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(1)

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.

(2)

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 = 1 , ω 2 , ω 3 , ω 4 } be the probability set and

C t

1

(ω) =

 

 

 

 

 

 

 

g u t

0

·C t

0

if ω = ω 1 b u t

0

·C t

0

if ω = ω 2

g d t

0

·C t

0

if ω = ω 3 b d t

0

·C t

0

if 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.

(3)

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

0

if ω ∈ {ω 1 , ω 2 } S t

0

(ω) = S t

0

for ω

S t

1

(ω) = d t

0

· S t

0

if 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

0

x B t

0

=

 

b

ut0

−b

dt0

u

t0

−d

t0

C

t0

S

t0

−b

ut

0

d

t0

+b

dt

0

u

t0

u

t0

−d

t0

C

t0

B

t1

 

=

 1 0 0 d

t0

u

t0

B

−1t0

B

t1

 

b

ut0

−b

td0

u

t0

−d

t0

C

t0

S

t0 bdt0

dt0

butut0

u

t0

−d

t00

C

t0

B

t0

 

3

This confinement is for the sake of simplicity only.

(4)

comprising a number of x S t

0

risky assets and x t B

0

money market fund certificates (MMF) gen- erates cash flows

P t

1

(x t

0

) =

 

b u t

0

·C t

0

if ω ∈ {ω 1 , ω 2 }

b d t

0

·C t

0

if ω ∈ {ω 3 , ω 4 } . (4)

Regarding the definitions

q t

0

: = 1 + r t

0

d t

0

u 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

0

u t

0

d t

0

+ 1 B −1 t

0

B t

1

−b t u

0

d t

0

+ b d t

0

u t

0

u t

0

d t

0

! C t

0

= 1

B t −1

0

B t

1

Ã

b u t

0

B −1 t

0

B t

1

d t

0

u t

0

d t

0

+ b d t

0

u t

0

B −1 t

0

B t

1

u t

0

d t

0

! C t

0

= 1

B t −1

0

B t

1

³

b u t

0

· q t

0

+ b d t

0

· (1 q t

0

)

´ C t

0

= 1 + b ¯ t

0

1 + r t

0

C t

0

.

From (2) and (4) it follows that buying the enterprise i.e. cash flow C t

1

at price Π t

0

(C t

1

) while at the same time short selling portfolio x t

0

will payoff

C t

1

(ω) + P t

1

(−x t

0

)(ω) 0 for each 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 .

(5)

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

0

is 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

0

1 + r t

0

C 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

0

x 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

0

x 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

0

x 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

1

C t

0

³

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)

(6)

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

0

with 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

0

q(ω 3 ) + q(ω 4 ) = 1 q t

0

so that (8) simplifies to

B t −1

1

C t

0

³

(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

1

C t

0

= B −1 t

0

P t

0

(x t

0

)

and that x t

0

is the most expensive weakly dominated portfolio indeed. Moreover it shows that ¯ b t

0

and ¯ g t

0

might 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

0

C t c

1

|F t

0

¢

1 (c ∈ {b, g}) .

(7)

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

0

with payoff P t

1

(y t

0

) at time t 1 weakly dominates the cash flow C t

1

generated 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

0

is 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

0

is the portfolio with cash flows

C t

1

(ω) =

 

g u t

0

·C t

0

if ω ∈ {ω 1 2 }

g d t

0

·C t

0

if ω ∈ {ω 3 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

tu0

u

t0

= b

dt0

d

t0

, we get

x

t0

=

Ã

bdt0Ct0

dt0St0

0

!

and the lower bounds simplifies to

V

t0

(C

t1

) = b

dt0

C

t0

d

t0

S

t0

S

t0

= b

dt0

d

t0

C

t0

.

(8)

and composition

y t

0

: =

y S t

0

y B t

0

=

 

g

ut0

−g

dt0

u

t0

−d

t0

C

t0

S

t0

−g

ut0

d

t0

+g

td0

u

t0

u

t0

−d

t0

C

t0

B

t1

 

=

 1 0 0 d

t0

u

t0

B

−1t0

B

t1

 

g

tu0

−g

dt0

u

t0

−d

t0

C

t0

S

t0 gdt0

dt0

gutut0

u

t0

−d

t00

C

t0

B

t0

 

.

From this we get

V t

0

(C t

1

) = P t

0

(y t

0

) = 1 + g ¯ t

0

1 + r t

0

C t

0

for 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 := 1 , ω 2 , ω 3 , ω 4 } n and the portfolios x t

0

, y t

0

being 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

i

for ω S t −1

i

(m · S t

i−1

) := ©

ω |S t

i

(ω) = m · S t

i−1

ª

be a short cut for the price of portfolio x t

i

you 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

(9)

t i . By analogy to the one-period case we know that

x t

n−1

: =

x t S

n−1

x t B

n−1

=

 

b

tn−1u

−b

tn−1d

u

tn−1

−d

tn−1

C

tn−1b

S

tn−1

−b

tn−1u

d

tn−1

+b

dtn−1

u

tn−1

u

tn−1

−d

tn−1

C

tn−1b

B

tn

 

=

  1 0 0 d

tn−1

u

tn−1

B

tn−1−1

B

tn

 

 

b

utn−1

−b

dtn−1

u

tn−1

−d

tn−1

C

tn−1b

S

tn−1

bdtn−1 dtn−1

butn−1

utn−1

u

tn−1

−d

tn−1

C

tn−1b

B

tn−1

 

 (11)

has market price

P t

n−1

(x t

n−1

) =

à b u t

n−1

b t d

n−1

u t

n−1

d t

n−1

+ 1 B t −1

n−1

B t

n

−b t u

n−1

d t

n−1

+ b t d

n−1

u t

n−1

u t

n−1

d t

n−1

! C t b

n−1

= 1 + b ¯ t

n−1

1 + r t

n−1

C t b

n−1

at 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−1

1 + r t

n−1

b t u

n−2

b d t

n−2

u t

n−2

d t

n−2

C t b

n−2

S t

n−2

and

x t B

n−2

= P t b,m

n−1

x S t

n−2

· m · S t

n−2

B t

n−1

= 1 + b ¯ t

n−1

1 + r t

n−1

C t b

n−2

B t

n−1

Ã

b m t

n−2

m · b u t

n−2

b t d

n−2

u t

n−2

d t

n−2

!

= 1 + b ¯ t

n−1

1 + r t

n−1

d t

n−2

u t

n−2

B t −1

n−2

B t

n−1

b

tn−2d

d

tn−2

b

tn−2u

u

tn−2

u t

n−2

d t

n−2

C t b

n−2

B t

n−2

(m ∈ {d, u})

(10)

or equivalently

x t

n−2

= 1 + b ¯ t

n−1

1 + r t

n−1

 

b

utn−2

−b

dtn−2

u

tn−2

−d

tn−2

C

tn−2b

S

tn−2

d

tn−2

u

tn−2

1+r

tn−2

bdtn−2 dtn−2

butn−2

utn−2

u

tn−2

−d

tn−2

C

tn−2b

B

tn−2

 

.

Further recursion leads to

x t

n−i

=

à n−1 j=n−i+1

1 + b ¯ t

j

1 + r t

j

! 

 

b

utn−i

−b

tn−id

u

tn−i

−d

tn−i

C

tn−ib

S

tn−i

d

tn−i

u

tn−i

1+r

tn−i

bdtn−i dtn−i

butn−i

utn−i

u

tn−i

−d

tn−i

C

tn−ib

B

tn−i

 

 (12)

and

P t

n−i

(x t

n−i

) =

n−1

j=n−i+1

1 + b ¯ t

j

1 + r t

j

à b t u

n−i

b t d

n−i

u t

n−i

d t

n−i

+ 1 B −1 t

n−i

B t

n−i+1

−b u t

n−i

d t

n−i

+ b t d

n−i

u t

n−i

u t

n−i

d t

n−i

! C t b

n−i

=

n−1

j=n−i

1 + b ¯ t

j

1 + r t

j

C t b

n−i

.

Rearranging terms according to

B t −1

n−i

P t

n−i

(x t

n−i

) = B t −1

n−i

B t −1

n−i

B t

n−i+1

n−1

j=n−i+1

1 + b ¯ t

j

B t −1

j

B t

j+1

³

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+1

³

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−i

is 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

0

we finally arrive at

V t

0

(C t

k

) =

k−1

j=0

1 + b ¯ t

j

1 + f t

j

C t

0

5

Cf. H

ARRISON AND

K

REPS

[4].

(11)

and

V t

0

(C t

k

) =

k−1

j=0

1 + g ¯ t

j

1 + f t

j

C 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

i

implied 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

j

and d t

j

then 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

j

B t

j+1

= 1 + r for all j i

6

It is well known since C

OX

, I

NGERSOLL

,

AND

R

OSS

[1] that future spot rates must be equal to implied

forward rates to preclude arbitrage if the interest rate evolves deterministically.

(12)

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 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

h

1 + f t

h

µ 1 + b ¯ 1 + r

k−i C t

0

and

V t

0

(C t

k

) =

i−1

h=0

1 + g ¯ t

h

1 + f t

h

µ 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

0

³

(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

0

and

V t

0

³

(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=1

θ

i

= C

t0

· θ

−1

1 θ

−1

1 ·

n i=1

θ

i

= C

t0

· θ

−1

·

n

i=1

θ

i

n

i=1

θ

i

θ

−1

1 = C

t0

·

n−1

i=0

θ

i

n

i=1

θ

i

θ

−1

−1 = C

t0

· 1 θ

n

θ

−1

1 .

(13)

For f t

g

= r and ¯ c t

g

= c ¯ (g = 0, . . . , k 1 ; c ∈ {b, g}) the above formulae simplify to

V t

0

³

(C t

i

) k i=1

´

= 1 + b ¯ r b ¯

³

1 θ( b, ¯ r) k

´ C t

0

and

V t

0

³

(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

0

³

(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

0

and

k→∞ lim V t

0

³

(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

i

applicable 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

ti

(14)

payments 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

i

x 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

i

by 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−1

S t

n−1

= b u t

n−1

b t d

n−1

u t

n−1

d t

n−1

C t b

n−1

S 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−1

C t b

n−1

S t

n−1

= (1 t t

n

)x S t

n−1

(15)

(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−1

C t b

n−1

x S t

n−1

· (m (m 1)t n ) · S t

n−1

B 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

n

B t

n−1

x t B

n−1

t t

n

(1 t t

n

)B t

n

+ t t

n

B t

n−1

x S t

n−1

S t

n−1

= x t B

n−1

t t

n

(1 t t

n

)B t

n

+ t t

n

B t

n−1

(x B t

n−1

B t

n−1

+ x t S

n−1

S t

n−1

) (16)

and

x B,I t

n−1

= (1 t t

n

)b t m

n−1

C t b

n−1

(1 t t

n

)x t S

n−1

· m · S t

n−1

B t

n

(B t

n

B t

n−1

)t t

n

= B t

n

(1 t t

n

)B t

n

+ t t

n

B t

n−1

(1 t t

n

)x B t

n−1

= B −1 t

n−1

B t

n

(1 t t

n

)B t −1

n−1

B t

n

+ t t

n

(1 t t

n

)x B t

n−1

= 1 + r t

n−1

1 + 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

i

u t

i

d t

i

b ¯ 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−1

and x I t

n−1

at 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

n

B t

n−1

(1 t t

n

)B t

n

+ t t

n

B t

n−1

= P t

n−1

(x t

n−1

)

à (1 t t

n

)B −1 t

n−1

B t

n

(1 t t

n

)B −1 t

n−1

B t

n

+ t t

n

!

= 1 + r t

n−1

1 + r t tax

n−1

(1 t t

n

)P t

n−1

(x t

n−1

) (18)

(16)

and

P t

n−1

(x t I

n−1

) = (1 t t

n

)

à b u t

n−1

b t d

n−1

u t

n−1

d t

n−1

+ 1 1 + r t tax

n−1

−b u t

n−1

d t

n−1

+ b d t

n−1

u t

n−1

u t

n−1

d t

n−1

! C t b

n−1

= 1 + b ¯ tax t

n−1

1 + b ¯ t

n−1

1 + r t

n−1

1 + r tax t

n−1

(1 t t

n

)P t

n−1

(x t

n−1

)

= 1 + b ¯ tax t

n−1

1 + b ¯ t

n−1

P 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

i

α j : = 1 + f t

j

1 + f t tax

j

β j : = 1 + b ¯ tax t

j

1 + b ¯ t

j

γ j : = 1 + g ¯ tax t

j

1 + g ¯ t

j

finally 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

j

has two instructive implications: Firstly, it implies q tax t

i

< q t

i

which again implies the equivalence

¯

c tax t

i

c ¯ t

i

c t d

i

c u t

i

(c ∈ {b, g})

(17)

and thus

β j ≶ 1 b t d

j

b t u

j

γ j ≶ 1 g t d

j

g t u

j

.

Secondly, as can be seen from

α j β j =

1+ b ¯

taxt j

1+ f

t jtax

1+ b ¯

t j

1+ f

t j

=

b

ut j

−b

dt j

u

t j

−d

t j

+ 1+ 1 f

tax t j

−b

ut j

d

t j

+b

dt j

u

t j

u

t j

−d

t j

b

t ju

−b

t jd

u

t j

−d

t j

+ 1+ 1 f

t j

−b

ut j

d

t j

+b

t jd

u

t j

u

t j

−d

t j

=

b

ut j

−b

dt j

u

t j

−d

t j

+ 1+ u

t j

d f

taxt j t j

bdt j dt j

but j

ut j

u

t j

−d

t j

b

t ju

−b

t jd

u

t j

−d

t j

+ u 1+

t j

d f

t j

t j bdtj dt j

butj

ut j

u

t j

−d

t j

(20)

in combination with the fact that neither the nominator nor the denominator of (20) can get negative, it implies

α j β j ≶ 1 b d t

j

d t

j

b t u

j

u t

j

and

α j γ j ≶ 1 g t d

j

d t

j

g u t

j

u t

j

.

Thus, if the payoff characteristic is convex for any trading interval up to t k that is if c d t

i

d t

i

< c t u

i

u t

i

for each i ∈ {0, . . . ,k 1}, c ∈ {b, g}

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