enterprise value
Thomas Braun
Department of
Business Administration and Economics Bielefeld University
P.O. Box 10 01 31 D-33501 Bielefeld
Germany
tbraun@wiwi.uni-bielefeld.de
October 2005
Discussion Paper No. 544
Abstract
This paper derives and draws on simple formulae for the upper and lower bounds to the value of a series of risky cash flows in order to provide some instructive insights in the impact of taxation on these bounds.
The formulae are based on no-arbitrage conditions in a setting that is a straightforward extension of the C OX , R OSS , AND R UBINSTEIN [2] option-pricing model to an incom- plete market model and look exactly like the popular G ORDON growth formula.
Although based on stylized facts concerning the tax scheme the results promise to be a reliable guide for further research in this field.
Keywords: arbitrage theory, incomplete markets, taxes, enterprise value JEL Classification: G12
∗
I thank Ariane Reiß and Christoph Wöster for valuable comments. The usual disclaimer applies.
1 Introduction
It is really astonishing that so many financial analysts hang on to the practice of pinning down the value of a business sharply to one point. They should know for better that this practice requires undue restrictions as for example assuming complete markets or that the T OBIN -separation holds although it is an empirical fact that is does not. 1
This paper avoids the above mentioned problem adding the least possible complexity by reference to an incomplete market model that is a straightforward extension of the C OX , R OSS , AND R UBINSTEIN [2] option-pricing model. Within this setting the upper and lower bounds to the enterprise value implied by no-arbitrage conditions simply focus on the best- case and the worst-case scenario, respectively, and convexity becomes a crucial determinant for the impact of taxation. To be more precise: What counts is how the enterprise cash flows behave relative to the price of some exchange-traded reference asset in the best and in the worst case, respectively. If in principal it makes no difference as to this behavior whether you are in the best or the worst of all worlds, that is to say if the payoff characteristic is convex or concave in any of these worlds, the upper and lower bound always move to the same direction as long as you switch between symmetric 2 tax schemes. This does not hold however for asymmetric tax schemes.
2 The one-period case
Let Ω = {ω 1 , ω 2 , ω 3 , ω 4 } be the probability set and
C t
1(ω) =
g u t
0·C t
0if ω = ω 1 b u t
0·C t
0if ω = ω 2
g d t
0·C t
0if ω = ω 3 b d t
0·C t
0if ω = ω 4
(1)
1
This is really bad because you can miss the value to a specific investor by far if this investor’s portfolio is significantly different from the reference portfolio your calculations are based on. As a remedy W
ILHELM[5] has proposed to value uncertain cash-flows as good as possible by replication with traded assets so that only the residuum remains susceptible to undue restrictions of investor-specific preferences and endowments.
2
This means that short positions and long positions induce exactly the same tax payments in absolute terms.
with (‘b’ stands for ‘bad’ and ‘g’ stands for ‘good’ )
b m t
0≤ g t m
0(m ∈ {d, u}) (2)
be the payoff characteristic of the enterprise to be valued. Further let
S t
1(ω) = u t
0· S t
0if ω ∈ {ω 1 , ω 2 } S t
0(ω) = S t
0for ω ∈ Ω
S t
1(ω) = d t
0· S t
0if ω ∈ {ω 3 , ω 4 }
be the price movement of some exchange traded risky asset which together with some money market fund certificate (MMF) with market prices
B t
0, B t
1: = B t
0(1 + r t
0) (3)
are the only exchange-traded instruments the enterprise value shall refer to. 3
2.1 Lower Bound
From C OX , R OSS , AND R UBINSTEIN [2] it is well known that portfolio
x t
0: =
x S t
0x B t
0
=
b
ut0−b
dt0u
t0−d
t0C
t0S
t0−b
ut0
d
t0+b
dt0
u
t0u
t0−d
t0C
t0B
t1
=
1 0 0 d
t0u
t0B
−1t0B
t1
b
ut0−b
td0u
t0−d
t0C
t0S
t0 bdt0dt0
−
butut0u
t0−d
t00C
t0B
t0
3
This confinement is for the sake of simplicity only.
comprising a number of x S t
0risky assets and x t B
0money market fund certificates (MMF) gen- erates cash flows
P t
1(x t
0) =
b u t
0·C t
0if ω ∈ {ω 1 , ω 2 }
b d t
0·C t
0if ω ∈ {ω 3 , ω 4 } . (4)
Regarding the definitions
q t
0: = 1 + r t
0− d t
0u t
0− d t
0(5)
b ¯ t
0: = b t u
0· q t
0+ b t d
0· (1 − q t
0) − 1 (6)
and
¯
g t
0: = g t u
0· q t
0+ g t d
0· (1 − q t
0) − 1 (7)
the market price of this portfolio at t 0 might be written as
P t
0(x t
0) =
à b t u
0− b t d
0u t
0− d t
0+ 1 B −1 t
0B t
1−b t u
0d t
0+ b d t
0u t
0u t
0− d t
0! C t
0= 1
B t −1
0B t
1Ã
b u t
0B −1 t
0B t
1− d t
0u t
0− d t
0+ b d t
0u t
0− B −1 t
0B t
1u t
0− d t
0! C t
0= 1
B t −1
0B t
1³
b u t
0· q t
0+ b d t
0· (1 − q t
0)
´ C t
0= 1 + b ¯ t
01 + r t
0C t
0.
From (2) and (4) it follows that buying the enterprise i.e. cash flow C t
1at price Π t
0(C t
1) while at the same time short selling portfolio x t
0will payoff
C t
1(ω) + P t
1(−x t
0)(ω) ≥ 0 for each ω ∈ Ω
at time t 1 and thus would be an arbitrage opportunity if
Π t
0(C t
1) + P t
0(−x t
0) ≤ 0 .
Hence in the absence of any impediment to trade such as transaction costs or taxes the enter- prise must have a price
Π t
0(C t
1) > −P t
0(−x t
0) = P t
0(x t
0)
in order to prevent arbitrage.
Nevertheless there might be other arbitrage opportunities. To preclude any such arbitrage op- portunity Π t
0(C t
1) must be higher than the most expensive portfolio with a cash flow that is weakly dominated by payoff characteristic (1). In what follows we will use linear program- ming to show that portfolio x t
0is the most expensive weakly dominated portfolio indeed, and thus the lower bound to the enterprise value is
V t
0(C t
1) = P t
0(x t
0) = 1 + b ¯ t
01 + r t
0C t
0. Taking the MMF certificates as a numeraire the objective is
z P := x B t
0+ x S t
0· B −1 t
0· S t
0→ max
x
Bt0,x
St0∈ !
and the constraints for this portfolio to be weakly dominated by (1) are
x B t
0+ x S t
0· u t
0· B −1 t
1· S t
0≤ g u t
0· B −1 t
1·C t
0x B t
0+ x S t
0· u t
0· B −1 t
1· S t
0≤ b u t
0· B −1 t
1·C t
0x B t
0+ x S t
0· d t
0· B −1 t
1· S t
0≤ g d t
0· B −1 t
1·C t
0x B t
0+ x S t
0· d t
0· B −1 t
1· S t
0≤ b d t
0· B −1 t
1·C t
0.
Duality theory says that if there is a solution to the above program there is also one for the following program
z D := B −1 t
1C t
0³
g t u
0· q(ω 1 ) + b u t
0· q(ω 2 ) + g t d
0· q(ω 3 ) + b d t
0· q(ω 4 )
´
→ min
q(ω
1),...,q(ω
4)≥0 ! (8)
s.t.
q(ω 1 ) + q(ω 2 ) + q(ω 3 ) + q(ω 4 ) = 1 B t −1
1· [S t
0· u t
0· (q(ω 1 ) + q(ω 2 )) + S t
0· d t
0· (q(ω 3 ) + q(ω 4 ))] = B t −1
0· S t
0with exactly the same objective value. In combination with (3) and (5) the two constraints are equivalent to
q(ω 1 ) + q(ω 2 ) = q t
0q(ω 3 ) + q(ω 4 ) = 1 − q t
0so that (8) simplifies to
B t −1
1C t
0³
(g u t
0− b u t
0) · q(ω 1 ) + b t u
0· q t
0+ (g t d
0− b t d
0) · q(ω 3 ) + b d t
0· (1 − q t
0) ´
→ min
q(ω
1),q(ω
3)≥0 ! Regarding that (2) implies
0 ≤ g m t
0− b t m
0(m ∈ {d, u}) (9)
it becomes obvious that both objective functions have optimal value
z ∗ P = z ∗ D = (1 + b ¯ t
0)B t −1
1C t
0= B −1 t
0P t
0(x t
0)
and that x t
0is the most expensive weakly dominated portfolio indeed. Moreover it shows that ¯ b t
0and ¯ g t
0might be interpreted as expected worst-case and best-case growth rates under measure S designed to make the price of the exchange-traded risky asset measured in units of the numeraire a martingale. They will be referred to as pseudo growth rates
¯
c t
0:= E S ¡
C t −1
0C t c
1|F t
0¢
− 1 (c ∈ {b, g}) .
in what follows. 4
2.2 Upper Bound
The line of reasoning that leads to the upper bound rests on difference arbitrage: If portfo- lio y t
0with payoff P t
1(y t
0) at time t 1 weakly dominates the cash flow C t
1generated by the enterprise at time t 1 that is if
P t
1(y t
0)(ω) ≥ C t
1(ω) for each ω ∈ Ω,
a potential buyer would rather buy this portfolio than the enterprise if at time t 0 it would cost no more than the enterprise that is if
P t
0(y t
0) ≤ Π t
0(C t
1) .
Assume that y t
0is the cheapest dominating portfolio then its price P t
0(y t
0) marks the upper bound V t
0(C t
1) to the price of a business with payoff C t
1.
Proceeding exactly as above leads to the conclusion that y t
0is the portfolio with cash flows
C t
1(ω) =
g u t
0·C t
0if ω ∈ {ω 1 ,ω 2 }
g d t
0·C t
0if ω ∈ {ω 3 ,ω 4 } (10)
4
One obvious implication of our central result is that if the lower of the two possible cash flows generated by the enterprise contingent on the asset price and the asset price itself are perfectly correlated, that is if
b
tu0u
t0= b
dt0d
t0, we get
x
t0=
Ã
bdt0Ct0dt0St0
0
!
and the lower bounds simplifies to
V
t0(C
t1) = b
dt0C
t0d
t0S
t0S
t0= b
dt0d
t0C
t0.
and composition
y t
0: =
y S t
0y B t
0
=
g
ut0−g
dt0u
t0−d
t0C
t0S
t0−g
ut0d
t0+g
td0u
t0u
t0−d
t0C
t0B
t1
=
1 0 0 d
t0u
t0B
−1t0B
t1
g
tu0−g
dt0u
t0−d
t0C
t0S
t0 gdt0dt0
−
gutut0u
t0−d
t00C
t0B
t0
.
From this we get
V t
0(C t
1) = P t
0(y t
0) = 1 + g ¯ t
01 + r t
0C t
0for the upper bound to the enterprise value.
Summary 2.1 The upper and lower bounds may be derived by referring to the best- and worst-case scenarios and valuing them as if the market were complete.
3 The multi-period case
The n-period case is a straightforward extension of the one-period case with
ω ∈ Ω := {ω 1 , ω 2 , ω 3 , ω 4 } n and the portfolios x t
0, y t
0being replaced by trading strategies (x t
i) n−1 i=0 , (y t
i) n−1 i=0 that have to be determined recursively as follows: Let
P t b,m
i: = P t
i(x t
i(ω))(ω)
= x t S
i(ω) · S t
i(ω) + x B t
i(ω) · B t
ifor ω ∈ S t −1
i(m · S t
i−1) := ©
ω |S t
i(ω) = m · S t
i−1ª
be a short cut for the price of portfolio x t
iyou need at time t i in order to replicate C t b
n(ω) at
time t n depending on the price move m of the exchange-traded risky asset from time t i−1 to
t i . By analogy to the one-period case we know that
x t
n−1: =
x t S
n−1x t B
n−1
=
b
tn−1u−b
tn−1du
tn−1−d
tn−1C
tn−1bS
tn−1−b
tn−1ud
tn−1+b
dtn−1u
tn−1u
tn−1−d
tn−1C
tn−1bB
tn
=
1 0 0 d
tn−1u
tn−1B
tn−1−1B
tn
b
utn−1−b
dtn−1u
tn−1−d
tn−1C
tn−1bS
tn−1bdtn−1 dtn−1
−
butn−1utn−1
u
tn−1−d
tn−1C
tn−1bB
tn−1
(11)
has market price
P t
n−1(x t
n−1) =
à b u t
n−1− b t d
n−1u t
n−1− d t
n−1+ 1 B t −1
n−1B t
n−b t u
n−1d t
n−1+ b t d
n−1u t
n−1u t
n−1− d t
n−1! C t b
n−1= 1 + b ¯ t
n−11 + r t
n−1C t b
n−1at time t n−1 . Hence, if the short term interest rate evolves deterministically the numbers of risky assets and MMF certificates needed at time t n−2 are
x t S
n−2= P t b,u
n−1− P t b,d
n−1(u t
n−2− d t
n−2) · S t
n−2= 1 + b ¯ t
n−11 + r t
n−1b t u
n−2− b d t
n−2u t
n−2− d t
n−2C t b
n−2S t
n−2and
x t B
n−2= P t b,m
n−1− x S t
n−2· m · S t
n−2B t
n−1= 1 + b ¯ t
n−11 + r t
n−1C t b
n−2B t
n−1Ã
b m t
n−2− m · b u t
n−2− b t d
n−2u t
n−2− d t
n−2!
= 1 + b ¯ t
n−11 + r t
n−1d t
n−2u t
n−2B t −1
n−2B t
n−1b
tn−2dd
tn−2− b
tn−2u
u
tn−2u t
n−2− d t
n−2C t b
n−2B t
n−2(m ∈ {d, u})
or equivalently
x t
n−2= 1 + b ¯ t
n−11 + r t
n−1
b
utn−2−b
dtn−2u
tn−2−d
tn−2C
tn−2bS
tn−2d
tn−2u
tn−21+r
tn−2bdtn−2 dtn−2
−
butn−2utn−2
u
tn−2−d
tn−2C
tn−2bB
tn−2
.
Further recursion leads to
x t
n−i=
à n−1 j=n−i+1 ∏
1 + b ¯ t
j1 + r t
j!
b
utn−i−b
tn−idu
tn−i−d
tn−iC
tn−ibS
tn−id
tn−iu
tn−i1+r
tn−ibdtn−i dtn−i
−
butn−iutn−i
u
tn−i−d
tn−iC
tn−ibB
tn−i
(12)
and
P t
n−i(x t
n−i) =
n−1 ∏
j=n−i+1
1 + b ¯ t
j1 + r t
jà b t u
n−i− b t d
n−iu t
n−i− d t
n−i+ 1 B −1 t
n−iB t
n−i+1−b u t
n−id t
n−i+ b t d
n−iu t
n−iu t
n−i− d t
n−i! C t b
n−i=
n−1 ∏
j=n−i
1 + b ¯ t
j1 + r t
jC t b
n−i.
Rearranging terms according to
B t −1
n−iP t
n−i(x t
n−i) = B t −1
n−iB t −1
n−iB t
n−i+1n−1 ∏
j=n−i+1
1 + b ¯ t
jB t −1
jB t
j+1³
b u t
n−i· q t
n−i+ b d t
n−i· (1 − q t
n−i)
´ C t
n−i= B −1 t
n−i+1³
P t b,u
n−i+1· q t
n−i+ P t b,d
n−i+1· (1 − q t
n−i)
´
reveals that the market price of portfolio x t
n−iis a martingale if measured in units of MMF certificates. Thus, rebalancing this portfolio from time to time must be self financing. 5 Hence, regarding the identity
C t b
0≡ C t
0we finally arrive at
V t
0(C t
k) =
k−1 ∏
j=0
1 + b ¯ t
j1 + f t
jC t
05
Cf. H
ARRISON ANDK
REPS[4].
and
V t
0(C t
k) =
k−1 ∏
j=0
1 + g ¯ t
j1 + f t
jC t
0,
respectively, by substituting the short rates for future periods with the respective forward rates
f t
i:= f (t 0 ,t i ,t i+1 ) = r(t i ,t i+1 ) =: r t
iimplied by the term structure of interest rate in t 0 . 6
If predictability beyond time t i is limited so that it does not make sense to be too specific about the scenario from that time on the following assumptions seem adequate
b t m
j= b m for all j ≥ i
and
u t
j· d t
j= 1 for all j ≥ i .
The growth factors u t
jand d t
jthen can be traced back to the implied volatility σ of the exchange-traded asset according to
u t
j= e σ for all j ≥ i .
Adding the assumption of a constant spot rate r that is
B −1 t
jB t
j+1= 1 + r for all j ≥ i
6
It is well known since C
OX, I
NGERSOLL,
ANDR
OSS[1] that future spot rates must be equal to implied
forward rates to preclude arbitrage if the interest rate evolves deterministically.
we get
b ¯ t
j= b ¯ : = b d + (b u − b d ) · q − 1
= b d − 1 + (b u − b d ) · (1 + r) · e σ − 1
e 2σ − 1 for all j ≥ i + 1 and thus for any given i , 1 ≤ i ≤ k
V t
0(C t
k) =
i−1 ∏
h=0
1 + b ¯ t
h1 + f t
hµ 1 + b ¯ 1 + r
¶ k−i C t
0and
V t
0(C t
k) =
i−1 ∏
h=0
1 + g ¯ t
h1 + f t
hµ 1 + g ¯ 1 + r
¶ k−i C t
0.
Given the definition
θ(x, y) : = 1 + x 1 + y
and ¯ b , r the upper and lower bounds to the value of a series of cash flows (C t
i) k i=1 at times t 1 , . . . ,t k are 7
V t
0³
(C t
i) k i=1
´
= Ã i−1
h=0 ∑
∏ h g=0
θ ¡ b ¯ t
g, f t
g¢ +
i−1 ∏
h=0
θ ¡ b ¯ t
h, f t
h¢ 1 + b ¯ r − b ¯
³
1 − θ( b,r) ¯ k−i
´ ! C t
0and
V t
0³
(C t
i) k i=1
´
= Ã i−1
h=0 ∑
∏ h g=0
θ ¡
¯ g t
g, f t
g¢ +
∏ i−1 h=0
θ ( g ¯ t
h, f t
h) 1 + g ¯ r − g ¯
³
1 − θ( g, ¯ r) k−i
´ ! C t
0.
7
It is well known that
C
t0·
∑
n i=1θ
i= C
t0· θ
−1− 1 θ
−1− 1 ·
∑
n i=1θ
i= C
t0· θ
−1· ∑
ni=1
θ
i− ∑
ni=1
θ
iθ
−1− 1 = C
t0·
n−1
∑
i=0
θ
i− ∑
ni=1
θ
iθ
−1−1 = C
t0· 1 − θ
nθ
−1− 1 .
For f t
g= r and ¯ c t
g= c ¯ (g = 0, . . . , k − 1 ; c ∈ {b, g}) the above formulae simplify to
V t
0³
(C t
i) k i=1
´
= 1 + b ¯ r − b ¯
³
1 − θ( b, ¯ r) k
´ C t
0and
V t
0³
(C t
i) k i=1
´
= 1 + g ¯ r − g ¯
³
1 − θ( g, ¯ r) k
´ C t
0.
For
¯
c < r ⇔ c u − c d
u − d < 1 + r − c d
1 + r − d (c ∈ {b, g}) (13) the limits for an infinite series of cash flows are
k→∞ lim V t
0³
(C t
i) k i=1
´
= 1 + b ¯ r − b ¯
µ
1 − lim
k→∞ θ( b, ¯ r) k
¶ C t
0= 1 + b ¯ r − b ¯ C t
0and
k→∞ lim V t
0³
(C t
i) k i=1
´
= 1 + g ¯ r − g ¯ C t
0.
The right hand sides of the above formulae perfectly resemble the structure of the so called G ORDON 8 growth formula.
4 The impact of taxes
We will analyze the impact of three tax schemes T S ∈ {GI, I, I + } using stylized facts to gain some useful insights without adding too much complexity. The tax rates t t
iapplicable at time t i are the same for any kind of income. 9 Tax schemes GI and I require taxes on realized and unrealized gains with MMF certificates to be paid immediately and grant immediate
8
Cf. G
ORDON[3].
9
So they also apply to the cash flows C
tipayments on realized and unrealized losses with MMF certificates (‘I’ stands for ‘Interest’).
Tax scheme GI treats realized and unrealized gains and losses with stocks (‘G’ stands for
‘Capital Gain’) the same way as tax schemes GI and I treat gains and losses with MMF certificates. Tax schemes GI and I are symmetric in the sense that short positions and long positions induce exactly the same tax payments in absolute terms. Thus, given the definition
x T S t
i: =
x t S,T S
ix B,T S t
i
the fundamental equation
V t T S
0= −P t
0(−x T S t
0) = P t
0(x T S t
0) (T S ∈ {GI, I})
still holds. This is not true for tax scheme I + that is inspired by non deductability of interest paid and therefore subjects only long positions in MMF certificates to taxation. Given the definition
r t tax
i: = (1 − t t
i+1)r t
i= (1 − t t
i+1) f t
i=: f t tax
iby analogy to a world without taxes we get
x S,GI t
n−1= (1 − t t
n)(b t u
n−1− b d t
n−1)
u t
n−1− (u t
n−1− 1)t n − (d t
n−1− (d t
n−1− 1)t n ) C t b
n−1S t
n−1= b u t
n−1− b t d
n−1u t
n−1− d t
n−1C t b
n−1S t
n−1= x S t
n−1(14)
and
x S,I t
n−1= (1 − t t
n)(b t u
n−1− b d t
n−1) u t
n−1− d t
n−1C t b
n−1S t
n−1= (1 − t t
n)x S t
n−1(15)
for the number of risky assets needed at time t n−1 . Hence, again by analogy the required numbers of MMF certificates may be derived from the residual according to
x B,GI t
n−1= (1 − t t
n)b t m
n−1C t b
n−1− x S t
n−1· (m − (m − 1)t n ) · S t
n−1B t
n− (B t
n− B t
n−1)t t
n= (1 − t t
n)B t
n(1 − t t
n)B t
n+ t t
nB t
n−1x t B
n−1− t t
n(1 − t t
n)B t
n+ t t
nB t
n−1x S t
n−1S t
n−1= x t B
n−1− t t
n(1 − t t
n)B t
n+ t t
nB t
n−1(x B t
n−1B t
n−1+ x t S
n−1S t
n−1) (16)
and
x B,I t
n−1= (1 − t t
n)b t m
n−1C t b
n−1− (1 − t t
n)x t S
n−1· m · S t
n−1B t
n− (B t
n− B t
n−1)t t
n= B t
n(1 − t t
n)B t
n+ t t
nB t
n−1(1 − t t
n)x B t
n−1= B −1 t
n−1B t
n(1 − t t
n)B t −1
n−1B t
n+ t t
n(1 − t t
n)x B t
n−1= 1 + r t
n−11 + r tax t
n−1(1 − t t
n)x B t
n−1. (17)
Given the definitions
q tax t
i: = 1 + r tax t
i− d t
iu t
i− d t
ib ¯ tax t
i: = b d t
i+ q tax t
i(b u t
i− b t d
i) − 1
the market prices of portfolios x GI t
n−1and x I t
n−1at time t n−1 are
P t
n−1(x t GI
n−1) = P t
n−1(x t
n−1) µ
1 − t t
nB t
n−1(1 − t t
n)B t
n+ t t
nB t
n−1¶
= P t
n−1(x t
n−1)
à (1 − t t
n)B −1 t
n−1B t
n(1 − t t
n)B −1 t
n−1B t
n+ t t
n!
= 1 + r t
n−11 + r t tax
n−1(1 − t t
n)P t
n−1(x t
n−1) (18)
and
P t
n−1(x t I
n−1) = (1 − t t
n)
à b u t
n−1− b t d
n−1u t
n−1− d t
n−1+ 1 1 + r t tax
n−1−b u t
n−1d t
n−1+ b d t
n−1u t
n−1u t
n−1− d t
n−1! C t b
n−1= 1 + b ¯ tax t
n−11 + b ¯ t
n−11 + r t
n−11 + r tax t
n−1(1 − t t
n)P t
n−1(x t
n−1)
= 1 + b ¯ tax t
n−11 + b ¯ t
n−1P t
n−1(x GI t
n−1) . (19)
Calculating backwards in the same manner as in the absence of taxes in combination with the definitions
C t tax
i: = (1 − t t
i)C t
iα j : = 1 + f t
j1 + f t tax
jβ j : = 1 + b ¯ tax t
j1 + b ¯ t
jγ j : = 1 + g ¯ tax t
j1 + g ¯ t
jfinally leads to the following expressions for the lower and upper bounds in the multi-period case
V I t
0¡ C t tax
k¢
=
k−1 ∏
j=0
β j ·V t GI
0¡ C tax t
k¢
=
k−1 ∏
j=0
α j β j · V t
0¡ C t tax
k¢
V I t
0¡ C t tax
k¢
=
k−1 ∏
j=0
γ j · V GI t
0¡ C t tax
k¢
=
k−1 ∏
j=0
α j γ j · V t
0¡ C t tax
k¢
.
Notice that f t tax
j< f t
jhas two instructive implications: Firstly, it implies q tax t
i< q t
iwhich again implies the equivalence
¯
c tax t
i≶ c ¯ t
i⇔ c t d
i≶ c u t
i(c ∈ {b, g})
and thus
β j ≶ 1 ⇔ b t d
j≶ b t u
jγ j ≶ 1 ⇔ g t d
j≶ g t u
j.
Secondly, as can be seen from
α j β j =
1+ b ¯
taxt j1+ f
t jtax1+ b ¯
t j1+ f
t j=
b
ut j−b
dt ju
t j−d
t j+ 1+ 1 f
tax t j−b
ut jd
t j+b
dt ju
t ju
t j−d
t jb
t ju−b
t jdu
t j−d
t j+ 1+ 1 f
t j
−b
ut jd
t j+b
t jdu
t ju
t j−d
t j=
b
ut j−b
dt ju
t j−d
t j+ 1+ u
t jd f
taxt j t jbdt j dt j
−
but jut j
u
t j−d
t jb
t ju−b
t jdu
t j−d
t j+ u 1+
t jd f
t jt j bdtj dt j
−
butjut j