Arbitrage And Nonlinear Taxes

Marcus Becker and Andreas Löffler∗

November 2, 2016

AbstractIncorporating a progressive or convex income tax into valuation prob-lems raises the question of the appropriate tax rate to use in commonvaluation formulas. We apply arbitrage theory in a riskless (multi-period) economy to answer this question. It turns out that the appro-priate tax rate depends on the marginal tax rate of the investor’s initialendowments. In case of no endowments this automatically leads to amarginal tax rate at a base of zero. We are able to give an intuitiveexplanation for the latter result.

With tax liabilities that are a convex function of the tax base weidentify a new kind of arbitrage: trading strategies where the gainsfrom trades remain unchanged if this strategy is applied multiple times.We call these strategies ‘bounded’ arbitrage opportunities because thegain is bounded by a constant.Going beyond earlier research, we are able to give a complete charac-

terization based on properties of the tax liability function as to whetherbounded as well as unbounded arbitrage opportunities will occur.

KeywordsNo-Arbitrage with Taxation, Fundamental Theorem of Asset Pricing, Non-Constant Tax Rates, Application of Convex Optimization Problems

∗Department of Banking and Finance, Freie Universität Berlin. We thank Ralph T. Rockafellar, Gün-ter Bamberg, Marcel Fischer, Lutz Kruschwitz, Jochen Hundsdoerfer, Caren Sureth-Sloane, RainerNiemann, Corinna Treisch, Stephan Burggraef, Martina Rechbauer and the participants of the 12th

arqus workshop 2016 in Munich as well as the participants of the 3rd tax workshop 2016 in Viennafor their helpful remarks.

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JEL Classification NumbersC61, E62, G12, H24

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Contents

1. Introduction 41.1. The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2. Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2. Elements of a convex tax liability function 82.1. Marginal and average tax rate . . . . . . . . . . . . . . . . . . . . . . . . . 82.2. Convex tax functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3. Subdifferential, subgradient, and conjugate tax function . . . . . . . . . . 12

3. Arbitrage-free asset pricing in the presence of convex tax functions 133.1. The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2. Characterization of all possible arbitrage opportunities . . . . . . . . . . . 163.3. Three explanatory examples . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4. Intuition of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4. Conclusion 24

A. Proof of theorem 10 28

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1. Introduction

1.1. The problem

This paper looks at the effect of a progressive (personal) income tax on valuation. Forlinear taxes, i.e. when the tax liability is a constant multiple of the tax base, the taximpact can be easily identified.1 However, worldwide national tax rates are progressive.To simplify matters, often the tax burden is nonetheless determined as a product of a(fixed) tax rate with the tax base. But with progressive taxes there is not one but anentire interval of tax rates that could be used and choosing the appropriate rate forvaluation purposes is not straightforward. Intuitively, marginal as well as average taxrates seem plausible.To illustrate the impact of taxes on company value, it is helpful to consider a simple

numerical example. Suppose a company exists forever and distributes its (riskless) cashflow of $100 to its owners. We assume that the cash flows grow by 1% p.a. and theriskless rate is 3%. Ignoring taxes, the company should be valued at $5,000.2 If thegovernment introduces a personal income tax of 25% on cash flows, then the discountrate would amount to 1.25% (= 3% · (1 − 0.25) − 1%) and the company would nowbe valued at $6,000. Further increasing the income tax rate to 50% will decrease thediscount rate down to 0.5% – now the company would be worth $10,000 which is twicethe amount in the case of no taxes. Hence, we cannot neglect the effect of income taxeswithout good reason.Tax laws all over the world have progressive tax rates with a wide range. The statutory

federal tax rate in the US on interest income varies today between 0% and 39.6% and wasas high as 90% between 1953 and 1963.3 The German income tax offers a progressivetax rate going from 0% to almost 50%. It is very difficult to find empirical evidence

1Linear transformations do not change the analysis and therefore a linear tax is straightforward tohandle. Two examples may illustrate this point. Brennan (1970) writes “. . . we assume for simplicitythat each investor has marginal tax rates on dividend and capital gains income tdi, and tgi which areconstant and independent of their portfolio choice”. Similar to this, Bradford (2000) states “linearityis a desideratum of a tidy tax system.” We were only able to identify very few papers dealing with anon-linear tax liability as a function of the tax base; see section 1.2.

2Hereby we utilize the Gordon-Shapiro equation

V0 =F (1− τ)

rf (1 + τ)− g ,

where V0 states the (fair) company value today, τ is the income tax rate and rf , g display the riskfreereturn and growth rate respectively.

3See taxfoundation.org for an overview of the statutory federal tax rate between 1913 and 2013.

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for a natural tax rate in this case.4 Thus the identification of the appropriate tax ratebecomes inescapable. Taking into account the large sums of money involved in mergersand acquisitions as well as court disputes, it is a remarkable fact that this issue has notyet been fully resolved.To tackle our problem, we examine a multi-period market model with riskless bonds.

We do not allow for uncertainty because we have been unable to generalize our resultsso far.5

We consider an artificial tax liability function that comprises a very broad class ofexisting national tax codes. Apart from other minor technical conditions, we will relyon convex tax liabilities as a function of the tax base. Examples of convex tax liabilitiesinclude piecewise linear functions with two or more different tax rates (cf. the Americanfederal income tax in 2015), or tax rates that are affine as in the case of German incometax system in 2015. Tax allowances, as is the case with the German capital income tax2015 (“Abgeltungsteuer”), ensure convexity as well. Convexity in the tax liability forcesus to generalize the classical arbitrage approach by augmenting the terminology.Typically an arbitrage opportunity is a riskfree gain which can be increased to an arbi-

trary scale. Once we find a trading strategy paying out a positive amount today withoutany expenses in the future, we can repeat this strategy over and over again and there-fore become endlessly rich. When tax liabilities are non-linear, the situation is different.It may be the case that we find an arbitrage opportunity with a positive payment to-day and no cash outflow tomorrow, but the maximum amount of money obtained usingthis strategy multiple times is still limited to a constant K < ∞. Therefore, we callsuch an arbitrage opportunity bounded in contrast to the classic, (unbounded) arbitrageopportunity.Our main result will be a complete characterization of prices that are compliant with

the no-arbitrage principle, for bounded as well as unbounded arbitrage opportunities.Moreover, we show that the absence of arbitrage is closely connected to the propertiesof tax liability - a link that has not been made in field literature so far. It turns out thatthe assumption of an arbitrage-free market will provide a clear guideline to which taxrates shall be used for valuation purposes.To formulate necessary and sufficient conditions for arbitrage-free markets, we intro-

duce the concept of implied tax rates deduced from current market prices. Let pt be themarket price of an asset at time t with cash flow payments xt. The allowance at has to

4Sialm (2009) tried to identify marginal investment tax rates for US companies. The Institute ofAuditors in Germany agreed on a standardized tax rate of 35% without reasoning, for details seeHeintzen et al. (2008, in German).

5We do not expect uncertainty to affect our results.

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be deduced from xt in order to form the tax base of the income tax function T (·). If thetax is linear with a constant tax rate τ , it is a well-known outcome that the market isfree of arbitrage iff for all t

pt =pt+1 + xt+1 − τ · (xt+1 − at+1)

1 + rf · (1− τ)(1)

where rf is the riskfree interest rate.6 For convex tax liabilities we use this equation inorder to define an implied tax rate. Given observed market prices pt, the particular taxrate τ that satisfies (1) is called the implied tax rate. Implied tax rates can be seen asa linearization of the tax liability function. The implied tax rate will be unique at eachpoint in time t = 0, . . . , T − 1. However, it is not clear that the tax rates, being theresult of a simple linear equation, coincide with the marginal tax rate or average tax rateof the tax liability function; nor, without further restrictions on the parameters, can westate that τ lies in the interval of [0, 1) which is a natural requirement for a tax system.Suppose an investor with initial endowments wt that are due to income tax payments.Our main result is then the following:

– Asset prices pt are free of arbitrage iff for all t the implied tax rates equal themarginal tax rate of T ′(wt).

– Asset prices pt admit unbounded arbitrage opportunities iff there is at least oneimplied tax rate which is not equal to any possible marginal tax rate T ′(wt) (forarbitrary tax bases wt).

– In all other cases, asset prices pt are said to admit bounded arbitrage.

If the investor has no initial endowment (wt = 0 for all t), then we show that arbitrageopportunities do not exist until marginal and average tax rates coincide. This offers astrong analogy to the discussion of marginal and average costs in classical productiontheory: there, the optimal output choice is given by the quantity where marginal costsequals average costs.7 We transfer that thought to taxation. Market participants areable to improve their portfolio choices by moving taxable income through the tax base,lowering the overall tax due. With arbitrage-free markets, all investors must be marginalon their prices, meaning that the asset prices they pay depend on their marginal tax rateT ′(wt). Ultimately, this is the only tax rate that should be used for valuation purposes.

6For an example, see Ross (1987, p. 380).7See Exercise 5.D.1 in Mas-Colell, Whinston, and Green (1995, p. 143-144).

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1.2. Literature

The impact of non-linear taxation on asset prices (and therefore on investment decisions)has been examined to some extent.8

Schaefer (1982) is one of the first to analyze a non-linear tax liability. Concentrating onriskfree assets, he uses the example of two investors; a necessary condition for equilibriumprices to exist is for the marginal tax rates of both investors to overlap. If net discountfactors are not unique, arbitrage opportunities will exist and therefore equilibrium isunattainable. To overcome this dilemma, Schaefer puts constraints on short sales.From our point of view, Schaefer’s model reveals a fundamental weakness: marginal

tax rates “[t]o avoid pointless complication” are unique, meaning for two different taxbases there are two different marginal tax rates, see Schaefer (1982, p. 168). With thisassumption Schaefer implies that marginal tax rates are strictly monotone, which weprove to be equivalent to strict convexity – a property that certainly does not hold forevery tax code throughout the world.One important work taking arbitrage theory into account originates from Ross (1987),

proving a fundamental theorem for arbitrary riskless as well as risky assets. The tax basecomprises cash flows as the argument of a convex tax liability function. Ross differentiatesbetween local and global arbitrage. An arbitrage is local if it offers a classic arbitrage inthe sense of higher net income compensation and non-positive costs for creating this newportfolio position given the initial portfolio of one single investor. For another investorwith a different initial endowment, such a portfolio change does not necessarily have toresult in an arbitrage opportunity. An arbitrage is said to be global if it offers a localarbitrage at every endowment.At first, the definition of a local arbitrage seems to be more general and less restrictive

than the usual arbitrage condition. But because Ross forbids such an opportunity, theabsence of a local arbitrage is a stronger assumption than the classical one. Ross showsthe existence of Arrow-Debreu state prices and therefore the existence of risk-neutralprobabilities is necessary and sufficient for the existence of a local arbitrage. Withoutproof, Ross claims that his results can be transferred into the multi-period or continuousframework. But the associated problems are not easy to deal with. For example, Ross’assumption of a piecewise linear tax rules out the case of the German income tax of2015 having quadratic tax liability functions for small- to medium-sized income or any(partially) strictly convex tax function, see Ross (1987, p. 387 as well as p. 392).Prisman (1986) investigates convex tax liabilities and includes transaction costs in

8There is a long list of literature focusing on linear taxation which we will disregard here. For recentworks on this topic see, e.g. Gallmeyer and Srivastava (2011).

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the model (which we will neglect in our argument). By using a standard dual theoryof convex optimization, he is able to derive a non-linear and investor-specific pricingoperator. Unlike us, Prisman does not work out the exact link between arbitrage-freeasset prices and the structure of tax liability functions.Dybvig and Ross (1986) as well as Dammon and Green (1987) reveal how the intro-

duction of non-linear tax liabilities in asset pricing enables clientele effects, i.e., allowinginvestors in different tax brackets to hold different amounts of assets and/or pay differ-ent prices for the same assets. Dammon and Green (1987) extend the model of Dybvigand Ross by permitting short sales and proving an existence theorem for a competitiveequilibrium. The definition of no-tax arbitrage that Dammon and Green use is closelyconnected (in the one-period model) to our definition of bounded arbitrage. But againthe authors fail to give a clear statement of the structure of prices that prevent arbitrage.Dermody and Rockafellar (1991) analyze non-linear taxes in a multi-period framework,

including transaction costs. They assume future tax payments depend solely on pricesrather than the specific amount that an investor holds. This assumption is critical.Investors can easily avoid paying taxes, which Dermody and Rockafellar have to ruleout. Because long and short prices do not coincide in their model, the authors concludethat net discount factors (term structure) are not unique, and further “[t]here are strongmathematical reasons for believing that the nonuniqueness indicates underlying non-linearities in the behavior of value that cannot be captured by a single term structure”,Dermody and Rockafellar (1991, p. 32). We will give a complete analysis how to identifythese term structures.

2. Elements of a convex tax liability function

2.1. Marginal and average tax rate

In the analysis of taxation we have to define three basic concepts: tax liability, taxbase, and tax liability. Let x denote the tax base; then the tax liability is a functionT (x) : R −→ R that is also applied for a negative tax base, thus including the possibilityof taxable losses. We use the following assumptions about T (·).First, we assume that the tax function is continuous.9 This rules out the possibility of

an exemption limit, where at a certain income level of F the whole income is taxed and

9This assumption may be dropped later on, since a convex function defined on R is continuous.

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not just the excess amount over F .10

Additionally we assume that no tax is paid without income, or conversely

T (0) = 0. (2)

The condition is appropriate insofar as for typical tax functions the property T (x) ≤ x

holds for all x.11 If T (0) < 0, we have an arbitrage for doing nothing. This obviousarbitrage opportunity cannot hold in our analysis.12

The following is a reasonable definition of a tax rate:

Definition 1 (Average tax rate). An average tax rate (also tax scale) is every functiont(·) : R \ {0} → R for which

T (x) = t(x) · x (3)

holds.

At x = 0 the average tax rate is undefined. We assume t(x) ∈ [0, 1) from now on.Marginal tax rate is the coefficient of incremental tax over incremental income. With-

out further assumptions, we cannot guarantee the differentiability of T (·) at all x. Thishappens to be the case when there are jumps in the tax rate or in the presence of taxallowances. Thus the following definition only holds for bases in which the limits belowexist.

Definition 2 (Marginal tax rate, left and right derivative). The marginal tax rateT ′(·) : R→ R at a base x0 is

T ′(x0) := limx→x0

T (x)− T (x0)x− x0

,

whenever the limit exists and is unique. Further we call

T ′−(x0) := limx↑x0

T (x)− T (x0)x− x0

= supx<x0

T (x)− T (x0)x− x0

10A typical tax with exemption limit has the structure

T (x) =

0 if x ≤ F

τ x otherwise.

Note that the function is neither continuous at x = F nor is it convex, but marginal tax equalsaverage tax at all x 6= 0.

11This follows from the fact that (theoretically) tax rates should be smaller than 1.12If T (0) < 0 holds, we refer to a negative income tax which is discussed, for example, in Friedman

(1962, chapter XII).

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andT ′+(x0) := lim

x↓x0

T (x)− T (x0)x− x0

= infx>x0

T (x)− T (x0)x− x0

left and right derivative respectively.

We call a tax function differentiable if the marginal tax rate exists at all x0 ∈ R orequivalently if T ′−(x0) = T ′+(x0) holds. Differentiability clearly does not hold for a varietyof tax functions, for example if marginal tax rates increase stepwise as in case of the USfederal income tax system.We require that all average tax rates are nonnegative and lower than 100%. The same

often holds true for the marginal tax rate, however we observe particular cases in nationaltax law in which this is not the case.13 Therefore, we do not restrict the first derivativeto the interval [0, 1).

2.2. Convex tax functions

We focus on convex tax liability functions. It is rather sensible to assume convexity.From the economic perspective convexity means that tax payers with higher taxableincomes are charged at higher marginal tax rates in order to obtain the so called “verticaltax justice”. Therefore, many tax codes are convex by law. As Graham and Smith Jr.(1999) point out, investors try to obtain the highest degree of convexity by exploiting thevariety of interdependent tax laws and thereby flattening their tax payments which mayresult into strict convex tax liability functions. Vice versa, if we allow for non-convextax functions, we allow for cases in which higher taxable income is marginally taxed ata lower rate than for smaller taxable incomes leading to “tax injustice”.

Definition 3 (Convex tax function). A tax liability function T (·) is convex if for all0 ≤ λ ≤ 1 and x, y ∈ R we have

T (λx+ (1− λ)y) ≤ λT (x) + (1− λ)T (y). (4)

For continuous tax functions convexity is equivalent to the somehow weaker condition

T (x) + T (y)− 2T(x+ y

2

)≥ 0,

meaning that it is sufficient to test the inequality in definition 4 only for λ = 12 . This

inequality is quite intuitive when we look at the regulation of splitting of marital income13For example, the OECD (2006, p. 65, in German) shows that in Germany, after an increase in income,

individuals (single parent or single-income families with two children) will face a marginal tax rateabove 100% (all transfer payment included) due to an increase of the average tax rate from 50% to55%.

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as it can be found in many European countries as well as in the US. Instead of taxing eachindividual, the average income of the couple is taxed, thus resulting in tax savings dueto progression in the income tax system. The latter inequality gives the exact amountof tax savings.Many tax systems have the property of monotonically increasing average tax rates t(x)

in x in order to tax higher income with higher rates, sometimes called the ability-to-payprinciple or vertical tax justice. Such average tax rates will be denoted as progressive.The two concepts of convexity and progression are not logically equivalent. Convexity

of a tax implies progression due to the increasing slope property since

T (x)− T (0)x− 0

= t(x)

is increasing on R\{0}.14 The opposite implication is not true. For an example considerT : R→ R with

T (x) =

0, x ≤√2,

12x−

1x , otherwise.

Here, the average tax rate t(x) is given by (undefined at x = 0)

t(x) =

0, x ≤√2, x 6= 0

12 −

1x2, otherwise.

√2

x

T (x)

(a) Nonconvex tax function (the dashed lineis linear).

√2

x

0.5t(x)

(b) Progressive tax rate.

Figure 1: Example of a progressive tax function which is not convex.

As can be seen in Figure 1, the average tax t(·) is increasing but the tax liability T (·)is not convex. In this respect the assumption of convexity of the tax liability is somehowstronger than the assumption of a progressive tax scale.14For a revision of the increasing slope property, see for example proposition 6.1 in Hiriart-Urruty and

Lemaréchal (2001, p. 15).

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2.3. Subdifferential, subgradient, and conjugate tax function

In many countries we observe tax systems with different tax brackets, meaning that themarginal tax rate takes different values.15

The simplest case is of two tax brackets where gains are differentially-taxed, as arelosses. Such a tax function is given by

T (x) = τ− ·min(x, 0) + τ+ ·max(x, 0) (5)

Figure 2 shows an income tax of type (5) where losses are taxed at a rate of τ− = 5%

and gains are taxed by τ+ = 25%. The tax function is convex and not differentiable atx = 0.16

−2 0 1x

−0.1

+0.25

T (x)

∂T

(a) Losses and gains are taxed differentially.

−2 0 2x

5%

25%

T ′(x),t(x)

(b) Marginal and average tax coincide.

Figure 2: Example of a piecewise linear income tax with τ− = 5% and τ+ = 25%.

As we observe in Figure 2a, there is not one but multiple straight lines forming atangent of T (·) at x = 0. The set of all possible slopes is given by the interval [τ−, τ+].We call the union of all possible tangent slopes subdifferential of T (·) at x denoted by∂T (x), which represents a generalization of the first derivative in the context of convexfunctions. In the case of non-differentiability, ∂T (x) is not a single number but rather aset of numbers, and in our one-dimensional case it is always an interval. In the settingof Figure 2a, ∂T (0) is equal to [T ′−(0), T

′+(0)] = [τ−, τ+].

If the tax function is differentiable at x, the subdifferential is a single element, givenby ∂T (x) = {T ′(x)}. In the example above it is ∂T (x) = {τ−} for all negative x and∂T (x) = {τ+} for all positive x as seen in Figure 2b. We borrow the formal definition of

15We also call such tax systems piecewise linear or affine which is formally not the same as linear althoughin some literature these terms are used interchangeably.

16Since the kink lies in the origin, we can observe the identity of marginal and average tax rate for allx 6= 0.

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a subdifferential from the theory of convex analysis.17

Definition 4 (Subdifferential and Subgradient). The subdifferential of a convex tax func-tion T (·) at x is given by the following set of real numbers

∂T (x) := {g ∈ R | ∀y T (y) ≥ T (x) + g(y − x)}. (6)

An element g ∈ ∂T (x) is called a subgradient of T (·) at x.

Finally we need the definition of a conjugate tax function in order to explain possiblearbitrage opportunities. In particular we are interested in the domain, the set of numbersfor which the conjugate is finite.

Definition 5 (Conjugate tax function). The conjugate tax function T ∗(·) of a tax liabilityfunction T (·) is given by

T ∗(τ) := supx∈R

τx− T (x).

The domain is given by dom(T ∗) = {τ | T ∗(τ) <∞}.

The conjugate tax function is convex on its domain.18 Lastly, we need another generalresult from the theory of conjugates.

Lemma 6. It is ∂T (x) ⊂ dom(T ∗) for all x.

Proof. In Hiriart-Urruty and Lemaréchal (2001, Theorem 1.4.1, p. 220) it is shown thatτ ∈ ∂T (x) is equivalent to

T ∗(τ) + T (x)− τ · x = 0.

Therefore, T ∗(τ) is finite and it holds τ ∈ dom(T ∗).

3. Arbitrage-free asset pricing in the presence of convex taxfunctions

3.1. The model

We assume a multi-period framework with different points in time denoted by t =

0, 1 . . . , T . In time t = 1, . . . , T the investor’s initial endowments are given by wt. Fur-ther she receives cash flows xt in case of security trading. In t = 0, . . . , T −1 it is possible17For standard literature see Rockafellar (1997, p. 214-215), as well as Hiriart-Urruty and Lemaréchal

(2001, p. 165). It is also proven that the subdifferential of a convex function, defined on R, alwaysexists and is represented by a closed interval.

18See Rockafellar (1997, p. 104).

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to trade these securities on the capital market. The theory of arbitrage-free asset pric-ing is characterized by a relative valuation: the price of one asset or trading strategy isdetermined by the price of another asset or trading strategy with known market prices,offering the same cash flows as the asset to be valued. Both prices have to be identical;otherwise arbitrage can be realized. This differs from equilibrium theory insofar as wedo not need to assume investor-specific utilities or market clearing conditions.For this reason we have to presume the existence of some asset prices. The theory of

arbitrage then explains how to compute prices of other assets. The same is true whenwe take taxes into account. We will assume the existence of a standard coupon bondwith constant interest payments of the riskfree rate rf > 0; the price after taxes for onecoupon bond is normalized to one. We now have to introduce several symbols.19

Cash flows of the coupon bond are given by a vector x0 = (rf , . . . , rf , 1+rf ) ∈ RT . Theprice vector for one unit of coupon bond is p0 = (1, . . . , 1, 0) ∈ RT+1. The terminal price,for reasons of convenience, is set to zero, since there is no trade going on carried out at thelast point in time. Allowing for dynamic trading, we can achieve any arbitrary payoutstructure after taxes by building a portfolio comprising of coupon bonds. Therefore,under certainty it is sufficient to assume one single asset with given prices.We introduce a second asset with arbitrary payout structure and investigate which

properties of prices lead to no-arbitrage in our market. Cash flows of the second bondare denoted by x1 = (x11, . . . , x

1T ) ∈ RT where the index states the time of capital inflow.

Prices for one unit are p1 = (p10, . . . , p1T−1, p

1T ) ∈ RT+1, again with p1T = 0.

Shareholders are subject to payment of income taxes on cash flows in t = 1, . . . , T .Tax liability comprises of a (time independent) convex function T (·) which is due to ayet-to-be-defined tax base. Furthermore, it is T (0) = 0 and the average tax rate t(·) liein the interval of [0, 1).In most countries throughout the world, taxable income is not similar to actual cash

flows because very detailed legal provisions must be observed. Owners of securitiesmight receive payments that do not have the character of dividends or interest, suchas repayment of capital. Such payments do not incur income tax payments. So as notmix such payments with dividends, we introduce another variable a1t that comprises thedifference of cash flow and income tax base for the second asset. At time t, the assetpossesses an income tax base of

x1t − a1t .

Similarly, the capital repayment of the coupon bond is a0 = (0, . . . , 0, 1). The total taxbase of a portfolio amounts to the sum of all tax bases, weighted by portfolio holdings

19The assumption of riskfree rates to be constant is not necessary, but simplifies our calculation.

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which we denote by h = (h00, h01, . . . , h

0T−1) for the first title and analogously h1 for the

second. We simplify h0−1 = h0T = 0. Then the investor’s net income stream dependingon wt is

nwt := wt − T (wt).

Note that we assume (for simplicity) that all endowments are fully taxable.20 Aftertrading the securities our investor may realize an after-tax-withdrawal for consumptionpurposes of

δwt (h) := −p1th1t − p0th0t +(p1t + x1t

)h1t−1 +

(p0t + x0t

)h0t−1 + wt

− T((x1t − a1t )h1t−1 + (x0t − a0t )h0t−1 + wt

)t=1,. . . ,T.

Note that this definition differs from the usual model without taxation only by theintroduction of endowment specific tax payments T ( · + wt). The cost of the tradingstrategy h is independent of wt and given by

−δ0(h) := p10h10 + p00h

00

We define arbitrage as follows.

Definition 7 (Arbitrage). A trading strategy h is called arbitrage opportunity iff δwt (h) ≥nwt for all t = 1, . . . , T and −δ0(h) ≤ 0 where for at least one t = 0, . . . , T the inequalityis strict. If no arbitrage opportunities exist, we call the market free of arbitrage.

In the theory of arbitrage without taxation, the initial endowments wt will be ruledout of the above definition and the investor trades as if her initial income is zero. Eacharbitrage opportunity h in a setting without taxation can be multiplied by a constantλ > 0 so that the arbitrage profits of h are multiplied by λ, meaning that arbitragegains are theoretically unbounded. In the presence of convex taxes, it is possible torealize arbitrage profits that cannot be increased on an arbitrary scale, i.e. the optimalarbitrage gain remains constant at a certain level. To see this, we focus on self-financingstrategies. We particularly focus on the following primal problem (P) given by

infh

− δ0(h) (P)

s.t. δwt (h) ≥ nwt t = 1, . . . , T,

where p∗ denotes the optimal value of (P).From this point of view, our goal is to achieve the lowest cost possible in t = 0 given

by −δ0(h). Since h ≡ 0 is feasible, the optimal value must be non-positive (p∗ ≤ 0). If an20The argumentation for partial taxable income αtwt with αt ∈ [0, 1] is straightforward.

15

arbitrage opportunity exists, without loss of generality, we can assume that the optimalvalue is negative (p∗ < 0).21 By definition, the arbitrage gain is −p∗. Arbitrage-freemarkets correspond to an optimal value of p∗ = 0.Without taxes, arbitrage opportunities always imply p∗ = −∞. If a convex tax function

exists, it might be that the optimal solution of (P) is finite (−∞ < p∗). This gives riseto the following definition.

Definition 8 (Bounded and Unbounded Arbitrage). We call an arbitrage strategy h

bounded iff for the optimal value p∗ in (P) holds p∗ > −∞. An arbitrage is said to beunbounded iff p∗ = −∞ in (P).

In the presence of linear taxes, bounded arbitrage opportunities do not exist.

3.2. Characterization of all possible arbitrage opportunities

We now state our main result. Using standard duality arguments in convex optimization,we can identify prices leading to bounded as well as unbounded arbitrage.For this purpose we introduce the concept of implied tax rates τt. If the tax were

linear, prices p1t−1 would satisfy

p1t−1!=p1t + x1t − τt(x1t − a1t )

1 + rf (1− τt)t = 1, . . . , T.

If p1t−1 6=x1t−a1trf

holds, we can solve for τt in the above formula and get

τt =(1 + rf )p

1t−1 − p1t − x1t

rfp1t−1 − (x1t − a1t )

. (7)

That gives reason for the following definition.

Definition 9 (Implied tax rates). Suppose x1t − a1t 6= rfp1t−1 for all t. The numbers τt

that fulfill (7) are called implied tax rates of prices p1t in time t.

We will look at the case in which x1t −a1t = rfp1t−1 holds for all t in a separate analysis

later on.Let us consider a single implied tax rate of τt. We will look at three different (mutually

exclusive) possibilities for τt to depend on the tax function. The sets of interest involve21Suppose there are riskfree profits in a subsequent period t with δt(h) > 0 and zero price today

(δ0(h) = 0). By selling short some amounts of the coupon bond in t−1 we can realize a self-financingstrategy with net payoffs of zero in t and positive net-payoffs in t − 1. We can repeat our strategyand defer money in the prior period t− 2 and so on, resulting in a positive withdrawal in t = 0. Thisis the case whenever rf > 0 and average tax rates are lower than 100%.

16

∂T (wt) and dom(T ∗). Since ∂T (wt) ⊂ dom(T ∗), by using Lemma 6, it is only possiblefor τt to lie in ∂T (wt) or in dom(T ∗) \ ∂T (wt) or in neither of the two sets. Usingthis observation, we can establish a complete characterization of all existing arbitrageopportunities in the market that are liable to income taxes. We moved the proof of thetheorem to the appendix.

Theorem 10. Let τt denote the implied tax rates of prices pt in t = 0, . . . , T − 1. Thenonly one of the following holds true:

1. The market is free of arbitrage iff τt ∈ ∂T (wt) for all t.22

2. There are unbounded arbitrage opportunities iff for at least one implied tax rateτt /∈ dom(T ∗).

3. In all other cases, there are bounded arbitrage opportunities.

The latter is true iff τt ∈ dom(T ∗) for all t and for at least one t′ the conditionτt′ ∈ dom(T ∗) \ ∂T (wt) holds.

The results are even more comprehensible if we meet a further assumption on the taxfunction. To justify the assumption, we look at the following example of a tax function:

T (x) =

x+ 1−√x+ 1 x ≥ 0,

x2 x < 0.

For positive income the above tax function has marginal tax rates of 1 − 12√x+1

, thusmonotone increasing towards 100% without ever reaching its limit. For national taxsystems such a behavior is atypical. Moreover, we observe that all national income taxsystems have the property of constant tax rates at higher income, i.e. the tax functionis an affine function on the boundaries. Including this assumption will allow a muchsimpler characterization of all arbitrage opportunities.23

Definition 11. A tax function T (·) is affine on the boundaries iff there is a sufficientlylarge tax base x0 � 0 and two numbers T+, T− such that for all x > x0 we have

T (x) = T+ + τmax · x

22Notice that ∂T (wt) can be a single number if the tax liability function is differentiable at wt. Otherwiseit is a closed interval comprising of left- and right-derivative using the fact that ∂T (wt) is alwaysconvex and closed for arbitrary wt (see Rockafellar (1997, p. 217ff.)).

23We are grateful to Tyrrell Rockafellar who suggested these boundary conditions.

17

and for all x < −x0T (x) = T− + τmin · x

with 0 ≤ τmin ≤ τmax < 1 denoting the minimal and maximal marginal tax rate.

In this special case, the domain of the conjugate is easy to compute since it coincideswith the union of all possible marginal tax rates. Because ∂T (wt) is convex and closed (onR), it represents an interval given by ∂T (wt) = [T ′−(wt), T

′+(wt)]. If T (·) is differentiable

at wt, the interval degenerates into a single element, i.e. ∂T (wt) = {T ′(wt)}.We can nowshow the following.

Corollary 12. Let τt be the implied tax rates for given market prices pt. Let the taxliability function be affine on the boundaries. Then the following results hold true.

1. The market is free of arbitrage iff T ′−(wt) ≤ τt ≤ T ′+(wt) for all t.

2. There are bounded arbitrage possibilities iff τmin ≤ τt ≤ τmax for all t and for atleast one t′ the condition τt′ < T ′−(wt) or T ′+(wt) < τt′ holds.

3. There are unbounded arbitrage opportunities iff for at least one implied tax rate wehave τt < τmin or τt > τmax.

The results can be summarized as shown in Figure 3.

τt

0% τmin T ′−(wt) T ′+(wt)τmax 100%

no arbitrage bounded arbitragebounded arbitrage arbitragearbitrage

Figure 3: Illustration of Corollary 12. The range of values τt permits different types ofarbitrage.

Proof. First we proof that under piecewise linearity dom(T ∗) = [τmin, τmax]. Since

T ∗(τ) := supx

τx− T (x) =⇒ ∀x T ∗(τ) ≥ τx− T (x)

Fenchel’s inequality holds. Now suppose τ > τmax and T ∗(τ) finite, then by definitionτ ∈ dom(T ∗). Using Fenchel’s inequality for sufficiently large x, exploiting piecewiselinearity, we have

T ∗(τ) ≥ (τ − τmax) · x− T.

18

Since this inequality holds for all sufficiently large x and by assumption τ − τmax > 0 weget T ∗(τ) = +∞ which contradicts our assumptions. We conclude for all τ ∈ (τmax,∞)

that τ /∈ dom(T ∗).Analogously, it is shown that for all τ < τmin we have T ∗(τ) =∞ resulting in

dom(T ∗) ⊂ [τmin, τmax].

Further from lemma 6 we have⋃x

∂T (x) ⊂ dom(T ∗)

and because of Rockafellar (1997, Theorem 24.1, p. 227-229 i.c.w. Theorem 24.3, p. 232)accounting for differentiability at the boundaries we have

[τmin, τmax] =⋃x

∂T (x).

Together with inequality (3.2) this results in

[τmin, τmax] ⊂ dom(T ∗) ⊂ [τmin, τmax].

and finally dom(T ∗) = [τmin, τmax].The rest of the statement follows directly from theorem 10.

We now consider the case in which x1t − a1t = rfp1t−1 holds for all t. This refers to the

commonly known economic gains taxation presented by Preinreich (1951) and Samuelson(1964).24

Theorem 13. Let x1t − a1t = rfp1t−1 for all t. Then the market is free of arbitrage iff

pt−1 =pt + xt1 + rf

t = 1, . . . , T.

If there is at least one t′ for which the above equality does not hold, unbounded arbitrageprofits exist.

Note that in the above theorem the valuation formula is independent of any marginaltax rate τt and does not differ from arbitrage-free bond prices in a setting without taxa-tion. Under economic gains taxation no bounded arbitrage opportunities do exist.

24Although the authors assume a linear tax system. The following result also holds true for nonlinear(convex) tax systems fulfilling the assumptions in chapter 2.1.

19

Proof. As shown in the proof of theorem 10, our primal problem (P) can be solved bysolving the dual problem (D*) given by

− infτt∈dom(T ∗),t=1,...,T

T∑t=1

T ∗(τt)− τtwt + T (wt)∏ts=1

(1 + rf (1− τs)

) (D*)

s.t. pt−1 =pt + (1− τt)xt + τtat

1 + rf (1− τt)t = 1, . . . , T.

Let now x1t −a1t = rfp1t−1 hold for all t. A little bit of algebra shows that in this case the

constraints are independent of τt. Since the objective function in (D*) is non-negativeand we have ∂T (wt) ⊂ dom(T ∗) for all t, we get (D*) is equivalent to

sup 0

s.t. pt−1 =pt + xt1 + rf

t = 1, . . . , T

with p∗ = d∗ = −∞ in case of infeasibility and nil otherwise.

3.3. Three explanatory examples

For simplicity we assume in the following that wt = 0 holds for all t. We analyze threedifferent tax liability functions (see Figure 4).

Fx

T (x)

(a) Tax with allowance.

x

T (x)

τmax

τmin

(b) Gains differentially taxedthan losses.

−A

Ax

T (x)

τmax

τmin

τ

(c) Three tax brackets.

Figure 4: Three examples of tax liability functions.

Tax with allowance We start analyzing a tax with allowance (see Figure 4a). Thecorresponding tax liability function is

T (x) = τ max(x− F, 0), τ ∈ [0, 1).

20

Tax payments apply when income exceeds F > 0, otherwise the tax is nil. Based on thenotation of Corollary 12 we have

τ0− = τ0+ = 0, τmin = 0, τmax = τ,

and ∂T (0) = {T ′(0)} = {0}. Corollary 12 states that the only prices compatible withno-arbitrage are those for which τt = 0 holds, meaning that taxes dissolve. Note thatthe result remains the same for tax functions which are not necessarily linear beyond Fbut rather when they are quadratic (as is the case in Germany, for example). For allarbitrage opportunities to be bounded, one must have an implied tax rate τt ∈ (0, τ ].For all other tax rates, the arbitrage opportunity is unbounded.

Different taxation of gains and losses The second example represents a tax in whichgains and losses are taxed differently (see Figure 4b). This type of tax has alreadybeen discussed in equation (5) to illustrate the definition of a subdifferential. Unlike theprevious tax example, we have

τ0− = τmin, τ0+ = τmax.

There are no arbitrage opportunities if the implied tax rates are contained in [τmin, τmax].For all implied tax rates that do not fall into this interval we have unbounded arbitrageopportunities. Note that there are no bounded arbitrage opportunities.

Three tax brackets The third example shows a tax function with three different valuesof marginal rates τmin < τ < τmax (see Figure 4c):

T (x) =

τmin · x+ (τmin − τ) ·A if x < −A

τ · x if −A ≤ x ≤ A

τmax · x+ (τ − τmax) ·A if A < x.

Thenτ0− = τ0+ = τ.

Thus, the only implied tax rate offering no arbitrage is τt = τ . For all τt ∈ [τmin, τmax] \{τ} there are bounded arbitrage opportunities. For all other values of τt the arbitrageopportunity is unbounded.

21

3.4. Intuition of the results

At first glance, our results may seem difficult to grasp since for investors with no initialincome (wt = 0) a marginal tax rate at a base of zero is, in many tax systems worldwide,usually nil, for example if there are tax offsets. But then, a valuation in the presenceof non-linear taxation as in (1) implies a neglect of taxes. In order to understand theintuition of the result, we draw an analogy from production and cost theory. It is wellknown that the marginal cost curve crosses average (variable) costs at its minimum point.The same idea can be applied to taxes. Taxes are minimal if the average tax rate is equalto the marginal tax rate. The convexity of the tax functions implies that in this case thetax base must be zero, as we will show below. This explains our result intuitively.Convex functions have the basic property of being nondifferentiable only at a countably

infinite number of points.25 For the next theorem, we assume that the number of non-differentiabilities is finite.Let ∂T (0) = [τ0−, τ0+] and τmin, τmax denote the minimal and maximal marginal tax

rates; x0 ≥ 0 is the largest tax base for which τ0+ ∈ ∂T (x0) still holds. For simplicity,we focus on positive tax bases.

Theorem 14. Let the tax liability function be convex, affine at the boundaries and non-differentiable at a finite number of points. The tax liability function then has the followingproperties:

1. If x0 > 0, then the tax liability function is linear on [0, x0]. Then, marginal andaverage tax rates coincide on (0, x0).

2. For all x ∈ (x0,∞), marginal tax rates exceed average tax rates, i.e. ∂T (x) > t(x)

and marginal tax rates are in (τ0+, τmax].

3. For x→∞ average tax rates approach marginal tax rates or, equivalently, limx→∞ t(x) =

τmax.

The first property defines the set of implied tax rates offering no arbitrage opportunitiesin prices if wt = 0 for all t. That is the case in which marginal and average tax ratescoincide. For those tax rates there is no arbitrage.The second property defines the set of implied tax rates with bounded arbitrage. Here,

marginal and average tax rates differ. Unbounded arbitrage opportunities only exist formarginal tax rates outside of [τmin, τmax]. As a conclusion, a typical tax function can berepresented as in Figure 5.25We call a set S countably infinite if there is a one-to-one-mapping from S to the natural numbers N.

This statement about convex functions is proved, for example, in Zajíček (2007).

22

x0x

T (x)

T (x)

x0x

T ′(x),t(x)

T ′(x)

t(x)

T ′(x)=t(x)

Figure 5: Typical course of a convex tax liability function (left) and it’s marginal andaverage tax rate (right). The tax function is linear on [0, x0] and marginaland average tax rates coincide at this interval (T ′(x) = t(x)). Outside of thisinterval the marginal rate is always greater than the average rate (T ′(x) > t(x)).

Proof. We start with the first claim. Let x0 > 0 and

∂T (0) = [τ0−, τ0+], τ0+ ∈ ∂T (x0).

We choose x ∈ (0, x0). Because of monotonicity of the subdifferential (see Hiriart-Urrutyand Lemaréchal (2001, Theorem 6.1.1, p. 199)) we have for all τ ∈ ∂T (x)

(τ0+ − τ) · (x0 − x) ≥ 0, (τ − τ0+) · (x− 0) ≥ 0,

which results in τ = τ0+. It follows ∂T (x) = {τ0+} that is T (·) is differentiable on (0, x0),especially linear with constant tax rate τ0+.We prove the second claim26

∂T (x) ≥ t(x),

where the inequality holds for all subgradients. To see this, let g ∈ ∂T (x). From property(6) for x and y = 0 we get

T (0) ≥ T (x) + g · (0− x) ⇐⇒ g · x− T (x) ≥ 0

which proves the claim.The third claim

limx→±∞

T ′(x)− t(x) = 0

is shown as follows. We prove that tax liabilities move to infinity when there is a taxbase x with positive subgradient 0 < g ∈ ∂T (x). Suppose tax liabilities are bounded

26Note that the second statement holds true for negative tax bases. In that case it is ∂T (x) ≤ t(x).

23

with T (y) ≤ M for all y. Since there is a tax base x with 0 < g ∈ ∂T (x) it follows forall y > M−T (x)

g + x using the definition of a subgradient that

T (y) ≥ T (x) + g(y − x) > M,

which contradicts boundedness of T (·).Since T (·) is non-differentiable at a finite number of points and T (x) is monotone

increasing, we can assume differentiability of T (·) for higher x � 0 and by using del’Hospital we get

limx→∞

t(x) = limx→∞

T (x)

x= lim

x→∞T ′(x) = τmax.

The limit exists since from convexity the average tax rate t(·) is bounded above by100%.

4. Conclusion

We investigate arbitrage opportunities in a capital market model under certainty withmultiple trading periods and convex tax liability functions. By deducing period-specificimplied tax rates from prevailing market prices of bonds, we can give a complete char-acterization of arbitrage depending on the investor’s initial income. If all implied taxrates equal marginal tax rates at the tax base of the initial income, then the no-arbitragecondition holds. This condition is necessary as well as sufficient.In addition, we show that arbitrage opportunities can be bounded or unbounded.

Arbitrage opportunities are unbounded iff there is at least one implied tax rate whichis not equal to any possible marginal tax rate (for an arbitrary tax base). In all othercircumstances, bounded arbitrage opportunities exist.From our point of view, the opportunity to realize unbounded arbitrage gains still leads

to unbearable contradictions in a market model. The ability to create financial resourcesout of nothing makes it easy to exceed the investor’s budget constraint and therefore willnot result in an optimal solution to the individual’s utility maximization problem. Westrongly believe that the same applies to bounded arbitrage opportunities. Let us assumethat such an opportunity exists: then, at least one investor would demand more than herendowments allow, even after tax. If all other investors meet their budget restrictions,then supply cannot equal demand. But this contradicts the market-clearing condition.Therefore, bounded arbitrage opportunities are inconsistent with equilibrium as well.Our results conform with classical production and cost theory if we assume an investor

with no initial income. By identifying taxes as costs, we show that in this case theoptimum marginal and average tax rates coincide in a market free of arbitrage.

24

In future studies we will extend our theory to uncertainty.

25

References

Bradford, David F. (2000): Taxation, wealth, and saving. MIT Press, Cambridge, Mass.Brennan, Michael J. (1970): “Taxes, Market Valuation and Corporate Financial Policy”.National tax journal (23).4, 417–427.

Dammon, Robert M. and Richard C. Green (1987): “Tax Arbitrage and the Existence ofEquilibrium Prices for Financial Assets”. The Journal of Finance (42).5, 1143–1166.

Dermody, Jaime Cuevas and R. Tyrrell Rockafellar (1991): “Cash Stream Valuation Inthe Face of Transaction Costs and Taxes”. Mathematical Finance (1).1, 31–54.

Dybvig, Philip and Stephen A. Ross (1986): “Tax Clienteles and Asset Pricing”. TheJournal of Finance (41), 751–62.

Friedman, Milton (1962): Capitalism and Freedom. University of Chicago Press.Gallmeyer, Michael and Sanjay Srivastava (2011): “Arbitrage and the Tax Code”. Math-ematics and Financial Economics (4), 183–221.

Graham, John R. and Clifford W. Smith Jr. (1999): “Tax Incentives to Hedge”. Journalof Finance (54).6, 2241–2262.

Heintzen, Markus et al. (2008): “Die typisierende Berücksichtigung der persönlichenSteuerbelastung des Anteilseigners beim squeeze-out”. Zeitschrift für Betriebswirtschaft(78), 275–287.

Hiriart-Urruty, Jean-Baptiste and Claude Lemaréchal (2001): Fundamentals of convexanalysis. Grundlehren Text Editions. Springer, Berlin.

Mas-Colell, Andreu et al. (1995): Microeconomic Theory. Oxford University Press, NewYork.

OECD (2006): “OECD-Beschäftigungsausblick”. url: /content/book/empl_outlook-2006-de.

Preinreich, Gabriel A.D. (1951): “Models of taxation in the theory of the firm”. EconomiaInternazionale (4), 372–397.

Prisman, Eliezer Z. (1986): “Valuation of Risky Assets in Arbitrage Free Economies withFrictions”. Journal of Finance (41).3, 545–557.

Rockafellar, Ralph Tyrrell (1997): Convex analysis. 10. print and 1. paperback print.Princeton landmarks an mathematics and physics. Princeton Univ. Press, Princetonand N.J.

Ross, Stephen A. (1987): “Arbitrage and martingales with taxation”. Journal of PoliticalEconomy (95), 371–393.

Samuelson, Paul A. (1964): “Tax deductibility of economic depreciation to insure invari-ant valuations”. Journal of Political Economy (72), 604–606.

26

Schaefer, Stephen M. (1982): “Taxes and security market equilibrium”. In: FinancialEconomics: Essays in Honor of Paul Cootner. Ed. by William F. Sharpe and CathrynM. Cootner. Prentice-Hall, Englewood Cliffs (NJ), 159–178.

Sialm, Clemens (2009): “Tax Changes and Asset Pricing”. American Economic Review(99).4, 1356–1383.

Zajíček, Luděk (2007): “On sets of non-differentiabilty of Lipschitz and convex functions”.Mathematica Bohemica (132), 75–85.

27

A. Proof of theorem 10

The prove bases on standard arguments from dual convex analysis. We start with thefollowing Lemma.

Lemma 15. For the conjugate tax function T ∗(·) holds

(i) T ∗(τ) ≥ τ x− T (x) for all x ∈ R (Fenchel’s Inequality).

(ii) T ∗(τ) = τ x∗ − T (x∗) iff τ ∈ ∂T (x∗).

(iii) 0 < T ∗(τ)− τx∗ + T (x∗) <∞ iff τ ∈ dom(T ∗) \ ∂T (x∗).

Proof. (i) follows directly from the definition of the conjugate function.

(ii) follows immediately by Corollary 1.4.4 in Hiriart-Urruty and Lemaréchal (2001,p. 221).

(iii) By definition of dom(T ∗) as well as property (i) and (ii).

Starting with the primal problem (P) we analyze the dual Lagrange-Problem (D) givenby

supλ

infh

L(h, λ) (D)

s.t. λt ≥ 0 t = 1, . . . , T,

with λ = (λ1, . . . , λT )′ and the Lagrangian27

L(h, y, λ) : = p′0h0 −T∑t=1

λt(δwt (h)− nwt )

= p′0h0 −T∑t=1

λt{(pt + xt)

′ht−1 + T (wt)− T((xt − at)′ht−1 + wt

)− p′tht

},

using vector notation for ht, pt and xt as well as at for both assets (standard couponbond and bond to be valued).Let d∗ be the optimal value of (D). It is well known that (D) is concave and d∗ ≤ p∗

(weak duality). Since marginal tax rates are less than one, we find a parameter µ � 0,such that for h∗t = µ (1, 0)′

δwt (h∗) > nwt

27(·)′ denotes the transpose of a vector.

28

holds for all t. Thus strong Slater condition is fulfilled meaning that there is a portfolioconsisting of standard coupon bonds which represents an inner solution of (P). Therefore,we have d∗ = p∗ (strong duality).We transform the variable in the dual problem

yt := (xt − at)′ht−1 + wt

and rewrite (D) which leads to the following maximization problem

sup(λ,ν)

infh,y

L̂(h, y, λ, ν)

s.t. λt ≥ 0 t = 1, . . . , T,

where ν = (ν1, . . . , νT )′ are new Lagrange multipliers with Lagrangian

L̂(h, y, λ, ν) := (p0)′h0 −

T∑t=1

λt((pt + xt)

′ht−1 + T (wt)− T (yt)− (pt)′ht)

+

T∑t=1

νt((xt − at)′ht−1 + wt − yt

).

Setting λ0 = 1 and using that (pT )′hT = 0 for both bonds, we can combine sums to

L̂(h, y, λ, ν) =T∑t=1

λt−1 (pt−1)′ht−1

−T∑t=1

(λt(pt + xt)

′ht−1 − νt(xt − at)′ht−1 −T∑t=1

(νtyt − λtT (yt)− νtwt + λtT (wt)

),

which results in

L̂(h, y, λ, ν) =

T∑t=1

(λt−1pt−1 −

(λt(pt + xt)− νt(xt − at)

))′ht−1−

T∑t=1

(νtyt−λtT (yt)−νtwt+λtT (wt)

).

Note that some infima and suprema in the above problem can be simplified such that(D) writes as

sup(λ,ν)

T∑t=1

infht−1

(λt−1pt−1 − λt(pt + xt) + νt(xt − at)

)′ht−1 −

T∑t=1

supyt

(νtyt − λtT (yt)− νtwt + λtT (wt)

)s.t. λt ≥ 0 t = 1, . . . , T. (D1)

By just looking at the objective of (D1) one has

infht−1

(λt−1pt−1−λt(pt+xt)+νt(xt−at)

)′ht−1 =

0, λt−1pt−1 − λt(pt + xt) + νt(xt − at))

−∞, else.

29

Hence, (D1) is equivalent to

sup(λ,ν)−

T∑t=1

supyt

(νtyt − λtT (yt)− νtwt + λtT (wt)

)(D2)

s.t. λt ≥ 0, λt−1pt−1 − λt(pt + xt) + νt(xt − at) = 0 t = 1, . . . , T.

We now prove that all feasible λt in (D2) are positive. Suppose there is a t′ with λt′ = 0

in (D2). Then d∗ = −∞ unless vt′ = 0 in (D2). Due to the recursion property in theconstraints we get for all t < t′ immediately λt = νt = 0 and especially for t = 1 we have

λ0p0 = p0 = 0

which contradicts the existence of a riskfree coupon bond with p00 = 1. Thus, we concludepositivity of all feasible Lagrange multipliers in (D1).In this case

supyt

(νtyt − λtT (yt)− νtwt + λtT (wt)

)= sup

yt

(λtνtλtyt − λtT (yt)−

νtλtwt + T (wt)

)= λt sup

yt

(νtλtyt − T (yt)

)− νtλtwt + T (wt)

= λt

(T ∗( νtλt

)− νtλtwt + T (wt)

).

Plugging in this term into (D2) we get an equivalent problem

− inf(λ,ν)

T∑t=1

λt

(T ∗( νtλt

)− νtλtwt + T (wt)

)(D3)

s.t. λt > 0, pt−1 =λtλt−1

(pt + xt)−νtλt−1

(xt − at) t = 1, . . . , T,

with usual convention inf ∅ =∞. Thus infeasibility of (D3) results in p∗ = d∗ = −∞.By substituting τt := νt

λtand looking at the standard coupon bond it follows for the

optimal Lagrange multipliers

λt =λt−1

1 + rf (1− τt)t = 1, . . . , T.

Using an iteration argument we get

λt =1∏t

s=1

(1 + rf (1− τs)

) t = 1, . . . , T.

30

We conclude that (D3) is equivalent to

− infτt∈dom(T ∗),t=1,...,T

T∑t=1

T ∗(τt)− τtwt + T (wt)∏ts=1

(1 + rf (1− τs)

) (D*)

s.t. pt−1 =pt + (1− τt)xt + τtat

1 + rf (1− τt)t = 1, . . . , T.

Using (D*) we can finish the proof. Obviously τt are implied tax rates of the secondbond. Then it holds:

1. By using Fenchel’s inequality (see property (i) in 15) all summands in (D*) arenonnegative and p∗ = d∗ = 0 iff T ∗(τt) = τtwt − T (wt) for all t being equivalent,after using (ii) in lemma 15, to τt ∈ ∂T (wt) for all t.

2. By using (iii) in lemma 15 it follows −∞ < p∗ < 0 iff τt ∈ dom(T ∗) for all t andfor at least one t′ it holds τt′ ∈ dom(T ∗) \ ∂T (wt).

3. Since the sum in (D*) is finite with nonnegative elements it follows p∗ =∞ iff forat least one t we have τt /∈ dom(T ∗).

31