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# 2.1. Different expressions of the Value of the tax shield and of the required return to equity

Table 1 contains the 8 most important theories. For each theory, the table contains the formula for

calculating the VTS and the equation that relates the required return to equity, Ke, with the required return to

assets (or required return to unlevered equity), Ku.

7 According to Fernandez (2001), VTS = PV[Ku; D(KuT+ RF - Kd)]

Pablo Fernánde z Financial literature about discounted cash flow valuation

Table 1. Competing theories for calculating the value of the tax shields

Theories VTS Ke

1 Fernandez (2004) PV[Ku; DTKu] Ke = Ku+

D(1- T)

E

(Ku - Kd)

2 Damodaran (1994) PV[Ku; DTKu - D (Kd- RF) (1-T)] Ke = Ku+

D (1- T)

E

(Ku -RF )

3 Practitioners PV[Ku; T D Kd - D(Kd- RF)] Ke = Ku+

D

E

(Ku - RF )

4 Harris-Pringle (1985),

Ruback (1995)

PV[Ku; T D Kd ] Ke = Ku +

D

E

(Ku -Kd)

5 Myers (1974) PV[Kd; T D Kd ] Ke = Ku +

D -DVTS

E

(Ku -Kd)

6 Miles-Ezzell (1980) PV[Ku; T D Kd] (1+Ku)/(1+Kd0) Ke = Ku +

D

E

(Ku - Kd) 1 -

T Kd

1+ Kd

7 Miller (1977) 0 Ke = Ku+

D

E

[KuKd(1- T)]

8 Fernandez (2001) PV[Ku; D(KuT+ RF- Kd) Ke = Ku+

D

E

[Ku(1- T)+ KdT - RF ]

PV = Present value; T = Corporate tax rate; Ku = Cost of unlevered equity (required return of unlevered

equity); Ke = Cost of levered equity (requi red return of levered equity); Kd = Required return of debt = cost of

debt; D = Value of debt; E = Value of equity; RF = Risk free rate; WACC = weighted average cost of capital;

According to the Fernandez (2004) theory, the VTS is the present value of DTKu (not the interest tax

shield) discounted at the unlevered cost of equity (Ku). This theory implies that the relationship between the

leveraged beta and the unlevered beta is

[7] L = u

D(1T)

E

(u d)

The second theory is that of Damodaran (1994). Although Damodaran does not mention what should be

the discounted value of the tax shield, his formula relating the levered beta with the asset beta

[8] L = u

D(1T)

E

u implies that VTS = PV[Ku; DTKu - D (Kd- RF) (1-T)]

It is important to notice that formula [8] is exactly formula [7] assuming that d = 0. Although one

interpretation of this assumption is that “all of the firm’s risk is borne by the stockholders (i.e., the beta of the

debt is zero)”8, we think that it is difficult to justify that the return on the debt is uncorrelated with the return on

assets of the firm. We rather interpret formula [8] as an attempt to introduce leverage cost in the valuation: for

a given risk of the assets (u), by using formula [8] we obtain a higher L (and consequently a higher Ke and a

lower equity value) than with formula [7].

We label the third theory as being that of practitioners. The formula that relates the levered beta with the

asset beta

Pablo Fernánde z Financial literature about discounted cash flow valuation

[9] L = u

D

E

u implies that VTS = PV[Ku; T D Ku - D(Kd- RF)]

It is important to notice that formula [9] is exactly formula [8] eliminating the (1-T) term. We interpret

formula [9] as an attempt to introduce still higher leverage cost in the valuation: for a given risk of the assets

(u), by using formula [9] we obtain a higher L (and consequently a higher Ke and a lower equity value) than

with formula [8].

Harris and Pringle (1985), and Ruback (1995) propose that the value creation of the tax shield is the

present value of the interest tax shield discounted at the unlevered cost of equity (Ku). One straight

interpretation of this assumption is that “the interest tax shields have the same systematic risk as the firm’s

underlying cash flows”9. But another interpretation comes from analyzing the formula that relates the levered

beta with the asset beta:

[10] L = u

D

E

(u d)

It is important to notice that formula [10] is exactly formula [7] eliminating the (1-T) term. We

interpret formula [10] as an attempt to introduce still higher leverage cost in the valuation: for a given risk of

the assets (u), by using formula [10] we obtain a higher L (and consequently a higher Ke and a lower equity

value) than with formula [7].

According to Myers (1974), the value creation of the tax shield is the present value of the interest tax

shield discounted at 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.

The sixth theory is that of Miles and Ezzell (1980). Although Miles and Ezzell do not mention what

should be the discounted value of the tax shield, his formula relating the required return to equity with the

required return for the unlevered company [Ke = Ku + (D/E) (Ku - Kd) [1 + Kd (1-T)] / (1+Kd)] implies that

PV[Ku; T D Kd] (1+Ku)/(1+Kd0). 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 Dt -1) is Kd for the tax saving during the first year, and Ku for the

tax saving during the following years.

The seventh theory is Miller (1977). The value of the firm is independent of its capital structure, that is,

VTS = 0.

The eighth theory is Fernandez (2001). It quantifies the leverage cost (assuming that Fernandez (2004)

provides the VTS without leverage costs) as PV[Ku; D (Kd- RF)]. One way of interpreting this assumption is that

the reduction in the value of the firm due to leverage is proportional to the amount of debt and to the difference

of the required return on debt minus the risk free rate. The cost of leverage does not depend on tax rate. 10