Optimal Investment with Corporate Tax Payments

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IN

DEGREE PROJECT

MATHEMATICS,

SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2017

Optimal Investment with

Corporate Tax Payments

EMIL TINGSTRÖM

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Optimal Investment with

Corporate Tax Payments

EMIL TINGSTRÖM

Degree Projects in Financial Mathematics (30 ECTS credits) Degree Programme in Applied and Computational Mathematics KTH Royal Institute of Technology year 2017

Supervisor at Ampfield AB: Torbjörn Hovmark Supervisor at KTH: Thomas Önskog

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TRITA-MAT-E 2017:14 ISRN-KTH/MAT/E--17/14--SE

Royal Institute of Technology

School of Engineering Sciences

KTH SCI

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Optimal Investment with Corporate Tax Payments

Abstract

This Master’s thesis examines the problem of optimal investment when corporate taxes have to be paid on capital gains. Tax payments share a lot of similarities with payoff from a call option where the underlying is the firm’s capital. How to optimally deal with this tax-option is a non-trivial problem that is complicated by the feedback between the value of the tax-option and the strategy the firm uses to handle it.

For the case with only one tax payment it is possible to derive an explicit expression for the optimal strategy using a martingale method. The tax payment introduces some interesting properties to the optimal wealth process, such as a non-zero probability of ending up exactly at the tax basis at the terminal date.

The optimal strategy is then generalized to the problem with multiple tax payments using the martingale framework. A numerical method of calculating the optimal strategy based on trinomial trees is also presented and implemented. This is then used to verify the theoretical results and to calculate the optimal strategy with an increasing number of tax payments.

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Optimal Investering med Företagsbeskattning

Sammanfattning

Den här Masteruppsatsen undersöker problemet med optimal investering för ett företag som måste betala skatt på kapitalvinster. Skattebetalningar har många likheter med utfallet för en köpoption där det underliggande är företagets kapital. Att hantera denna skatteoption optimalt är ett icke-trivialt problem, som kompliceras av återkopplingen mellan värdet på skatteoptionen och den strategi företaget använder för att hantera den.

För fallet med enbart en skattebetalning är det möjligt att härleda ett explicit uttryck för den optimala strategin med en martingalmetod. Skattebetalningen introducerar några intressanta egenskaper hos den optimala kapitalprocessen, såsom en nollskild sannolikhet att hamna exakt på omkostnadsvärdet vid slutdatum.

The optimala strategin generaliseras sedan till problemet med flera skattebetalningar inom ramverket för martingalmetoden. En numerisk metod för att beräkna den optimala strategin baserad på trinomialträd presenteras också och implementeras. Den används sedan för att verifiera det teoretiska resultatet och beräkna den optimala strategin med ett ökande antal skattebetalningar.

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Acknowledgements

I would like to thank Torbjörn Hovmark, who not only provided the original topic of this thesis but also gave continuous input and ideas. This thesis was written at Ampfield AB, which provided office space and equipment during the project. I would also like to thank my supervisor Thomas Önskog for valuable comments and feedback.

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List of variables

Variable Description Defined in section

Πt The firm’s trading portfolio 2.5

Xt The total capital available to the firm 2.5

ϑt A trading strategy determining the capital

ex-posure to an asset at the time t

2.5

θt The portion of the available capital exposed

with a trading strategy at the time t, ϑt= θtXt

2.5

τ Tax rate 3.0

K The tax basis 3.0

P The real-world probability measure 2.5 Q The risk-neutral probability measure 2.5

WQ

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1 Introduction

The first chapter introduces the subject and main objective of the thesis. Some necessary limitations of the scope are discussed as well as the disposition of the report.

1.1 Optimal investment and taxes

Corporate tax payments have a clear option-like structure. The government will take a portion of the firm’s profits, but are rarely willing to share any part of the firm’s losses. The firm could be seen as the writer of a call option with the firm’s capital as the underlying asset and the government would be the holder of this option. The delta of the tax-option will give the firm negative exposure to its own profits. From the perspective of the firm’s shareholders this exposure is undesirable. Ideally, the firm would want to reduce the negative exposure by investing more than usual in the underlying business while maximizing the efficient use of its capital. This is however not a trivial problem to solve. Altering the exposure to handle a potential tax payment will in turn affect the taxes that might be paid, and hence again necessitate further alterations in the exposure. For these reasons, contrary to most derivatives commonly analyzed in mathematical finance, the value of the tax-option is determined mostly by endogenous factors. Or put differently, the value of the taxes paid will be determined by the choices made by the firm on how to deal with them.

The possibility of the firm to alter its exposure to tax payments depends on its ability to replicate the profits by taking a position in the underlying assets. For most firms it is usually not possible to take a position in the underlying profits without incurring significant overhead costs or other restrictions. There is however one type of firm for which the profits are completely replicable and where the firm has a significant amount of freedom in determining its business exposure. A proprietary trading firm is a company which trades its own capital in financial markets for profit. Today the trading is usually made in a high-frequency fashion and with automated trading systems. For a proprietary trading firm its business is the underlying trading portfolio and the firm chooses its exposure to the portfolio to maximize the efficient use of its available capital, while maintaining an acceptable level of risk. Through the use of financial instruments, such as futures contracts, the firm is able to adjust its leverage on a trade-by-trade basis which creates the opportunity to implement a strategy to deal with the tax-option in an optimal manner.

1.2 Purpose and main objectives

The purpose of this thesis is to derive an optimal strategy that allows a proprietary trading firm to reduce the negative effects of tax payments and to maximize the efficient use of its capital. The objectives can be formulated, in no particular order, in the following points.

• Derive a theoretical foundation of the optimal strategy and gain an understanding of how it relates to the value of the tax-option.

• Develop and implement numerical methods to calculate the optimal strategy with multiple future tax payments.

• Gain an understanding of the sensitivity of the optimal strategy to changes in the parameters. • Approximate the optimal strategy with an infinite time horizon and future tax payments. • Calculate the impact on the utility of using the tax-optimal strategy compared to the

optimal strategy when taxes are disregarded.

1.3 Limitations of the scope

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a single asset with constant volatility and growth rate. The assumptions of constant volatility and growth rate are necessary in order to derive an optimal strategy with a reasonable level of complexity. The sensitivity of the results with respect to changes in these two parameters will be analyzed with some numerical examples. The trading portfolio will represent the aggregate of the contracts and claims held by the firm as a result of its trading. The model will also include a risk-free asset, which is assumed to yield a constant interest rate.

The taxation modelled here will be a somewhat simplified version of the real world situation. Usually the taxable income will depend on the rules regarding tax deduction which will vary with the jurisdiction. There is also the possibility of interperiod tax allocations, which may allow the firm to delayed recognition of taxable income by allocation part of the pre-tax income to a tax allocation reserve. The situation modelled here will disregard the effect of such possibilities, and will consider tax as a fixed fraction of any positive net result paid at the end of the year. Some other important aspects faced by the firm that can be handled within the framework presented will also be analyzed. One such example is the costs paid by the firm that are not directly related to the trading portfolio. Costs not directly proportional to the capital will introduce the possibility of bankruptcy and a liquidation of the firm’s capital. This will affect the business decisions and the optimal strategy besides the effect of the tax payments. Since the main focus here is the effect of taxation on the optimal strategy, a solution for a simplified case with costs is instead given in Appendix A.

1.4 Disposition of the report

Chapter 2 will review some important theory and results from mathematical finance that will be used throughout the thesis. Some previous research on the subject in presented and its relation to the material. After that the framework for the thesis is developed and used to derive the optimal strategy without taxes.

In Chapter 3 the the optimal strategy for the one-year problem with a tax is derived using the martingale method. An explicit expression for the optimal strategy is presented and given an intuitive interpretation.

In Chapter 4 the martingale method is applied to the extended problem with multiple tax payments over several years. Some properties of the solution is analyzed and discussed to provide some intuition for the numerical results.

Chapter 5 provides a numerical method to calculate optimal strategies using a trinomial tree model. An algorithm based on dynamic programming is then introduced which allows the optimal strategy to be calculated with multiple future tax payments.

The results of the numerical implementation is presented in Chapter 6. Some relevant properties of the optimal strategy is exemplified and discussed, including the convergence of the N -years problem.

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2 Previous research and theoretical background

This chapter starts out by presenting some previous research related to the subject of investment under taxation and familiarizes the reader with some basic results that will be used in the thesis. After this comes a section that describes and motivates the models used in the thesis as well as the utility maximization problem problem at hand.

2.1 Previous research

Since the seminal papers of Merton [8, 9] there have been multiple attempts to extend the problem of optimal consumption and investment decisions in a financial market to include practical considerations, such as incomplete markets, portfolio constraints and transaction costs. Although taxes represent a significant portion of the friction an investor will face in the real world, the literature on the subject is comparatively scarce. Capital gains are only taxed after they have been realized, which creates the option to defer the payment. Further complications come from the precise calculation of the tax, which will depend on both the selling price and price for which the securities have been purchased, creating a path-dependency in the optimal policy.

The existing literature mainly focuses on optimal timing decision involving tax bases and liquidation. Constantinides [2] shows that, if shortselling was costless and unconstrained, the optimal policy would be to realize all losses immediately and to defer all gains as long as possible. Deferral of the gains is achieved by shortselling those securities with an embedded capital gain instead of realizing the gain, avoiding any tax payment and thus capturing the time value of taxes. Also, losses are realized in order to get a tax rebate and then re-balancing the portfolio by buying securities at the current market price. The sale of a security at a loss only to repurchase the same or a similar security is know as a wash sale.

One significant strand of literature is based on dynamic programming. DeMiguel and Uppal [4] formulated the problem using the exact tax basis and showed how to determine the optimal portfolio policy via nonlinear programming. The authors examine the optimal portfolio composed of a risk-free asset and one risky asset modeled in discrete time with the objective to maximize the expected utility at some terminal date. Tahar et al. [6] formulates the problem in continuous-time using an averaging rule to compute the tax basis and derive a first order explicit approximation of the value function of the problem. None of these papers relates much to the problem as formulated in this thesis. Here optimal timing decisions will not be considered, whereas the focus will rather be on the more fundamental problem of determining how much to risk on investment when taxes will always have to be paid on capital gains at the end of the year.

A previous Master’s thesis written at Ampfield by Danell and Eriksson [3] attempted to derive the arbitrage-free value of future tax payments under the assumption that the delta from the tax-option was hedged. The authors managed to numerically value the next N annual tax payments and approximate the value of an infinite number of future tax payments for a firm that pays a dividend of the same size as the tax payments. The framework used considers the firm’s total capital denoted by Vtseparated into

Vt= Ut+ ϕt (1)

where ϕt is the value of the tax-option and Ut is the remaining capital which belongs to all

the stakeholder in the firm excluding the government. The firm is assumed to pay a constant continuous cost of Cdt, and the process Utis modelled with the Q-dynamics

dUt= −Cdt + σUtdWtQ (2)

until the time of bankruptcy. By specifying the process Ut as unaffected by the tax-option,

the value of ϕt(Ut) can be calculated from arbitrage reasoning as if it was hedged. Since Ut

is now unaffected by the tax payments, optimal portfolio allocations can be done for Utwhile

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While this delta-hedging will eliminate the negative exposure caused by the tax-option, it will also increase the total exposure to the underlying asset and hence increase the total amount of taxes paid. The implicit assumption in the thesis was that the expected utility for the firm would increase if the strategy of delta-hedging the tax-option was used. This assumption was however not verified, only a quick Monte Carlo simulation that indicated an increase in expected utility was included in the appendix. As will be seen, the method of delta-hedging the tax payments presented in the thesis is in fact not optimal. However, it does share similar characteristics to the optimal strategy.

Seifried [10] introduces a novel approach compared to the existing literature by considering the optimal investment problem with taxes due on the gains in total wealth only at the end of a time horizon. This model applies directly to so called tax-deferred pension accounts, available in both the U.S. and Germany, where taxes are due exclusively on total capital gains as accrued until the date of retirement, i.e., the liquidation of the entire portfolio, and with no transactions costs for portfolio re-allocations. Using a Brownian framework and a modification of the standard martingale method applied to the after-tax utility function, he derives explicit formulas for the optimal strategies with the help of Clark’s formula; an important result in Malliavin calculus. This is similar to the setting used in Chapter 3, although in this thesis the derivation of the optimal strategy is based on the martingale method as presented by Björk [1] using standard results in stochastic calculus.

2.2 Stochastic calculus

This section aims to give an overview of the main results of stochastic calculus, which can be found in [1].

Definition 1. A Geometric Brownian Motion (GBM) is a process defined by the stochastic

differential equation

(

dXt= αXtdt + σXtdWt

X0= x0

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with the solution

Xt= x0· exp n α −1 2σ 2_{t + σW} t o (4)

The expected value is given by

E[Xt] = x0· eαt (5)

with the variance

V ar[Xt] = x20· e

2αt_{· (e}σ2_t

− 1) (6)

Theorem 1. Itô’s lemma (one dimension)

For a twice differentiable scalar function f (t, Xt) where Xt is defined by the GBM in definition

(1) the differential can be expressed as

df (t, Xt) = ∂f ∂t + αXt ∂f ∂x + 1 2σ 2_X2 t ∂2_f ∂x2 ! dt + σXt ∂f ∂xdWt (7)

Theorem 2. Radon-Nikodym derivative

Let P and P0 be probability measures on (Ω, F ) with P0  P. Then there exists a non-negative

random variable L that is unique P-almost surely such that

P0(A) = EP[1AL], A ∈ F (8)

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Adapting this to the filtration F = {Ft}t≥0, the process L is written as LT = dP 0_|F T dP|FT (9)

analogous to a derivative in calculus. By definition L is a P-martingale

Lt= EP[LT|Ft], t ∈ [0, T ] (10)

Here P0|Ft denoted the measure P0 restricted to Ft. If the measures are not only absolutely

continuous with respect to one other but also equivalent, then every semi-martingale under P will also be a semi-martingale under P0. Changing from one probability measure to another is done using the Girsanov theorem.

Theorem 3. The Girsanov theorem

Let WP _{be a Wiener process on (Ω, F , P, F) and let ϕ be any process adapted to F}

t. Choose a

fixed T and define the process L on [0, T ] by

dLt= ϕtLtdWtP (11)

L0= 1 (12)

with the solution

Lt= e Rt 0ϕsdW P s− Rt 0ϕ 2 sds ₍₁₃₎ Assume that EP_[L T] = 1 (14)

and define a new probability measure Q on FT by

LT =dQ|F T dP|FT (15) Then dWP t = ϕdt + dWtQ (16)

where WQ _{is a Q-Wiener process.}

We will refer to L as the likelihood process, which can be used to compare expectations under different measures. A significant application of this is in arbitrage-free pricing.

2.3 Arbitrage-free pricing

Some standard results in mathematical finance that will be used throughout the thesis are presented here. Starting with some definitions and using the results of stochastic calculus we are able to derive the fair, arbitrage-free price of a contingent claim in a complete market.

Definition 2. Self-financing portfolio. Let hi

t denote the number of shares of stock i in the

portfolio at time t, and Si

t the price of stock i in a frictionless market with trading in continuous

time. Let V_th= n X i=1 hi_tS_ti (17)

Then the portfolio ht= (h1t, . . . , hnt) is self-financing if

dV_th=X

i

hi_tdS_ti (18)

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Definition 3. The numéraire is an asset with a strictly positive price that can be used as a

basic standard to express the relative prices of other assets.

An example of a numéraire is a standard bank account, which can be used to form a discounted price process.

Definition 4. An arbitrage opportunity in a financial market is a self-financing portfolio h

such that

V₀h= 0 (19)

P (V_Th≥ 0) = 1 (20)

P (VTh> 0) > 0 (21)

A market is said to be arbitrage-free if there are no arbitrage opportunities.

Arbitrage gives the opportunity to construct a portfolio that from zero initial investment will almost never give a negative payoff but with a strictly positive probability for a strictly positive payoff. Making money out of nothing without any risk is unreasonable from an economic standpoint, hence models of financial markets in mathematical finance are usually assumed to be efficient and arbitrage-free.

A convenient way of characterizing arbitrage-free markets is by means of martingale theory.

Definition 5. An Equivalent Martingale Measure (EMM), denoted Q, is a probability

measure equivalent to the real-world measure P for which each tradable asset’s discounted price process is a Q-martingale.

An equivalent martingale measure is also called a risk-neutral measure since in the Q-world, the price of an asset is simply the discounted expected value at a future date, without regard for the risk.

Theorem 4. The First Fundamental Theorem of Asset Pricing (FTAP1)

There exists an Equivalent Martingale Measure Q if and only if the market is arbitrage-free.

Since an EMM will reproduce an arbitrage-free price of an asset, it is useful to ask if the measure and hence the price is also unique. A stronger statement can be made using the definition of attainable and a complete market.

Definition 6. A given contingent claim X is said to be attainable is there exists a self-financing

portfolio h such that

V_Th= X (22)

If any conceivable claim based on the possibles states of the world is attainable we say that the market is complete.

Theorem 5. The Second Fundamental Theorem of Asset Pricing (FTAP2)

An arbitrage-free market that includes a risk-free numéraire is complete is and only if the equivalent martingale measure is unique.

An important achievement of arbitrage theory is the development of valuation models for common derivatives. Most famous is perhaps the Black–Scholes model for the valuation of options. The value of a European call option on a non-dividend paying stock St in the Black–Scholes model is

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Here Φ(·) is the cumulative distribution function of the standard normal distribution.

While arbitrage can be used to characterize efficient markets, it does not help guide investment decisions in an efficient market. To this end utility functions are used.

2.4 Utility theory and utility maximization

This section gives some basic overview of utility functions and preferences, most of which can be found in Hult et al. [5].

Definition 7. A Utility function U : R 7→ R measures the utility of a random portfolio value

X from the investor’s perspective. Suitable assumptions of the function U are that it is strictly increasing, representing that more money if preferred to less, and strictly concave, representing that the increase in utility from additional monetary units is decreasing with increasing wealth.

A seminal result in decision theory is the von Neumann-Morgenstern utility theorem, which states that any individual whose preferences satisfies four axioms will, when faced with risky (probabilistic) outcomes of different choices, behave as if he is maximizing the expected value of some utility function defined over the potential future outcomes. The theorem has been the basis for expected utility theory, which concerns peoples preferences and subjective valuation regarding choices that have uncertain outcomes (e.g., gambles). While rational behaviour dictates that a utility function exists, it is not immediately clear what that function looks like. Also, utility functions are not uniquely defined since any positive affine transformation of the utility function will lead to the same decision with regard to the expected utility. To this end, utility functions are conveniently parametrized using a measure that stays constant with respect to these transformations.

Definition 8. The Arrow–Pratt measure of absolute risk-aversion (ARA), also know

as the coefficient of absolute risk aversion, is defined as

A(x) = −U

00_(x)

U0_(x) (26)

Note that by definition A(x) has the dimension money−1. Hence a natural parametrization is

A(x) = 1

b + ηx, b + ηx > 0 (27)

where η is a dimensionless constant and the constant b has the dimension money.

This can be solved to get popular HARA (Hyperbolic absolute risk aversion) family of utility functions. U (x) =      1 η−1(b + ηx) 1−1/η_, _{η 6= 1, 0} ln(b + x), η = 1 −bx−x/b_, _{η = 0} (28)

A special case in this family is the isoelastic, or power utility function

U (x) =

(_xγ

γ , γ < 0

ln(x), γ = 0 (29)

The parameter γ relates to the constant relative risk aversion with −xU_U000_(x)(x) = 1 − γ. This

often has the implication that decision-making is unaffected by scale. For instance, in a market with both risk-free and risky assets, under constant relative risk aversion the fraction of wealth optimally placed in risky assets is independent of the level of wealth.

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2.5 Modelling the firm’s capital

The trading portfolio will be modelled as a single asset and denoted by Π. The portfolio will usually consist of a collection of multiple individual financial contracts with allocations decided by the firm’s trading strategies. By focusing on the portfolio as an aggregate of the individual contracts that constitute the portfolio, the problem relating to the firm-wide decisions (e.g., dealing with taxes) can be decoupled from the portfolio-specific decisions (i.e., when to buy and sell a given contract).

Let Xt denote the firm’s total capital available to be exposed to risk at some time t. The trading

portfolio will be modelled under the real-world probability measure P with a stochastic differential of the form

dΠt= µΠtdt + σΠtdWtP (30)

where WP _{is a P-Wiener process and µ > 0 and σ > 0 are known constant. Transaction costs}

will not be included in the problem formulation, however if the trading is assumed to occur with sufficient frequency they can be seen as included in the drift µ. This is since any fixed trading cost will be split over a large number of trades to such a degree that the fixed cost per trade can be neglected. The variable transactions costs will be proportional to the capital traded and hence, form a part of the drift.

We also assume that there exists a risk-free asset Btwith a constant interest rate r and with the

dynamics

dBt= rBtdt (31)

The asset B will serve as our numéraire and together with the trading portfolio form a complete market. Standard arbitrage-theory applies and we have a unique equivalent martingale measure which can be used to price claims on the trading portfolio.

The firm is assumed to invest all capital in the risk-free asset to be used as collateral, or margin, to assume risk in the trading portfolio. The firm decides on some amount ϑt as the capital

exposed to the trading portfolio at time t, and continuously either receives or pays an amount given by the changes in value of the claims in the trading portfolio. This means that the dynamics for the firm’s capital Xtwill be

dXt= µϑtdt + σϑtdWtP+ rXtdt (32)

Here ϑt= ϑt(Xt, t) is assumed to be a predictable process, adapted to the filtration generated by

Πt. The process ϑ will be referred to as a strategy since it gives the amount of capital that the

firm decides should be exposed to the trading portfolio. Another relevant assumption is that ϑt

is admissible in the sense that the total capital is non-negative almost surely. This implies that it can be written as ϑt= θtXtfor some θtfor all t. This precludes any doubling strategies where

the firm would be able to assume risk without having any collateral.1

To ease later calculations we will from here on chose to denote capital Xtin the risk-free numéraire. 2 _{The discounted dynamics will be}

dXt= µϑtdt + σϑtdWtP (33)

With everything denoted in the numéraire, the equivalent martingale measure Q is given by

dWP

t = ϕdt + dWtQ (34)

where ϕ := −µ_σ_{. Under this measure Q, the discounted firm capital X}t will be a martingale with

dXt= σϑtdWtQ (35)

1_{Usually there are strict limits on the amount of risk that can be assumed with a given margin. We disregard} this temporarily, and simply assume that the optimal strategies do not violate the margin requirements.

2_{At a given time t, it makes no difference if X}

tis said to be denoted in the numéraire or not since its value is

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With a model formulation in place, we can start to solve the problem of maximizing the use of the firm’s capital by maximization of expected utility, starting with the tax-free case.

2.6 Optimal leverage for a tax-free firm

Using the dynamics in equation (35) the value of the firms capital at some time t ≥ 0 given initial capital x and strategy ϑ is given by

X_tϑ= x + µ Z t 0 ϑsds + σ Z t 0 ϑsdWsP (36)

Let Θ(x) denote the set of admissible strategies with an initial capital x such that, for any

ϑ ∈ Θ(x), it holds almost surely that Xϑ

t ≥ 0 for all t.

By fixing a terminal date T , the objective is to solve the utility-maximization problem given by

sup ϑ∈Θ(x) EPh_U_Xϑ T i (37)

The optimal strategy is found by optimizing the criteria of expected utility at time T over the set Θ(x). It is not obviously clear how to solve this without further specification of Θ(x). The problem can however be solved rather easily if we restrict the search to optimal strategies of the form ϑt= θXt, where θ ∈ R. As we will see this is not an all too unreasonable restriction for a

specific choice of utility function.

The portfolio Xtwill now be driven by a geometric Brownian motion, with the dynamics

dXt= µθXtdt + σθXtdWtP (38)

Solving this gives the portfolio value as

XT = x exp µθ − 1 2σ 2_θ2_{T + σθW}_P T (39)

where we use the notation WP T :=

RT

0 dWsP.

Assuming the firm uses a power utility function of the form

U (x) =x

γ

γ (40)

the utility of the portfolio at the future date T will be

U (XT) = xγ γ exp µθ −1 2σ 2_θ2_{γT + σθγW}P T (41)

Taking the expectation gives the expected utility as

E[U (XT)] = xγ γ exp µθ −1 2σ 2_θ2_{γT +} 1 2σ 2_θ2_γ2_T (42)

The optimization problem is now

sup θ EPh_U_Xϑ T i (43)

But this is easy, taking the derivative with respect to θ of the exponent and setting it zero gives

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hence the optimal exposure θ is

ˆ

θ = µ

σ2_{(1 − γ)} (45)

A more detailed analysis will show that this is also the optimal solution to (37); this is in fact the solution to the well-known Merton’s portfolio problem [8]. The strategy ˆϑt=_σ2_(1−γ)µ Xtwill be

referred to as the Merton strategy. As a result of the constant relative risk-aversion when using power utility, the optimal strategy is to apply a constant proportional leverage, independent of the level of wealth. Intuitively, the investor should invest a fraction corresponding to the ratio of the expected return divided by the variance scaled by the relative risk aversion. Letting

γ → 0 corresponds to the case with log-utility, U (x) = ln(x) and maximization of the geometric

mean of the terminal wealth. In this case ˆθ = _σµ2 which is commonly referred to as the Kelly

criterion.

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3 Optimal strategy with the martingale method

In this chapter a method based on martingale theory is used to derive the optimal strategy for the one-year problem. First the problem of maximizing the expected utility of the terminal wealth after tax has been paid is introduced as an optimization problem. The problem is then decoupled into the simpler problem of finding the optimal wealth at the terminal date given initial capital constraints. This is then solve and used to derive the optimal strategy.

As in the previous chapter, Θ(x) is the set of admissible strategies available with the initial wealth x ∈ R+, where for each ϑ ∈ Θ(x) the firm’s capital is given by

X_tϑ= x + µ Z t 0 ϑsds + σ Z t 0 ϑsdWsP≥ 0, a.s ∀t ∈ [0, T ] (46)

We begin by amending the optimization problem to include a tax payment at the terminal date

T . At time T the firm will pay a tax of τ (Xϑ T − K)

+ _{where K is the predetermined tax basis}

denoted in the numéraire and τ is the tax rate. Here K is assumed to be arbitrary, but if the firm starts out with no tax deductions and capital x at time t = 0 then the tax basis for the year is simply x re-quoted into the value of the numéraire at the end of the year, i.e., K = e−rTx.

3.1 The optimal terminal wealth profile

The objective is to find the optimal strategy ˆϑ that maximizes the expected utility of the capital

after taxes have been paid

sup ϑ∈Θ(x) EP_{[U (X}ϑ T − τ (X ϑ T − K) +_)] ₍₄₇₎

Define the after-tax utility function ˜U by

˜

U (x) := U (x − τ (x − K)+) (48)

The result of the tax payment can be seen in Figure 1, which compares U (x) with ˜U (x).

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From the definition in (46), the firm’s capital Xtϑ is a self-financing portfolio and hence also a

Q-martingale with

EQ_[Xϑ

T] = x (49)

Define the Radon-Nikodym derivative as

Lt=dQ|F t

dP|Ft

(50)

where Q|Ftdenotes the measure Q restricted to Ft.

The expectation can now be written as

EP_[L

TXT] = x (51)

From the Girsanov theorem the L-dynamics are given by

dLt= ϕLtdWtP, L0= 1 (52)

where ϕ = −µ_σ < 0 is the constant Girsanov kernel.

The problem of determining the optimal terminal wealth can now be decoupled from the problem of determining the optimal strategy ˆϑ by introducing the static problem

sup XT∈XT EPh ˜_{U (X} T) − λ(LTXT− x) i (53)

Here XT = {XTϑ|ϑ ∈ Θ(x)} denotes the set of wealth profiles at T that can be obtained by an

admissible strategy with initial capital x. Solving this can be done by using first-order conditions and the result is stated in the following proposition.

Proposition 1. (Optimal terminal wealth profile) Given constants x ∈ R+ and λ ∈ R+ such

that EP_[L

TXˆT] = x, the random variable ˆXT(LT) = ˜I(λLT) satisfies

EPh ˜_{U ( ˆ}_X T i = sup XT∈XT EPh ˜_{U (X} T) i (54) where ˜ I(y) =        1 1−τ h I_1−τy − τ Ki, y ∈ (0, (1 − τ )U0_(K)) K, y ∈ [(1 − τ )U0(K), U0(K)] I(y), y ∈ (U0(K), ∞) (55) and I := U0.

Proof. The Lagrangian can be written in integral form

L = Z ω∈Ω n ˜_{U (X} T(ω)) − λ(LT(ω)XT(ω) − x) o dP (ω) (56)

Since there are no constraints, maximizing this can be done separately for each ω. Note that the derivative ˜U0( ˆXT) is discontinuous for ˆXT = K, hence there will be three cases depending on

ˆ

XT. For ˆXT < K we have ˜U0( ˆXT) = U0( ˆXT) hence first order condition gives

U0( ˆXT) − λLT = 0 (57)

which implies ˆXT = I(λLT) where I := (U0)−1. The second case is ˆXT = K which is trivial. For

ˆ

XT > K we have ˜U ( ˆXT) = U (XT − τ (XT − K)), hence taking the derivative with respect to

ˆ

XT gives

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Solving for ˆXT gives ˆ XT = 1 1 − τ h IλLT 1 − τ − τ Ki (59)

Looking at the first case, the inequality ˆXT = I(λLT) < K can be solved using that I−1= U0 is

a strictly decreasing function (since the utility function is assumed to be concave) to get

λLT > U0(K) (60)

For the third case, ˆXT > K implies that

1 1 − τ h IλLT 1 − τ − τ Ki> K (61)

which again can be solved to get the condition on λLT that

λLT < (1 − τ )U0(K) (62)

With this we seen that for λLT ∈ [(1 − τ )U0(K), U0(K)] the function is maximized with ˆXT = K.

This motivates us to define the (pseudo) inverse marginal utility of the after-tax utility function, ˜ I, as ˜ I(y) =        1 1−τ h I_1−τy − τ Ki, y ∈ (0, (1 − τ )U0(K)) K, y ∈ [(1 − τ )U0(K), U0(K)] I(y), y ∈ (U0(K), ∞) (63) with ˆXT(LT) = ˜I(λLT).

3.2 The optimal wealth process

Note that optimal terminal wealth ˆXT is obtained by some strategy ˆϑ. The problem has now

been reduced to finding the strategy ˆϑ such that ˆXT = X ˆ ϑ

T. The aim is to first find the optimal

wealth process, then use the martingale property to find the strategy from its dynamics. First note that since ˆXT is a Q-martingale

ˆ

Xt= EQ[ ˆXT|Ft] (64)

The L-dynamics can be solved to get

LT = exp n ϕWP T − 1 2ϕ 2_To ₍₆₅₎

Changing the probability measure with

dWP t = ϕdt + dWtQ (66) allows us to write LT = exp n ϕWQ T + 1 2ϕ 2_To ₍₆₇₎

We rewrite LT into two factors, one measurable with respect to Ft and one stochastic factor for

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Note that WT− Wt|Ft∼ N (0, √ T − t), hence we write LT = Ltexp n − ϕ√T − tz +1 2ϕ 2 (T − t)o (69) with z ∼ N (0, 1). The minus sign is used simply for later convenience.

The condition λLT < (1 − τ )U0(K) can now be written as (using that λ > 0)

expn− ϕ√T − tz +1 2ϕ 2_{(T − t)}o_< U0(K) λLt (70) −ϕ√T − tz +1 2ϕ 2_{(T − t) < ln} (1 − τ )U0(K) λLt ! (71) −ϕ√T − tz < ln (1 − τ )U 0_(K) λLt ! −1 2ϕ 2_{(T − t)} ₍₇₂₎

Note that ϕ < 0 by assumption hence −ϕ > 0 and we can divide both side by this without changing the sign of the inequality. The inequality can now be written

z < − ln (1−τ )U_λL0(K) t ! −1 2ϕ 2_{(T − t)} ϕ√T − t (73) Define d1:= − ln (1−τ )U_λL0(K) t ! −1 2ϕ 2_{(T − t)} ϕ√T − t (74)

Similarly λLT > U0(K) can be solved to find that this implies

z > − ln U_λL0(K) t ! −1 2ϕ 2_{(T − t)} ϕ√T − t (75) where we define d2:= − ln U_λL0(K) t ! −1 2ϕ 2_{(T − t)} ϕ√T − t (76) Note that d2= d1+ ln(1−τ ) ϕ√T −t.

For the random variable ˆXT(LT)|Ft with ˆXT(LT) = ˜I(λLT) we have

˜ I(λLT(z)) =        1 1−τ h IλLT(z) 1−τ − τ Ki, z ∈ (−∞, d1) K, z ∈ [d1, d2] I(λLT(z)), z ∈ (d2, ∞) (77)

We can now calculate the conditional expectation ˆXt= EQ[ ˆXT|Ft] by integrating for the standard

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ˆ Xt= Z ∞ d2 I(λLT(z)) e−12z 2 √ 2π dz + Z d2 d1 Ke −1 2z 2 √ 2π dz + 1 1 − τ Z d1 −∞ I λLT(z) 1 − τ ! e−1 2z 2 √ 2π dz − τ 1 − τ Z d1 −∞ Ke −1 2z 2 √ 2π dz (78)

This gives the following result.

Theorem 6. (Optimal wealth process with power utility) For an investor with power utility and

a risk-aversion parameter γ < 1 the optimal wealth process ˆXtis given by

ˆ Xt= λ− 1 1−γ_L− 1 1−γ t exp γϕ2(T − t) 2(1 − γ)2 ! Φ ϕ √ T − t (1 − γ) − d2 ! + KΦ(d2) − 1 1 − τKΦ(d1) + 1 1 − τ λ 1 − τ !−1−γ1 L− 1 1−γ t exp γϕ2_{(T − t)} 2(1 − γ)2 ! Φ d1− ϕ√T − t (1 − γ) ! (79)

where Φ(x) is the cumulative density function of a standard normal variable.

Proof. Power utility implies that U (x) = x_γγ hence the inverse marginal utility is given by

I(y) = y−1−γ1 _{. It is easy to note that, looking at the second integral above,}

Z d2 d1 Ke −1 2z 2 √ 2π dz = K Z d2 −∞ e−12z 2 √ 2π dz − K Z d1 ∞ e−12z 2 √ 2π dz = KΦ(d2) − KΦ(d1) (80) using the definition of Φ(x). The other integrals can be handled in a similar manner, by completing the square and a change of variable.

3.3 The optimal strategy

The expression for the optimal wealth process ˆXtwill allow us to compute the optimal strategy.

By assumption the Q-dynamics of ˆXt should be

d ˆXt= σ ˆϑtdWtQ (81)

The previous expression gives ˆXt= ˆXt(Lt, t). By Itô’s lemma

d ˆXt(Lt, t) = ∂ ˆXt ∂t dt + ∂ ˆXt ∂Lt dLt+ 1 2 ∂2_Xˆ_t ∂L2 t (dLt)2 (82)

With dLt= ϕLtdWtP= ϕ2Ltdt + ϕLtdWtQ this becomes

d ˆXt= " ∂ ˆXt ∂t + ϕ 2∂ ˆXt ∂Lt Lt+ 1 2ϕ 2∂ 2_X_ˆ t ∂L2 t L2t # dt + ϕ∂ ˆXt ∂Lt LtdWtQ (83)

Under Q the drift terms will sum to zero hence,

d ˆXt= ϕ

∂ ˆXt

∂Lt

LtdWtQ (84)

Comparing this to the expression for the Q-dynamics above implies that

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hence we are interested in the expression for ∂ ˆXt

∂LtLt. First note that

∂d1 ∂Lt = ∂d2 ∂Lt = 1 ϕ√T − t 1 Lt (86) and that _dxd Φ(x) =e− 1_√2x2

2π which both follow from definition. Taking the derivative with respect

to Lt gives −∂ ˆXt ∂Lt Lt= 1 1 − γλ − 1 1−γL− 1 1−γ t exp γϕ2_{(T − t)} 2(1 − γ)2 ! Φ ϕ √ T − t (1 − γ) − d2 ! + 1 1 − γ 1 1 − τ λ 1 − τ !−1−γ1 L− 1 1−γ t exp γϕ2_{(T − t)} 2(1 − γ)2 ! Φ d1− ϕ√T − t (1 − γ) ! − K ϕ√T − t e−12d 2 2 √ 2π + 1 1 − τ K ϕ√T − t e−12d 2 1 √ 2π +√1 2π 1 ϕ√T − tλ − 1 1−γ_L− 1 1−γ t exp γϕ2_{(T − t)} 2(1 − γ)2 ! exp ( −1 2 ϕ√T − t (1 − γ) − d2 !2) −√1 2π 1 ϕ√T − t 1 1 − τ λ 1 − τ !−1−γ1 L− 1 1−γ t exp γϕ2_{(T − t)} 2(1 − γ)2 ! exp ( −1 2 d1− ϕ√T − t (1 − γ) !2) (87)

The last two terms can be simplified significantly. Looking at the second last term above gives 1 √ 2π 1 ϕ√T − tλ − 1 1−γ_L− 1 1−γ t exp γϕ2_{(T − t)} 2(1 − γ)2 ! exp ( −1 2 ϕ√T − t (1 − γ) − d2 !2) = 1 ϕ√T − tλ − 1 1−γ_L− 1 1−γ t exp γϕ2(T − t) 2(1 − γ)2 ! exp ( −ϕ 2_{(T − t)} 2(1 − γ)2 + d2 ϕ√T − t (1 − γ) ) e−12d 2 2 √ 2π (88) In the last expression we have

d2 ϕ√T − t (1 − γ) = − lnU_λL0(K) t −1 2ϕ 2_{(T − t)} ϕ√T − t ϕ√T − t (1 − γ) = − lnU_λL0(K) t (1 − γ) + ϕ2_{(T − t)} 2(1 − γ) (89) Now most of the exponents cancel out since

γϕ2(T − t) 2(1 − γ)2 − ϕ2(T − t) 2(1 − γ)2 + ϕ2(T − t) 2(1 − γ) = −(1 − γ) ϕ2(T − t) 2(1 − γ)2 + ϕ2(T − t) 2(1 − γ) = 0 (90)

The remaining exponent gives

exp ( − lnU_λL0(K) t (1 − γ) ) = U 0_(K) λLt !−(1−γ)1 = Kλ1−γ1 _L 1 1−γ t (91)

using the inverse of U0(x). We can conclude that the second last term is

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Computing the last term in equation (87) can be done in a similar manner. Now with d1instead we get exp ( − ln(1−τ )U_λL0(K) t (1 − γ) ) = (1 − τ )U 0_(K) λLt !−(1−γ)1 = Kλ1−γ1 _L 1 1−γ t (1 − τ ) − 1 1−γ

hence we can conclude that the last term is

− 1 1 − τ K ϕ√T − t e−12d 2 1 √ 2π (93)

We notice that after inserting these expressions most of the terms cancel out and we are left with −∂ ˆXt ∂Lt Lt= 1 1 − γλ − 1 1−γL− 1 1−γ t exp γϕ2_{(T − t)} 2(1 − γ)2 ! Φ ϕ √ T − t (1 − γ) − d2 ! + 1 1 − γ 1 1 − τ λ 1 − τ !−_1−γ1 L− 1 1−γ t exp γϕ2_{(T − t)} 2(1 − γ)2 ! Φ d1− ϕ√T − t (1 − γ) ! (94)

These results can now be collected into the following theorem.

Theorem 7. (Optimal strategy with power utility) For an investor with power utility and a

risk-aversion parameter γ < 1 the optimal portfolio ˆXt has the P−dynamics

d ˆXt= µ ˆϑtdt + σ ˆϑtdWtP (95)

The optimal strategy ˆϑtis given by

ˆ ϑt= µ σ2_{(1 − γ)} ∆A_t + ∆B_t (96) where ∆A_t = (λLt)− 1 1−γ _exp γϕ 2_{(T − t)} 2(1 − γ)2 ! Φ ϕ √ T − t (1 − γ) − d2 ! (97) and ∆B_t = (1 − τ )1−γγ (λL_t)− 1 1−γexp γϕ 2_{(T − t)} 2(1 − γ)2 ! Φ d1− ϕ√T − t (1 − γ) ! (98)

Proof. Inserting the expression for −∂ ˆXt

∂LtLtin (94) into the expression for ˆϑtin (85) and simplifying

gives the desired result.

Remark 1. Note that ∆A

t + ∆Bt = ˆXt+_1−τ1 KΦ(d1) − KΦ(d2). This follows from the expression

for ˆXt in Theorem 6.

Remark 2. The assumption that µ > 0 does not actually narrow the original premise. If µ < 0

simply set µ → |µ| and the optimal strategy for the original problem will be − ˆϑ.

Retracing the steps taken to derive the optimal strategy provides some intuition on what the components in ˆϑ represent. Note that the expression for ∆A

t came from the integral for the

case when ˆXT < K, i.e., no tax was paid, while ∆Bt came from the integral where ˆXT > K.

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Corollary 1. (Derivatives in the optimal strategy) In the expression for the optimal strategy,

∆At is the arbitrage-free price of a derivative which pays ˆXT at time T if ˆXT < K, while ∆Bt is

the price of a derivative which pays ˆXT +_1−ττ K at time T if ˆXT > K. Hence

∆A_t = EQh ˆ_X T1{ ˆXT<K} Ft i (99) and ∆Bt = EQh ˆXT + τ 1 − τK 1{ ˆXT>K} Ft i (100)

The interpretation of this is as follows. If it is clear that ˆXT < K the investor can invest

while disregarding taxes and hence the optimal strategy is simply the standard solution with ˆ

ϑt= _σ2_(1−γ)µ Xˆt.

If it is clear that ˆXT > K the investor has to pay tax and will therefore only receive part of the

ˆ

XT at time T , hence will alter the leverage to make sure that that his part has optimal exposure.

The term _1−ττ K can be seen as a tax refund that the investor receives at time T , but will invest

in the portfolio at time t to increase its utility.

Note that ({ ˆXT < K} ∪ { ˆXT > K}) = { ˆXT = K}, hence we would naturally expect there to be

a third term, ∆C

t, which represents the event that ˆXT is exactly equal to the tax basis K. This

is in fact consistent with the derived result. If it is clear that ˆXT = K, the expected return on

any investment would be negative and hence the investor will refrain from investing at all. The term ∆C_t is therefore the price of a derivative which pays zero at time T , hence ∆C_t = 0.

Remark 3. Note that the tax-refund term is _1−ττ K and not τ K which one may expect since the investor will receive XT − τ (XT − K) = (1 − τ )XT + τ K at time T . This is due to the

recursive nature between the leverage and the expected tax payment. For each unit of K invested in the portfolio the investor will receive a rebate of τ , which in turn can be reinvested and hence introduce an additional rebate of τ2_{, which again can be reinvested and so on. In total this leads}

to

τ K(1 + τ + τ2+ τ3+ τ4+ . . . ) = τ

1 − τK (101) It should be made clear that the equations (99) and (100) does not uniquely determine the optimal strategy. For example it has a trivial solution Xt= K for all t with ϑt= 0. Only after

the static optimization problem where ˆXT is determined is resolved does the above representation

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4 Optimal strategy for multiple years

In this chapter the martingale method introduced in the previous chapter is applied to the extended problem of determining the optimal strategy given multiple years with potential tax payments. Some useful properties of the optimal strategy are derived and used to characterize the solution to the N -years problem.

The solution in the previous chapter for the one-year problem can be generalized to multiple years with potential tax payments. Since tax payments will introduce a discontinuity in the process Xt, the firm’s capital, it will no longer be a martingale for every time t. We assume that

the firm will operate for N years then shut down, liquidate and return all capital to the investors. Introduce the set

TN := {1, . . . , n, n + 1, . . . N } ⊂ N (102)

where at any t ∈ TN a tax payment can be made.

The dynamics of the wealth process will be

dXt= σϑtdWtQ, t ∈ [n − 1, n), ∀n ∈ TN (103)

As before, the problem of determining the optimal strategy ˆϑtwill be decoupled into the static

problem of determining the optimal wealth at the end of a year. Since there are now multiple potential tax payments, there will also be multiple static problems to be solved. Rather than solving the full multiple-year, constrained problem, we can solve the static problem for each year and use solutions to solve the full problem with dynamic programming.

4.1 The optimal end-of-year wealth profile

The problem of determining the optimal end-of-year wealth profile ˆXn with an arbitrary number

of years left will involve maximizing the expected utility at some future date N . To solve this we introduce the indirect utility function.

4.1.1 Indirect expected utility

Define the indirect utility function as the expected utility at the terminal date N given current capital x and tax basis k,

υ(N )_t (x, k) := EP_{[U ( ˆ}_X

N)|Ft] (104)

with t ∈ [0, N ] and assuming that the firm shuts down at t = N . By the Law of total expectations we have

EP_[υ(N )

t (x, k)] = EP[EP[U ( ˆXN)|Ft]] = EP[U ( ˆXN)] (105)

hence maximizing the expected utility at some terminal date N is equivalent to maximizing the expected value of the indirect utility function at some time t < N . Note that υ_N(N )(x, k) =

U (x).

For n ∈ TN any potential tax payment will have been deducted with

Xn= Xn−− τ (X_n−− K_n)+ (106)

where Xn− denoted the left limit of the capital at the time n and K_n is the tax basis for the

year.

The tax basis will be updated and re-quoted into the value of the numéraire for the end of the following year

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For any year n ≤ N the optimal end-of-year wealth profile ˆXn is found by solving the problem of

maximizing

EP_[υ(N )

n (Xn, Kn)|Fn] (108)

subject to the constraint

Xn−1= EQ[Xn−|F_n] (109)

As before we have the likelihood process

Lt=dQ|F t

dP|Ft

(110)

The constraint can now be written as

Ln−1Xn−1= EP[LnXn−|F_n] (111)

which leads to the following optimization problem sup

Xn

EP_[υ(N )

n (Xn, Kn) − λn(LnXn−− Ln−1Xn−1)] (112)

Solving this problem will require some more information about υ(N )t (x, k). Unfortunately, it is

not possible to get an analytical expression for the indirect utility since it depends on a Lagrange multiple which is only defined by an implicit equation. It is however possible to gain some understanding of the solution using the properties of υ_t(N )(x, k). With power utility, the decisions are unaffected by scale and the portion of capital exposed to risk is independent of the level of wealth. When taxes are included, this implies that for the optimal strategy the portion of capital exposed to risk depends on the ratio between the capital and the tax basis. Hence the general economy of the problem, suggests that the function should be homogeneous in its arguments, with υ(N )_t (x, k) = kγ_G(N )

t (x/k), for some function G (N ) t (·).

Consider the case when Xn−> Kn. The problem is now to maximize

υ_n(N )(Xn, Kn) − λn(LnXn−− L_n−1X_n−1) (113)

in terms of the process Ln. But this can be done in the same manner as before, we have

X_nγe−rγG(N )_n (er) − λn(LnXn−− L_n−1X_n−1) (114)

here using that Kn+1= e−rXn. Since Xn = (1 − τ )Xn−+ τ Kn, first order condition now gives

(1 − τ )γ((1 − τ ) ˆXn−+ τ K_n)γ−1e−rγG(N )_n (er) − λ_nL_n= 0 (115)

and hence the optimal wealth profile before the tax is deducted at the end of year n is

ˆ Xn− = 1 1 − τ " I λnLne rγ (1 − τ )γG(N )n (er) ! − τ Kn # , Xˆn− > K_n (116)

where as before I(y) := y−1−γ1 _.

This is similar to the solution for the one-year problem, except for the constant factor erγ

γG(N )n (er)

, which comes from future potential tax payments. As we will see subsequently, this will imply that the optimal strategy right before a tax payment is made will be the same regardless of the year.

There will be two other solutions, ˆXn− = Kn and

ˆ

Xn− = F_n(L_n), Xˆ_n− < K_n (117)

for some function Fn(·). While we may not be able to further specify Fn(·) we can infer some

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4.2 The martingale property of the optimal strategy

Recall that the dynamics of the optimal wealth process was assumed to be of the form

d ˆXt= σ ˆϑtdWtQ (118)

for t ∈ [n − 1, n) within a year.

The optimal strategy for the one-year problem was given by

ˆ ϑt= µ σ2_{(1 − γ)}E Q " ˆ X1−1 { ˆX₁₋<K}+ ˆX1−+ τ 1 − τK 1{ ˆX₁₋>K} Ft # (119)

Note that this seems to be the expected value of the optimal strategy close to the terminal date

t = 1. This property is generalized to multiple years in the following lemma.

Lemma 1. (Martingale property of the optimal strategy) Let ˆϑtbe the strategy for t ∈ [0, 1) with

N years left. Then it hold that

ˆ

ϑt= EQ[ ˆϑ1−|F_t] (120)

where ˆϑ1− is the left-limit of the optimal strategy at the end of the year.

Proof. As seen in the previous chapter, by using Itô’s lemma on ˆXt as a function of Lt the

optimal strategy was given by

ˆ θt= − µ σ2 ∂ ˆXt ∂Lt Lt (121)

which is valid for t ∈ [n − 1, n), ∀n ∈ TN. Hence the martingale property for ˆθtholds if and only

if ∂ ˆXt

∂LtLtwas also a Q-martingale for t ∈ [n − 1, n).

Fix an n ∈ TN and let t ∈ [n − 1, n). Using the martingale property for the wealth process

ˆ Xt= EQ[ ˆXn−|F_t] we have ∂ ˆXt ∂Lt = ∂ ∂Lt EQ_{[ ˆ}_X n−|F_t] = EQ " ∂ ˆXn− ∂Lt Ft # (122)

Here we assume that the differentiation and expectation can be interchanged. This is valid under the assumption that the derivative is bounded and integrable, using a lemma given in Appendix B.

Note that, from the solution, the likelihood process can be written as Ln = LtD where D is

independent of Lt. Furthermore, EQ " ∂ ˆXn− ∂Lt Ft # = EQ " ∂ ˆXn ∂Ln ∂Ln ∂Lt Ft # (123) Using that ∂Ln ∂Lt = D = Ln Lt now gives ∂ ˆXt ∂Lt Lt= EQ " ∂ ˆXn− ∂Ln Ln Ft # (124)

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4.3 Characterizing the optimal strategy for multiple years

While it may not be possible to derive an analytical expression for the optimal strategy for multiple years, we can us the martingale property to characterize the optimal solution. We begin by introducing some new notation. Let ˆϑ(N )_t denote the optimal exposure at time t ∈ [0, 1) within a year and with N years and potential tax payments left until the terminal date t = N . With N = 1 we have ˆ ϑ(1)_t (Xt, K1) = µ σ2_{(1 − γ)} ∆A_t + ∆B_t (125) with ∆A_t = EQh ˆ_X 1−1 { ˆX₁₋<K1} Ft i (126) and ∆Bt = EQh ˆX1−+ τ 1 − τK1 1{ ˆX₁₋>K1} Ft i (127)

Recall that the case with Xn−> K_n gave the solution as

ˆ Xn− = 1 1 − τ " I λLne rγ (1 − τ )γG(N )t (er) ! − τ Kn # (128)

The optimal strategy close to the boundary can be computed from ∂ ˆXn−

∂Ln Ln. Set n = 1 and some

differentiation gives ∂ ˆX1− ∂L1 L1= − 1 1 − γ ˆ X1−+ τ 1 − τK1 ! (129) hence ˆ ϑ(N )₁− = µ σ2_{(1 − γ)} Xˆ1−+ τ 1 − τK1 ! , Xˆ1−> K₁ (130)

For X1− = K₁ we will have ˆϑ₁−= 0, and for ˆX₁− < K₁

ˆ ϑ(N )₁− = − µ σ2 ∂F1 ∂L1 L1, Xˆ1− < K₁ (131)

If no tax payment is made then ˆX1− = ˆX₁ almost surely, hence it should hold that

ˆ

ϑ(N )₁− = ˆϑ (N −1)

0 , Xˆ1− < K₁ (132)

Consider the case with a two-year investment horizon. By specifying ˆϑ(2)₁−, the optimal strategy

at the end of the year as seen above, the martingale property of the optimal strategy now gives

ˆ ϑ(2)t (Xt, K1) = µ σ2_{(1 − γ)}E Qh ˆ_X 1−+ τ 1 − τK1 1{ ˆX₁₋>K1}+ ˆϑ (1) 0 ( ˆX1, e−rK1)1_{{ ˆ}X₁₋<K1} Ft i (133) Following from this argument, the optimal strategy for the N -year investment horizon can be characterized as follows.

Proposition 2. The optimal strategy for t ∈ [0, 1) with N years left is given by the recursive

formula ˆ ϑ(N )_t (Xt, K1) = µ σ2_{(1 − γ)}E Qh ˆ_X 1−+ τ 1 − τK1 1{ ˆX₁₋>K1}+ ˆϑ (N −1) 0 ( ˆX1, e−rK1)1{ ˆX₁₋<K1} Ft i (134)

where we define ˆϑ(0)₀ ( ˆX1, e−rK1) := ˆX1 to make the formula conform with the solution to the

(37)

4.4 The optimal strategy for the infinite-horizon problem

Definition 9. The infinite-horizon optimal strategy ˆϑt at some time t ∈ [n − 1, n) is defined as

the limiting value of a sequence of solutions for the N -years problem

ˆ

ϑt= ˆϑt(Xt, K, t) = lim N →∞

ˆ

ϑ(N )_t (Xt, K, t) (135)

It is not clear under what conditions ˆϑtactually exists.

Note that ˆϑ(N )_t (Xt, K, t) depends on probability of a tax payment occurring under Q at some

time during N years. Since this in turn will depend on Xt for any fixed K we can expect the

convergence of limN →∞ϑˆ (N )

t (Xt, K, t) to be pointwise in Xt. Naturally we can expect symmetry

in time

ˆ

ϑ(X, K, t) = ˆϑ(X, K, t + n), ∀n ∈ N (136)

Optimal Investment with Corporate Tax Payments

IN

DEGREE PROJECT

MATHEMATICS,

SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2017

Optimal Investment with

Corporate Tax Payments

EMIL TINGSTRÖM

Optimal Investment with

Corporate Tax Payments

EMIL TINGSTRÖM

Optimal Investment with Corporate Tax Payments

Optimal Investering med Företagsbeskattning

Contents

List of variables

1

Introduction

1.1

Optimal investment and taxes

1.2

Purpose and main objectives

1.3

Limitations of the scope

1.4

Disposition of the report

2

Previous research and theoretical background

2.1

Previous research

2.2

Stochastic calculus

2.3

Arbitrage-free pricing

2.4

Utility theory and utility maximization

2.5

Modelling the firm’s capital

2.6

Optimal leverage for a tax-free firm

3

Optimal strategy with the martingale method

3.1

The optimal terminal wealth profile

3.2

The optimal wealth process

3.3

The optimal strategy

4

Optimal strategy for multiple years

4.1

The optimal end-of-year wealth profile

4.2

The martingale property of the optimal strategy

4.3

Characterizing the optimal strategy for multiple years

4.4

The optimal strategy for the infinite-horizon problem