Tax and the Present Value of Future Income

Transcription

1 Tax and the Present Value of Future Income Scott Gilbert SIUC May 11, 2012

2 Overview Impact of tax on income lost to injury or death. Simple case: constant income stream. More (FE) relevant: growing income streams. Anderson and Barber (2010) tax model. Problem in model, a solution. Main result: tax effect (+ or -) depends on expected worklife. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

3 Overview Impact of tax on income lost to injury or death. Simple case: constant income stream. More (FE) relevant: growing income streams. Anderson and Barber (2010) tax model. Problem in model, a solution. Main result: tax effect (+ or -) depends on expected worklife. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

4 Overview Impact of tax on income lost to injury or death. Simple case: constant income stream. More (FE) relevant: growing income streams. Anderson and Barber (2010) tax model. Problem in model, a solution. Main result: tax effect (+ or -) depends on expected worklife. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

5 Overview Impact of tax on income lost to injury or death. Simple case: constant income stream. More (FE) relevant: growing income streams. Anderson and Barber (2010) tax model. Problem in model, a solution. Main result: tax effect (+ or -) depends on expected worklife. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

6 Overview Impact of tax on income lost to injury or death. Simple case: constant income stream. More (FE) relevant: growing income streams. Anderson and Barber (2010) tax model. Problem in model, a solution. Main result: tax effect (+ or -) depends on expected worklife. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

7 Overview Impact of tax on income lost to injury or death. Simple case: constant income stream. More (FE) relevant: growing income streams. Anderson and Barber (2010) tax model. Problem in model, a solution. Main result: tax effect (+ or -) depends on expected worklife. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

8 Literature Forensic economics: Franz (1982, 1989) Goodwin and Paul (1988), Ciecka (1989) Journal of Forensic Economics (1994) Anderson and Barber (2010) Economics: Modigliani and Miller (1963) Samuelson (1964) Adams (1977) Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

9 Literature Forensic economics: Franz (1982, 1989) Goodwin and Paul (1988), Ciecka (1989) Journal of Forensic Economics (1994) Anderson and Barber (2010) Economics: Modigliani and Miller (1963) Samuelson (1964) Adams (1977) Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

10 Literature Forensic economics: Franz (1982, 1989) Goodwin and Paul (1988), Ciecka (1989) Journal of Forensic Economics (1994) Anderson and Barber (2010) Economics: Modigliani and Miller (1963) Samuelson (1964) Adams (1977) Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

11 Literature Forensic economics: Franz (1982, 1989) Goodwin and Paul (1988), Ciecka (1989) Journal of Forensic Economics (1994) Anderson and Barber (2010) Economics: Modigliani and Miller (1963) Samuelson (1964) Adams (1977) Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

12 Literature Forensic economics: Franz (1982, 1989) Goodwin and Paul (1988), Ciecka (1989) Journal of Forensic Economics (1994) Anderson and Barber (2010) Economics: Modigliani and Miller (1963) Samuelson (1964) Adams (1977) Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

13 Literature Forensic economics: Franz (1982, 1989) Goodwin and Paul (1988), Ciecka (1989) Journal of Forensic Economics (1994) Anderson and Barber (2010) Economics: Modigliani and Miller (1963) Samuelson (1964) Adams (1977) Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

14 Literature Forensic economics: Franz (1982, 1989) Goodwin and Paul (1988), Ciecka (1989) Journal of Forensic Economics (1994) Anderson and Barber (2010) Economics: Modigliani and Miller (1963) Samuelson (1964) Adams (1977) Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

15 Death and Taxes Situation: injury or death at time 0 Lost future incomes: E 1, E 2,... Present value: P t Interest rate: r Tax rate: τ Future Value equation: (1 + r)p t = (1 τ)e t+1 + τrp t + P t+1 (1) Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

16 Perpetuity Constant future earnings E 1 = E 2 =... Present value: P = E r (2) Damages P invariant to tax rate τ. Reason: constant price P t over time, so from FV: (1 + r)p = (1 τ)e + τrp + P (3) Economics literature: Modigliani and Miller (1963) Conclusion: no tax effect on PV of perpetual income. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

17 Perpetuity Constant future earnings E 1 = E 2 =... Present value: P = E r (2) Damages P invariant to tax rate τ. Reason: constant price P t over time, so from FV: (1 + r)p = (1 τ)e + τrp + P (3) Economics literature: Modigliani and Miller (1963) Conclusion: no tax effect on PV of perpetual income. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

18 Perpetuity Constant future earnings E 1 = E 2 =... Present value: P = E r (2) Damages P invariant to tax rate τ. Reason: constant price P t over time, so from FV: (1 + r)p = (1 τ)e + τrp + P (3) Economics literature: Modigliani and Miller (1963) Conclusion: no tax effect on PV of perpetual income. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

19 Perpetuity Constant future earnings E 1 = E 2 =... Present value: P = E r (2) Damages P invariant to tax rate τ. Reason: constant price P t over time, so from FV: (1 + r)p = (1 τ)e + τrp + P (3) Economics literature: Modigliani and Miller (1963) Conclusion: no tax effect on PV of perpetual income. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

20 Annuity Constant future earnings E 1 = E 2 =... = E N Present value: P = E r ( ) 1 1 (1 + (1 τ)r) N (4) Damages P decreasing in tax rate τ. Reason: FV and P N = 0 Economics literature: Samuelson (1964) Conclusion: negative tax effect on PV of constant income. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

21 Annuity Constant future earnings E 1 = E 2 =... = E N Present value: P = E r ( ) 1 1 (1 + (1 τ)r) N (4) Damages P decreasing in tax rate τ. Reason: FV and P N = 0 Economics literature: Samuelson (1964) Conclusion: negative tax effect on PV of constant income. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

22 Annuity Constant future earnings E 1 = E 2 =... = E N Present value: P = E r ( ) 1 1 (1 + (1 τ)r) N (4) Damages P decreasing in tax rate τ. Reason: FV and P N = 0 Economics literature: Samuelson (1964) Conclusion: negative tax effect on PV of constant income. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

23 Annuity Constant future earnings E 1 = E 2 =... = E N Present value: P = E r ( ) 1 1 (1 + (1 τ)r) N (4) Damages P decreasing in tax rate τ. Reason: FV and P N = 0 Economics literature: Samuelson (1964) Conclusion: negative tax effect on PV of constant income. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

24 Growing Income Incomes: E 1, E 2,..., E N Present value, at time 0: P 0 = N t=1 (1 τ)e t (1 + r(1 τ)) t (5) Standard PV, based on after-tax income & interest rate Tax effect unclear Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

25 Growing Income Incomes: E 1, E 2,..., E N Present value, at time 0: P 0 = N t=1 (1 τ)e t (1 + r(1 τ)) t (5) Standard PV, based on after-tax income & interest rate Tax effect unclear Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

26 Anderson and Barber (2010) Constant income growth: E 0 = base income g = growth rate Present value: E t = E 0 (1 + g) t (6) [ PV earningsaftertax = E 0(1 τ e ( ) ] )(1 + g) 1 + g N (r(1 τ i 1 ) g) 1 + r(1 τ i (7) ) [ PV earningswithouttax = E ( ) ] 0(1 + g) 1 + g N 1 (8) (r g) 1 + r Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

27 Mixed Effects of Tax Anderson and Barber breakeven point : where D is the constant: τ e = τ i rd (1 + r) (9) (1 + r) N D = (r g) + [ 1 For small N, LH > RH and tax effect < 0 For large N, LH < RH and tax effect > 0 ( ) N 1+g 1+r ( 1+g 1+r ) N ] (10) Conclusion: tax lowers economics damage if worklife short, otherwise raises it. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

28 Mixed Effects of Tax Anderson and Barber breakeven point : where D is the constant: τ e = τ i rd (1 + r) (9) (1 + r) N D = (r g) + [ 1 For small N, LH > RH and tax effect < 0 For large N, LH < RH and tax effect > 0 ( ) N 1+g 1+r ( 1+g 1+r ) N ] (10) Conclusion: tax lowers economics damage if worklife short, otherwise raises it. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

29 Mixed Effects of Tax Anderson and Barber breakeven point : where D is the constant: τ e = τ i rd (1 + r) (9) (1 + r) N D = (r g) + [ 1 For small N, LH > RH and tax effect < 0 For large N, LH < RH and tax effect > 0 ( ) N 1+g 1+r ( 1+g 1+r ) N ] (10) Conclusion: tax lowers economics damage if worklife short, otherwise raises it. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

30 Example τ e = τ i = 0.1 g = r = 0.05 and: PV earningswithouttax = N t=1 E 0 (1 + g) i (1 + r) i (11) D = Breakeven condition: D = 21, N = 40. Tax lowers PV if N < 40, raises it if N > 40. Or does it? N te 0 (1+g) t t=1 (1+r) t N (12) E 0 (1+g) t t=1 (1+r) t Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

31 Example τ e = τ i = 0.1 g = r = 0.05 and: PV earningswithouttax = N t=1 E 0 (1 + g) i (1 + r) i (11) D = Breakeven condition: D = 21, N = 40. Tax lowers PV if N < 40, raises it if N > 40. Or does it? N te 0 (1+g) t t=1 (1+r) t N (12) E 0 (1+g) t t=1 (1+r) t Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

32 Example, details Table: Present values, Tax Rate = 10 percent PV PV N without tax with tax Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

33 Other Examples Table: Present values, Tax Rate = 1 percent PV PV N without tax with tax Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

34 Table: Present values, Tax Rate = 50 percent PV PV N without tax with tax Scott Gilbert (SIUC) 53 Tax and the Present 53 Value of Future Income May 11, / 23

35 Breakeven Possibilities Table: Breakeven Possibilities tax rate breakeven point Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

36 Forensics Macaulay s duration (Maucaulay 1938, Hicks 1939): a weighted average of income arrival dates 1, 2,..., N. For a bond, coupon I, face value F : Generally: D = I R + 2I + + NI R 2 R N I R + I + + I R 2 R N + NF R N + F R N (13) D = N t=1 t PV t PV (14) PV t = E t (1 + r) t (15) Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

37 Macaulay s duration is then: D = N te t t=1 (1+r) t N (16) E t t=1 (1+r) t Hick s interpretation: write PV as: PV = with R = gross interest rate. N R t E t (17) t=1 D is elasticity of PV with respect to R: D = PV R R PV (18) Anderson & Barber s D is negative, Macaulay s duration is positive. Doesn t resolve the modelling problem. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

38 Macaulay s duration is then: D = N te t t=1 (1+r) t N (16) E t t=1 (1+r) t Hick s interpretation: write PV as: PV = with R = gross interest rate. N R t E t (17) t=1 D is elasticity of PV with respect to R: D = PV R R PV (18) Anderson & Barber s D is negative, Macaulay s duration is positive. Doesn t resolve the modelling problem. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

39 More Forensics If the negative effect of income tax just offsets the positive effect of interest tax on present value then, at the margin, the change dpv in present value equals 0: dpv = PV τ e dτ e + PV τ i dτ i = 0 (19) Suppose, moreover, that the earnings tax rate τ e grows at the same instantaneous rate as does the tax rate τ i on interest: dτ e τ e = dτ i τ i (20) Then we can then interpret the marginal condition (19) on present value in terms of elasticities: PV τ e τ e PV = PV τ i τ i PV (21) Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

40 The elasticity of (after-tax) PV with respect to the earnings tax rate is: PV τ e τ e τ e PV = 1 τ e (22) For a small tax rate τ e on earnings, we can approximate this elasticity as follows: PV τ e τ e PV τ e (23) Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

41 Elasticity of PV with respect to the interest tax rate: PV τ i τ i PV = PV (1 + r(1 τ i )) (1 + r(1 τ i )) τ i rτ i PV = 1 + r(1 τ i ) (1 + r(1 τ i )) τ i PV 1 + r(1 τ i ) PV (24) (25) = rτ i 1 + r(1 τ i ) D aftertax (26) with D aftertax the variant of Macaulay-Hicks duration D based on after-tax earnings and interest: D aftertax = N t=1 N t=1 te t (1+(1 τ i )r) t (27) E t (1+(1 τ i )r) t Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

42 If the tax rate τ i is close to zero then after-tax duration is about the same as before-tax duration: D aftertax D (28) If tax rates τ i and τ e are both small then, applying the small-tax approximations (23) and (28), the balancing condition (19) is approximately: Interpretation: τ e τ i r 1 + r D (29) Anderson and Barber breakeven condition (9) restated as a small-tax approximation. A minus (-) sign appears on the right-hand side of (9) but not (29). This difference in sign reflects the difference between the Anderson and Barber constant D and Macaulay-Hicks duration D. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

43 If the tax rate τ i is close to zero then after-tax duration is about the same as before-tax duration: D aftertax D (28) If tax rates τ i and τ e are both small then, applying the small-tax approximations (23) and (28), the balancing condition (19) is approximately: Interpretation: τ e τ i r 1 + r D (29) Anderson and Barber breakeven condition (9) restated as a small-tax approximation. A minus (-) sign appears on the right-hand side of (9) but not (29). This difference in sign reflects the difference between the Anderson and Barber constant D and Macaulay-Hicks duration D. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

44 Small-Tax Theory τ PV earningsaftertax = N t=1 (1 + g) t N (1 + r(1 τ)) t + tr(1 τ)(1 + g) t (1 + r(1 τ)) t+1 (30) t=1 Evaluating the derivative at τ = 0, and denoting the result as PV, yields: PV = N ( ) ( ) r 1 + g t t 1 + r 1 (31) 1 + r t=1 Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

45 PV = = r 1 + r N ta t t=1 r a 1 + r 1 a N a t (32) t=1 ( 1 a N 1 a NaN ) a(1 an ) 1 a (33) where: a = 1 + g 1 + r (34) Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23

46 Proposition: Suppose that an earnings stream grows at a constant positive rate, and that earnings and interest are taxed at equal rates. Let U be any finite upper bound for the earnings horizon N, such that U > N with N the closest integer approximation to the breakeven condition. Then, for all tax rates sufficiently small, the present value of the earnings stream is lower after-tax than without tax when horizon N is less than N, but is higher after tax when N is greater than N yet less than U. Scott Gilbert (SIUC) Tax and the Present Value of Future Income May 11, / 23