Pension fund taxation and risk-taking: Should we switch from the EET to the TEE regime?
Katarzyna Romaniuk1
2
Abstract: Most countries tax retirement savings according to the EET (Exempt contributions, Exempt accumulations, Taxable withdrawals) regime. Relevant literature recommends the use of TEE (Taxable contributions, Exempt accumulations, Exempt withdrawals) or EET systems and emphasizes their near equivalence. We show that this near equivalence breaks down when considering the tax e¤ects on risk-taking. This paper proves that the TEE regime is risk-taking neutral, while the EET regime does not, in general, respect this property. The argument of risk-taking neutrality thus calls for broadening the use of the TEE con…guration.
Keywords: tax; EET regime; TEE regime; pension funds; de…ned contribution; de…ned bene…t; risk-taking.
JEL Classi…cation: C61; G11; G23; G28; H22; H39.
1
Universidad de Santiago de Chile, Department of Economics, and Université de Paris 1 Panthéon-Sorbonne, PRISM. Universidad de Santiago de Chile, Department of Economics, Lib. B. O’Higgins 3363, Estacion Central, Santiago, Chile. E-mail:
[email protected],
[email protected]. 2 I am indebted to an anonymous referee for extensive discussions. I am grateful to Anne Villamil (the Editor) and an anonymous Co-Editor. For helpful comments, I wish to thank Jaime Casassus, Pablo Castañeda, Paul Klumpes, Yul Lee, Roberto Rigobón, François Salanié, Facundo Sepúlveda, Salvador Valdés Prieto, Radu Vranceanu, Stephen Zeldes, Felipe Zurita and the participants in the 2008 FMA European Conference in Prague, the 2008 EGRIE Seminar in Toulouse, the 2008 LACEA Meeting in Rio de Janeiro and a seminar at the Ponti…cia Universidad Católica de Chile. All remaining errors are mine.
0
"We have the question of how voluntary pensions should be taxed, something on which there is little literature (...). How this tax favoring should be done is an important issue that I ‡ag as needing research rather than o¤ ering an answer." Diamond (2009)
"The impact of taxation on household portfolio behavior is an issue that already attracts attention in both applied tax-policy debates and in the academic disciplines of public economics and …nancial economics. But this issue is likely to become even more important prospectively." Poterba (2002)
Deciding which tax regime to apply to retirement savings largely remains an unresolved issue. One has the possibility to tax contributions, investment income or withdrawals. Literature on this subject, albeit scarce, calls for taxing contributions or withdrawals. Then, either the TEE (Taxable contributions, Exempt accumulations, Exempt withdrawals) or the EET (Exempt contributions, Exempt accumulations, Taxable withdrawals) system prevails. The argument supporting this choice is that the two systems are neutral to the decision of whether to consume now or save and consume in retirement (Whitehouse 1999). Actual tax regimes generally follow the given advice, with the majority of countries opting for the EET con…guration. Following Yoo and de Serres (2005), 22 of 30 OECD countries use the latter regime. Yet, the general classi…cations that are encountered obviously cover speci…cities. In particular, the United States are EET-classi…ed, while traditional (EET) and Roth-style (TEE) tax incentives are available, and this is found both in individual retirement accounts (IRAs) and 401(k)-type plans (CBO 2011). In the United Kingdom, pensions are broadly EET-taxed; Individual Savings Accounts (ISAs) are TEE-taxed (PPI 2011). The equivalence of EET and TEE systems is a widely shared view because these grant equal amounts of tax bene…ts, provided the same tax rate applies when contributing and withdrawing
1
(Whitehouse 1999; CBO 2004, 2011; Huang 2008).3
"Either EET or TEE is acceptable", and
the choice between the two is made by essentially weighing political and budgetary considerations (Robalino et al. 2005). The TEE treatment may not be credible if there is fear that the policy will revert to taxing investment income or withdrawals; conversely, shifting tax revenues toward the future, as implied by the EET system, may not be …nancially viable for the government.4 Robalino et al. (2005) emphasize that "any di¤erence in the economic impacts of either option will be of a second order of magnitude". This …nancial and economic near equivalence of EET and TEE systems will be challenged in our paper. The argument will be based on the taxation impact on risk-taking. Analyses of the e¤ects taxation has on risk-taking date from Domar and Musgrave (1944), who prove that, in the full o¤set case, introducing income taxation enhances the investor´s risk-taking. In the expected utility framework, Stiglitz (1969) shows that the e¤ects of wealth, income and capital gains taxation on risk-taking crucially depend on the investor´s risk aversion characteristics. In a dynamic setting, the e¤ect of capital gains taxes on optimal portfolio decisions is …rst analyzed by Constantinides (1983). Dammon et al. (2001), in a more restrictive environment, and DeMiguel and Uppal (2005), using the exact tax basis, show that the introduction of capital gains taxes leads to an increased equity proportion. Yet, Seifried (2010), considering deferred taxes, concludes that the Merton (1971) strategy generates a nearly optimal result under taxation. In terms of empirical research, Feldstein (1976), Hubbard (1985) and Poterba and Samwick (2002) document a positive link between the income tax rate and household stock holdings. Following Agell and Edin (1990) and King and Leape (1998), taxes have a larger impact on the probability of investing in an asset than on its portfolio proportion. In a summary of the state of research concerning the tax e¤ects on household portfolio structure, Poterba (2002) concludes 3
Depending on whether the tax rate is lower or higher when withdrawing, compared to the contributing period, the agent prefers the EET or TEE treatment, respectively (CBO 2011). 4 Among other arguments supporting the TEE treatment, the latter limits tax avoidance and evasion because taxes are paid up-front, and collects more revenues from agents who are higher-rate taxpayers during working life and standard-rate taxpayers during retirement; as to the pros of the EET regime, psychological traits lead individuals to prefer up-front tax relief (Whitehouse 1999).
2
that "the empirical literature, while not o¤ering universal support, generally suggests that taxation plays an important role in determining (...) the amount that they [households] invest in each of the available assets (...)." The literature, which analyzes the taxation e¤ects on the pension fund portfolio, studies de…ned contribution (DC) and de…ned bene…t (DB) funds separately and focuses, for each type of fund, on a speci…c topic: the interactions between taxable and tax-deferred DC accounts and the taxarbitrage opportunities, when taking an integrated view of the sponsoring company and of its DB pension fund, respectively. For DC funds, the asset allocation and location decisions are jointly considered. Dammon et al. (2004) prove that there is a pronounced preference for holding taxable bonds in the taxdeferred account, and for holding equity in the taxable account. Yet, empirical studies show that this advice is not followed in practice (Ameriks and Zeldes 2004; Bergstresser and Poterba 2004). The literature provides arguments trying to resolve this apparent dissonance between theory and practice (Amromin 2003; Shoven and Sialm 2003; Garlappi and Huang 2006; Gomes et al. 2009; Zhou 2009). Huang (2008) de…nes the conditions under which allocation and location decisions remain separate. Regarding DB funds, pure tax considerations lead Black (1980) and Tepper (1981) to advocate for an all-bond pension fund portfolio. When additional dimensions are included, for example, with the Pension Bene…t Guaranty Corporation (PBGC) insurance, a mixed portfolio can become advisable (Bicksler and Chen 1985). The all-bond portfolio advice is not followed in practice (Bergstresser et al. 2006; Rauh 2009); yet, Frank (2002) documents a positive relationship between tax bene…ts and the bond proportion in pension assets. The fact that tax e¤ects on a pension fund portfolio are studied each time within an enlarged setting (combining taxable with tax-deferred DC accounts and the sponsoring …rm …nances with its DB fund …nances) leads to results that are clearly interesting, yet do not respond to the main question asked, and responded to in the general non-pension setting: What are the tax e¤ects 3
on risk-taking per se, without placing other related problems at the heart of the analysis? The literature has until now analyzed the impact of the choice between taxable and tax-deferred DC accounts on the agent´s risk-taking decision, and the DB optimal portfolio form in a tax-arbitrage approach within a consolidated balance sheet setting. Yet, the literature still lacks a response to these fundamental questions: How does the fact that a tax is introduced impact on the pension fund risk-taking decision? How do di¤erent types of taxes in‡uence risk-taking behavior? Is the tax regime choice, as it is made in practice, justi…ed, considering the tax e¤ects on risk-taking as the decision criterion? These are the questions to which we will respond in this paper. The setting we choose, contrary to the literature, is unique to DC and DB funds, allowing us to make easy comparisons between these two types of funds. In this paper, we will study the TEE and EET system e¤ects on risk-taking, by analyzing the DC and DB pension fund optimal portfolio rules, in a continuous-time setting à la Merton (1971). Our objective is to judge the appropriateness in light of the tax e¤ects on risk-taking of the frequently assumed, near equivalence of TEE and EET regimes and of the EET choice in practice. While taxation types comparable to the EET regime have previously been discussed in the literature, the TEE system has not yet received its due attention, at least in a continuous-time setting. The main reason for this appears to be the fact that in simpler environments the TEE and EET regimes operate through identical mechanisms, and thus induce the same e¤ect. Huang (2008) emphasizes the equivalence, within a constant factor, of TEE- and EET-taxed retirement accounts under deterministic tax rates. Yet, the underlying assumption is only an initial contribution, while in practice contributions are paid through the entire accumulation period. Furthermore, if a continuous-time setting is considered, contributions acquire a particular - cash ‡ow - nature, which leads to a speci…c TEE system impact on the optimal portfolio rule. This paper´s main results are the following. The TEE regime is risk-taking neutral. Its e¤ect is only on the contribution hedging demand, for both the DC and DB plans. Only the EET regime 4
can a¤ect risk-taking behavior. Yet, this e¤ect on risk-taking does not materialize for DB plans because the necessity to respect the solvency constraint precludes the emergence of the tax rate in the utility function argument. For DC plans, risk-taking can become distorted by EET taxation. Yet, the direction of the e¤ect is not obvious. Intuition suggests that this direction is strongly driven by risk aversion properties, in line with Stiglitz´s (1969) conclusion. The paper is organized into the following three sections: the …rst section builds and solves the model, the second section analyzes the tax e¤ects on risk-taking, and the conclusion is provided in the last section.
1 1.1
The model The TEE system
In a TEE system, only contributions are taxed. These contributions, brought continuously into the fund assets, are de…ned as a constant proportion of the employee´s wage Y . The employee contributes a proportion
1,
and the employer contributes a proportion
2,
with
1; 2
0. The par-
ticipant’s contributions are taxed at a constant rate t1 , and the …rm´s contributions are taxed at t2 , also constant, with 0
1.2
t1 ; t2 < 1. The contributions thus amount to ((1
t1 )
1
+ (1
t2 ) 2 ) Y .
The EET system
In a EET system, the tax only applies to the pension. The constant tax rate, denoted by , respects 0 1.2.1
< 1.
DC plan
In a DC plan, the pension is de…ned as the asset value A at the retirement date T . The after-tax pension is thus (1
)A(T ). The DC plan manager solves the following optimization program on
a date t preceding T :
M axEt [U ((1
5
)A(T ))]
(1)
where Et denotes the expectation, conditional on the information available in t, and U the utility function, whose properties will be discussed later. 1.2.2
DB plan
A DB plan o¤ers a pension depending primarily on the participant’s wage and the number of years of service. The pension promise forms the plan liabilities, which we denote by L. The after-tax pension is written as (1
)L(T ).
In a world without taxes, standard asset-liability management principles induce the DB plan manager to follow the program (Basak 2002; Romaniuk 2007):
M axEt [U (A(T ) where the constraint A(T )
L(T ))]
(2)
L(T ) is implicitly incorporated provided the utility function
respects the Inada condition limA!L+ U 0 (A
L) = +1, a property met by CRRA preferences
(Karatzas et al. 1986; Merton 1990). We will further impose this property on the utility function. Let us introduce the EET tax. One is tempted to consider program M axEt [U (A(T )
(1
)L(T ))]. Yet, this con…guration would be incompatible with the solvency constraint A(T ) L(T ). One has to generate, at the …nal date, an asset value that is at least equal to the pre-tax liability value, not to its after-tax value. If the program M axEt [U (A(T ) used, one would be certain to only meet A(T )
(1
(1
)L(T ))] were
)L(T ). Consequently, the DB fund program
remains the choice for an environment without taxation, de…ned by (2). The solvency constraint reveals itself to be a determining factor for the program form. 1.2.3
Comparing DC and DB cases
One observes a fundamental di¤erence between the DC and DB plans. In the DC case, the EET tax manifests itself in the utility function argument, however, this is no longer true when DB funds are considered. Whether the EET tax emerges in the utility function argument has a decisive e¤ect on the impact of EET taxation on optimal portfolio rules. We will prove that if the EET tax is
6
not present in the utility function argument, it has no e¤ect on the optimal investment policy. A less abstract, and more intuitive, explanation for this con‡icting outcome in the DC and DB con…gurations follows from how pensions are de…ned in these two types of plans. While the DC plan o¤ers a pension that is directly dependent on the …nal asset value, and thus on portfolio decisions taken, the DB plan proposes a pension that is primarily determined by the …nal salary value, and thus is independent of the …nal asset value and the investment strategies that the plan followed.
1.3
The relevant variables dynamics
The fund assets A are invested in the risky assets Si (i = 1; 2; :::; N ) and in the riskless asset . The dynamics of the risky assets vector S obeys:
dS(t) = IS where IS denotes an (N (N
S (t; Z(t))dt
S (t; Z(t))dB(t)
(3)
N ) diagonal matrix valued function of S(t),
1) vector valued function of t and Z,
of t and Z, B(t) an (M
+ IS
S (t; Z(t))
a bounded (N
S (t; Z(t))
a bounded
M ) matrix valued function
1)-dimensional Wiener process in RM , Z(t) a (K
1)-dimensional vector
of state variables. The ith risky asset Si follows the stochastic di¤erential equation (SDE):
dSi (t) = Si (t) with
Si (t; Z(t))
Si (t; Z(t))dt
+ Si (t)
Si (t; Z(t))dB(t)
de…ned as a bounded function of t and Z,
Si (t; Z(t))
(4) a bounded (1
M)
vector valued function of t and Z. The state variables vector Z has the following dynamics:
dZ(t) = IZ
Z (t; Z(t))dt
7
+ IZ
Z (t; Z(t))dB(t)
(5)
where IZ stands for a (K (K
K) diagonal matrix valued function of Z(t),
1) vector valued function of t and Z,
Z (t; Z(t))
a bounded (K
Z (t; Z(t))
a bounded
M ) matrix valued function
of t and Z. The riskless asset
evolves according to the dynamics:
d (t) = (t)r(t; Z(t))dt
(6)
with r(t; Z(t)) the instantaneously riskless interest rate, a function of t and Z. The dynamics of the employee´s wage Y , whose proportion continuously accumulates to the fund assets in the form of contributions, follows the SDE:
dY (t) = Y (t) where
Y
Y
(t; Z(t))dt + Y (t)
Y
(t; Z(t)) denotes a bounded function of t and Z,
(t; Z(t))dB(t) Y
(7)
(t; Z(t)) a bounded (1 M ) vector
valued function of t and Z. The dynamics of the DB fund liabilities L obeys:
dL(t) = L(t) where
L (t; Z(t))
L (t; Z(t))dt
+ L(t)
L (t; Z(t))dB(t)
stands for a bounded function of t and Z,
L (t; Z(t))
(8) a bounded (1
M)
vector valued function of t and Z.
1.4
The optimization program
Let us denote the utility function argument by X. The DC and DB optimization programs are de…ned by (1) and (2), respectively. For the DC plan, X is thus (1
)A, while it is A
L for
the DB plan. A general X formulation follows:
X = (1
)A
8
L
(9)
where the subcases L = 0 and
= 0 represent the DC and DB con…gurations, respectively.
The utility function takes the terminal form U (X(T )). It is assumed to be increasing and concave in X and to respect the Inada conditions: limX!1 UX = 0 and limX!0 UX = 1, with UX the derivative of U with respect to X. The respect of the Inada condition limX!0 UX = 1 is critical to the DB fund case because it guarantees the respect of the solvency constraint A(T )
L(T ). For the DC fund con…guration,
imposing this condition is important because it insures the …nal date utility function argument non negativity; yet, the role played by the Inada property is not as vital as in the DB case. The manager´s optimization program is written as:
M axxS Et [U (X(T ))] u:c:dX(t) dZ(t)
= X(t) = IZ
(10)
X (t; xS ; X(t); Z(t))dt
Z (t; Z(t))dt
+ IZ
+ X(t)
X (t; xS ; X(t); Z(t))dB(t)
Z (t; Z(t))dB(t)
X(0) > 0 X(T )
0
where xS denotes an (N
1) vector of the proportions of X to invest in the risky assets Si ,
the variable X dynamics is written in a general form, X and Z, and
X (t; xS ; X(t); Z(t))
a (1
X (t; xS ; X(t); Z(t))
is a function of t, xS ,
M ) vector valued function of t, xS , X and Z.
The two last constraints guarantee the X strict positivity at the initial date 0 and its non negativity at the retirement date T .
1.5
The model solution
The optimal portfolio rules for DC and DB plans in the TEE and EET regimes are presented in Table 1. Appendix A develops the proof.
9
Table 1 : Optimal portfolio rules in di¤erent plan and tax con…gurations
TEE regime DC plan
DB plan
EET regime JX r1N ) JXX X
(
SS )
1
(
SS )
1
(
SS )
1
(
SS )
1
(
SS )
1
+(
SS )
1
L SL X
(
SS )
1
1 0 SZ IZ JXZ JXX X
(
S SY
((1
t1 )
1
SS )
1
(
SS )
1
(
SS )
1
(
SS )
1
(
SS )
1
+(
SS )
1
L SL X
(
SS )
1
1 0 SZ IZ JXZ JXX X
Y t2 ) 2 ) X
+ (1
1 0 SZ IZ JXZ JXX X
(
JX r1N ) JXX X
S SY
((1
t1 )
1
Y t2 ) 2 ) X
+ (1
(
S SY
2
(
1
+
Y 2) X
1 1 0 SZ IZ JXZ JXX X 1
(
JX r1N ) JXX X
S SY
Subscripts on J, the indirect utility function, denote partial derivatives. the covariance matrix of the variables i and j. 1N is an (N
JX 1 r1N ) JXX X 1
(
(
1
+
ij
Y 2) X
0 i j
stands for
1) vector of all ones.
Tax e¤ects on the optimal portfolio rules
2.1
Optimal portfolio rules analysis
Our focus is on the nature of the taxation e¤ect. The fundamental question we seek to answer is whether there is a tax impact on risk-taking. The element of the optimal portfolio rule that represents risk-taking behavior is the …rst component: the speculative fund. This fund translates the risk-reward arbitrage. The agent´s aim is to …nd the …nancially most interesting couple ( of relative risk tolerance
S
r1N ;
SS ),
while considering her degree
JX JXX X .
The remaining portfolio components are hedge funds. They belong to the risk management area. Hedge funds provide covers against the evolutions of the corresponding variables, in our model, the wage Y , the liabilities L and the state variables Z.5
These hedge funds take the
5 The liability hedge emerges only in the DB fund case. Asset-liability managers (when compared to asset managers) are bound to additionally cover against liability variations.
10
standard covariance to variance form, where
SS
stands for the variance, and
,
SY
SL
and
SZ
are the covariance in the Y , L and Z hedges, respectively. The existing literature generally opted for the entire risky asset proportion to be the variable to represent risk-taking (see, e.g., Stiglitz 1969). This choice is due to the fact that the settings considered by this literature were simpler than the one developed here, and there was, therefore, no hedging demand. 2.1.1
The TEE regime e¤ect
For the TEE regime, risk-taking is not a¤ected by taxation because the tax rate is not present in the speculative fund formulation. The TEE tax impact, which is similar for DC and DB plans, falls only on the contribution hedge. Instead of taking the no-taxation form (
SS )
1
SY
((1
t1 )
1
+ (1
(
SS )
1
SY
(
1
+
Y 2) X ,
it is written as
Y when the TEE tax applies. The employee’s and emt2 ) 2 ) X
ployer’s contribution rates become multiplied by one less their respective tax rates. The contribution hedge term´s absolute value thus decreases with the tax rates t1 and t2 . The weaker importance of contributions in the optimal portfolio rule in the presence of TEE taxes has a simple interpretation. When compared with the no-taxation case, these contributions now play a less important role in wealth accumulation because a portion of them is initially subtracted in the form of taxes. Consequently, the contribution hedging demand decreases. 2.1.2
The EET regime e¤ect
In the EET regime, one observes a tax e¤ect for DC plans and no e¤ect for DB plans. Because the tax rate
did not enter the DB utility function argument, it also does not emerge in the
DB optimal portfolio policy. The DC optimal investment strategy becomes distorted by EET taxation: the speculative fund and the state variables hedge are divided by 1
.
Among the four plan and tax con…gurations under consideration in this analysis, the only case where a (potential) tax impact on risk-taking is observed is thus the DC-EET con…guration, where
11
a taxation-induced distortion is registered in the speculative fund formulation. It would be bene…cial to determine the direction of the DC-EET tax e¤ect. Regarding the direction of the wealth tax’s e¤ect in a general setting, which plays the role of the EET regime in the pension fund setting, Stiglitz´s (1969) proposition 1(a) states that "a proportional wealth tax increases, leaves unchanged, or decreases the demand for risky assets as the individual has increasing, constant, or decreasing relative risk aversion". The e¤ects on risk-taking behavior induced by wealth taxation thus crucially depend on the individual’s risk aversion characteristics. Analytically determining the direction of the tax e¤ect on risk-taking in di¤erent risk aversion con…gurations is not one of our objectives. This appears to be an extremely di¢ cult, probably impossible task in a setting as general as the one in which we are working. In Appendix B, however, we propose a simple approximation exercise on the speculative fund formulation. The results obtained are compatible with the direction of the tax e¤ect in Stiglitz’s proposition. Intuition thus suggests that Stiglitz’s conclusion concerning the primordial role that is played by risk aversion characteristics for the direction of the tax e¤ect on risk-taking in a one-period setting remains valid in continuous time. 2.1.3
Reason for the di¤erence in the e¤ect nature of TEE and EET regimes
The only taxes that have the potential to in‡uence risk-taking are those that can appear in the utility function argument, i.e., EET taxes. These taxes emerge in the utility function argument only in the DC fund case. In the DB fund case, the necessity to respect the solvency constraint precludes their emergence. Consequently, only DC-EET taxes can induce modi…cations in risktaking behavior. For TEE taxation, DC and DB funds are a¤ected in the same way, yet through a channel di¤erent from the one isolated for the EET system. TEE taxes never enter the utility function argument. They impact on the contribution process, which continuously adds funds to the plan assets, with these funds being instantaneously invested in the risky and riskless assets. The cash
12
‡ow nature of contributions leads to the TEE regime impacting on only the hedging demand. Previous research did not isolate the nature of the TEE taxation e¤ect because the initial contribution´s setting (that, considered by Huang 2008) is not characterized by this speci…c tax e¤ect nature, which we isolated for the continuous-contribution process. If we assumed for a moment that a constant tax rate t0 is paid on the initial contribution A(0), the optimization program would be written as M axEt [U ((1
t0 )A(T ))], and would thus reproduce the structure
of the EET system program M axEt [U ((1
)A(T ))]. The tax mechanisms involved in the cases
of the initial contribution´s TEE regime and of the EET regime would thus be equivalent.
2.2
Summary of the main results
The main results can be summarized as follows: The TEE regime is risk-taking neutral. The TEE e¤ect is only on the contribution hedging demand. The same e¤ect is observed on both the DC and DB optimal portfolio rules. Only the EET regime can a¤ ect risk-taking behavior. The EET e¤ ect on risk-taking does not materialize for DB plans, due to the necessity to respect the solvency constraint. For DC plans, risk-taking can become distorted by EET taxation. Intuition suggests that the non-obvious direction of the tax e¤ect is strongly driven by risk aversion properties.
2.3
The US case in light of the paper´s results
In the US, DC plans can be either TEE- or EET-taxed, while DB plans are EET-taxed (CBO 2011). The introduction of TEE-type incentives is recent. These were introduced for 401 (k) plans only in 2006.6 The evolution in pension fund taxation principles in the US is compatible with this paper´s recommendation: our results show that, when considering the argument of …scal policy risk-taking 6
TEE-type IRAs were created earlier, in 1998.
13
neutrality, DC funds should be TEE-taxed, while DB funds can be either TEE- or EET-taxed. The consequences of the recent introduction of TEE-type incentives in 401 (k) plans on participants´ behavior are beginning to attract attention in the empirical literature. In particular, Beshaers et al. (2012) analyze contribution patterns. More research is certainly needed in this area. The analysis of contribution behavior should be extended with an analysis of portfolio choice, in a setting that jointly studies TEE- and EET-type plans.
3
Conclusion
This paper compared the e¤ects of TEE and EET regimes on risk-taking behavior of DC and DB plans. We proved that the TEE regime is risk-taking neutral, while the EET system can a¤ect risk-taking, however, only in the case of DC funds. When taking account of the tax e¤ects on risk-taking, the frequently assumed near equivalence of TEE and EET regimes thus breaks down. The risk-taking neutrality argument calls for broadening the use of the TEE regime. This paper´s results provide a rationale for the taxation trend observed in the US pension savings market, where TEE-taxed retirement savings vehicles were recently introduced and have been progressively extended. Our results also constitute a response, in part, to the debate surrounding the possible generalization of the TEE approach in the United Kingdom (PPI 2011). When making decisions concerning the pension fund taxation system, policy makers should bear in mind the risk-taking dimension. Obviously, risk-taking is only one element in the larger picture to consider. Some aspects to take into account were noted in the introduction. Other concern, for example, the extensively debated issue of the tax relief e¤ects on savings and specifically, whether savings are e¤ectively created or merely diverted, and the associated topic of the budgetary costs of tax breaks.7
Policy makers should think multidimensionally and (in part)
intuitively, as described by Diamond (2009), who calls for "seeking inferences from an individual 7 The e¤ect on savings is analyzed by Engen et al. (1996), Poterba et al. (1996) or Huberman et al. (2007). CBO (2004) focuses on the impact on budget projections. Both issues are dealt with in the studies included in OECD (2005).
14
study to be combined with inferences from other studies that consider other aspects of a policy question, as well as with intuitions about aspects of policy that are not in the models".
15
APPENDICES
APPENDIX A : SOLUTION TO THE OPTIMIZATION PROGRAM
We work with a TET regime to be able to solve the TEE and EET cases in one step. In the TET system, taxes apply on both the contributions and the pension. The DC plan manager follows program (1) and the DB plan manager program (2), while contributions accumulate on an after-tax basis. The TET system reduces to the TEE (EET) system by assuming
= 0
(t1 = t2 = 0). Replacing A by its asset composition in the X de…nition, as de…ned by Eq. (9), yields:
X = (1
)
"
N X
X S i Si + X
i=1
#
L
(11)
where XSi and X the number of assets Si and , respectively. Relative to X, Si is held in the proportion xSi (negatively) in the proportion xL
XSi Si X ,
X
in the proportion x , and L "N #X X L ) xSi + x xL = 1 X . The following identity: (1 i=1
is met. Di¤erentiating Eq. (11) and dividing by X, one obtains:
dX X
=
(1 xL
with ((1
t1 )
1
+ (1
"
N X
dSi d ) xSi +x + ((1 S i i=1 dL L
t1 )
1
+ (1
Y dY t2 ) 2 ) X Y
#
Y dY t2 ) 2 ) X Y appearing due to the continuous accumulation of the con-
tribution process. When one replaces the dynamics (4), (6), (7) and (8), and uses the identity "N # X (1 ) xSi + x xL = 1, the following dynamics are obtained: i=1
16
2
6 (1 = 6 4
dX X
)
x0S ( S
r1N ) + ((1 +r
2
6 (1 +6 4
)
x0S S
t1 )
xL (
+ ((1
t1 )
1
xL
L
1
+ (1
t2 ) 2 )
Y X
Y
r)
L
+ (1
t2 ) 2 )
with the prime ’standing for a transpose and 1N an (N
Y X
Y
3
3
7 7 dt 5
(12)
7 7 dB 5
1) vector of all ones.
Let the indirect utility function J, increasing and strictly concave in X, be de…ned by:
J(t; X(t); Z(t))
max Et [U (X(T ))] xS
The Hamilton-Jacobi-Bellman (HJB) equation takes the form:
max DJ = 0
(13)
xS
with D the Dynkin operator. Deriving the Dynkin of J with respect to xS , one obtains the system of …rst order conditions:
0N
=
(
S 2
r1N )(1
6 6 6 +6 6 + 6 4
+
)JX X SS xS (1
SY
((1
0 SZ IZ JXZ X(1
t1 )
1
+ (1
3
)2 Y t2 ) 2 ) X (1
SL xL (1
)
)
where subscripts on J denote partial derivatives, and variables i and j. The optimal vector of proportions xS follows:
17
ij
0 i j
(14)
7 7 7 2 7 JXX X 2 ) 7 7 5
the covariance matrix of the
xS
(
SS )
1
(
SS )
1
+(
SS )
1
(
SS )
1
=
(
S
SY
r1N ) ((1
SL xL
JX 1 JXX X 1
t1 )
1
+ (1
(15) t2 ) 2 )
Y X
1 1
0 SZ IZ JXZ
1 1 JXX X 1
Recalling that the X de…nition, as given by Eq. (9), is valid for DC and DB funds, provided that L = 0 and
= 0, respectively, one concludes that, in a TET regime, the DC fund optimal
policy takes the form:
xTS ET;DC
=
(
SS )
1
(
SS )
1
(
SS )
1
(
r1N )
S
SY
((1
JX 1 JXX X 1
t1 )
0 SZ IZ JXZ
1
+ (1
(16) t2 ) 2 )
Y X
1
1 JXX X 1
while the DB fund optimal policy is de…ned by:
xTS ET;DB
=
(
SS )
1
(
SS )
1
+(
SS )
1
SL xL
(
SS )
1
0 SZ IZ JXZ
(
S
SY
r1N ) ((1
JX
(17)
JXX X
t1 )
1
+ (1
t2 ) 2 )
Y X
1 JXX X
The Table 1 optimal portfolio rules in the DC-TEE, DB-TEE, DC-EET and DB-EET con…gurations are obtained by recalling that the TET system reduces to the TEE (EET) system by assuming
= 0 (t1 = t2 = 0).
18
APPENDIX B: SIMPLE APPROXIMATION EXERCISE ON THE DC-EET SOLUTION
xS is de…ned as the vector of the proportions of X to invest in risky assets, while X = (1
)A
for DC funds. To be able to compare the taxation and no-taxation environments, one should focus on the vector of the proportions of A to invest in risky assets, which we denote by xSi
XSi Si X
and
Si
XSi Si A ,
one directly concludes that
the optimal vector of proportions
(
S
Si
S.
As
= xSi X A . The speculative fund in
follows:
SS )
1
(
S
r1N )
1 JX JXX A 1
(18)
Two benchmark utility functions are considered - CRRA and CARA.8 We …rst consider the CRRA case, with coe¢ cient . In the simplest setting, characterized by a geometric Brownian motion (GBM) stock and the riskless asset earning the constant riskless interest rate, Merton (1971) proved that the term
JX JXX X
simply translates to 1 . Using this result
to approximate the speculative fund formulation (18), one obtains (
1
SS )
(
S
r1N )
1X 1 A 1
.
Now, turning to the CARA case, with coe¢ cient , recall that Pliska (1986) proved, in the simplest setting just described, that the term
JX JXX
represents 1 e
r(T
t)
.
Table 2 presents the speculative fund approximations in the two utility settings.
8 Dealing with CARA preferences could be seen as inadequate, as the Inada conditions imposed on the utility function preclude the possibility of CARA utility. Yet, as we emphasized earlier, Inada conditions were fundamentally imposed for the DB fund case.
19
Table 2 : Speculative fund approximations with CRRA and CARA preferences
CRRA preferences speculative fund approximation
(
SS )
1
(
S
r1N )
CARA preferences 1
(
SS )
1
(
S
r1N ) 1 e
r(T
t) 1 1 A1
In the CRRA con…guration, the speculative fund approximation is independent of the tax rate . In the CARA case, the proposed formulation´s absolute value is an increasing function of . Noting that a CARA function is characterized by an increasing relative risk aversion, one understands that the results obtained are compatible with the direction of the tax e¤ect in Stiglitz’s proposition.
20
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