Modern Economy, 2010, 1, 47-50
doi:10.4236/me.2010.11004 Published Online May 2010 (http://www.SciRP.org/journal/me)
Is it Rational to Minimize Tax Payments?
Andreas Löffler, Lutz Kruschwitz
Universität Paderborn, Lehrstuhl für Finanzierung und Investition, Paderborn, Germany
Freie Universität Berlin, Institut für Bank-und Finanzwirtschaft, Berlin, Germany
Received March 15, 2010; revised April 5, 2010; accepted April 25, 2010
Abstract
The opinion is occasionally voiced that investors should avoid paying tax at all costs. In this paper it is being
investigated, using a simple portfolio model with taxes, whether avoiding tax really leads to more µ-σ-effi-
cient solutions. It is demonstrated for four different concepts of tax-minimising policy that they are a far cry
from an efficient solution.
Keywords: Decision, Taxes, CAPM, Tax-CAPM
1. Problem Outline
Considering the question on how to increase one’s net
income, an investor has two options. He or she will ei-
ther look for opportunities to increase the pre-tax income,
or alternatively to diminish tax payments. Many people
are assumed to especially favor tax saving schemes. A
few years ago, a German economist, Ekkehard Wenger,
has written on this predilection, which he considers thor-
oughly reasonable. Rationally acting investors would
either make legal use of tax incentives in their invest-
ments, or even attempt tax evasion. Anyone acting dif-
ferently would be “maximizing stupidity” [1]. Regardless
of whether one agrees or not with such drastic views, the
rationale behind such a position seems to make sense. As
a scientist however, one should be armed with a funda-
mentally skeptical outlook; easily comprehended matters
do not always fit in with reality, as one readily learns in
school from the conundrum involving Achilles and the
turtle.
Wenger’s considerations are relevant in view of em-
pirical analyses of the stock market. Fair evaluations of
companies’ values are normally performed by using
Capital Asset Pricing Models (CAPM). This involves the
comparison with similar companies, in order to deter-
mine the so-called “beta factor” of the firm in question.
Particularly the financial auditor profession is adamant
that for this determination not the basic model of CAPM
be used, but rather its tax-enabled version, the Tax-
CAPM [2]. The values resulting from the respective uses
of CAPM and Tax-CAPM can sometimes vary signifi-
cantly. If one subscribes to the opinion that tax payments
reflect irrational behavior, it follows that Tax-CAPM
will yield incorrect company valuations. Thus, the ques-
tion of whether saving taxes constitutes rational behavior,
directly affects what a company is worth.
In our present contribution we will, while using the
framework of portfolio theory, examine whether a policy
of minimizing taxes can be regarded as advantageous.
For this purpose, we consider a simple portfolio problem
from a tax perspective and attempt a characterization of
all portfolios which are efficient under uncertainty. Such
portfolios can be characterized as desirable even without
more precise knowledge on the risk averseness of inves-
tors. On this basis we will examine the question, how the
concept of “tax minimization” can be formalized in our
context and if such concepts can be efficient. It will
emerge that tax minimizers, while maybe considering
themselves to be particularly clever, end up looking less
than smart.
We will concentrate our analysis exclusively on port-
folio theory and will not employ the CAPM. This proce-
dure follows from the goal we have set for ourselves. We
would like to show that even rational investors can be
willing payers of taxes, as they prefer a balanced after-
tax structure of payments. What kind of macroeconomic
result would be the upshot of such behavior is outside the
scope of our work.
2. The Model
We use a formal model which is based on definitions and
simplifying assumptions deemed appropriate. On this
basis we will use logical operations to draw conclusions
whose validity can be checked by an expert third person
at any time. Given that we do not make mistakes during

A. LÖFFLER ET AL. 48
J

the logical operations, our results can only be legiti-
mately criticised by referring to a possible use of inap-
propriate assumptions.
2.1. Assumptions
We are looking at a one period model under uncertainty.
An investor has the option to invest in risky securities
. The price vector of these investments is
called
1, ,j

1,,
J
pp p. We notate future payments
connected with the securities with
j
Y


. We use the sym-
bol for the vector of the ex-
pected cashflows. The covariance matrix of the cash-
flows is . It is assumed that the covariance matrix is
regular. Therefore there are no redundant investments. A
risk free asset will not be introduced. But we concentrate
on the edge of an “eggshell-like surface” which may be
interpreted as the geometric place of all desirable
risk-return-positions and uses to be called the efficient
set.

1,,Y E
 
 

J Y


EE


We imply an investor with nominal assets of

, who
has already financed his momentary consumption and
has to decide how to invest the amount

in the capital
market. For that purpose he chooses a portfolio consist-
ing of

1,,

J
NN N units of the risky assets. Short
selling is not excluded. Therefore some entries in this
vector may also be negative. When deciding on his in-
vestment the investor orients himself by the expectation
value and variance. He thus has a utility function of the
type

2
,U

 .
According to our requirements the expectation value
and variance of the future payments of a securities port-
folio are determined to NE

 and .
The price of the portfolio is given by the vector product
.
2NN
 
Np
In order to model the taxation an assessment basis as
well as a tax-rate function is required. As assessment
basis for a unit of the-th financial asset we us the dif-
ference between the risky cashflow and a riskless depre-
ciation which in the simplest case corresponds to the
purchase price, but not necessarily. We term the vector
of the depreciations of the risky assets as
j

1,,
J
A
AA.
The rate function is linear, the tax rate

is secure.
2.2. Efficient Portfolios before and after Taxes
First we look at the decision under the simplifying assu-
mption that no taxes are being levied. Then we deal with
the classic problem of portfolio selection, introduced into
the literature by Markowitz more than 50 years ago [3].
The investor can choose between positions charted on an

-2

-diagram laid out on an eggshell-shaped surface.
The most interesting points on this surface are those
around the edge, as it is there that for any given expected
wealth the variance will be minimal. All positions which
are not situated on the edge are denoted as non-efficient.
Any rational investor will attempt to occupy an

-2

-
efficient position. In order to determine these edge posi-
tions, a maximization problem
max
NNE (1)
under budget constraints
2
andNpN N

   (2)
has to be solved. The first condition is a budget restric-
tion which ensures that the investor’s wealth is fully ex-
hausted; the second condition makes sure that the cash
flow variance of a portfolio will reach an exogenously
demanded level 2

. In order to fully determine the effi-
cient set, the optimization problem has to be solved for
all possible 2


.1 Because short sales are not ex-
cluded, the optimization can be accomplished by using a
Lagrange function. The specific solution is irrelevant to
this discussion.
We now focus our attention on after-tax-efficient port-
folios. Here, when dealing with cash flows, we must note
that taxes are due. An investor holding one unit of asset
will have to deliver the amount
j

j
j
YA  
 to the
tax authority. Efficient portfolios are those, which for
any given variance and nominal wealth

, will maxi-
mize the expected value e. The expected cash flows
after taxes then amount to

1AE

EE A

  .
The pre-tax covariance matrix  changes to the
post-tax covariance matrix



2
Cov1, 1
1Cov,
j
jk k
jk
YAY
YY
 

A


 



 


Thus, we now have to maximize the function




1NE A

 
Np
under the budget constraints


 and

22
1NN

 . So long as the
amount of write-offs
A
is not specified, there are many
solutions to this problem. However, in a one-period
model,
A
p

represents a plausible-even natural-
choice. The function to be maximized now takes the
1If cash flows are not perfectly negatively correlated there are
no portfolios having zero variance. Therefore, there is a mini-
mum variance that cannot be undercut.

A. LÖFFLER ET AL.49
form

1NE

 . Since a positive linear transfor-
mation will affect the target function but not the solution
itself, we can similarly denote the optimization problem
after taxes in the form
N
max NE

(3)
under auxiliary conditions


2
2
anNp d1 N N


(4)


2
  
We recognize at once: The maximization problem af-
ter taxes differs from that before taxes in only one, and
furthermore irrelevant, respect. In the second budget con-
straint,

became the parameter

2
2
1


. It follows
that the set of all parameters for both maximization
problems must be identical. The efficient set before taxes
coincides with the efficient set after taxes.
2.3. Tax-Minimizing Portfolios
We begin by stating that, in the context of our discussion,
it is not immediately clear what tax avoidance or tax
minimization mean. We see several possible ways to
specify the concept of tax minimization, if we are deal-
ing with identifying a risky portfolio. Four specific al-
ternatives shall be considered. In each case, we will as-
sume that the risky projects will be completely written
off, thus
A
p.
2.3.1. Minimizing Taxes in the Worst-Case Scenario
To formalize this concept of tax minimization, we as-
sume that the number of relevant states at time 1t

is
finite and that the state-dependent cash flows of assets
can be characterized by . Then

,1,,Ys sS


NYs pN

Ys

    describes the tax pay-
ments coming due for an investor if the state
s
occurs
in . In the worst case, this payment will amount to
1

NYs
t
maxs

 
min
N
. Our first version of tax minimiza-
tion may amount to choosing portfolio in such a
manner, as to minimize the highest tax amount possible
N

max
sNYs
 
under auxiliary constraint Np


. The solution can
be found by means of an appropriate algorithm of linear
programming. It is obvious that such a solution is not

-2

-efficient.
2.3.2. Absolute Minimization of Expected Taxes
The expected tax payments of portfolio amount to N

NEp  

. Due to the budget restriction Np

, the
optimization problem reads


min
NNE

  ,
if no additional auxiliary conditions are taken into ac-
count. The solution is not lower-bound. The expected tax
payment tends towards infinite minus. This result does
not make economic sense.
2.3.3. Fading Expected Taxes
In order to avoid the just mentioned outcome, we can
restrict the search for a portfolio, for which the expected
tax payments disappear, i.e.,

!0NE


.
For a positive tax rate, the expected tax payment will
tend towards zero if the basis of assessment disappears.
This is the case, if the expected payments from the port-
folio are equal to the invested capital, a slightly surpris-
ing, but equally disappointing outcome. Without earning
no taxes are due. But would someone cancel one's net
earnings in order to save taxes? This would be as unrea-
sonable as if a firm maximized its wages in order to
minimize corporate taxes.
2.3.4. Minimizing Expected Taxes on a Given Variance
As a final version of a tax-minimizing policy, we con-
sider


min
NNE

 
under the constraints

2
2
and 1 NpN N



  
this version of tax minimization we do not have any
economics-driven intuition, excluding maybe that the
determination of an efficient portfolio without taking
variance into account does not seem possible. As the
constants in the target function can be ignored, we can
rewrite it in the form . Comparing this opti-
mization problem with the procedure involving Equa-
tions (1) and (2), it becomes immediately clear, that no
minNNE

-2

-efficient portfolios can be determined in this
manner. A rational investor does not minimize the ex-
pected taxes at a given variance. He or she will instead
maximize them.
3. Conclusions
In the framework of a simple portfolio model with taxes
it was considered whether the strategy of tax avoidance
will lead to

-2

-efficient solutions. The examination

A. LÖFFLER ET AL. 50
of four distinct concepts of tax minimization strategy has
proven that each of them (by far) misses an efficient so-
lution. Thus, when evaluating companies, one must
avoid the classic CAPM for determining beta factors and
should instead employ a Tax-CAPM.
4. References
[1] E. Wenger, “Verzinsungsparameter in der Unterneh-
mensbewertung: Betrachtungen aus theoretischer und em-
pirischer Sicht,” in German, Die Aktiengesellschaft, Vol.
50, 2005, pp. 9-22.
[2] M. Jonas, A. Löffler and J. Wiese, “Das CAPM mit
deutscher Einkommensteuer,” in German, Die Wirtsch-
aftsprüfung, Vol. 57, No. 17, 2004, pp. 898-906.
[3] H. M. Markowitz, “Portfolio Selection,” The Journal of
Finance, Vol. 7, No. 1, 1952, pp. 77-91.