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3. The basic problem: the value of the tax shield due to the payment of interest (VTS)

Fernandez (2005) shows that in a world without leverage cost, the discounted value of the tax shields for

a perpetuity is DT. It is assumed that the debt’s market value (D) is equal to its book value 15 (N).

Table 2 reports the implications that each of the 7 theories has for the case of perpetuities. Column [1]

contains the general formula for calculating the VTS according to the 8 theories. Column [2] contains the

formula for calculating the VTS for perpetuities according to the 8 theories when the tax rate is positive.

Column [3] contains the formula for calculating the VTS for perpetuities according to the 8 theories when

there are no taxes.

It may be seen that only 2 theories accomplish formula [15], which implies VTS = DT. The 2 theories

are: Fernandez (2004) and Myers.

The other 6 theories provide a VTS lower than DT. The difference could be attributed to the leverage

cost. These 6 theories could be applicable in a “real world”, where the leverage cost do exist. But if this is the

case, leverage cost exist also when there are no taxes. In this situation (column [3] of table 2) these theories

should provide a negative VTS. It only happens in 3 theories of the 6: Damodaran, Practitioners and Fernandez

(2001).

With these two conditions, we are able to eliminate 3 theories that not only do not provide us with a

value of the tax shield of DT (as the candidates for a world without cost of leverage should), nor do they

provide us with a negative VTS when there are no taxes (as the candidates for a world with leverage cost should). The 3 candidates eliminated due to a lack of consistent results are the following: Harris-Pringle

(1985) or Ruback (1995), Miles-Ezzell (1980, and Miller (1977).

The 8 candidate theories provide a value of VTS = 0 if D = 0.

Table 2. Perpetuity. Value of the tax shield (VTS) according to the 8 theories.

VTS VTS in perpetuities

Theories General formula T>0 T = 0

[1] [2] [3]

1 Fernandez PV[Ku; DTKu] DT 0

2 Damodaran PV[Ku; DTKu - D(Kd-RF) (1-T)] DT-[D(Kd-RF)(1-T)]/Ku< DT - D(Kd-RF)/Ku < 0

3 Practitioners PV[Ku; T D Kd - D(Kd- RF)] D[RF-Kd(1-T)]/Ku < DT - D(Kd-RF)/Ku < 0

4 Harris-Pringle PV[Ku; T D Kd ] T D Kd/Ku< DT 0

5 Myers PV[Kd; T D Kd ] DT 0

6 Miles-Ezzell PV[Ku; T D Kd](1+Ku)/(1+Kd0) TDKd(1+Ku)/[(1+Kd0)Ku] <

DT

0

7 Miller 0 0 0

8 Fernandez PV[Ku; D(KuT+ RF- Kd)] D(KuT+RF- Kd)/Ku < DT - D(Kd-RF)/Ku < 0

necessary conditions with leverage cost without leverage cost

T > 0 < DT DT

T = 0 < 0 0

Number of theories: 3 2

Damodaran, Practitioners and Fernandez

(2001)

Fernandez (2004) and Myers

3 theories do not accomplish the necessary conditions to be considered:

Harris-Pringle (1985) or Ruback (1995), Miles-Ezzell (1980), and Miller (1977).

3.5. Analysis of competing theories in a world without cost of leverage and with constant growth

It is clear that the required return to levered equity (Ke) should be higher than the required return to

assets (Ku). Table 3 shows that only Fernandez (2004) provides us always with Ke > Ku.

Table 3. Problems of the candidate formulas to calculate the VTS in a world without cost-of-leverage

and with constant growth

Ke < Ku

Fernandez (2004) never

Myers If g>Kd(1-T) 16

Another problem of Myers is that Ke < Ku for high g and/or high T. VTS independent of unlevered

beta. On top of that, according to Myers, Ke decreases when T (tax rate) increases. According to Fernandez

(2004) Ke increases when T increases.

When the cost of debt (r) is not equal to the required return to debt (Kd), the value of the tax shield

according to Fernandez (2004) should be calculated as follows 17:

VTS = PV[Ku; DTKu + T(Nr-DKd)]

We would point out again that this expression is not a cash flow’s PV, but the difference between two

present values of two cash flows with a different risk: the taxes of the company without debt and the taxes of

the company with debt.