This article shows how the existing forward-looking measures of the effective tax rate may be biased when firms operate in a dynamic context. Using option pricing techniques we thus propose a measure of the effective tax rate which embodies future business changes.

Paolo M. Panteghini

English

Research Papers in Economics

A dynamic measure of the effective tax rate 
Paolo M. Panteghini
University of Brescia and CESifo
Abstract
This article shows how the existing forward−looking measures of the effective tax rate may
be biased when firms operate in a dynamic context. Using option pricing techniques we thus
propose a measure of the effective tax rate which embodies future business changes.
I wish to thank Gianni Amisano for helpful comments. Usual disclaimers hold.
Citation: Panteghini, Paolo M., (2003) "A dynamic measure of the effective tax rate." Economics Bulletin, Vol. 8, No. 15 pp.
1−7
Submitted: September 17, 2003.  Accepted: November 21, 2003.
URL: http://www.economicsbulletin.com/2003/volume8/EB−03H20004A.pdfA Dynamic Measure of the Eﬀective Tax Rate
Paolo M. Panteghini
University of Brescia and CESifo
Dipartimento di Scienze Economiche,
Via San Faustino 74B, 25122 Brescia, ITALY.
E-mail: panteghi@eco.unibs.it
Phone number: 0039-030-2988816;
Fax number: 0039-030-2988837
September 17th, 2003
Abstract
This article shows how the existing forward-looking measures of
the eﬀective tax rate may be biased when ﬁrms operate in a dynamic
context. Using option pricing techniques we thus propose a measure
of the eﬀective tax rate which embodies future business changes.
Jel classiﬁcation: H25.
Keywords: eﬀective taxation, options.
1I n t r o d u c t i o n
Business projects can usually be altered during their lifetime. Since ﬁrms
are aware that they can defer, expand, contract or abandon a project, their
interest on real options is growing up1. Despite of this, the existing literature
usually disregards future strategy changes when measuring a forward-looking
eﬀective tax rate (ETR)2.
1Graham and Harvey (2001) show that about 26% of the companies surveyed always or
almost always incorporate real options when evaluating a project. Furhermore, McDonald
(2000) argues that even when ﬁrms use standard techniques, it is possible that they apply
ad hoc rules of thumb which proxy for optimal timing behaviour.
2See e.g. Devereux (2003), Devereux and Hubbard (2003), and Sφrensen (2003).
1The aim of this article is therefore to show how the existing measures of
the ETR are biased when ﬁrms operate in a dynamic context. Using option
pricing techniques we show that volatility may have an asymmetric eﬀect on
the ETR.
The paper is structured as follows. Section 2 shows how the ETR should
account for timing. Section 3 introduces a continuous-time model and dis-
cusses a dynamic measure of the ETR. Section 4 summarizes the ﬁndings
and identiﬁes some topics for further research.
2 The ETR in a dynamic context
The ETR is usually measured as the scaled wedge between the pre- and
post-tax net present value of a project, say NPV (t) − NPVT (t), given the
investment time t. Dividing the wedge by NPV (t) we thus obtain:
ETRS (t)=
NPV (t) − NPVT (t)
NPV (t)
. (1)
Though ETRS (t) is a forward-looking measure, as it accounts for future prof-
itability, it is inherently static. In fact, it is based on the implicit assumption
that ﬁrms’ strategies do not change over time. On the contrary, evidence
shows that managerial ﬂexibility is a fairly important input for business ac-
tivities. In this case, the standard NPV approach fails to yield a reliable
valuation of future projects. The intuition behind this point is straightfor-
ward. Suppose that, at time t,t h eﬁrm can delay investment until time T.
If the ﬁrm invests immediately, it will enjoy the proﬁt stream between time
t and time T. If it waits until time T, it has the possibility of enjoying better
market conditions. Thus, investing at time t implies the exercise of the op-
tion to delay and entails paying an opportunity cost for the ﬂexibility lost in
the ﬁrm’s strategy3. This implies that the NPV must exceed the summation
of the investment cost and of the opportunity cost. Otherwise the loss in
ﬂexibility would not be compensated. As will be shown, in this case, the
ETRS is a biased measure.
In a dynamic context, the ﬁrm’s problem is one of choosing the optimal
exercise timing for its business option4,i . e .
V (t)=m a x
t E
n
NPV (t)e
−rt
o
. (2)
3For further details see Trigeorgis (1996).
4Without any loss in generality, we assume that the ﬁrm owns one option. Actually,
2Similarly, the post-tax value of the project can be expressed as
V
T (t)=m a x
t E
nh
NPV
T (t)
i
e
−rt
o
. (3)
As shown in Panteghini (2002), the standard NPV rule yields the same rank-
ing as the real-option one only if:
1. delaying the decision is impossible or the initial time t is the optimal
timing for any project;
2. taxation is neutral.
The ﬁrst condition is trivial. When the value of ﬂexibility is nil, t is the
optimal timing and the standard NPV rule is correct. The second condition
can be explained as follows. Suppose we have an increase in the tax rate.
Its eﬀect is twofold. On the one hand, the present value of future discounted
proﬁts is reduced: this induces the ﬁrm to delay investment. On the other
hand, the option value decreases. As the opportunity cost drops, investment
is stimulated. Neutrality means that these two contrasting eﬀects perfectly
oﬀset each other.
When neither of the above conditions holds, the solutions of (2) and (3)
are diﬀerent. This implies that distortive taxation has a diﬀerent impact
on the net present value and on the ﬁrm’s option, thereby aﬀecting the
intertemporal decision. To show the importance of the intertemporal eﬀect,
we thus propose the following measure of the ETR:
ETRD (t)=
maxt E {NPV (t)e−rt} − maxt E
n
NPVT (t)e−rt
o
maxt E {NPV (t)e−rt}
, (4)
As can be seen ETRD (t), takes into account the intertemporal decision. It is
straightforward to show that when the value of the option is nil, (4) collapses
to (1). Therefore, ETRS (t) can be regarded as a special case of ETRD (t).
It is worth pointing out that the above formula can be used to compute
both average and marginal taxation. In this latter case, in fact, it is suﬃcient
to set economic rents equal to zero5.
ﬁrms may hold more than one option, e.g. regarding new locational choices, R&D invest-
ments, new marketing programs or the deﬁnition of the type and quality of the goods
produced.
5For further details see Devereux (2003).
33 An application: the basic investment tim-
ing problem
To have a feeling of how ETRD (t)i sa ﬀected by dynamic strategies, we
present a basic investment timing problem with risky payoﬀ. Using a continuous-
time framework, we study the behavior of a risk-neutral representative ﬁrm,
who must decide whether and when to undertake an investment project. We
assume that risk is fully diversiﬁable, the risk-free interest rate r is ﬁxed, and
that the project’s payoﬀ follows a geometric Brownian motion
dΠ(t)=σΠ(t)dz,
where σ is the variance parameter and z is a Wiener process. The ﬁrm starts
to earn the payoﬀ once a non-depreciable sunk cost, say I,i sp a i d . I nt h e
absence of taxation the NPV is therefore
NPV (Π(t)) =
Π(t)
r
− I. (5)
Let us next introduce taxation. Without any loss of generality, we deﬁne the
tax base as the ﬁrm’s current income, net of an imputation rate ρ6.T h u s ,
given the tax rate τ, current tax payments are T (t)=τ [Π(t) − ρI], and the
post-tax income is
Π
T(t)=( 1− τ)Π(t)+ρτI. (6)
Given (6), the post-tax NPV is
NPV
T (Π(t)) =
(1 − τ)Π(t)
r
−
µ
1 −
ρτ
r
¶
I. (7)
The optimal investment timing T c a nb ea s s o c i a t e dt oap r o ﬁt level Π.
This entails that whenever current payoﬀ reaches ¯ Π,t h eﬁrm invests7.T h u s
the ﬁrm’s problem (3) can be rewritten as8
Ψ(Π(t),β1)=m a x
¯ Π>0


Ã
Π(t)
¯ Π
!β1
NPV
T(¯ Π)

, (8)
6This assumption is in line with Boadway and Bruce (1984).
7For further details see Dixit and Pindyck (1994, Ch.6).
8A full derivation can be found in Panteghini (2002).
4where β1 > 1a n dNPVT(¯ Π)i st h eﬁrm’s static project value when Π(t)=¯ Π.
Hereafter, for simplicity we will omit the time variable t.
Using (7) and solving the problem (8) yields the trigger point above which
investment is proﬁtable
¯ Π ≡
"
1 −
ρ
rτ
1 − τ
#
e Π, (9)
where e Π ≡
β1
β1−1rI > rI is the laissez-faire trigger point.
The tax treatment of the investment cost is crucial for the sign of the
diﬀerence
³
¯ Π − e Π
´
. Easy computations show that
³
¯ Π − e Π
´
∝ (r − ρ). When
therefore ρ <r ,the inequality ¯ Π > e Π holds, namely the ﬁrm’s propensity to
invest is reduced by taxation. This can be regarded as the underinvestment
case. When, conversely, ρ >roverinvestment takes place. When, ﬁnally,
ρ = r, the tax system is neutral, i.e. ¯ Π = e Π.
Since underinvestment is the most common case, in the following discus-
sion we will assume that ρ <r 9. Depending on the current level of payoﬀ
we can ﬁnd three cases. Firstly, when e Π < ¯ Π < Π, investing immediately is
the optimal choice irrespective of taxation. According to Panteghini’s (2002)
ﬁndings, timing does not aﬀect the investment decision and ETRD collapses
to ETRS. This entails that volatility does not matter and that ETRS is
unbiased.
In the following two cases, instead, a bias emerges. When e Π < Π <
¯ Π, investing immediately is the better choice in the absence of taxation,
whereas delaying is preferable under taxation. This implies that, in the
absence of taxation, the standard NPV rule holds, whereas, under taxation,
the intertemporal eﬀect matters. Thus substituting (5) and (8) into (4) yields
ETRD =1−
Ψ
³
Π,β1
´
Π
r − I
when e Π < Π < ¯ Π. (10)
If Π lies between e Π and ¯ Π,b o t hc u r r e n t( Π) and expected proﬁtability (¯ Π)
aﬀect ETRD. Moreover, volatility aﬀects ETRD. By applying the Envelope
Theorem, it is easy to ascertain that
dETRD
dσ ∝−
∂Ψ(Π,β1)
∂β1
dβ1
dσ < 0.
When Π < e Π < ¯ Π, immediate investment is non-proﬁtable in either case.
Substituting (7) and (8) into (4) yields
9Under the assumption that ρ >r ,the sign of results would be reverted.
5ETRD =1− (1 − τ)
Ã
1 − τ
1 −
ρ
rτ
!β1−1
when Π < e Π < ¯ Π. (11)
It is straightforward to show that ∂ETR
∂σ ∝ (ρ − r).
A ss h o w nb y( 1 0 )a n d( 1 1 ) ,a ni n c r e a s ei nσ reduces ETRD. This entails
that the higher the standard deviation the greater is the bias caused by
the use of ETRS. The intuition behind this result is straightforward. On
the one hand, an increase in the tax rate reduces both the present value
of future discounted proﬁts and the option value. On the other hand, an
increase in volatility makes the timing option more valuable, thereby raising
the opportunity cost of investing immediately. Therefore, the distortive eﬀect
of taxation on the option value is partially compensated by higher volatility.
This reduces the eﬀective impact of taxation on investment decisions, namely
the ’true’ ETR turns to be lower.
4 Concluding remarks
In this paper we have proven that the standard measure of the ETR may
be biased, when managerial ﬂexibility is a worthy input. In particular, we
have shown that if taxation is distortive and proﬁtability is low enough, then
volatility aﬀects the ETR. If, instead, taxation is neutral and/or proﬁtability
is high enough the ETR is unaﬀected. This asymmetry implies that market
conditions may deeply aﬀect the measurement of eﬀective taxation.
This article should be considered as the starting point for further research.
In particular, the introduction of a more general model, dealing with diﬀerent
kinds of options and stochastic processes, and the deﬁnition of algorithms
able to proxy for the eﬀect of volatility are left to future work.
References
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