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1 Università Commerciale Luigi Bocconi Econpubblica Centre for Research on the Public Sector WORKING PAPER SERIES Measuring Effective Tax Rates on Risky Assets Luigi Pascali Working Paper n 94 January 2004 wwweconpubblicauni-bocconiit

2 Measuring effective tax rates on risky assets Luigi Pascali January 2004 Abstract This paper proposes an integrated framework for measuring the fiscal burden on risky assets Auerbach (1981) demonstrates that the expected tax rate, normally used for this purpose in the economic analysis, can be a misleading measure: it confuses changes in risk characteristics with changes in the fiscal burden For this reason, we propose a new methodology for calculating a risk-adjusted tax rate This methodology is then used to evaluate different kinds of tax systems KEYWORDS: Effective tax rate, Fiscal burden, Portfolio choices JEL CLASSIFICATION: H22 I would like to thank Giampaolo Arachi and Alan Auerbach for some helpful comments Contacts: Luigi Pascali, Econpubblica, Università Commerciale Luigi Bocconi, Istituto di Economia Politica Ettore Bocconi, Via Gobbi 5, 20100, Milano - LuigiPascali@uni-bocconiit 1

3 Introduction An issue which has been largely overlooked in the literature regarding the imposition of taxation under conditions of uncertainty, is how to measure the effective tax rate on risky assets Risky assets may be viewed as bundles of contingencies Tax systems typically impose different tax rates in different circumstances For example, when loss offset provisions are imperfectly applied, the effective tax rates are higher in the positive states of nature than in the negative ones In order to compare the burden imposed by different tax systems on the same asset or the burden imposed by a given tax system on different assets, a method for aggregating tax rates across states is required In economic literature, the use of the expected tax rate is widely accepted (Alworth et al 2002; Mintz and Smart, 2002) This measure implicitly combines the state contingent tax rates using the probability of each state of nature as weight It has the advantage of being relatively easy to calculate; however, as Auerbach (1981) demonstrates, it can provide a seriously misleading estimate of the tax burden imposed on financial incomes This point is perhaps most easily recognized in the case of a tax on the excess return This tax doesn t change the investor s consumption possibilities but the expected return from this tax is positive and the expected tax rate is therefore positive Auerbach (1981) has already defined a method for calculating a risk-adjusted effective tax rate This effective tax rate has some optimal properties; however it was conceived for an economy with only two hypothetical states of nature and cannot pratically be applied to the real world This paper will describe Auerbach s approach first and will then propose to broaden some of the more restrictive aspects of his hypotheses in order to achieve the following objectives: the definition of a method for computing an effective tax rate on financial yields in order to compare the fiscal burden on different assets; the definition of a method for computing an effective tax rate on the entire financial market in order to compare the fiscal burden imposed by different tax systems The paper is divided into four sections The first one illustrates the critique of Auerbach with respect to the use of the expected tax rate, as a measure of the effective tax burden The second section describes a new methodology for computing effective tax rates on uncertain financial yields The third section provides two empirical applications: we compare the risk adjusted tax rate and the expected tax rate for a wealth tax and for a tax with limited loss offset The fourth section describes some extensions of the model Some concluding remarks will close the paper 1 The expected tax rate: a critique Taxation has two effects on the yield of financial assets: a reduction of the expected value and a reduction of volatility Expected tax rates are not able to measure the last effectandthereforearenotabletomeasuretheeffects of 2

4 Wealth in state 1 W(1+i) A W(1+x 1 ) P B W(1+i) W(1+x 2 ) Wealth in state 2 Figure 1: Portfolio possibilities where two outcomes to investment (two states of nature) taxation on the investor s consumption possibilities This point is illustrated by Auerbach (1981) The simplest contest in which one can analyze the intertemporal investment decisions of a single individual, is the two period model which was first introduced by Irving Fisher (1930) Afterwards, the model has been extensively used in public finance (Musgrave, 1959; Atkinson and Stiglitz, 1980; Sadmo, 1985; Myles, 1955) In the simplest version of this model, the individual is endowed with some amount of savings (W ) In the first period, the individual purchases financial assets In the second period, the individual totally consumes his final wealth Suppose that there are only two assets in the financial markets: a riskfree asset that yields the same positive return (i) in the two states of nature and a risky one that yields an higher return (x 2 ) in the positive state of nature and a lower return (x 1 ) in the negative one The individual is risk-averse and his wealth in the second period is a direct function of his consumption He determines what fraction of savings to invest in the risky asset in order to maximize the expected utility in the second period Figure 1 depicts the portfolio allocation chosen by the investor Points A and B indicate the combination of consumption in the two states of nature if savings are totally invested in either the safe asset or the risky one By changing the fraction of savings invested in the risky asset, the individual can reach all the combinations of consumption represented by the straight line (budget line) The points along each indifference curve represent all the combinations of 3

5 consumption that give the same expected utility to the investor Those curves are decreasing and concave The concavity is due to the risk aversion of the investor The individual chooses the portfolio allocation represented by the point where the budget line is tangent to the indifference curve Wealth in state 1 A P B B Wealth in state 2 Figure 2: Effects of a tax on excess return Figure 2 depicts the effects of a tax on that portion of the return on risky assets that exceeds the risk-free rate The combination of consumption generated by asset B shifts towards A along the original budget line, to B, on an amount equivalent to the fraction of tax on excess returns The result is an unchanged budget line The investor can react to the tax by increasing the fraction of his portfolio invested in the risky asset, until he reaches the same combination of consumption he had before Thus, a tax on excess return doesn t change the wealth of the investor However, this tax changes the expected yields on risky portfolios If m B and m B0 are the expected return pre-tax and post tax of asset B, then the Expected Tax Rate (ETR) on asset B is: ETR B m B m B 0 (1) m B Since the ETR is positive even if the tax doesn t change the investor s wealth, the expected tax rate is a misleading measure This result suggests that a more logical way of computing effective tax rates is by comparing the budget line pre-tax and after-tax instead of the expected yield pre tax and after tax 4

6 Auerbach computed a Risk-adjusted Effective Tax rate (RET) on asset B by comparing the vertical distance between the post tax budget line and the line parallel to it and passing through the original bundle B (see Figure 3): RET B z B i P z B (2) Wealth in state 1 W(1+z B ) W(1+i) W(1+i(1-t)) A A A W B B Wealth in state 2 Figure 3: Effects of an income tax The effective tax rate so computed has some optimal properties: it is nil for taxes on excess burden; it is equal to the nominal rate if the tax doesn t change the risk premium and has the same burden in all the states of nature; it is higher on those assets which have higher yields in the most taxed state of nature 2 A Risk-adjusted tax rate One of the most serious issues regarding Auerbach s approach is that it is a two states model In this section, the analysis is extended to an economy with infinite states of nature using the standard model mean-variance (Markovitz, 1952; Benninga, 2002) Suppose, as before, that there are only two assets This hypothesis is made in order to simplify the analysis but it is not crucial Asset A yields a certain return (Y A ) equal to the interest rate (i), while the asset B yields a risky return (Y B ) which is normally distributed with an expected value m B and a volatility σ 2 B 5

7 If x is the fraction of savings invested in the certain asset and (1 x) isthe fraction invested in the risky one, then the yield of the generic portfolio P will be: Y P = xy A +(1 x)y B (3) The mean (m) and the volatility (σ 2 )ofthefinancial yield of the generic portofolio P are: m = xi +(1 x)m B (4) σ 2 =(1 x) 2 σ 2 B (5) Combining the equations 4 and 5, we obtain: m = i + m B i σ (6) σ B The equation 6 represents the combinations of expected yield-risk that the individual can reach investing his savings in the financial market If the initial wealth is normalized to 1, then the equation: c =1+i + m B i σ (7) σ B represents the budget line of the investor in a world with infinite states of nature (where c is the consumption in the second period) This budget line is depicted in Figure 4 Now taxation is introduced into the model The taxation of financial yields shiftsdownwardthebudgetlinebecauseitreduces the expected yield for each level of risk Normally, taxation not only shifts downward the budget line but also reduces its slope: imperfect loss offset and other features of tax systems impose a higher fiscal burden on risky activities Taxes reduce the risk premium and consequently the slope of the budget line It is relatively straightforward to demonstrate that, also in this case, the expected tax rate does not properly measure the fiscal burden The taxation of risky assets not only reduces the expected yield but also reduces the risk of the assets Therefore, the expected yield decreases for two reasons: because of the fiscal burden and because of the reduction of risk An effective tax rate should measure only the reduction of the expected yield due to the fiscal burden The expected tax rate confuses changes in the fiscal burden with changes in risk characteristics Thus, the more accurate method for calculating an effective tax rate on uncertain yields requires translating everythingintorisk-adjustedterms 6

8 Expected consumption 1+m B B 1+i A Standard deviation of consumption Figure 4: Portfolio possibilities with infinite states of nature Consider Figure 5 The straight lines AB and A 0 B 0, different from the previous case, do not reflect the combinations of consumption available to the individual but the combination of financial yields This makes more intuitive the graphical representation of the model The point A 00 (0; z B )istheintercept of the straight line parallel to the budget line post-tax and passing through point B Inwordsz B is the certainty equivalent of the financial yield post tax computed using the risk characteristics after tax The fiscal burden on asset B can be represented by an effective tax rate calculated as follows: RET B z B i P (8) z B where i P is the post-tax interest rate It is straightforward to demonstrate that: z B = m B m B,p i p σ B (9) σ B,p where m B,p and σ B,p are the expected value and the standard deviation of the post-tax yield of asset B So the Risk-adjusted Effective Tax rate is: RET B = m B m B,p i p σ B i P σ B,p m B m (10) B,p i p σ B σ B,p 7

9 Expected consumption B z B A i A i P A B Standard deviation of consumption Figure 5: Effects of an income tax The Risk-adjusted Effective Tax rate, so computed, has some optimal properties: It is nil for a tax on excess return because this kind of tax has no effect on the budget line It is equal to the nominal tax rate for a proportional tax Since a proportional tax does not change the risk premium, the RET is simply equal to the relative variation of the certainty equivalent of the investment yield (in other words, to RET is equal to the relative difference between the risk free rate of return post tax and pre tax) It is lower than the nominal tax rate when taxation reduces the expected value of risky revenues more than their volatility 3 The Risk-Adjusted Tax Rate: two applications 31 The generalized cash flow taxation Under a realization based tax system, realization of positive capital gain is discouraged because of the advantage of deferring, without interest, the taxation of gains Vicrey (1939) proposed to capitalize virtual tax payments on accrued capital gains The problem with this solution is that it requires the path prices 8

10 will take, to be known Auerbach (1991; 2001) demonstrates that the deferral advantage could be offset if the tax liability at realization is: µ s 1+i(1 t) T s = 1 A s (11) 1+i where A s is the value of the asset at date s and t istheincometaxrate This tax scheme is easy to apply because it does not require any information on the actual path of the asset s value An investor will be indifferent ex ante between this tax and a tax based on accruals with a constant fiscal rate t Ifhe can offset changes in tax rates on the excess return, with portfolio shifts between safe and risky assets, he will not be simply indifferent ex ante but also ex post Thus, the effective tax rate on a tax described by equation 11 should be equal to the effective tax rate on a proportional tax based on accruals with nominal tax rate t Both the expected tax rate and the risk adjusted tax rate on a proportional tax based on accruals are equal to the nominal rate Thus, both the effective tax rates should be equal to the nominal rate for the tax described by equation 11 The problem is that, while this is true for the risk adjusted tax rate, it is not for the expected tax rate In order to demonstrate this, we have computed the expected tax rate and therisk-adjustedtaxrateforthetaxscheme represented by the equation 11, on a financial investment that generates no cash flow or tax liabilities until it is sold The investment is realized after 5 years; its price changes following an Ito s process (equations 12, 13 and 14) with different values of mean δ and standard deviation σ P t = δp t dt + σp t ξ dt (12) ξ N(0; 1) (13) The yield before and after tax are: P 0 = 1 (14) Y 5 = P 1 1+i(1 t) + P 2 [1 + i(1 t)] P 5 [1 + i(1 t)] 5 (15) Y5 P P 1 = 1+i(1 t) + P 2 [1 + i(1 t)] P 5 T 5 [1 + i(1 t)] 5 (16) " µ # 5 1+i(1 t) T 5 = 1 P 5 (17) 1+i where Y 5 is the yield pre tax and Y P 5 is the yield post tax The results for the risk adjusted tax rate and for the expected tax rate, computed on the tax scheme described by equation 17, are depicted in Figure 9

11 6 While the risk adjusted tax rate equals the nominal tax rate, the expected tax rate depends on the drift of prices The tax is similar to a wealth tax: it depends on the amount invested and not on the economic results Therefore, the higher the expected yield, the lower the expected tax rate Expected Tax Rate and Risk adjusted Tax Rate: a comparison 0,5 0,4 0,3 0,2 0, ,05 0,1 0,15 0,2 Expected yield Expe cte d Tax Rate Risk Adjusted Tax Rate Figure 6: A comparison between the Risk-adjusted Tax Rate and the Expected Tax Rate in evaluating the fiscal burden on financial investments with different values of expected yield The investments generate no cash flow until they are sold (after 5 years) It is applied the tax scheme proposed by Auerbach (with a nominal tax rate of 40%) 32 Limited loss offset As already mentioned, if losses are not completely deductible, taxation of financial yields not only reduces the expected yield for each level of risk but also changes the risk characteristics Taxation is heavier on risky assets and therefore reduces the risk premium in the financial market Also in this case, expected yield is a misleading measure because it confuses changes in risk characteristics with changes in the fiscal burden The Risk- Adjusted Tax Rate solves this problem by putting everything in risk-adjusted terms We have measured the differences between the values of the expected tax rate and the values of the risk adjusted tax rate when they are used for evaluating the fiscal burden imposed on risky assets by a tax system with limited loss 10

12 offset We have considered tax systems with different nominal tax rates (t) and different levels of tax deductibility (α) and assets that yield returns normally distributedwithdifferent levels of expected value (δ) andvariance(σ 2 ) If we indicate with Y the return before tax and with Y P the return post tax then: Therefore: Y P =(1 t)y I(Y 0) + (1 at)y [1 I(Y 0)] (18) Z + E(Y P )=(1 t) Y 0 Z + E(YP 2 )=(1 t) 2 Y πσ 2 1 2πσ 2 Z (Y δ)2 0 e 2σ 2 dy +(1 at) Y 1 2πσ 2 Z (Y δ)2 0 e 2σ 2 dy +(1 at) 2 Y 2 (Y δ)2 e 2σ 2 dy 1 2πσ 2 (19) (Y δ)2 e 2σ 2 dy (20) V (Y P )=E(YP 2 ) [E(Y )] 2 (21) where E( ) andv ( ) indicate the operators expected value and variance and I(Y ) is a dummy variable that assumes the value 1 when Y 0and0when Y<0 Thebasecasecontemplatesataxsystem with a nominal tax rate of 25% and a percentage of loss offset of 50% and a yield normally distributed with expected value 4% and standard deviation 6% After we have computed the expected value and the standard deviation of the post tax return, we use the formulas 1 and 10 for computing the expected tax rate and the risk adjusted-tax rate Figures 7, 8 and 9 plot the spread between the two measures As expected the spread between the Risk-adjusted Tax Rate and the Expected Tax Rate is nil when there is completely loss offset In this case, the tax burden is equal, in each state of nature, to the nominal tax rate: therefore both the effective tax rates are equal to the nominal one The spread is also almost nil for high values of expected yield and for low values of yield s standard deviation In these cases, the probability in incurring losses is not relevant and therefore the limited loss offset has no relevant effects 11

13 Effective tax rates for different levels of loss offset 10,00% Ret-Etr 7,50% 5,00% 2,50% t=25% t=30% t=40% 0,00% 0 0,2 0,4 0,6 0,8 1 Levels of loss offset Figure 7: Spread between the Risk-adjusted Tax Rate and the Expected Tax Rate in evaluating the fiscal burden on a financial yield normally distributed with expected value 4% and standard deviation 6% We consider tax systems with different levels of loss offsets and different nominal rates Effective tax rates for different levels of expected yield 10,00% Ret-Etr 7,50% 5,00% 2,50% t=25% t=30% t=40% 0,00% 0% 5% 10% 15% 20% Expected yield Figure 8: Spread between the Risk-adjusted Tax Rate and the Expected Tax Rate in evaluating the fiscal burden on a financial yield We suppose that the yield is normally distributed with standard deviation 6% and different values of expected value Moreover we consider tax systems with different nominal rates and loss offset percentage of 50% 12

14 Effective tax rates for different levels of standard deviation of yield 10,00% Ret-Etr 6,00% 2,00% t=25% t=30% t=40% 0% 5% 10% 15% 20% 25% -2,00% Standard deviation of yield Figure 9: Spread between the Risk-adjusted Tax Rate and the Expected Tax Rate in evaluating the fiscal burden on a financial yield We suppose that the yield is normally distributed with expected value 4% and different values of standard deviation Moreover we consider tax systems with different nominal rates and loss offset percentage of 50% 4 An extension of the model In this section the analysis is extended in order to take into consideration that different assets of varying degrees of risk are negotiated in the financial market The effective tax rate, that will be defined, can be used for measuring the fiscal burden imposed on the entire financial market A simple context for extending our analysis is the portfolio selection model (Markovitz, 1952 and 1959; Merton, 1973) Consider a financial market with n risky assets Call m R n the column vector of the expected yields of each risky asset, x R n the column vector of the fractions of saving invested in each asset, V R n n the matrix with the variances-covariances of the yields of the assets m m 1 m n ; x x 1 x n ; e 1 1 ; V σ 1,1 σ 1,2 σ 1,n σ 2,1 σ n,1 The yield of the generic portfolio P is given by the following equation: σ n,n (22) Y P = nx (x s Y s ) (23) 1 13

15 and presents expected value and variance equal to: m P = x T m (24) σ 2 P = x T Vx (25) The portfolio s set that presents the minimum variance for each level of expected yield m, is derived solving the following minimization problem The result is: min(x T x) x T m = m e T x =1 (26) where: r a m 2b m + a σ P = ± ac b 2 (27) a m T V 1 m (28) b m T V 1 e (29) c e T V 1 e (30) Equation 27 is represented in Figure 10 All the points contained in the parabola represent the combinations expected yield-risk of all the portfolios that can be made in the market The efficient portfolios are the ones represented by the upper part of the parabola Suppose that a riskless asset is introduced in the financial market The portfolios that present the higher expected yield, for each level of volatility, can be now represented by the half-line that starts from the combination expected yield-risk that represents the riskless asset and is tangent to the upper part of the parabola (Capital Market Line) This new efficient frontier is depicted in Figure 12 It is possible to demonstrate that M(σ M ; m M ), the point of tangency, represents the combination risk-expected yield ofaportfoliomadeupbytheassetsin the same proportion they circulate in the financial market The Capital Market Line will have the following equation: m = i + m M i σ (31) σ M If a tax on capital income is introduced in the model, it is likely to reduce the maximum expected yield in the market for each level of risk and to reduce 14

16 Expected yield A Standard deviation of yield Figure 10: Efficient portfolio possibilities with n risky assets Expected yield i A M Standard deviation of yield Figure 11: Efficient portfolio possibilities with n risky assets and a riskless one 15

17 Expected yield i M M i P Standard deviation of yield Figure 12: Effects of income tax the risk premium Graphically, the effects of taxation move down the Capital MarketLineandreduceitsslope(seeFigure11) If i P, m M,p, σ M,p are the post tax interest rate, the post tax expected return of portfolio M and the post tax standard deviation of portfolio M, the equation of the Post Tax Capital Market Line is: m = i P + m M,p i p σ (32) σ M,p We can define two different effects of an income tax on the financial revenues: 1 a reduction of the expected revenue due to the fiscal burden; 2 a reduction of the expected revenue due to a reduction of the risk premium A correct effective tax rate should measure only the reduction of revenues due to the fiscal burden The average tax rate is based on a simple evaluation of the reduction of the expected revenue and therefore confuses the two effects In order to evaluate the fiscal burden on the entire financial market, we concentrate on the shift of the Capital Market Line due to taxation For each levelofrisk,weproposetocomputethewedgebetweenthepretaxandthe post tax certainty equivalent, pretending that the premium for unit of risk is not affected by the tax Consider the financial revenues associated with the efficient portfolio B(m B ; σ B ), depicted in the figure 13 The pre-tax certainty equivalent of the financial yield B, computed by using the post-tax risk premium, is: 16

18 µ mm i z B = σ M µ mm,p i p σ M,p σ B + i (33) The post-tax certainty equivalent computed by using the post-tax risk premium, is obviously i P Therefore, the risk adjusted tax rate on this efficient portflio can be computed as: RET B z B i P (34) z B µ µ mm i mm,p i p σ B +(i i p ) σ M σ M,p RET B = µ µ (35) mm i mm,p i p σ B + i σ M σ M,p Equation 35 is the equivalent of equation 10 for a financial market with n risky assets The effective tax rate, computed by equation 35, has the same optimal properties of the effective tax rate computed by equation 17 Differently than before there is not a unique effective tax rate for the financial market but there is a different effective tax rate for each level of risk Obviously, if there is only one asset in the market, the yield of the portfolio M has the same expected yield and standard deviation of this asset and equation 35 became the same as equation 17 Expected consumption B z B i i P Standard deviation of consumption Figure 13: The Risk-adjusted Tax Rate 17

19 The RET so computed, can be used for comparing the effects of different tax systems on the returns to financial investors or on the opportunity cost of real investments 5 Conclusions This paper has proposed an integrated framework for measuring the fiscal burden on risky assets The expected tax rate, normally used for this purpose in the economic analysis, can be a misleading measure It confuses changes in risk characteristics with changes in the fiscal burden For this reason, we have proposed a methodology for calculating a risk-adjusted tax rate We have compared the expected tax rates and the risk-adjusted tax rates computed on different tax schemes: the differences are relevant Consider a tax system with a nominal tax rate of 40% and a percentage of loss offset of 35% 1 If the interest rate is 2%, the spread between the risk-adjusted tax rate and the expected tax rate on a yield, normally distributed with expected value 5% and standard deviation 7%, is 10,04% One restrictive hypothesis of our model is that only the mean and the variance of the financial yields are relevant for the Portfolio Selection This hypothesis has some empirical evidence but relies on the assumption that the variance of yields is a good measure of risk and that the investor displays a quadratic utility function However, this restrictive hypothesis is widely held in the financial analysis and it is crucial assumption for several well-accepted and highly regarded pricing models (CAPM, Black and Scholes Model) Its relaxation presents no logical problems and may be subject of further research 1 This value is consistent with the losses reported in 1988 by the italian companies that were not recovered under IRPEG by 1993 (Alworth and Arachi, 2001) 18

20 References: 1 Alworth Jand Arachi G (2001), The effects of taxes on corporate financing decisions: evidence from a panel of italian firms, International Tax and Public Finance, 8: Alworth J, Arachi G and Hamaui R (2002), Aiming at perfection: the italian experience adjusting capital gains taxation, mimeo 3 Atkinson, AB, Stiglitz JE (1980), Lectures on Public Economics, McGraw-Hill, New York 4 Auerbach AJ (1981), Evaluating the taxation of risky assets, Working Paper, NBER n Auerbach AJ (1991), Retrospective capital gains taxation, American Economic Review, 81: Auerbach AJ and Bradford D (2001), Generalized cash flow taxation, Working Paper, NBER n 8122 (forthcoming in the Journal of Public Economics) 7 Benninga S (2002), Financial Modeling, second edition, McGraw-Hill 8 Don Fullerton (1983), Which effective Tax Rate?, Working Paper, NBER n Fisher, I (1930), The theory of interest, MacMillan, New York 10 Gordon, RK (1985), Taxation of corporate capital income: tax revenues versus tax distorsions, Quarterly Journal of Economics, 100: Markowitz, HM (1952), Portfolio Selection Journal of Finance, 7: Markowitz, HM (1959), Portfolio Selection: Efficient Diversification of Investments, Wiley, New York 13 Merton, RC (1973), An analytic derivation of the efficient portfolio frontier, Journal of Financial and Quantitative Analysis, 7: Myles, GD (1995), Public Economics, Cambridge University Press 15 Mintz, J and Smart M (2002), Tax exempt investors and the asset allocation puzzle, Journal of Public Economics, 83: Musgrave, R (1959), The theory of Public Finance, McGraw Hill, New York 19

21 17 Sadmo, A (1985), The effects of taxation on savings and risk taking, Hanbook of Public Economics, Auerbach A J and Feldstein M, Vol 1, Chapter 5, North Holland 18 Vicrey, W (1939), Averaging income for income tax purposes Journal of Political Economy, 47: