equity (E) of the levered company is equal to the value of the equity of the unlevered company (Vu)

plus the value of tax shields due to interest payments (VTS).

[2] E + D = Vu + VTS

Knowing the value of tax shields (DT), and considering that Vu = FCF/Ku, we may rewrite

equation [2] as [3]

[3] E + D (1 – T) = FCF/Ku

Taking into consideration that the relationship between ECF and FCF for perpetuities is:

[4] FCF = ECF + D Kd (1 – T), and that E = ECF/Ke, we find that the relationship between the required

return to assets (Ku) and the required return to equity (Ke) in a world without leverage costs is:

[5] Ke = Ku + (Ku – Kd) D (1 – T) /E

The formulas relating the betas with the required returns are:

[6] Ke = RF + βL PM

[7] Ku = RF + βu PM

[8] Kd = RF + βd PM

RF is the risk-free rate and PM is the market risk premium. Substituting [6], [7] and [8] in [5], we

get:

[9] RF + βL PM = RF + βu PM + (RF + βu PM – RF – βd PM) D (1 – T) /E

Then, the relationship between the beta of the levered equity (βL), the beta of the unlevered equity

(βu) and the beta of debt (βd) for perpetuities in a world without leverage costs is:

[10] βL = βu + (βu – βd) D (1 – T) /E

3. Literature review: 7 main theories

There is a considerable body of literature on the discounted cash flow valuation of firms. We will

now discuss the most salient papers, concentrating particularly on those that proposed different expressions

for the relationship between levered beta and unlevered beta.

Before discusing the theories, it is useful to get the relationship between Ke, Ku, Kd, E, D, VTS and g

(growth) for growing perpetuities. As Vu = FCF / (Ku – g), we can rewrite equation [2] as

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[11] E + D = FCF / (Ku – g) + VTS

The relationship between the equity cash flow and the free cash flow is

[12] FCF = ECF + D Kd (1 – T) – g D

By substituting [12] in [11], we get:

[13] E + D = [ECF + D Kd (1 – T) – g D] / (Ku – g) + VTS

As the relationship between the equity cash flow (ECF) and the equity value is ECF = E (Ke – g) we

may rewrite [13] as:

[14] E + D = [E (Ke – g) + D Kd (1 – T) – g D] / (Ku – g) + VTS

Multiplying both sides of equation [14] by (Ku – g) we get:

[15] (E + D) (Ku – g) = [E (Ke – g) + D Kd (1 – T) – g D] + VTS (Ku – g)

Eliminating – g (E + D) on both sides of the equation [15]:

[16] (E + D) Ku = [E Ke + D Kd (1 – T)] + VTS (Ku – g)

Equation [16] may be rewritten as:

[17] D [Ku – Kd (1 – T)] – E (Ke – Ku) = VTS (Ku – g)

In the constant growth case, the relationship between Ke, Ku, Kd, E, D, VTS and g is Equation [17]:

Modigliani and Miller (1958) studied the effect of leverage on the firm's value. In the presence of

taxes and for the case of a perpetuity, they calculate the value of tax shields (VTS) by discounting the

present value of the tax savings due to interest payments of a risk-free debt (T D RF) at the risk-free rate

(RF). Their first proposition, with taxes, is transformed into Modigliani and Miller (1963, page 436, formula

3):

[18] E + D = Vu + PV[RF; D T RF] = Vu + D T

DT is the value of tax shields (VTS) for a perpetuity. This result is the same as our equation [1]. But

as will proven later on, this result is only correct for perpetuities. Discounting the tax savings due to interest

payments of a risk-free debt at the risk-free rate provides inconsistent results for growing companies.

Modigliani and Miller’ purpose was to illustrate the tax impact of debt on value. They never addressed the

issue of the riskiness of the taxes and only treated perpetuities. Later on, it will be seen that if we relax the

no-growth assumption, then new formulas are needed.

For a perpetuity, the relationship between levered beta and unlevered beta implied by [18] is [10] as

we have seen in section 2. But for a growing perpetuity, the value of tax shields for a growing perpetuity,

according to Modigliani and Miller (1963), is:

[19] VTS = D T RF / (RF – g)

Substituting [19] in [17], we get:

D [Ku – Kd (1 – T)] – E (Ke – Ku) = D T RF (Ku – g) / (RF – g)

Then, the relationship between the levered and the unlevered required return to equity according to

Modigliani and Miller (1963) is:

[20] Ke = Ku + (D / E) [Ku – Kd (1 – T) – T RF (Ku – g) / (RF – g)] =

= Ku + (D / E) [Ku – Kd (1 – T) – VTS (Ku – g) / D]

And the relationship between levered beta and unlevered beta is

[21] βL = βu + (D / E) [βu – βd +(T Kd/ PM) – VTS (Ku – g) / (D PM)]

Myers (1974) introduced the APV (adjusted present value). According to it, the value of the levered

firm is equal to the value of the firm with no debt (Vu) plus the present value of the tax saving due to the

payment of interest (VTS). Myers proposes calculating the VTS by discounting the tax savings (D T Kd) at

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Pablo Fernández. IESE Business SchoolBeta levered and beta unlevered

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the cost of debt (Kd). The argument is that the risk of the tax saving arising from the use of debt is the

same as the risk of the debt. Then, according to Myers (1974):

[22] VTS = PV [Kd; D T Kd]

It is easy to deduct that the relationship between levered beta and unlevered beta implied by [22] for

growing perpetuities is [23]:

[23] βL = βu + (D / E) (βu – βd) [1 – T Kd / (Kd – g)]

Luehrman (1997) recommends to value companies by using the Adjusted Present Value and

calculates the VTS as Myers. This theory provides inconsistent results for companies other than

perpetuities as will be shown later.

According to Miles and Ezzell (1980), a firm that wishes to keep a constant D/E ratio must be

valued in a different manner from the firm that has a preset level of debt. For a firm with a fixed debt target

[D/(D+E)] they claim that the correct rate for discounting the tax saving due to debt (Kd T D) is Kd for the

tax saving during the first year, and Ku for the tax saving during the following years.

The expression of Ke is their formula 22:

[24] Ke = Ku + D (Ku – Kd) [1 + Kd (1 – T)] / [(1 + Kd) E]

And the relationship between levered beta and unlevered beta implied by [24] is:

[25] βL = βu + (D / E) (βu – βd) [1 – T Kd / (1 + Kd)]

Lewellen and Emery (1986) also claim that the most logically consistent method is Miles and

Ezzell.

Harris and Pringle (1985) propose that the present value of the tax saving due to the payment of

interest (VTS) should be calculated by discounting the tax saving due to the debt (Kd T D) at the rate Ku.

Their argument is that the interest tax shields have the same systematic risk as the firm’s underlying cash

flows and, therefore, should be discounted at the required return to assets (Ku).

Then, according to Harris and Pringle (1985), the value of tax shields is:

[26] VTS = PV [Ku; D Kd T]

Substituting [26] for growing perpetuities in [17], we get:

[27] D [Ku – Kd (1 – T)] – E (Ke – Ku) = D Kd T

Substituting [26] for growing perpetuities in [17], we get:

[27] D [Ku – Kd (1 – T)] – E (Ke – Ku) = D Kd T

Then, the relationship between the levered and the unlevered required return to equity according to

Harris and Pringle (1985) is:

[28] Ke = Ku + (D / E) (Ku – Kd)

And the relationship between levered beta and unlevered beta implied by [28] is:

[29] βL = βu + (D / E) (βu – βd)

Ruback (1995) reaches formulas that are identical to those of Harris-Pringle (1985). Kaplan and

Ruback (1995) also calculate the VTS “discounting interest tax shields at the discount rate for an all-

equity firm”. Tham and Vélez-Pareja (2001), following an arbitrage argument, also claim that the

appropriate discount rate for tax shields is Ku, the required return to unlevered equity.

Taggart (1991) gives a good summary of valuation formulas with and without personal income tax.

He proposes that Miles & Ezzell’s (1980) formulas should be used when the company adjusts to its target