In his recent article âValuation with or without personal income taxesâ Richter analyzes the .... the uncertainty case without further (restrictive) assumptions.
Marc Steffen Rapp and Bernhard Schwetzler*
PERSONAL TAXES AND CIRCULARITY PROBLEMS IN CORPORATE VALUATION – A NOTE (NOT ONLY) ON „VALUATION WITH OR WITHOUT PERSONAL INCOME TAXES?“ BY FRANK
RICHTER, SBR 2004 VERSION: August 13, 2004 ABSTRACT In his recent article “Valuation with or without personal income taxes” Richter analyzes the impact of introducing income taxes into the calculus of corporate valuation. His major result is that, if correctly specifying the model, the income tax rate is only of minor importance for corporate values and asset prices. In this note we want to demonstrate that the sensitivity analysis of Richter (2004) has to be treated with some caution and highlight some problems occurring when introducing personal income taxes that are not fully reflected in Richter’s analysis. In a first step we show, that presuming that the analysis of Richter (2004) is theoretical correct, it is highly sensitive with respect to the parameters of the dividend process and choosing slightly different parameters suggests opposite conclusions. Second, we show that in general the analysis of taxation effects is characterized by circularity problems, since the empirical as well as the theoretical analysis of Richter requires the assumption that the market price of the market portfolio is independent from the tax rate. Our analysis based on a consumption based asset pricing model shows that equilibrium asset prices depend on the income tax rate even if we presume an idealized tax system with uniform tax rates that taxes the economic income. Key-words: valuation, personal taxes JEL-Classification: G 32
*
Dipl. Math. Marc Steffen Rapp, Prof. Dr. Bernhard Schwetzler, Lehrstuhl für Finanzmanagement und Banken, Handelshochschule Leipzig (HHL), Jahnallee 59, 04109 Leipzig
2
1
INTRODUCTION
The impact of personal income taxes upon corporate values and asset prices is intensely discussed in the literature. The debate can be split of in two separate problems: a)
The impact of a homogenous tax rate that differs for different income categories Income from assets and securities fall in different categories: dividends from stocks, capital gains from bonds and stocks and interest income from bonds. Usually tax regimes apply different income tax rates upon this different categories. A strand of research analyses the impact of different tax rates on different income categories upon equilibrium asset prices and on corporate values.1 A common assumption is that within a given income category the tax rate is homogenous over all investors. Approaches that analyze the impact of a general change of the overall income tax rate upon corporate values also belong into that category. The background for the current discussion in Germany is the publication of new guidelines for corporate valuation (IDW S1) by the German Institute of certified public accountants (IDW) in 2000.2 There the IDW gives up its former position that income taxes would cancel out of the valuation equation and thus are irrelevant. The new position that income taxes have to be included in the valuation calculus fueled a debate on the impact of the introduction of income taxes upon corporate values. Several changes of the German statutory income tax rate also propelled the discussion.
b)
The treatment of inter-individually different income tax rates In most tax regimes worldwide the income tax rate is subject to a progressive tax scheme. Thus different investors with different taxable income are expected to have different marginal income tax rates within a given income category. Some researchers analyze the effect of inter-individually different income tax rate on the calculus for corporate valuation.3 Empirical background is the problem of calculating a homogenous fair value for shareholders with different marginal income tax rates, e.g. in the case of a compensation of frozen out minority shareholders. IDW in this case recommends to use a representative income tax rate of 35%.4
1
E.g Sick (1990); Taggart (1990); Clubb/Doran (1992); Ballwieser (1997); Siepe (1997); Ollmann/Richter (1999); König/Wosnitza (2000); Laitenberger (2000); Doobs/Miller (2002); Löffler (2003); Schwetzler/Piehler (2004). 2 See IDW (2000) and Siepe/Dörschell/Schulte (2000) 3 See e.g. Jensen (2003a),(2003b). 4 IDW (2000) No. 41; Siepe/Dörschell/Schulte (2000), p. 959.
3 Frank Richter’s article “Valuation with or without personal income taxes?” belongs into the first category5 and tackles the problem of how to reflect differentiating taxation into a capital market based valuation calculus. The proposed method is an interesting and innovative approach to analyze the taxation effects, that are due to the tax-induced risk-sharing mechanism between government and taxed investors, and a first step to overcome the shortcomings of analyses based on the ad-hoc assumption that the risk neutral probabilities are independent of the prevailing tax rate.6 However, as we will show, the results of Richter (2004) are very sensitive with respect to the parameter choice, and – perhaps more serious – his analysis does not fully reflect the fact that the current capital market environment reflects the currently executed tax regime. The aim of this note is threefold: (A) Presuming that the analysis of Richter (2004) is theoretical correct, we show that it is highly sensitive with respect to the parameters of the dividend process. Choosing slightly different parameters suggests opposite conclusions. (B) We show that the discussion of taxation effects in general, not only in Richter (2004), is characterized by circularity problems: the empirical approach of Richter (2004) is subject to a circularity problem since it requires the market price of the market portfolio M to be independent from the tax rate (and thus the irrelevance of income taxes on aggregate level) as a condition for the analysis of the income tax effect on the particular asset’s level, the claimed irrelevance under the idealized tax regime that taxes the economic rent is subject to the same circularity problem. (C) Even presuming the idealized tax regime that applies the economic income as a tax base consumption based equilibrium asset prices depend on the income tax rate.7 2
THE ANALYSIS OF RICHTER
In this section we want to show that presuming the analysis of Richter (2004) is theoretical correct, it is very sensitive with respect to the underlying model for the dividend process. Richter (2004) makes very strong assumptions on the dividend process. These assumptions imply that there is only one degree of freedom left when specifying the valuation model. This 5
The analysis is based on the assumption of identical income tax rates for all individuals within the same income category but different tax rates for different categories. See Richter (2004), pp. 24. 6 Analysis based on the ad-hoc assumption are found in Leon/Gamba/Sick (2003), p. 4 and Löffler (2003), p. 9. 7 This observation immediately implies, that the irrelevance result of Samuelson (1964) may not be transferred to the uncertainty case without further (restrictive) assumptions.
4 last degree of freedom is used in order to capture the expected growth rate of dividends. However, a stochastic process is not well described by its first moment (the expected growth rate). At least incorporating the variance of dividends would be desirable.8 2.1
Richter’s approach
The analysis of Richter (2004) is based on a binominal model for future dividends of the market portfolio. In general a binominal process is characterized by three parameters in each period: (i) the probability pt for the up-movement, (ii) the growth-factor ut for the up-move, and (iii) the growth factor dt for the down-move. Therefore when specifying a general binominal process one has three degrees of freedom. In his analysis Richter makes two special assumptions: (i)
he assumes that the down-factor is the reciprocal of the up-factor, i.e. dt=ut-1, 9
(ii)
he assumes that the probability pt is equal to 50% for all periods of time.
With these two assumptions Richter cuts down the degrees of freedom in his valuation model to one, namely to the magnitude of ut. In order to meet the given expected growth rates g ptm for the dividends he chooses ut according to u t = 1 + g ptm +
(1 + g )
p 2 tm
−1
(1’).
Core of Richter’s valuation analysis is the risk-neutral dividend discount model for the market portfolio, that cumulates in the following equation10
P0, m = C 0, m 1 −
where
8
T −1
s 2
T −1
(1 + g )∏ (1 + g ) 1+ g ∏ 1 + r (1 − s ) + (r (1 − s ) − g )(1 + r (1 − s)) τ
τ =1 t =1
q Tm
q tm
f
f
q tm
t =1
q Tm
T
(1)
f
P0,m:
the current market capitalization of the market portfolio,
C0,m:
the current dividend payment of the market portfolio,
rf:
the current riskless interest rate
s:
the income tax rate
Richter (2004) tries to unify his model with the Tax-CAPM, see Richter (2004), section 4. In the standard CAPM as well as the Tax-CAPM asset returns are characterized by their mean and (co-)variance. 9 For time-independent parameters u and d this assumption implies a recombining binominal tree. However, in the example of Richter (2004) the dividend tree is non-recombining the first periods since the growth rates for each period differ. See figure 1. 10 Eq. (1) is equivalent to eq. (10) in Richter (2004).
5 g qtm :
the risk neutral growth rate of the market portfolio’s dividends in period t
T-1
the number of explicitly planned periods (phase I).
Eq. (1) derives the current market capitalization P0,m as the present value of the risk neutral future dividends on the market portfolio M. However, the risk-neutral probability measure needed to calculate the risk adjusted growth rate g qtm is not observable. The basic idea of Richter is now to link the market data on P0,m, C0,m, rf, and the markets estimates on g ptm as well as the tax rate in order to derive an implicit risk neutral probability measure q that solves eq. (1). By varying the different input parameters dividend yield, estimated terminal growth rate and income tax rate he then analyses the sensitivity of the risk neutral probability with respect to these parameters. In a second step the derived risk neutral probabilities are used for a sensitivity analysis of the value of a particular corporation with respect to changes in the income tax rate. By comparing the results with results of sensitivity analyses that ignore the adjustment of the valuation calculus, i.e. without an adjustment of q, Richter concludes the sensitivity of corporate values is substantially lower than the unadjusted analyses would suggest.11 2.2 The sensitivity of Richter’s approach If we use the same valuation model but apply it to a binominal dividend process with (i)
a down-factor that is the reciprocal of the up-factor, i.e. d=u-1, but
(ii)
p=75% instead of p=50%,
the result is quite different. It would lead us to conclude, that personal taxes should be included into the valuation calculus. Figure 1 gives the dividend trees for Richter’s model with p=50% and for our model with p=75%.12
11
Richter, F. (2004), p. 38, p. 41.
12
The expected growth rate
g ptm is equal to the one used in Richter’s analysis. See Richter (2004), section 3.2.
6
Growth rates of dividends for p=50%
Growth rates of dividends for p=75% 3.23560
1.50690
2.25253
1.33876 1.56815
1.18938
1.45673
1.13515 1.35323
1.08340
0.94208
0.96252 0.65585
0.85512
1
1 1.52474
1.16943
1.06148
1.03894 0.73897
0.92302
0.68647
0.88094 0.63770
0.84078
0.44394
0.74696 0.30906
0.66362
Figure 1: Dividend trees for different up-probabilities p
Figure 1 indicates that for given values of g ptm the second moment (variance) of our p=75%process is lower, than that of Richter’s process. This is also shown in figure 2. Figure 2 plots the up-factor u and the standard-deviation σ for an expected growth g ptm = 10% against different choices of p. 2.5
2
1.5
1
0.5
0 40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
subjective probability p u
σ
Figure 2: up-Factor and volatility with respect to the subjective probability at a given growth rate
g ptm = 10%
The decreasing dividend volatility now implies a increasing sensitivity of the implied riskneutral probability measure with respect to the tax rate s. Figure 3 shows the markets’ implied risk neutral probabilities q with respect to s for three different probabilities for the up-state p (30%, 50% and 75%) given gPTm=2.00% and d0m=5.00%
7
0.600
0.500
risk-neutral q
0.400
0.300
0.200
0.100
0.000 0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
Tax rate
30%
50%
75%
Figure 3: Sensitivity of the implicit risk-neutral probability for the up-movement
Figure 3 shows that for p = 75% an increase in the tax rate from 0% to 50% yields a decrease of q from 48.13% to 35.93%. Thus under this setting q is substantially more sensitive to changes in s than under Richter’s assumptions of p=50%. Furthermore the increasing sensitivity of the risk-neutral probability measure implies an increasing sensitivity of the valuation results for the corporation of Richter’s example. The numerical results for the corporation are given in table 1. p= δ0m 2% 3% 4% 5% 6% 7% 8% 9% 10% 11%
75.0% 1.0% 48.921 32.929 24.886 20.033 16.781 14.447 12.689 11.315 10.212 9.306
P TM
g s = 0.0% 2.0% 3.0% 33.613 28.868 27.437 25.194 23.191 22.339 20.089 20.055 17.720 18.185 15.850 16.626 14.335 15.306 13.083 14.173 12.030 13.191 11.131 12.330
=… 4.0% 26.531 23.944 21.797 19.987 18.439 17.102 15.934 14.906 13.994 13.180
P TM
5.0% 25.114 23.125 21.405 19.904 18.582 17.410 16.364 15.425 14.578 13.809
1.0% 49.409 33.191 25.047 20.141 16.856 14.501 12.728 11.344 10.233 9.322
g s = 35.0% 2.0% 3.0% 36.839 32.826 29.946 28.524 25.229 25.201 21.795 22.556 19.182 20.402 17.125 18.612 15.463 17.101 14.092 15.810 12.941 14.692 11.961 13.716
=… 4.0% 30.781 27.668 25.101 22.950 21.120 19.546 18.177 16.976 15.914 14.968
5.0% 29.499 27.063 24.972 23.158 21.569 20.168 18.922 17.808 16.806 15.901
Table 1: Valuation of Richter’s corporation with p=75%
Considering the p=75% scenario the company values differ up to 17.5% (e.g. 8.49% in the gray scenario), which is by far more critical than the 5% bound in the p=50% scenario of Richter (2004).13 Therefore, assuming that p=75% is a well fitting model for the dividend process one might come to the conclusion, that personal taxes have to be considered in the valuation calculus.
13
If we apply the model to a dividend process with p=80% the difference increases to 17.70%.
8 3
CIRCULARITY PROBLEMS WITH INCOME TAXES IN CORPORATE VALUATION
3.1 General Remarks As valuation always implies comparing the asset to be valued against other assets14 the tax treatment of alternative investments is relevant for the valuation. Usually the expected beforetax (after-tax) profitability of alternative investments is reflected in the before-tax (after-tax) discount rate to be applied on the cash flows of the asset to be valued. Thus imposing a particular tax treatment of the discount rate already implies certain assumptions about the effect of income taxes upon the return and thus upon market price of the alternative investment. We will show this for the common proposal to introduce personal taxes by applying the income tax factor upon the pre tax-discount rate r, thus yielding rS = r (1-s) as a net discount rate.15 Therefore, let Pt (PtAI) denote the price of the asset (the alternative investment AI that is risk equivalent on a before-tax base) in t, Dt (DtAI) the dividend of the asset (the alternative investment AI) in t, sCG and sD the capital gain and dividend tax rate for both assets. Furthermore let rt (rtAI) and rt,s (rt,sAI) denote the expected pre-tax and after-tax rate of returns. Applying the fact that AI is risk equivalent on a before tax base we have rt = r
AI t
AI D AI t +1 + Pt +1 = −1 . PtAI +1
AI What are sufficient conditions for rtAI to hold? Calculating the after-tax return of ,s = (1 − s ) rt
the alternative investment rtAI ,s as
AI t ,S
r
=
(
AI AI AI D AI t +1 (1 − s D ) + Pt +1,s − s CG Pt +1,s − Pt ,s AI t +1,s
P
) −1
we find, that if -
capital gains and dividends of the alternative investment are immediately taxed with the same tax rate s =sCG=sD (or equivalently there are no dividends or no capital gains of the AI in the pre-tax as well as in the after tax world) and
-
the current and the future market price of the alternative investment are independent of the income tax rate s (or equivalently there are no dividends and the current and the future market price of the alternative investment in the after-tax world are
14 15
Moxter (1983), p. 123. See e.g. Ballwieser (1997), p. 2394; Siepe (1997), p. 4; Copeland/Koller/Murrin (2000), p. 153; Brealey/Myers (2003), p. 495 and Ross/Westerfield/Jaffe (1999), p. 414.
9 multiples of the current and the future market price of the alternative investment in the pre-tax world) AI it follows that rtAI is to hold. We presume now, that there are dividends and ,s = (1 − s ) rt
(expected) capital gains/losses offering alternative investments that are risk equivalent on a before-tax as well as an after-tax base. Then rS = r (1-s) requires two important assumptions:16 (a)
for the alternative investment economic income is the relevant tax base. Capital gains and dividends are both immediately taxed with the same rate s =sCG=sD, and
(b)
current and the future market price of the alternative investment have to be independent from the income tax rate.
AI Going back to the asset to be valued rt ,s = rtAI = (1 − s)rt implies ,s = (1 − s) rt
Pt ,s =
(
D t +1 (1 − s D ) + Pt +1,s − s CG Pt +1,s − Pt ,s 1 + (1 − s )rt
)
Pt ,s =
D t +1 + Pt +1,s 1 + rt
.
Therefore s=sD=sCG implies the irrelevance of taxation for the asset, i.e. Pt,s = Pt, if (and only AI if) Pt+1,s = Pt+1 as well as rtAI ,s = (1 − s ) rt . However, note that Pt+1,s = Pt+1 is a non-trivial
assumption. Richter (2004) also discusses conditions for the irrelevance of income taxes. In his analysis a
tax regime that applies a uniform tax rate on the tax base “economic income” is claimed to imply the irrelevance of income taxes.17 If the CAPM, as an equilibrium model, is used to derive the discount rate the effect of income taxes on the market prices of all assets traded at the capital market is relevant. As these assets offer many different combinations of (differently taxed) dividends, capital gains and interest payments the impact of income taxes upon securities prices is far from obvious. The after tax CAPM of Brennan (1970) is a widely used model in this context.18 For the practical application of the model the parameters have to be estimated. As already mentioned, the immediate problem is that nobody knows the market price of the assets and thus the discount rate in a world without taxes, since empirically observed market prices and market returns come from a world with taxes at a given tax rate.19 Thus an analysis of effects on asset prices
16
See e.g. Richter (2003), p. 313. Richter (2004), p. 30; We define the term economic income of an asset in section 3.3.1. Roughly speaking, a tax regime that taxes the economic income taxes all kinds of income arising from an asset (dividends, interests and capital gains) with a uniform tax rate. 18 See also Ollmann/Richter (1999), pp. 163. For a critical view see Wiese (2004). Other models are found in Litzenberger/Ramaswamy (1979) or Lally (1992). 19 This problem is also addressed in Richter (2003), p. 313.
17
10 caused by changing tax rates that is based on empirical data is seriously hampered by this problem. 3.2
The empirical problem in Richter’s approach: Circularity and the derivation of capital markets’ risk neutral probabilities The problem with Richter’s approach becomes apparent when looking at his proposed
rearrangement of equ. (1). Dividing both sides of (1) by C0,m yields one over the dividend yield as a multiple with 1
δ 0,m
=
P0, m C 0, m
For the following sensitivity calculations 1/δ0,m is used as if it is a given, exogenous variable, thus assuming the dividend yield of the market portfolio to be independent from the income tax rate s. Following the common assumption in standard asset pricing theory of exogenously given assets’ payoffs (dividends) the pre tax dividend C0,m is independent from s.20 On the other hand, obviously the market price of the market portfolio P0,m comes from a world with taxes.21 Therefore, taking 1/δ0,m as an exogenous variable implies that the market price of M does not depend on the income tax rate s. That in fact points towards a circularity problem, since analyzing the impact of changes in the tax rate upon asset prices just requires the market price of the market portfolio M to be independent from the tax rate, thus imposing the irrelevance assumption about this impact on aggregate level. However, as long as we do not know the effect of tax rate changes on P0,m we might infer only the risk neutral probabilities for the currently observable combination of market prices and the current income tax rate s. Any variation of s without adjusting P0m requires the assumption of P0m being independent from s.22 In the following section we show based on a theoretical model that the equilibrium price of the market portfolio is affected by changes in the income tax rate even under an idealized tax regime using a uniform income tax rate and the economic rent as tax base. Thus even under rather ideal assumptions the necessary condition for the derivation of risk neutral probabilities for other income tax rates than the currently observable is not met. One might therefore expect that assuming the German tax regime with different income tax rates on capital gains,
20 Of course, this assumption neglects the effects of taxation on personal level on the investment policy of a corporation. However, it is common in standard asset pricing theory. See for example the models presented in LeRoy/Werner (2001). For a different approach see for example Constantinides (1982). 21 This “dilemma” is also noted by Richter himself in Richter (2003), p. 324. 22 Richter also points out that „q has (to be) determined on the basis of real world market prices (which reflect taxes).“ Richter (2004), p. 41.
11 dividends and interest income (and thus a less idealized tax regime) the condition of P0m being independent from s will not be met, too. 3.3
The theoretical problem: Taxation of the economic income, equilibrium asset prices and circularity
3.3.1 The irrelevance case and circularity problems We define the economic income of an asset as the sum of its cash flows plus the change in
value during a particular period.23 Referring to the seminal work of Samuelson (1964) tax regimes that apply a uniform tax rate on a tax base equal the economic income are often claimed to be neutral tax regimes.24,25 In this section we first show that the derivation of the neutrality result as shown in section 4.2 of Richter (2004) is also subject to a circularity problem. Then we analyze the impact of a tax rate change on the equilibrium price of the market portfolio using a consumption based asset pricing model with representative agents. A result of our anylsis is that Samuelson’s neutrality proposition in general may not be transferred to a model under uncertainty.26 We start at equ. (18) in Richter (2004), which shows the calculation of the theoretical value of the market portfolio M: Vt −1, m =
(
)
C t −1, m 1 + g ptm (1 − s C ) − E p [s V (Vtm − Vt −1, m ) + Vt , m Ft −1 ] 1 + rt , m
where
(3)
Vt-1m:
the theoretical value of the market portfolio in t-1,
Ct-1,m:
the current dividend payment of the market portfolio in t-1,
Ep[
Ft −1 ] : the expected theoretical value under the information of t-1
sC:
the income tax rate on dividends
s V:
the income tax rate on capital gains.
Equ. (3) is divided by Vt-1m to solve for the expected market return: rtm =
23
(
)
p C t −1, m 1 + g Tm (1 − s C )
Vt −1m
+ (1 − s V )
E p (Vtm Ft −1 ) Vt −1m
p* (1 − s V ) − 1 = δ tm (1 − s C ) + g Tm
(4)
This definition is similar to the one used in Richter (2004). See Richter (2004), p.30. Within our analysis a tax regime is neutral, if the equilibrium price of any asset does not depend on the prevailing tax rate. Elschen/Hüchtebrock (1983) discuss the use of the term “neutral” with respect to taxation in business and in economics. 25 See Richter (2004), p. 13; Leuthier (1988), pp. 163; Schwetzler/Piehler (2004), app. 3. 26 Samuelson (1964) covers the certainty case and therefore the discount rate is equal the risk-free interest rate. See Samuelson (1964), p. 604 24
12 In equ. (4) the pre tax cash flow Ctm is divided by the theoretical market value Vt-1m to derive the pre tax expected dividend yield δt-1m. For the further analysis Richter treats δt-1 as a constant and the income tax effect is supposed to be completely reflected by the tax factor (1sC) to be applied upon δt-1. Clearly Vt-1,m and Vt,m are market values of M in a world with taxes. Thus treating δt-1 as a constant requires that the (theoretical) market value Vt-1m is independent from the tax rates sC and sV.27 Therefore, Richter’s derivation of the irrelevance result requires the irrelevance result for the market portfolio as a necessary assumption. 3.3.2 Equilibrium asset prices and taxation of the economic income Considering taxation Atkinson/Stiglitz (1980) claim, that ”since it [taxation] must, of necessity, take income away from individuals, it makes them worse off. As a result of being worse off, they behave differently. That is, individuals typically make different decisions when their incomes change.”28 Therefore with respect to valuation there are two effects of taxation,
a direct one affecting the cash flow of an investment distributed to investors and an indirect one affecting the pricing mechanism of the economy. In general research analyzing taxation and its effect on corporate values concentrates on the direct tax effects caused by the asset or security to be valued and ignores the indirect effect. However, even models that try to capture the indirect effect often neglect the possible effects stemming from the governments’ spending of the collected tax payments.29 In Rapp/Schwetzler (2004) we analyze the impact of changes in the income tax rate using a consumption based asset pricing framework with agents having identical von NeumannMorgenstern utility functions and identical initial endowments.30 Furthermore we apply a two date binomial model with a boom state (b) and a recession state (r) in t=1. The boom state is characterized by an higher aggregate endowment compared to the recession state. The subjective probability for the boom state is denoted by p. Three assets i=0,1,2 are traded in the capital market of the model, each characterized by its state-dependent payoffs xi=(xib,xir).31 A (representative) agent acting on the capital market is characterized by her initial endowment (e0,e1) and her utility function U(c1 ) = E P [u (c1 )] =
27
p(ω) ⋅ u (c ω )
ω∈{r ,b}
(5).
Similar arguments hold for the capital gains component, since the pre-tax expected future market value is divided by the after-tax market value Vt-1m to calculate the pre-tax capital gains. 28 Atkinson/Stiglitz (1980), p. 27. 29 See for example Dammon/Green (1987). 30 For more details on the model see Rapp/Schwetzler (2004). 31 Since our model applies the economic income of an asset as its tax base there is no need to differentiate between the cash flow and the value component of the date-1 payoff.
13 Note, that U is a function in date-1 consumption c1 only. Therefore date-0, consumption does not enter agent' s optimization problem and date-0 initial endowment is assumed to be zero. Date-1 endowment e1 consists of a portfolio of the three tradable assets. We characterize a portfolio by a vector h=(h0,h1,h2), where hi denotes the number of asset i held in the portfolio. Equilibrium prices reflect demand and supply of the assets. We assume the supply of the riskless asset to be perfectly elastic whereas the supply of the risky assets is assumed to be perfectly inelastic.32 Imposing a market clearance condition, the equilibrium price for asset i in a world without taxes is Vi =
(e1 ) x i ] 1 E p [u ' p (e1 ) ] 1 + rf E [u '
(6)
where u’ denotes the agents’ marginal utility.33 According to equ. (6) pi may be characterized by the marginal utility in the initial endowment e1. The next step is to introduce the consequences of a tax system, that applies a uniform tax rate s on the economic income of an agent’s portfolio. Given a portfolio h an agent has to pay taxes of t ω (s ) = −s ⋅
2 i =0
h i (x iω − Vi ,s )
in state ω∈{b;r}. Note that tω0 if the agent receives a tax-subsidy. Within our current analysis we ignore government’s spending (labeled as “redistribution” in Rapp/Schwetzler (2004)) by assuming, that redistribution does not enter the individual investor’s optimization problem and the equilibrium pricing mechanism.34 Recalling the no-trade-property of representative agents models therefore yields the following equilibrium pricing formula35 Vi ,s =
(e + t (s )) x i,s u' 1 Ep p 1 (e1 + t (s )) ] E [u ' 1 + rf ,s
.
(7)
Note that the after-tax price of asset i in eq. (7) depends threefold on the tax rate
32
(i)
the tax rate determines the assets after-tax payoff
(ii)
the tax rate determines the marginal utilities in each of the two states
(iii)
the tax rate determines the riskless discount rate rf,s.
This assumption could be justified by the observation that the supply of riskless assets is coordinated by a authority like the European Central Bank or the Federal Reserve Bank in the US and therefore is likely to be elastic. Note, that this is the same set of assumptions as in the classical derivations of the CAPM. See e.g. Lintner (1965) and Sharpe (1964). 33 See Rapp/Schwetzler (2004), formula (20) on page 11. 34 That is, we only discuss the “no-redistribution regime” of Rapp/Schwetzler (2004). 35 Rapp/Schwetzler (2004) pp. 18. For the no-trade-property of representative agents models see LeRoy/Werner (2001), p. 11.
14
(e1ω + t ω (s )) is the state dependent risk neutral density to be applied on the subjective u' p E [u ' (e1 + t (s )) ] probabilities in order to derive the risk neutral probabilities q.36 Thus the risk neutral probability of state ω is q s (ω) = p(ω) ⋅
(e1ω + t ω (s )) u' p E [u ' (e1 + t (s )) ]
and the state price
πSω = q S (ω)
sω
.
for one unit of after-tax payoff in state ω is
(e + t (s )) 1 u' 1 = p(ω) ⋅ p 1ω ω (e1 + t (s )) ] 1 + rf ,s 1 + rf ,s E [u '
(8)
Assuming the aggregate tax base in the recession state to be negative, we derive the following results:37 -
The risk-neutral density for the recession state decreases in the tax rate, whereas for the boom state it increases with the tax rate. With a subjective probability measure that is ceteris paribus independent of the tax rate the same holds true for the risk-neutral probabilities and the state prices.
-
The impact of a tax rate change upon equilibrium asset prices depends on the correlation of the asset' s payoff with the aggregate endowment. -
If the correlation is is positive, i.e.the asset offers a higher payoff in the boom state than in the recession state, the assets equilibrium after-tax price increases in the tax rate.
-
If the correlation of the asset' s payoff with the aggregate endowment is negative, i.e. the asset offers a lower payoff in the boom state than in the recession state, the assets equilibrium after-tax price decreases in the tax rate.
-
If the correlation of the asset' s payoff with the aggregate endowment is zero, i.e. the asset offers a riskless payoff, the assets equilibrium after-tax price is independent of the tax rate.
Under the conditions stated above equilibrium asset prices depend on the prevailing tax rate. Thus in general the economic income as a tax base combined with a uniform tax rate on capital gains, dividends and interest income may not serve as a yardstick for tax neutrality under uncertainty. In order to achieve the irrelevance result further restrictive assumptions
36
The risk-neutral probabilities of the model are equal to the marginal utilities weighted subjective probabilities. See Rapp/Schwetzler (2004), p. 12. 37 We assume the negative aggregate tax base in the recession state for analytical convenience; it allows to derive the impact of tax rate changes without further restrictions on the representative agents’ utility function. See Rapp/Schwetzler (2004), pp. 21.
15 have to be made with regard to the redistribution of funds by the tax authorities. However, under the “ no redistribution” case assumed here the irrelevance property is not feasible.38 What can be said about the income tax effect on the market price of M, the market portfolio containing all risky assets? Our analysis starts at the after tax-pricing functional for risky assets with39 VS (x ) =
π S,b x − xr xr + b ⋅ 1 + rf 1 + rf πS,b + πS,b
(9)
where xr, xb denote the payoff to be valued in the recession and the boom state. As πSb increases and πSr decreases in the tax rate, the factor
π S, B in equ. (9) also π S , B + π S, R
increases in s. The impact of an increasing tax rate on the asset prices in (9) thus depends on the relation of xb to xr: an asset positively correlated with the total endowment offering payments xb > xr has an increasing equilibrium price in the tax rate whereas an asset negatively correlated with the total endowment offering payments xb < xr has a decreasing equilibrium price in s. In our simple model the market portfolio as understood in the classical CAPM consists only out of the two risky assets 1 and 2 which are pro- and counter-cyclical with respect to the total endowment. Thus the equilibrium market value of M can be calculated based on VM ,s = V1,s + V2,s =
π S,b π S, b x − x 1r x − x 2r x 1r x + 1b + 2r + 2b ⋅ ⋅ πS,b + π S,r 1 + rf π S,b + π S,r 1 + rf 1 + rf 1 + rf
(10)
Rearranging (10) yields VM ,s =
πS , b x1r + x 2 r [(x + x ) − (x + x )] + 1 + rf (πS,b + πS,r ) (1 + rf ) 1b 2b 1r 2 r
(10’ )
As the first term in (10’ ) does not depend on the income tax rate and the first factor in the second term is increasing in s, the total impact of an increase of the income tax on the equilibrium price of M depends on x1b + x2b being greater, smaller than or equal to x1r + x2r. As the two factors determine the distribution of the economies total endowment40 by definition (x 1b + x 2 b ) > (x 1r + x 2 r ) has to hold: the market portfolio’ s payoff in the boom state
38
Rapp/Schwetzler (2004), pp. 25. Rapp/Schwetzler (2004), p. 23. 40 The riskless assets´ payoff is state independent. 39
16 exceeds the market portfolio’ s payoff in the recession state. Thus the equilibrium price of the market portfolio M is increasing in the income tax rate. Equ. (10’ ) also highlights the necessary condition for the irrelevance of the income tax rate for the market price of M: (x 1B + x 2 B ) = (x 1R + x 2 R ) would require the market portfolio’ s
payoff in the boom state to be equal to the one in the recession state and thus the aggregated endowment of the economy in the boom state to coincide with the aggregated endowment in the recession state. By defining the total endowment to be state independent, such an assumption would rule out any uncertainty about the future development of the economy. 3.3.3 An example In this section we present a numerical example of a single agent economy in order to clarify
the effect of different tax rates on assets’ equilibrium prices and the equilibrium price of the market portfolio M containing all risky assets.41 The agent’ s preferences are characterized by a utility function (5) with u(x)=ln(x) and a subjective probability for the boom state of p=50% – similar to Richter (2004).42 Again three assets are traded in the market. The date-1 payoff structure of the assets is given by x0=(10;10), x1=(50;20) and x2=(12;15).43 The agent’ s portfolio is h=(1,1,1). Thus the economies’ total endowment is eb=x0b + x1b + x2b = 10 + 50 + 12 = 72 in the boom state and er=x0r + x1R + x2r = 10 + 20 + 15 = 45 in the recession state. In a world without taxation equilibrium asset prices are derived according to eq. (6). Assuming rf=0.10 we easily calculate V0=9.0909, V1=28.6713 and V2=12.5874 for the three assets and implicit state prices of πb=0.3497 and πr= 0.5594. The associated risk-neutral probabilities are qb=0.3846 and qr=0.6154. In the next step we calculate -
the risk-neutral probabilities qb and qr,
-
the state prices πr and πb,
-
the assets equilibrium prices V1,s and V2,s, and
-
the market portfolio’ s equilibrium price VM,s
for different tax rates s under the tax regime described above (tax base = economic income; redistribution not entering the optimization problem; uniform tax rate). The results are given in table 2.
41
According to the no-trade-property of representative agents models we can choose a single agent economy without loss of generality. 42 Note, that the agent’ s preferences are characterized by a HARA utility function. Therefore the optimal portfolio choice is independent of the level of endowment. See e.g. Niehaus (2003), p. 42. 43 Note that each pair of assets define a complete market, i.e. given the three assets of this example the market is over-complete.
17
s qb qb πb πb V1,s V2,s VM,s
0% 38.46% 61.54% 0.3497 0.5594 28.6713 12.5874 41.2587
10% 39.47% 60.53% 0.3621 0.5553 28.9472 12.5598 41.5071
20% 40.52% 59.48% 0.3752 0.5507 29.2331 12.5312 41.7644
30% 41.61% 58.39% 0.3888 0.5457 29.5288 12.5017 42.0304
40% 42.72% 57.28% 0.4031 0.5403 29.8337 12.4712 42.3048
50% 43.87% 56.13% 0.4178 0.5345 30.1473 12.4398 42.5871
60% 45.05% 54.95% 0.4332 0.5283 30.4691 12.4076 42.8767
70% 46.26% 53.74% 0.4491 0.5218 30.7980 12.3747 43.1727
80% 47.49% 52.51% 0.4656 0.5148 31.1332 12.3412 43.4744
90% 48.74% 51.26% 0.4825 0.5076 31.4737 12.3072 43.7808
99% 49.99% 50.01% 0.4998 0.5001 31.7552 12.2963 44.0515
Table 2: Equilibrium pricing parameter and asset prices of the economy
Table 2 shows that the equilibrium market price of portfolio M containing the two risky assets is increasing in the tax rate s as predicted by equ. (10’ ).44 Figure 4 plots the equilibrium prices for asset 1, 2 and M.
50.0 45.0 40.0
asset prices
35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0 0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Income tax rate s V1,s
V2,s
VM,s
Figure 4: After-tax equilibrium prices of the economy with respect to the income tax rate s
For the data of our example the impact of the tax rate change upon VM,s seems to be rather small. Increasing the income tax rate from 0% to 50% yields an increase of VM,s from 41.2587 to 42.5871. This corresponds to a a change of 3.22%. However, one may not conclude from this result that under an even stylized German tax system the deviation between the “ true” empirical values PM,s=0% and PM ,s =s0 might be negligible for any s0. Our model proves the relevance of taxes under a uniform tax rate applied to the economic income as a tax base. These are conditions that are commonly considered to be promoting the irrelevance of taxes. But the empirical approach of Richter (2004) requires the independence of PM,s from the tax rate under the German tax regime where dividends (s= s/2), capital gains (s = 0), and interest income (s = s) are taxed differently. Thus one might expect that this independence condition will not be met in the German tax system as well.
44 As the equilibrium price of the riskless security is independent from the tax rate the same is true for the equilibrium price of all securities including the riskless security.
18 Our finding that VM,s and Pm is increasing in the tax rate s implies that the “ pre tax” dividend yield δ 0, m is decreasing in the tax rate. Thus the sensitivity of the implicit risk neutral probabilities with respect to s is lower than that calculated in Richter (2004). This is shown in the 3rd column of table 3, where we calculated the implicit q given a 2001 dividend of € 33,741, million and an initial market capitalization of € 674,833 million. Starting in a world P without taxes for g Tm =2% and δ 0, m =5% the implicit risk neutral up-probability q is 43.86%.
Introducing a 35% tax rate without adjusting P0,m yields, in line with Richters results, a q of 42.16%. Now taking the increase of P0,m into account by assuming a 10% adjustment of the market capitalization caused by the introduction of taxation45 yields an implicit q of 43.02%. Richter (2004) also performs sensitivity analyses for corporate values based on multiples.46
The effect of adjusting P0,m in the sensitivity analysis of corporate values is not that straightforward: taxation affects the cash flows of the corporation and the market parameter q, whereas q affects the risk neutral growth rate of the market g qtm and of the individual stock P g qti as well.47 For the scenarios described above ( g Tm =2%; δ 0, m =5%) the right hand column
of table 3 shows that the tax rate sensitivity of the multiple used to derive the corporations value increases if an increase of P0,m in the tax rate is incorporated in the analysis. Corporate value multiple
0.4386
1/δ0,m (Multiple market portfolio M) 20.00
0%
0.4216
20.00
19.66
5%
0.4261
21.00
20.39
10%
0.4302
22.00
21.10
15%
0.4340
23.00
21.79
20%
0.4376
24.00
22.46
Implicit risk neutral up-probability q
0%
Assumed change of P0,m due to the change of s -
35%
Tax rate
19.07
Table 3: Analysis incorporating an increasing market capitalization
Presuming that the introduction of taxation with a tax rate s of 35% implies an increase in the market cap P0,m of 10%, then the increase on the multiple would be 10,6% (from 19.0748 to 45
As shown in section 3.3.3. the exact impact of an increase of s upon P0,m depends on different factors, especially on the total endowment of the economy in different states of the world. 46 Richter (2004), section 5.3. 47 In Richter’ s analysis the beta factor of the stock is expressed in term of the risk neutral growth rates of M and the individual stock i as
48
ß t ,i =
(1 + g )(1 + g ) − 1 . See Richter (2004), p. 30. (1 + g )(1 + g ) − 1 P t ,i
q t ,i
P m ,i
q m ,i
Unfortunately we were unable to exactly reconstruct Richters multiple for the case of s=0%.
19 21.10) compared to the 3.1% (from 19.12 to 19.66) in Richter’ s analysis. Now the impact of the tax rate might be above “ the bounds of error” of 5%. 4
CONCLUSION
The independence of the market portfolios equilibrium price from the tax rate is an important condition for the irrelevance of personal taxes under a tax regime using the economic income as a tax base. In this note we show that in a consumption based asset pricing model with representative agents the equilibrium prices of single assets and equilibrium price of the market portfolio of risky assets depend on the prevailing tax rate. With respect to the current discussion on the effect of income taxes on corporate values this result has two important consequences: -
The tax regime combining a uniform tax rate with the economic income as a tax base can not serve as a yardstick for the irrelevance case of personal taxes under uncertainty. In general Samuelson’ s irrelevance theorem can not be transferred to the uncertainty case.
-
Under the conditions of an idealized tax regime combining the economic income as tax base with a uniform tax rate the equilibrium price of the market portfolio of risky assets depends on the tax rate s. Thus the necessary condition for the treatment of the dividend yield as an exogenous variable when determining the markets’ risk neutral probability is not met. As Vtm itself depends on the tax rate, valid risk neutral probabilities q can only be derived for the combination of tax rates and market prices Ptm currently observed at the markets. The good news of this result is, that for given current tax rates and market prices the valuation formula in Richter (2004)49 may still be applied. The bad news is that sensitivity analyses varying the tax rate without adjusting the market price of M should be subject to caution.
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