This note discusses the neutrality conditions of a Firm Tax. In particular, it proves that the neutrality result found by Bond and Devereux (1995) holds under different default conditions.

Paolo M. Panteghini

English

Research Papers in Economics

Neutrality Properties of Firm Taxation under Default Risk 
Paolo M. Panteghini
University of Brescia and CESifo
Abstract
This note discusses the neutrality conditions of a Firm Tax. In particular, it proves that the
neutrality result found by Bond and Devereux (1995) holds under different default
conditions.
Citation: Panteghini, Paolo M., (2004) "Neutrality Properties of Firm Taxation under Default Risk." Economics Bulletin, Vol.
8, No. 4 pp. 1−7
Submitted: August 24, 2004.  Accepted: October 29, 2004.
URL: http://www.economicsbulletin.com/2004/volume8/EB−04H20003A.pdfNeutrality Properties of Firm Taxation under
Default Risk
Paolo M. Panteghini￿
University of Brescia and CESifo
August 24, 2004
Abstract
This note discusses the neutrality conditions of a Firm Tax. In
particular, it proves that the neutrality result found by Bond and
Devereux (1995) holds under di⁄erent default conditions.
1 Introduction
In a pioneering article Boadway and Bruce (1984) proposed a neutral corpo-
rate tax design, whose base is equal to current earnings, less depreciation and
the opportunity cost of ￿nance. According to this scheme, named Firm Tax
(FT), the opportunity cost of ￿nance is given by the nominal interest rate
on default-free bonds multiplied by the tax-written-down value of the ￿rm￿ s
depreciable assets. Bond and Devereux (1995) (hereafter BD) demonstrated
that, in the absence of credit market imperfections, the FT is neutral under
income, capital and default risk.
It is worth noting that BD do not explicitly introduce any default con-
dition. As we know, instead, there exist at least two di⁄erent default condi-
tions. First of all, default may be triggered when the ￿rms￿asset value falls
to the debt￿ s value. In this case debt is approximated with a positive net-
covenant and can be termed protected debt (see Leland, 1994). Alternatively,
￿Address: Department of Economics, Via S. Faustino, 74b, 25122 Brescia (ITALY);
panteghi@eco.unibs.it
1shareholders may have the opportunity to inject equity capital in order to
meet the ￿rm￿ s debt obligations. As long as they issue new capital and pay
the interest rate they can exploit future recoveries in the ￿rm￿ s pro￿tabil-
ity. In this case shareholders can decide when to default1. Following Leland
(1994), debt will then be named unprotected debt2. Since BD treat default
as an exogenous event, out of shareholders￿control, their result is implicitly
based on the assumption of protected debt. In this note we will prove that
neutrality holds even under unprotected debt ￿nancing.
2 The model
In this section we introduce a continuous-time model describing the invest-
ment decision by a representative ￿rm. At time t, the ￿rm invests if the
project￿ s expected Net Present Value (NPVt) is positive.
By assumption risk is fully diversi￿able, the risk-free interest rate r is
￿xed and the ￿rm is risk-neutral. Moreover, we assume that the ￿rm￿ s EBIT
(Earnings Before Interests and Taxes) follows a geometric Brownian motion
d￿t
￿t
= ￿dz; (1)
where ￿ is the variance parameter3. Moreover, the ￿rm￿ s project entails the
payment of a non-depreciable investment of given size4, say I.
Following Leland (1994) we assume that the ￿rm pays a coupon C to the
lender, and that the credit market is perfectly competitive. Thus, given C,
1The decision to default is thus equivalent to the exercise of a put option.
2Notice that both protected and unprotected debt are realistic ￿nancial instruments.
Leland (1994) argues that minimum net-worth requirements, implied by protected debt,
are common in short-term debt ￿nancing, whereas they are fairly rare in long-term debt
instruments. Moreover, Graham and Harvey (2001) ￿nd that ￿rms issue short-term debt
to time market interest rates. However more than 63% of the ￿rms surveyed state that
debt maturity is aimed at matching with assets￿lifetime. This entails that ￿rms use a
mix of short- and long-term debt istruments, which may be subject to di⁄erent default
conditions.
3The quality of results would not change if we introduced a drift in the Geometric
brownian motion.
4Notice that the introduction of depreciation would not a⁄ect the qualitative nature of
results.
2the market value of debt is computed by applying a non-arbitrage condition5.
In the event of default, the ￿rm is expropriated by the lender.
Let us next analyze the two default conditions. For simplicity, hereafter,
we will omit the time variable.
Protected debt When debt is protected, the threshold value of ￿ be-
low which default takes place is exogenous. To this end, we de￿ne ￿ as the
￿rm￿ s EBIT which leads to zero net cash ￿ ow, i.e. (1 ￿ ￿)￿ ￿ C + ￿rI = 0:
When, therefore, ￿ falls down to ￿; the ￿rm is expropriated and, therefore,
the value of equity falls to zero, namely
E
￿
￿;C
￿
= 0: (2)
Unprotected debt When debt is unprotected shareholders can choose
when to default. This entails that the threshold point below which default
takes place, say b ￿; is optimally chosen. Following Leland (1994), b ￿ is com-
puted by applying a Value Matching Condition (VMC) and a Smooth Pasting
Condition (SPC). The former condition requires that, when ￿ reaches b ￿; the
value of equity falls to zero, i.e.
E
￿
b ￿;C
￿
= 0; (3)
the latter condition requires that the marginal value of equity is nil, i.e.
@E
￿
b ￿;C
￿
@￿
= 0: (4)
As will be shown, b ￿ is a⁄ected by taxation and, thus, potentially distortive.
Despite this fact, we will prove that neutrality holds even under unprotected
debt ￿nancing.
3 The neutrality result
Under the FT, the base is given by the di⁄erence between ￿ and a tax
allowance, equal to the risk-free interest rate r times the book value of the
5Notice that this is equivalent to setting the value of debt and then computing the
relevant interest rate. For simplicity we also assume that debt cannot be renegotiated.
3asset I. Therefore, tax payments are equal to T = ￿ (￿ ￿ rI) and the ￿rm￿ s
net cash ￿ ow is
Y = ￿ ￿ C ￿ ￿ (￿ ￿ rI) = (1 ￿ ￿)￿ ￿ C + ￿rI (5)
Following BD, tax charges are assumed to be independent of the ownership
of the ￿rm. This entails that, after expropriation, the lender is subject to
the same tax treatment as shareholders.
Let us next analyze the e⁄ects of taxation on the project￿ s Net Present
Value (NPV (￿;C)). De￿ning E (￿;C) and D(￿;C) as the value of equity
and debt, respectively, yields
NPV (￿;C) = E (￿;C) + D(￿;C) ￿ I: (6)
De￿ning NPVLF as the laissez-faire Net Present Value, we can prove that:
Proposition 1 Given the tax rate ￿, the FT is neutral, i.e.
NPV (￿;C) = (1 ￿ ￿) ￿ NPVLF; (7)
irrespective of the default condition applied.
Proof. See the Appendix.
The intuition behind Proposition 1 is straightforward. In the Appendix,
we show that ￿ > b ￿: This entails that, under unprotected debt ￿nancing,
default does not take place when the net cash ￿ ow is nil. When ￿ lies
between b ￿ and ￿; shareholders face a negative cash ￿ ow. However, they
prefer to inject equity capital in order to exploit future recoveries in the
￿rm￿ s pro￿tability. The existence of such a put option entails that, coeteris
paribus, the value of equity is greater under unprotected debt ￿nancing. On
the other hand, for any C, the value of unprotected debt is less than that of
protected debt. This is due to the fact that the shareholders￿ability to delay
default reduces the value of the ￿rm in the event of default6. We can thus
argue that any switch from protected to unprotected debt ￿nancing entails
both an increase in the value of equity and a decrease in the value of debt.
As proven in Proposition 1, however, these two e⁄ects neutralise each other.
This implies that NPV (￿;C) is (1 ￿ ￿) times NPVLF irrespective of the
characteristics of debt.
6Recall that, when expropriation takes place, the ￿rm￿ s expected EBIT is lower (i.e. b ￿
< ￿).
44 Appendix: Proof of Proposition 1
Let us compute the value of debt. Before default, the lender receives C.
After default, the lender becomes shareholder and the value of debt turns to
be equity. Using dynamic programming one can write debt as
D(￿;C) =
8
<
:
Y dt + e￿rdt￿ [D(￿ + d￿;C)]
Cdt + e￿rdt￿ [D(￿ + d￿;C)]
if ￿ 2 [0; e ￿);
if ￿ 2 (e ￿;1):
(8)
where ￿ [:] is the expectation operator and e ￿ = ￿; b ￿: Expanding (8) and
using It￿￿ s Lemma, one obtains its closed-form solution
D(￿;C) =
8
<
:
(1￿￿)￿
r + ￿I +
P2
j=1 Bj￿￿j
C
r +
P2
j=1 Dj￿￿j
if ￿ 2 [0; e ￿);
if ￿ 2 (e ￿;1);
(9)
where ￿1 and ￿2 are, respectively, the positive and negative roots of the
characteristic equation7   (￿) ￿ ￿2
2 ￿(￿ ￿ 1) ￿ r = 0:
In the absence of any ￿nancial bubbles, we have B1 = D1 = 08. If,
moreover, the lender￿ s claim is null when ￿ = 0; the boundary condition
D(0;C) = 0 holds as well. This implies that B2 = 0, irrespective of the
quality of debt. To compute D2; let us stitch together the two branches of
function (9) at point ￿ = e ￿: Using (9) and solving for D2 one easily obtains
D2 =
"
(1 ￿ ￿) e ￿ ￿ C
r
+ ￿I
#
e ￿
￿￿2;
which yields
D(￿;C) =
8
> <
> :
(1￿￿)￿
r + ￿I if ￿ 2 [0; e ￿);
C
r +
h
(1￿￿)e ￿￿C
r + ￿I
i￿
￿
e ￿
￿￿2
if ￿ 2 (e ￿;1):
(10)
As can be seen, the value of D2 depends on the default condition applied
(namely on the trigger point e ￿).
7￿1;2 = 1
2 ￿
q￿1
2
￿2
+ 2r
￿2; with ￿1 > 1 and ￿2 < 0:
8For details on this condition see Dixit and Pindyck (1994).
5Using dynamic programming we can write the value of equity as
E (￿;C) =
8
<
:
0 if ￿ 2 [0; e ￿);
Y dt + e￿rdt￿ [E(￿ + d￿;C)] if ￿ 2 (e ￿;1):
(11)
Substituting (5) into (11), expanding and using It￿￿ s Lemma yields
rE (￿;C) = [(1 ￿ ￿)￿ ￿ C + ￿rI] +
￿2
2
￿
2E￿￿ (￿;C): (12)
Solving (12) one can rewrite (11) as
E (￿;C) =
8
> > > <
> > > :
0 if ￿ 2 [0; e ￿);
(1￿￿)￿￿C+￿rI
r +
2 X
i=1
Ai￿￿i if ￿ 2 (e ￿;1):
(13)
4.1 Protected debt
Let us ￿rst compute the value of equity under protected debt ￿nancing.
Under the assumption that no ￿nancial bubbles exist, A1 is nil. To compute
A2; substitute (13) into (2). It is straightforward to show that A2 is nil and
that
E (￿;C) =
8
<
:
0 if ￿ 2 [0;￿)
(1￿￿)￿￿C+￿rI
r if ￿ 2 (￿;1):
(14)
4.2 Unprotected debt
Let us next turn to the unprotected-debt case. Substituting (13) into (3)
and (4), one obtains a two-equation set where b ￿ and A2 are the unknowns.
Rearranging yields b ￿ =
￿2
￿2￿1
C￿￿rI
1￿￿ < ￿, and A2 = ￿
(1￿￿)
r
1
￿2
b ￿1￿￿2 > 0: The
value of equity is
E (￿;C) =
8
> <
> :
0 if ￿ 2 [0; b ￿)
(1￿￿)￿￿C+￿rI
r ￿
(1￿￿)
r
1
￿2
b ￿
￿
￿
b ￿
￿￿2
if ￿ 2 (b ￿;1):
(15)
6Comparing (14) with (15), therefore, one can see that the value of equity
is higher under unprotected debt ￿nancing. Using (10), (14), and (15), one
￿nally shows that condition (7) holds under both protected and unprotected
debt ￿nancing. This concludes the proof.￿
References
[1] Boadway R. and N. Bruce (1984), A General Proposition on the Design
of a Neutral Business Tax, Journal of Public Economics, v. 24, iss. 2, pp.
231-39.
[2] Bond S.R. and Devereux M.P. (1995), On the Design of a Neutral Busi-
ness Tax under Uncertainty, Journal of Public Economics, 58, pp. 57-71.
[3] Graham J.R. and C.R. Harvey (2001), The Theory and Practice of Corpo-
rate Finance: Evidence from the Field, Journal of Financial Economics,
60, pp.187-243.
[4] Leland H.E. (1994), Corporate Debt Value, Bond Covenants, and Op-
timal Capital Structure, Journal of Finance, September, 49, iss. 4, pp.
1213-52.
7

Neutrality Properties of Firm Taxation under Default Risk

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