Non Parametric Estimation for Financial Investment Under Log-utility

8/9/2019 Non Parametric Estimation for Financial Investment Under Log-utility

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Nonparametric Estimation forFinancial Investment under Log-Utility

Von der Fakult at Mathematik der Universit at Stuttgartzur Erlangung der W urde eines Doktors der

Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung

Vorgelegt von

Dominik Sch afer

aus Pforzheim

Hauptberichter: Prof. Dr. H. WalkMitberichter: Prof. Dr. V. Claus

Prof. Dr. L. Gy or

Tag der m undlichen Pr ufung: 15. Juli 2002

Mathematisches Institut A der Universit at Stuttgart

2002


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dedicated to

My Parentsto whom I owe so much

Professor Paul Glendinningwithout him I might never have found my way

to mathematical nance and economics


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CONTENTS

Abbreviations 3

Summary 7

Zusammenfassung 19

Acknowledgements 31

1 Introduction: investment and nonparametric statistics 33

1.1 The market model . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.2 Portfolios and investment strategies . . . . . . . . . . . . . . . . 37

1.3 Pleading for logarithmic utility . . . . . . . . . . . . . . . . . . 40

2 Portfolio benchmarking: rates and dimensionality 47

2.1 Rates of convergence in i.i.d. models . . . . . . . . . . . . . . . 48

2.2 Dimensionality in portfolio selection . . . . . . . . . . . . . . . . 61

2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3 Predicted stock returns and portfolio selection 73

3.1 A strategy using predicted log-returns . . . . . . . . . . . . . . . 74

3.2 Prediction of Gaussian log-returns . . . . . . . . . . . . . . . . . 77

3.2.1 An approximation result . . . . . . . . . . . . . . . . . . 80

3.2.2 An estimation algorithm . . . . . . . . . . . . . . . . . . 81

3.3 Proof of the approximation and estimation results . . . . . . . . 86

3.4 Simulations and examples . . . . . . . . . . . . . . . . . . . . . 97


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4 A Markov model with transaction costs: probabilistic view 103

4.1 Strategies in markets with transaction fees . . . . . . . . . . . . 104

4.2 An optimal strategy . . . . . . . . . . . . . . . . . . . . . . . . 108

4.2.1 Some comments on Markov control . . . . . . . . . . . . 110

4.2.2 Proof of Theorem 4.2.1 . . . . . . . . . . . . . . . . . . . 111

4.3 Further properties of the value function . . . . . . . . . . . . . . 126

5 A Markov model with transaction costs: statistical view 129

5.1 The empirical Bellman equation . . . . . . . . . . . . . . . . . . 129

5.1.1 An optimal strategy . . . . . . . . . . . . . . . . . . . . 131

5.1.2 How to prove optimality . . . . . . . . . . . . . . . . . . 135

5.2 Uniformly consistent regression estimation . . . . . . . . . . . . 135

5.3 Proving the optimality of the strategy . . . . . . . . . . . . . . . 145

6 Portfolio selection functions in stationary return processes 151

6.1 Portfolio selection functions . . . . . . . . . . . . . . . . . . . . 152

6.2 Estimation of log-optimal portfolio selection functions . . . . . . 155

6.3 Checking the properties of the estimation algorithm . . . . . . . 161

6.3.1 Proof of the convergence Lemma 6.2.1 . . . . . . . . . . 161

6.3.2 Proof of the related Theorems 6.2.2 - 6.2.4 . . . . . . . . 169

6.4 Simulations and examples . . . . . . . . . . . . . . . . . . . . . 175

LEnvoi 180

References 181


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ABBREVIATIONS

| | absolute value of a number, cardinality of a set Euclidean scalar product

supremum normq q -norm (on IR d or Lq ) other norm

IN positive integers 1, 2, 3,...IN0 nonnegative integers 0 , 1, 2, 3,...IR real numbers

IR+ real numbers > 0IR+0 real numbers 0x integer part of x

x N the smallest kN (kIN) such that kN x 0.x x rounded toward innity

T transpose of a vector or matrixspr() spectrum of a matrix

exp exponential to the base elog logarithm to the base elb logarithm to the base 2

an = o(bn ) Landau symbol for: an /b n 0an = O(bn ) Landau symbol for: an /b n is a bounded sequence

AC complement of the set AA closure of the set A

conv(A) convex hull of the set A


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4 Abbreviations

1A characteristic function of the set Adiam( A) Euclidean diameter sup a,bA a b of the set A

(x, A) Euclidean distance inf aA x a from x to the set AH (A, B ) Hausdorff distance max {supaA (a, B ), supbB (b, A)}

between the sets A and B

B (S ) Borelian algebra on the topological space S f (x)|x= y f evaluated in y

f + , f + positive part of f , i.e., max{f, 0}f , f negative part of f , i.e., max{f, 0}supp f support {x : f (x) > 0}of the function f

argmax f solution of a maximization problem (in some contextsset-valued, i.e. {x : f (x) = sup y f (y)}, in others ameasurably selected solution x with f (x) = sup y f (y))

P probability measureP X distribution of X

P Y |X = x conditional distribution of Y given X = xf X () a density of P X w.r.t. the Lebesgue measure

f Y |X (|x) a density of P Y |X = x w.r.t. the Lebesgue measureQ 1 Q 2 Q 1 is absolutely continuous w.r.t. Q 2D(Q 1||Q 2) Kullback-Leibler distance of Q 2 and Q 1

a.s. P -almost surely, with probability oneP -a.a. P -almost all

E mathematical expectationE[Y |X ] conditional expectation of Y given X

E[Y |X = x] conditional expectation of Y given X = xVar VarianceCov Covariance

N (; ) normal distribution with mean and variance-covariance matrix

L1(P ) space of Lebesgue integrable functions w.r.t. PLq (P ) q th order Lebesgue integrable functions w.r.t. P


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const. a suitable constantGSM geometrically strongly mixing

hot. higher order terms of an expansioni.i.d. independent, identically distributedp.a. per annum

w.r.t. with respect toend of proof

All non-standard notation is explained when it occurs for the rst time. Therandom variables in this thesis are understood to be dened on a commonprobability space ( , A, P ). IRd-valued random variables are implicitly assumedto be measurable w.r.t. the Borelian -algebra B (IRd). If not stated otherwise,measurability of functions f : IRd IRd means measurability w.r.t. B (IRd) andB (IRd ).


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SUMMARY

In this thesis we aim to plead for the application of nonparametric statistical

forecasting and regression estimation methods to nancial investment problems.In six chapters we explore applications of nonparametric techniques to portfolioselection for nancial investment. Clearly, this cannot be more than a crudeand somewhat arbitrary selection of topics within this vast area, so we decidedto concentrate on some typical situations. Our hope is to be able to illustratethe benets of nonparametric estimation methods in portfolio selection.

Chapter 1Introduction: investment and nonparametric statistics

Investment is the strategic allocation of resources, typically of monetary re-sources, in an environment, typically a market of assets, whose future evolution

is uncertain. Investment problems arise in a huge variety of contexts beyond thenancial one. Resources may also take the form of energy, of data-processingresources, etc. Strategic investment planning helps to run many processes withhigher benet. In this thesis we focus our attention on nancial investment,which we think is the prototypical example of a resource allocation process.

The three ingredients of nancial investment are the market, the actions theinvestor may take and his investment goal (discussed in detail in Sections 1.1-1.3):

As to the market: We assume that there are m assets in our nancial market.The j th asset yields a return X i,n on an investment of 1 unit of moneyduring market period n (lasting from time n 1 to n, time being mea-sured, e.g., in days of trading). The ensemble of returns on the nth dayof trading is given by

X n = ( X 1,n ,...,X m,n )T IRm+ .

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To the investor, the return process {X n}n=1 appears to be a stochasticprocess which, in many real markets, is stationary and ergodic (Denition1.1.1). In some chapters we impose additional (but realistic) conditionson the distribution of the process. The key point is, however, that

we use nonparametric models, i.e. models that do not assume a para-metric evolution equations such as ARMA, ARCH and GARCH equa-tion to hold.

These models guarantee highest exibility in real applications. As to the investment actions: We are concerned with an investor who neither

consumes nor deposits new money into his portfolio. At the beginning of each market period n, our investor uses all his current wealth to acquirea portfolio bn of the stocks. It will be convenient to describe the portfoliobn by the proportion b j,n of the investors current wealth invested in asset j ( j = 1 ,...,m ) during market period n. Thus, bn is chosen at time n 1from the set S of all portfolios, consisting of the vectors (portfolios)

bn = ( b1,n ,...,bm,n )T

satisfying b j,n 0 andm

j=1 b j,n = 1. In some situations the set of in-vestment actions S may be further narrowed down by the occurence of transaction costs.

As to the investment goal: If W 0 is his initial wealth, an investor usingthe portfolio strategy {bi}n1i=0 manages to accumulate the wealth W n =n

i=1 W 0 during n market periods ( < , > is the Euclideanscalar product). Naturally, the investor aims to maximize W n . It is knownfrom literature that there is no essential conict between short run ( nnite) and long term ( n ) investment. In both cases investmentaccording to the conditional log-optimal portfolio

bn := argmaxbS E [log < b, X n > |X n1,...X 1]

at time n is optimal, outperforming any other strategy because of

EW nW n 1 and limsupn

1n

logW nW n 0 with probability 1


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hope that underperformance vanishes sufficiently fast when with increasingnumber of market periods his estimates for the distribution of the returnprocess and hence his idea of the market become more and more complete.Now, if the investor evaluates the historical returns X 1,...,X n leading to theportfolio choice bn+1 at time n, he will achieve a return Rn = < bn+1 , X n+1 > onhis investment during the next market period. This should be compared withthe return Rn = < bn+1 , X n+1 > of the conditional log-optimal portfolio.

From our log-utility point of view we suggest to measure underperfor-

mance of bn+1 in terms of the positivity of E logRnRn . The smaller this

expectation becomes, the better is the selection rule bn+1 .

Assuming that the return data arises from a process of independent and iden-tically distributed (i.i.d.) random variables, it is important to know at whatrate the underperformance E log R

n

Rnvanishes for typical portfolio selection rules.

Using notions from information theory we prove a lower bound on this rate inSection 2.1. Even in the simplest of all markets, a market with only nitelymany possible return outcomes,

no empirical portfolio selection rule can make underperformance van-ish in every market faster than 1

ntends to 0, i.e. there is always a

market for which the inequality E log RnRn const. 1n holds (Theorem 2.1.1).

There are empirical portfolio selection rules that achieve this rate. In particular,the empirical log-optimal portfolio

bn+1 := argmaxbS

1n

n

i=1

log < b, X i > (0.0.1)

proves to be rate optimal in as far as

the empirical log-optimal portfolio selection rule (0.0.1) attains the lower bound for the rate at which underperformance vanishes, whatever the number of stocks in the market (Theorem 2.1.3).

Loosely speaking, it compensates for wrong investment decisions as fast as pos-sible. Interesting enough, the ndings are largely unaffected by the number


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of stocks in the market, which is a rather untypical feature in nonparamet-ric estimation (Theorem 2.1.4 shows that this phenomenon perseveres in morecomplicated market settings).

This is why we discuss the effects of dimensionalityon the portfolio selectionprocess in more detail in Section 2.2. We argue that a reduction of the wholestock market to some pre-selected stocks is inevitable, e.g., because of compu-tational restrictions. In other words, the investor can only handle a smallishsubset of all stocks in the market for investment strategy planning. These stockshave to be selected in the planning phase, even before investment starts. Hence,criteria for the pre-selection of stocks from the market are needed. A commonway to do this is to pick the stocks whose chart promises high growth rates. Itwill turn out, however, that this is fallacious:

any selection algorithm that assesses the single stocks seperately, e.g.on the basis of single stock expected returns, is sure to pick the badstocks in some realistic market (Theorem 2.2.1).

This is a somewhat negative result, but it warns us that reasonable selectionschemes have to include further information about the market. We will showthat the variance-covariance structure of the stock returns provides sufficient

information in many markets (more precisely, in markets with log-normal re-turns). Section 2.3 illustrates the results with simulations and examples, demon-strating their practical relevance.

Chapter 3Predicted stock returns and portfolio selection

Having gained the insight that variance-covariance information about the mar-ket (inter-stock correlations as well as temporal correlations) are integral tosuccessful investment decisions, we move on to particular investment strate-gies. In Section 3.1 we consider a strategy which is particularly popular amonginvestors.

The strategy works in two steps, with the past logarithmic returns Y n , Y n1,...,Y 0(Y i := log X i) as input data for the investment decision at time n:

1. Produce forecasts of the market future. It is established that forecastsshould be based on conditional expectations of future log-returns given


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the observed past, i.e. on

Y n+1 := E[Y n+1 |Y n , Y n1,...].

2. Invest in those stocks whose forecast Y n+1 promises to beat a risklessinvestment in a bond with return rate r , i.e. invest in a stock iff

exp( Y n+1 ) r.

We will call this strategy a greedy strategy, because it tries to single outthe best possible stocks only. As we shall see, this provides us with a naturalstrategy which can be applied in markets with low log-return variance (Section3.1).

The major problem in implementing the greedy strategy is the fact that theforecasts Y n+1 can only be calculated if the distribution of the return processis known to the investor. Hence, we need to derive an estimate E (Y n ,...,Y 0)for the conditional expectation Y n+1 = E[Y n+1 |Y n , Y n1,... ] from the marketobservations Y n ,...,Y 0. It is known from literature that no such forecaster canbe strongly consistent in the sense of

limnE (Y n ,...,Y 0) E[Y n+1 |Y n , Y n1,... ] = 0 (0.0.2)

with probability 1 for any stationary and ergodic process {Y n}n (Bailey, 1976).This result is discouraging, but it does not rule out the existence of stronglyconsistent forecasting rules for log-return processes as they arise in real nancialmarkets. In particular, Gaussian log-return processes have been proven to bea good approximation for real log-return processes, but so far no answer hasbeen found to the question whether there exist forecasters that are stronglyconsistent in any stationary and ergodic Gaussian process. In Section 3.2 weprove that the answer is indeed affirmative. Under weak extra conditions onthe Wold coefficients of the process

we present a forecaster E (Y n ,...,Y 0) for stationary and ergodic Gaus-sian processes which satises the strong consistency relation (0.0.2)and which is remarkably easy to compute (Lemma 3.2.1 and Corollary 3.2.3).


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This results provides us with the necessary tools to implement the greedy strat-egy in Gaussian log-return processes. However, the algorithm is of interest verymuch in its own right, forecasting problems for Gaussian processes arising inmany areas.

Section 3.3 proves the convergence properties of the algorithm. Applicationexamples with simulated and real data in Section 3.4 are promising when thealgorithm is run as a mere forecasting algorithm as well as when the algorithmis run as a subroutine for the greedy strategy.

Chapter 4A Markov model with transaction costs: probabilistic view

In simple markets where returns arise as i.i.d. data, the investor should investin a constant log-optimal portfolio strategy. This requires him not to changethe proportion of wealth held in each stock during the investment process. Theproportions remain constant, however, the prices of the assets change relativelyto each other during each market period, so that the actual quantities of thesingle stocks in the portfolio vary from market period to market period. Thus,a large number of transactions are needed to follow a constant log-optimalstrategy. In practice, this is a huge drawback: Much of the wealth accumulatedby a log-optimal strategy has to be spent to settle transaction costs such as

brokerage fees, administrative and telecommunication expenses. The conclusionfor the investor must be to adapt his strategy to meet two requirements: tomake as few costly transactions as possible, but to make as many as necessaryto boost his wealth. The aim of Chapters 4 and 5 is to investigate how thesetwo conicting requirements can be balanced in one strategy.

To this end we shall assume that the returns arise from a d-stage Markov pro-cess. In Chapter 4 the distribution of the return process is known, an unrealisticassumption which we will drop in Chapter 5. Section 4.1 generalizes the mar-ket model from Chapter 1 to include transaction costs proportional to the totalvalue of the purchased shares. Not surprisingly, the investor can only afforda limited range of portfolio choices in presence of transaction costs, and as we

shall see,in d-stage Markovian return processes it suffices to consider strategies based on portfolio selection functions, i.e. portfolio selection schemes of the form bi = c(bi1, X id,...,X i1) with an appropriate function c(Denition 4.1.2).


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Hence, the next portfolio is a function of the last portfolio and the last d ob-served return vectors. The investor aims to maximize his expected mean loga-rithmic return as before by choosing an optimal selection function c.

In Section 4.2 we tackle the problem how to obtain an optimal selection functionc if the distribution of the return process were known. The main resultdemonstrates that

an optimal portfolio selection function c can be obtained from a solu-tion of the Bellman equation (Theorem 4.2.1, equation 4.2.2).

The Bellman equation is known from the theory of dynamic programming,but fundamental differences between classical dynamical programming and theportfolio selection problem will become evident. Further properties of solutionsof the Bellman equation will be derived in Section 4.3, results that will beneeded for the arguments in Chapter 5.

Chapter 5A Markov model with transaction costs: statistical view

The Bellman equation considered in Chapter 4 heavily depends upon the distri-bution of the return process {X n}n through a peculiar conditional expectation.Hence, the results of Chapter 4 are valid only under the assumption that the in-vestor knows the distribution of the stock return process. Of course, in practicethis is illusory. At best, the investor has an estimate of the return distributionat his disposal. This, in turn, allows him to produce an estimate of the con-ditional expectation in question and hence gives him an approximate Bellmanequation involving the observed empirical return data. Using nonparametricregression estimation techniques

we will show in Section 5.1 how a natural empirical counterpart of the Bellman equation from Chapter 4 can be found (equation 5.1.2).

With similar techniques as in Chapter 4 we will establish that this empiricalequation can be solved under realistic conditions.

This will lead us to a strategy that merely relies on observational data but has the same optimality properties as the (theoretical) optimal portfolio selection rule in presence of transaction costs (Theorems 5.1.1and 5.1.2).

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For this, we will fall back on generalizations of existing uniform consistency results in regression estimation, which will be provided in Section 5.2. In par-ticular, if {X n}n is a stationary geometrically strongly mixing process and g istaken from a class Gof Lipschitz continuous functions we estimate the condi-tional expectation

R(g,b,x) := E[g(X 1, b)|X 0 = x] (bS )by a kernel regression estimator Rn (g,b,x). Depending on the smoothness of adensity of X 0 (which we assume to exist) we determine the rate of convergenceof

supgG

E supxX ,bS |

Rn (g,b,x) R(g,b,x)| 0 (n ),i.e. of the expected uniform estimation error, uniformly in G(Corollary 5.2.2).This result is of interest in other areas of nonparametric statistics as well.

Finally, Section 5.3 is devoted to the proof of optimality and combines theresults from Chapter 4 with uniform consistent regression estimation techniques.

Chapter 6Portfolio selection functions in stationary return processes

Considering the fact that the investor may have reason to believe that the his-torical return data does not follow a d-stage Markov process in some cases,we should move on to even more general market models than in the previouschapters. Ignoring transaction costs, we consider a market whose returns aremerely stationary and ergodic. It is natural for the investor to take his invest-ment decisions on the basis of recently observed returns, say on the basis of thereturns during the last dIN market periods ( d xed). This leads us to thenotion of log-optimal portfolio selection functions.

We make this more concrete in Section 6.1, where we take our familiar log-utility approach again. The investor tries to nd a log-optimal portfolio selection function , i.e. a measurable function

b: IRdm+ S such that ( < , > denoting the Euclidean scalar product)

E (log < b(X 0,...,X d1), X d > ) E (log < f (X 0,...,X d1), X d > )

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for all measurable f : IRdm+ S . For the ( n + 1)st day of trading, badvisesthe investor to acquire the portfolio b(X nd+1 ,...,X n).Clearly, the concept of log-optimal portfolio selection functions does not reachthe same degree of generality as the concept of a conditional log-optimal port-folio (where d is such that the whole observed past is included in the portfoliodecision). In spite of being a simplication, this approach nevertheless gives usseveral advantages over the log-optimal strategy as far as computation, estima-tion and interpretation are concerned.

With log-optimal portfolio selection functions we face the same problem as withlog-optimal portfolios. Both can only be calculated if the true distribution of thereturn process happens to be known. A practitioner, however, needs to have anestimation procedure that evaluates observed past return data to approximatethe true log-optimal device.

In Section 6.2 we therefore develop an algorithm to produce estimates bn of a log-optimal portfolio selection function bfrom past return data.

We require very mild conditions beyond stationarity and ergodicity. More pre-cisely, we assume that the return process {X n}n=0 is an [a, b]m-valued station-ary and ergodic process (0 < a

b |X d1 =xd1,...,X 0 = x0] holds. The Lipschitz constant L is taken as a known marketconstant.

Using a stochastic gradient algorithm and combining it with nonparametricregression estimators,

we establish the strong convergence of the estimates bn to the true log-optimal portfolio selection function b, avoiding the usual mixing conditions (Theorem 6.2.2).

What is even more important in practical applications:

Selecting portfolios on the basis of the estimated log-optimal portfolioselection functions yields optimal growth of wealth among all other strategies that take their investment decisions on the basis of the last d observations.


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Indeed, let S n be the wealth accumulated during n market periods when onthe ( i + 1)st day of trading the portfolio bi(X id+1 ,...,X i) is selected using themost recent estimate bi of a log-optimal portfolio selection function. Then, if S n is the wealth accumulated during the same period using any other portfolioselection function of the last d observed return vectors,

limsupn

1n

logS nS n 0

with probability 1 (Corollary 6.2.3).

After an appropriate modication, the algorithms and the results remain valideven if the market constant L is unknown in real market applications (Theorem6.2.4). Section 6.3 proves the ndings, and the chapter is rounded off withseveral realistic examples in Section 6.4.

Chapters 2, 3 and 6 can be read independently from each other, they are self-contained. Chapters 4 and 5 are closely linked, however. Notation that goesbeyond common mathematical style is explained where it occurs for the rsttime. We also refer the reader to the list of abbreviations at the beginningof the thesis. The calculations and plots for the examples were generated us-ing Matlab 4.0 and 6.0.0.88, Minitab 11.2 and R 1.1.1 with historical stockquotes (daily closing prices) from the New York Stock Exchange provided bywww.wallstreetcity.com.


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ZUSAMMENFASSUNG

Diese Arbeit soll ein Pl adoyer sein f ur die Anwendung nichtparametrischerstatistischer Vorhersage- und Sch atzmethoden auf Probleme, wie sie bei derPlanung von Finanzanlagen und Investitionen auftreten.

In sechs Kapiteln werden verschiedene Anwendungsm oglichkeiten nichtparamet-rischer Techniken bei der Portfolioauswahl an Finanzm arkten analysiert. Dieskann nat urlich nur einen groben und zugegebenermaen willk urlichen Aus-schnitt aus diesem weiten Gebiet widerspiegeln wir hoffen jedoch, dadurch dieVorz uge nichtparametrischerSch atzmethoden bei der Portfolioauswahl aufzeigenzu konnen.

Kapitel 1Einf uhrung: Investment und nichtparametrische Statistik

Investment ist der strategisch geplante Einsatz von Ressourcen ( ublicherweisevon nanziellen Ressourcen) in einer Umgebung ( ublicherweise in einem Fi-nanzmarkt), deren zuk unftige Entwicklung zuf alligen Fluktuationen unterliegt.Investitionsprobleme treten in einer Vielzahl von Gebieten auch uber den -nanziellen Kontext hinaus auf. Dabei k onnen Ressourcen u. A. die Formvon Energie, von Datenverarbeitungskapazit aten, etc. annehmen. Die strate-gische Planung von Investitionen hilft, viele Prozesse mit h oherem Nutzen zubetreiben. Diese Arbeit konzentriert sich auf nanzielle Investitionen, welchegleichsam denPrototypf ur verschiedenste Prozesse bilden, bei denen System-ressourcen gewinnbringend einzusetzen sind.

Bei Investitionen nanzieller Natur spielen drei Komponenten eine Rolle: derMarkt, die Handlungsm oglichkeiten des Investors und sein Investitionsziel. DieseBausteine werden in den Abschnitten 1.1-1.3 im Detail diskutiert.

Zum Markt: Wir gehen von einem Finanzmarkt mit m Anlagem oglichkeiten(Aktien, festverzinsliche Wertpapiere, ...) aus. Die i. Anlagemoglichkeiterzielt in der Marktperiode n eine Rendite X i,n auf eine Investition von

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einer Geldeinheit. Die n. Marktperiode dauere vom Zeitpunkt n 1 biszum Zeitpunkt n, wobei die Zeit z.B. in Handelstagen gemessen wird. DieRenditen der einzelnen Anlagem oglichkeiten am n. Handelstag werden imRenditevektor

X n = ( X 1,n ,...,X m,n )T IRm+

zusammengefasst. In den Augen des Investors ist {X n}n=1 ein stochasti-scher Prozess, welcher in vielen realen M arkten station ar und ergodischist (Denition 1.1.1). In manchen Kapiteln dieser Arbeit werden (re-alistische) Zusatzannahmen uber die Verteilung des Prozesses getroffen.Entscheidend ist dabei jedoch,

dass wir nichtparametrische Modelle betrachten Modelle also, die nicht von der Existenz einer parametrischen Entwicklungsgleichung ausgehen, wie sie z.B. ARMA-, ARCH- und GARCH-Prozesse be-sitzen.

Diese Modelle garantierenh ochste Flexibilit at bei der Anwendung in realenFinanzm arkten.

Zu den Handlungsm oglichkeiten: Wir betrachten einen Investor, der wederTeile seines Verm ogens auf personlichen Konsum verwendet, noch seinemPortfolio im Verlauf des Investitionsprozesses neues Geld zuieen l asst.Am Beginn jeder Marktperiode n verwendet der Investor sein gesamtesVerm ogen darauf, ein Aktienportfolio bn zu erwerben. Ein solches Portfoliobn wird durch die Anteile b j,n am aktuellen Gesamtverm ogen des Investorsbeschrieben, welche in der n. Marktperiode in die Anlegem oglichkeit j =1,...,m investiert werden. Die Wahl von bn erfolgt dann aus der Menge S aller Portfolios, welche aus den Vektoren (Portfolios)

bn = ( b1,n ,...,bm,n )T

besteht, f ur die b j,n 0 und m j =1 b j,n = 1. In manchen Situationen wirdS weiter durch das Auftreten von Transaktionskosten eingeschr ankt.

Zum Investitionsziel: W 0 sei das anf angliche Verm ogen des Investors. Ver-wendet er die Portfoliostrategie {bi}n1i=0 , wird er nach n Marktperiodenuber das Verm ogen W n = ni=1 W 0 verf ugen (< , > be-zeichnet das Euklidische Skalarprodukt). Ziel des Investors ist es, f ur W n

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einen moglichst groen Wert zu erzielen. Aus der Literatur ist bekannt,dass dabei kein grundlegender Konikt zwischen nahen ( n endlich) undfernen (n ) Investitionshorizonten besteht. In beiden F allen ist eineInvestition zum Zeitpunkt n gema dem bedingt log-optimalen Portfolio

bn := argmaxbS E [log < b, X n > |X n1,...X 1]

optimal. Es ubertrifft jede andere Strategie indem

EW n

W n 1 und limsup

n

1

nlog

W n

W n 0 mit Wahrscheinlichkeit 1

(Cover and Thomas, 1991, Theorem 15.5.2). W n ist dabei das Verm o-gen zum Zeitpunkt n, das der Investor durch eine Serie von bedingt log-optimalen Investitionen erzielt, W n das Verm ogen mit einer beliebigen an-deren Portfoliostrategie, die nicht uber mehr Information verf ugt als ausvergangenen Marktbeobachtungen ableitbar (eine sogenannte kausaleStrategie).

Dies sollte f ur den Investor Grund genug sein, eine logarithmische Nutzenfunktion zu verwenden, d.h. mit dem Wissen um die in der Vergangenheit beobachteten Renditevektoren die Maximierung der er-

warteten zuk unftigen logarithmierten Rendite zu betreiben.

Das bedingt log-optimale Portfolio leitet sich aus der Verteilung des Rendite-prozesses {X n}n ab. In der Realit at ist die wahre Verteilung der Renditen unddamit auch die bedingt log-optimale Strategie dem Investor nicht bekannt. Andiesem Punkt bedarf die Finanzplanung der Statistik als Partner. Die Statistikdient dem Investor zur L osung des Problems,

eine Methode zu nden, die nur anhand historischer Renditedaten und ohne Kenntnis der wahren Renditeverteilung eine optimale kausale Portfoliostrategie

{bn}n erzeugt. Optimalit at wird hier in dem Sinn verwendet, dass die Strate-gie f ur jeden station aren und ergodischen Renditeprozess {X n}n das Verm o-gen W n :=

ni=1 < bi, X i > des Investors im Mittel genauso schnell wach-

sen l asst wie die log-optimalen Strategie {bn}n . Formal ausgedr uckt soll {bn}ngarantieren, dass mit Wahrscheinlichkeit 1

limsupn

1n

logW nW n 0.

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Es ist bekannt, dass solche Methoden existieren (Algoet, 1992). Diese brin-gen jedoch den Nachteil mit sich, h ochst komplex zu sein und zur Erzeugungpraktisch verwertbarer Ergebnisse eine Unmenge historischer Daten zu ben oti-gen. Ein Ziel dieser Arbeit ist es, vereinfachte, aber effiziente Algorithmen zurPortfolioauswahl zu entwickeln, die auf nichtparametrischen Vorhersage- undSchatzverfahren basieren. Die Algorithmen sollen so gestaltet sein, dass sie f urmoglichst groe Klassen von Markten anwendbar sind.

Kapitel 2Der Vergleich von Portfolios: Konvergenzraten und Dimension

Die Gute einer Methode zur Portfolioauswahl wird in der Regel durch denVergleich mit einer Referenzstrategie beurteilt. Unsere Referenzstrategie istdie log-optimale Portfolioauswahl, die wie wir in Kapitel 1 gesehen habeneine optimale Verhaltensregel darstellt. Dem Investor wird es nicht gelin-gen, letztere zu ubertreffen. Nat urlich wird er hoffen, dass der Mangel anLeistungsf ahigkeit seiner eigenen Strategie im Verlauf des Investitionsprozessesverschwindet, wenn n amlich seine Schatzungen f ur die Verteilung des Ren-diteprozesses mit zunehmender Menge verf ugbarer historischer Daten immerbesser werden. W ahlt der Investor zum Zeitpunkt n anhand der Beobachtun-gen X 1,...,X n sein Portfolio, wird er in der n achsten Marktperiode eine Renditevon R

n= erwirtschaften, w ahrend die log-optimale Strategie

Rn = < bn+1 , X n+1 > liefert. Der Vergleich beider Werte erm oglicht die Ein-schatzung, um wieviel bn+1 der log-optimalen Strategie bn+1 unterlegen ist.

Vom Standpunkt einer logarithmischen Nutzenfunktion ist es daher angebracht, die Unterlegenheit der Strategie bn+1 an der Positivit at der erwarteten Differenz der log-Renditen, an E log R

n

Rnzu messen. Je

kleiner dieser Wert, desto besser ist die Strategie bn+1 .

Zur Beurteilung der Qualit at der Strategie bn+1 ist also insbesondere zu analysie-ren, mit welcher Geschwindigkeit E log R

n

Rngegen Null strebt. Dabei wird davon

ausgegangen, dass die Renditen in einem Prozess von unabh angigen, identischverteilten Zufallsvariablen auftreten. Unter Verwendung von Konzepten derInformationstheorie wird in Abschnitt 2.1 eine untere Schranke f ur diese Kon-vergenzgeschwindigkeit abgeleitet. Diese besagt, dass selbst im einfachsten allerMarkte, einem Markt mit nur endlich vielen m oglichen Renditekonstellationengilt:

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Es gibt keine Portfolioauswahlregel, die ihre Unterlegenheit im Ver-gleich zur log-optimalen Strategie in jedem Markt schneller kompen-siert als 1n gegen Null strebt, d.h. es gibt stets einen Markt, f ur den E log R

n

Rn const. 1n (Theorem 2.1.1).

Es gibt jedoch Portfolioauswahlregeln, die diese Rate erreichen. Insbesonderedas empirisch log-optimale Portfolio

bn+1 := argmaxb

S

1

n

n

i=1

log < b, X i > (0.0.3)

erweist sich hier als gunstig:

Das empirische log-optimale Portfolio (0.0.3) erreicht die untere Schranke f ur die Konvergenzrate von E log R

n

Rn(Theorem 2.1.3).

Etwas leger ausgedr uckt k onnte man sagen, dass das empirisch log-optimalePortfolio seine Dezite mit optimaler Geschwindigkeit wettzumachen vermag.Die Ergebnisse gelten weitestgehend unabh angig von der Anzahl der Aktienam betrachteten Markt. Dies ist untypisch f ur nichtparametrische Sch atzver-fahren und bedarf daher genauerer Diskussion (Theorem 2.1.4 zeigt, dass diesesPh anomen auch in komplizierter gearteten M arkten auftritt).Aus diesem Grund schlieen wir in Abschnitt 2.2 eine detailliertere Diskus-sion der Auswirkungen der Dimension des Marktes auf die Portfolioauswahl an.Beschrankte rechnerische Kapazit aten werden den Investor bei seiner Investi-tionsplanung dazu zwingen, sich auf eine kleinere Teilmenge aller Aktien amMarkt zu beschr anken. Diese Teilmenge muss bereits in der Planungsphase,also vor dem eigentlichen Investitionsprozess ausgew ahlt werden. Es werdenKriterien f ur diese Vorauswahl ben otigt. Ublicherweise wurde man vorgehen,indem man einzelne Aktien ausw ahlt, deren Chart hohe Wachstumspotentialeversprechen. Es wird gezeigt werden, dass dieser Weg mit substantiellen Un-zulanglichkeiten behaftet ist:

Jedes Auswahlverfahren, das die einzelnen Aktien getrennt, z.B. an-hand ihrer erwarteten logarithmierten Rendite, beurteilt, wird mit Sicherheit in einem realistischen Markt die falsche Auswahl treffen (Theorem 2.2.1).


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Dieses negative Resultat zeigt, dass Portfolioauswahlverfahren uber die einzel-nen erwarteten log-Renditen hinausgehende Information ben otigen. Die Vari-anz-Kovarianz-Struktur der Renditen wird in M arkten mit log-normal verteiltenRenditen hinreichend viel Information vermitteln. In Abschnitt 2.3 werden dieResultate anhand von Simulationen und realen Beispielen illustriert und ihrepraktische Relevanz aufgezeigt.

Kapitel 3Renditevorhersagen und Portfolioauswahl

Mit der Erkenntnis, dass erfolgreiche Portfolioauswahl Information uber dieVarianz-Kovarianz-Struktur der Aktien am Markt bedarf (es spielen sowohlzeitliche Korrelationen als auch Korrelationen zwischen den einzelnen Aktieneine Rolle), wird in Abschnitt 3.1 eine Investmentstrategie vorgestellt, die sichunter den Investoren groer Beliebtheit erfreut.

Die Strategie ist zweistug und verwendet dabei die historischen log-RenditenY n ,Y n1,...,Y 0 (Y i := log X i) als Eingangsdaten f ur die Investitionsentscheidungzur Zeit n:

1. Erstelle eine Schatzung f ur die Zukunft des Marktes. Es wird gezeigtwerden, dass Vorhersagen f ur den Markt auf bedingten Erwartungen f ur

zukunftige log-Renditen bei gegebener Vergangenheit basieren sollten,d.h. auf Y n+1 := E[Y n+1 |Y n , Y n1,...].

2. Investiere ausschlielich in die Aktien, deren Vorhersagen Y n+1 eine bessereRendite verheien als ein festverzinsliches Wertpapier mit Rendite r . Ineine Aktie wird also investiert genau dann, wenn

exp( Y n+1 ) r.Wir nennen diese Strategie eine Strategie f ur den gierigen Investor, da sie da-rauf ausgerichtet ist, nur die bestm oglichen Anlagemoglichkeiten herauszupicken.

Die Einfachheit der Strategie besticht, und in M arkten mit geringer Varianz derlog-Renditen f uhrt sie zu sinnvollen Ergebnissen (Abschnitt 3.1).

Bei der Implementierung der Strategie sieht sich der Investor der Schwierigkeitgegenuber, dass die Vorhersagewerte Y n+1 nur unter Kenntnis der wahren Vertei-lung des Prozesses berechnet werden k onnen. Daher wird man sich auf die


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Berechnung einer Sch atzung E (Y n ,...,Y 0) f ur den bedingten ErwartungswertE [Y n+1 |Y n , Y n1,...] aus den Marktbeobachtungen Y n ,...,Y 0 beschr anken m ussen.Aus der Literatur ist bekannt, dass keine auf solche Weise gewonnene Sch atzungstark konsistent sein kann in dem Sinne, dass

limn

E (Y n ,...,Y 0) E[Y n+1 |Y n , Y n1,... ] = 0 (0.0.4)mit Wahrscheinlichkeit 1 f ur jeden station aren und ergodischen Prozess {Y n}ngilt (Bailey, 1976). Dieses Resultat ist einerseits entmutigend, andererseits

schliet es nicht aus, dass stark konsistente Vorhersagemechanismen f ur loga-rithmierte Renditeprozesse existieren, wie sie in realen Finanzm arkten auftreten.Dabei ist insbesondere an Gausche log-Renditeprozesse zu denken, die einegute Approximation f ur reale log-Renditeprozesse liefern. Bis jetzt jedoch wardie Frage unbeantwortet, ob f ur station are und ergodische Gausche Prozessestark konsistente Vorhersagealgorithmen existieren. Abschnitt 3.2 wird nuneine positive Antwort darauf geben k onnen. Unter schwachen Zusatzvorausset-zungen an die Wold-Koeffizienten des Prozesses

wird ein Vorhersagealgorithmus E (Y n ,...,Y 0) f ur station are und er-godische Gausche Prozesse entwickelt, der stark konsistent gem a (0.0.4) ist und der bemerkenswert einfach zu implementieren ist (Corollary 3.2.3).

Diese Ergebnisse geben uns die Subroutinen an die Hand, um die Strategie f urden gierigen Investor in Gauschen log-Renditeprozessen umzusetzen. DerAlgorithmus selbst ist jedoch auch unabh angig von seiner hier gegebenen An-wendung von Interesse, treten Vorhersageprobleme f ur Gausche Prozesse dochin einer Vielzahl von Gebieten auf.

Der Beweis der Konvergenzeigenschaften wird in Abschnitt 3.3 gef uhrt. Anwen-dungsbeispiele mit realen und simulierten Daten schlieen sich in Abschnitt 3.4an und zeigen vielversprechende Ergebnisse, wenn der Algorithmus zur reinenVorhersage, aber auch als Subroutine f ur die gierigeStrategie dient.

Kapitel 4Ein Markov-Modell mit Transaktionskosten: stochastische Aspekte

In den einfachsten M arkten, in denen die Renditen als unabh angige, iden-tisch verteilte Zufallsvariablen auftreten, sollte in ein zeitlich konstantes log-


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26

optimales Portfolio investiert werden. Bei Verwendung eines zeitlich konstan-ten Portfolios verwendet man auf jede Aktie einen gleichbleibenden Anteil desaktuellen Gesamtverm ogens. Der Anteil bleibt somit derselbe, bedingt durchdie Anderung der Aktienpreise zueinander andert sich jedoch die tats achlicheAnzahl an gehaltenen Aktien von Marktperiode zu Marktperiode. Zur Durch-f uhrung einer log-optimalen Strategie wird somit eine groe Anzahl an Transak-tionen notwendig. In der Realit at stellt dies einen nicht zu untersch atzendenNachteil dar. Was immer an Verm ogen anwachst, ein Groteil der Gewinne wirdzur Begleichung von Transaktionskosten wie Maklerprovisionen, Verwaltungs-

und Kommunikationskosten wieder abieen. Folglich muss der Investor seineStrategie diesen Gegebenheiten anpassen: Er muss so wenige kostenintensiveTransaktionen wie m oglich machen, aber doch so viele, um ein gutes Wertwachs-tum zu erzielen. Kapitel 4 und 5 widmen sich der Frage, wie diese beidenAnforderungen in einer Strategie miteinander vereinbart werden k onnen.

Zu diesem Zweck nehmen wir an, dass die Renditen sich gem a einem d-stugenMarkovschen Prozess entwickeln. In Kapitel 4 arbeiten wir unter der Pr amisse,dass die Verteilung des Renditeprozesses bekannt ist, eine unrealistische An-nahme, die wir in Kapitel 5 fallen lassen werden. Zun achst wird in Abschnitt4.1 das Marktmodell aus Kapitel 1 um Transaktionskosten erweitert, die pro-portional zum Volumen gekaufter Aktien anfallen. Es ist nicht uberraschend,

dass sich der Investor in einer solchen Situation nur eine eingeschr ankte Mengevon Portfoliozusammenstellungen leisten kann, ohne bankrott zu gehen. Eswird deutlich werden,

dass es in d-stugen Markovschen Renditeprozessen ausreicht, Strate-gien zu betrachten, die auf Portfolioauswahlfunktionen beruhen, d.h.Strategien der Form bi = (bi1, X id,...,X i1) mit einer geeigneten Funktion (Denition 4.1.2).

Das nachste zu wahlende Portfolio ist somit eine Funktion des letzten gew ahltenPortfolios und der letzten d am Markt beobachteten Renditevektoren. Wiezuvor strebt der Investor danach, sein zu erwartendes logarithmiertes Verm o-

genswachstum zu maximieren, hier nun indem er eine optimale Portfolioaus-wahlfunktion wahlt.

Abschnitt 4.2 legt dar, wie eine optimale Auswahlfunktion c konstruiert werdenkann alles unter der Pr amisse, dass die wahre Verteilung der Renditen bekanntware. Das Hauptresultat wird zeigen,


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dass eine optimale Portfolioauswahlfunktion aus einer L osung der Bellman-Gleichung konstruiert werden kann (Theorem 4.2.1, Glei-chung 4.2.2).

Die Bellman-Gleichung ist aus der Theorie der dynamischen Optimierung wohl-bekannt, dennoch werden sich fundamentale Unterschiede zwischen klassischerdynamischer Optimierung und dem Portfolioauswahl-Problem zeigen. Zur Vor-bereitung auf Kapitel 5 werden in Abschnitt 4.3 schlielich weitere analytischeEigenschaften der L osung der Bellman-Gleichung abgeleitet.

Kapitel 5Ein Markov-Modell mit Transaktionskosten: statistische Aspekte

Die Bellman-Gleichung, wie sie in Kapitel 4 aufgestellt wurde, h angt entschei-dend von der Verteilung des Renditeprozesses {X n}n ab. Diese Abh angigkeitbesteht in Form eines zu evaluierenden bedingten Erwartungswertes. Aus diesemGrund sind die Ergebnisse von Kapitel 4 nur unter der Pr amisse gultig, dassder Investor die wahre Verteilung des Renditeprozesses kennt, was in der Praxisnat urlich illusorisch ist. Bestenfalls verf ugt der Investor uber eine Schatzung derVerteilung der Renditen. Diese erm oglicht es ihm, eine Schatzung f ur bewusstenbedingten Erwartungswert zu berechnen, welche ihm dann eine N aherung derBellman-Gleichung liefert. Mit Hilfe von Techniken aus der nichtparametrischen

Regressionssch atzung

wird in Abschnitt 5.1 gezeigt, dass zur Bellman-Gleichung aus Kapi-tel 4 eine nat urliche empirische Entsprechung basierend auf Markt-beobachtungen existiert (Gleichung 5.1.2).

Ahnliche Schlussweisen wie in Kapitel 4 werden es uns erm oglichen, diese em-pirische Bellman-Gleichung unter realistischen Bedingungen zu l osen.

Das wird zu einer Strategie f uhren, die ausschlielich auf historischen Renditen basiert, dabei jedoch dieselben Optimalit atseigenschaften wie die (theoretisch) optimale Portfolioauswahlstrategie unter Transak-tionskosten hat (Theoreme 5.1.1 und 5.1.2).

In den Betrachtungen von Kapitel 5 werden wir auf Verallgemeinerungen vonbekannten Resultaten uber die gleichm aige Konvergenz von Regressionssch at-zern zuruckgreifen. Diese Verallgemeinerungen werden in Abschnitt 5.2 her-geleitet. Ist z.B. {X n}n ein station arer Prozess, welcher die geometrischen

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Mischungseigenschaft hat, und ist g aus einer Klasse Glipschitzstetiger Funk-tionen gewahlt, sch atzen wir den bedingten Erwartungswert

R(g,b,x) := E[g(X 1, b)|X 0 = x] (bS )durch einen Kernsch atzer Rn (g,b,x). In Abhangigkeit von der Glattheit einerDichte von X 0 (wir nehmen an, dass eine solche existiert) wird die Konvergenz-geschwindigkeit in der Limesrelation

supg

G

E supx

X ,b

S |Rn(g,b,x) R(g,b,x)| 0 (n )bestimmt. Dabei wird der zu erwartende maximale Sch atzfehler gleichm aig inder Klasse Gbetrachtet (Corollary 5.2.2). Das erhaltene Resultat ist nicht nurim Hinblick auf unsere Anwendung von Interesse, sondern auch dar uber hinausals unabh angiges Resultat in der nichtparametrischen Regressionssch atzung.

Abschnitt 5.3 schlielich widmet sich dem Beweis der Optimalit atseigenschaftendes Algorithmus und kombiniert dabei die Ergebnisse aus Kapitel 4 mit denErgebnissen zur gleichm aig konsistenten Regressionssch atzung.

Kapitel 6Portfolioauswahlfunktionen in station aren Renditeprozessen

An realen Finanzm arkten beobachtet man unter Umst anden eine Abweichungdes Renditeprozesses {X n}n von einem d-stugen Markov-Prozess. Deshalbwerden in diesem Kapitel noch allgemeinere Marktmodelle zu betrachten sein.Transaktionskosten werden dabei ignoriert, daf ur aber Renditeprozesse betrach-tet, f ur die im Wesentlichen nur Stationarit at und Ergodizit at vorausgesetztwird. Fur den Investor ist es naheliegend, seine Investitionsentscheidungen an-hand der letzten d am Markt beobachteten Renditevektoren ( d fest) zu treffen.Dies f uhrt zum Konzept von log-optimalen Portfolioauswahlfunktionen.

Dieses Konzept wird in Abschnitt 6.1 eingef uhrt. Der Investor verwendet wiedereine logarithmische Nutzenfunktion und versucht daher, eine log-optimale Port- folioauswahlfunktion zu ermitteln, d.h. eine messbare Funktion

b: IRdm+ S,so dass (< , > bezeichnet das euklidische Skalarprodukt)

E (log < b(X 0,...,X d1), X d > ) E (log < f (X 0,...,X d1), X d > )

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f ur alle messbaren Funktionen f : IRdm+ S . Fur die (n + 1). Marktperiodelegt bdem Investor nahe, das Portfolio b(X nd+1 ,...,X n) zu erwerben.Das Konzept log-optimaler Portfolioauswahlfunktionen bleibt in seiner Allge-meinheit hinter dem Konzept des bedingt log-optimalen Portfolios zur uck (dieseswahlt den Parameter d so, dass die ganze Vergangenheit des Prozesses in diePortfolioauswahl einbezogen wird). Obwohl es sich in diesem Sinn um eineVereinfachung handelt, vereinigen log-optimale Portfolioauswahlfunktionen imVergleich zum log-optimalen Portfolio einige Vorteile auf sich, insbesondere wasBerechnung, Sch atzung und Interpretation angeht.

Bei log-optimalen Portfolioauswahlfunktionen sieht sich der Investor demselbenProblem gegen uber wie bei der Verwendung log-optimaler Portfolios. Beidekonnen nur berechnet werden, wenn die wahre Verteilung des Renditeprozessesbekannt sein sollte. In der Praxis ist dies nicht der Fall, und man ben otigt wiedereine Schatzprozedur, die eine log-optimale Portfolioauswahlfunktion anhand inder Vergangenheit beobachteten Renditedaten ann ahert.

In Abschnitt 6.2 wird deshalb ein Algorithmus entwickelt, der Sch atzungen bn f ur eine log-optimale Portfolioauswahlfunktion baus historischen Renditedaten berechnet.

Uber Stationarit at und Ergodizit at hinaus werden dabei sehr milde Zusatzvo-raussetzungen getroffen, konkret wird davon ausgegangen, dass der Renditepro-zess {X n}n=0 ein [a, b]m-wertiger station arer und ergodischer stochastischer Pro-zess ist (0 < a b < brauchen nicht bekannt zu sein) und dass eineLipschitzbedingung f ur den bedingten Renditequotienten E[X d / < s, X d >

|X d1 = xd1,...,X 0 = x0] gilt. Die Lipschitzkonstante L sei dabei eine bekan-nte Marktkonstante.

Mit Hilfe eines stochastischen Gradientenverfahrens und Methoden der nicht-parametrischen Regressionssch atzung wird gezeigt,

dass die Sch atzungen bn mit Wahrscheinlichkeit 1 gegen die wahre log-optimale Portfolioauswahlfunktion bkonvergieren, wobei die in der Literatur typischerweise vorausgesetzten Mixing-Bedingungen ver-mieden werden (Theorem 6.2.2).

In der praktischen Anwendung spielt das folgende Resultat eine noch wichtigereRolle:


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Eine Portfolioauswahl anhand der gesch atzten log-optimalen Portfolio-auswahlfunktionen liefert ein optimales Verm ogenswachstum unter allen Strategien, die ihre Investitionsentscheidungen anhand der letz-ten d am Markt beobachteten Renditen treffen.

Sei S n das Verm ogen, das man nach n Marktperioden erzielt hat, wenn manam (i + 1). Handelstag die aktuelle Sch atzung bi verwendet, um das Portfo-lio bi(X id+1 ,...,X i) zu wahlen. Wenn S n das Verm ogen angibt, das man inderselben Zeit mit einer beliebigen anderen Auswahlstrategie basierend auf den

jeweils letzten d beobachteten Renditen erwirtschaftet, so ist

limsupn

1n

logS nS n 0

mit Wahrscheinlichkeit 1 (Corollary 6.2.3).

Nach einer geeigneten Modikation behalten die Algorithmen und die Resultateihre Gultigkeit, selbst wenn wie in der Anwendungspraxis die MarktkonstanteL dem Investor unbekannt ist (Theorem 6.2.4). Abschnitt 6.3 beweist die Resul-tate, und das Kapitel wird mit mehreren realistischen Beispielen in Abschnitt6.4 abgerundet.

Die Kapitel 2, 3 und 6 konnen unabh angig voneinander gelesen werden, sie sindin sich abgeschlossen. Kapitel 4 und 5 sind jedoch eng verzahnt. Notationen,die uber die mathematische Standardnotation hinausgehen, werden bei ihremersten Auftreten erkl art. Der Leser sei auch auf das Abk urzungsverzeichnis amAnfang dieser Arbeit verwiesen. Berechnungenund Schaubilder f ur die Beispielewurden mit Matlab 4.0 und 6.0.0.88, Minitab 11.2 sowie R 1.1.1 erzeugt, wobeidie historischen Kursnotierungen (t agliche Schlusskurse) der New Yorker B orsevon www.wallstreetcity.com Verwendung fanden.


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ACKNOWLEDGEMENTS

I am endebted to

Prof. Harro Walk, who suggested to investigate the subject. He advised me onmany points, always found the time to discuss the results, and more thanonce I benetted from his extensive knowledge.

Prof. Laszlo Gy or, for his hospitality during my visits to the Technical Uni-versity of Budapest. On several occasions he gave me the right impulseand really useful advice.

Prof. Volker Claus, for his interest in my work and for discussing the contentsof this thesis with me.

The DFG and the College of Graduates Parallel and Distributed Systems, for

funding my research with everything that involves.

Dr. Michael Kohler, who introduced me to nonparametric curve estimation.His expertise in this eld was an invaluable source.

Dr. J urgen Dippon, who never threw me out when I felt like discussing prob-lems in mathematical statistics and nance.

Prof. Adam Krzy zak, for being my host during a stay at Concordia University,Montreal, for many discussion about mathematical and other interestingsubjects.


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CHAPTER 1

Introduction: investment and nonpara-

metric statisticsInvestment is the strategic allocation of resources, typically of monetary re-sources, in an environment, typically a market of assets, whose future evolutionis uncertain. This denition leaves much room for subjective interpretation. Inparticular, the following points have to be made more precise:

What market is under consideration? This involves specifying and stan-dardizing the assets traded in the market (e.g. stocks, bonds, options,futures, currencies, gold, oil, ...) as well as setting up a reference systemfor pricing the assets (e.g. closing or opening prices at the New York

Stock Exchange, world market price for raw materials, ...).

What actions and instruments may be applied by the investor? Possibleactions may be restricted by exogenous terms and regulations of trade(e.g. transaction costs, brokerage fees, trading limitations) or personalpreferences (e.g. to rule out borrowing money or short positions in stocks).

What investment goal is pursued by the investor? Traditionally, the goalis the maximisation of a personal utility function of the returns on theallocated resources. The market being chancy, individual risk aversionpreferences may enter the form of the utility function, or restrictions areimposed on the set of possible investment actions.

Thus, investment becomes a highly subjective term, including investment asit is understood in this thesis. In the following we set up the specic invest-ment scenario as we shall consider it in this thesis. We believe this scenariois broadly accepted as the typical setting for investment analysis, although we


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34 Chapter 1. Introduction: investment and nonparametric statistics

do not deny that particular investment situations require further adaptationand modication. It should also be pointed out that, as future asset prices aresubject to random uctuations,investmentis a good deal about decision tak-ing under uncertainty, which makes mathematical statistics the natural partnerof investment (an observation that may be attributed to the groundbreakingwork of Bachelier, 1900, who used statistics to compare his theoretical modelwith real market data). An economist will nd the economic side of this thesisto be lacking. There are excellent books on investment science from a moreeconomic point of view (e.g. Francis, 1980; Luenberger, 1998), but most of

them are lacking in statistical depth. This thesis is about investment from adecisively statistical point of view we can therefore only supercially touchupon economic issues.

1.1 The market modelWe consider a market in which m assets (which we will think of as stocks andbonds) are traded. Taking a macroeconomic point of view, the prices of theassets (stock quotes, bond values) are generated under the authority of themarket as a whole, i.e. by the large ensemble of investors. We assume that for

the individual investor there is no way to inuence the prices by launching spe-cic investment actions or distributing insider or side information of whateverkind. In this situation, let P 1,n ,...,P m,n > 0 be the prices of the assets 1 ,...,mat the beginning of market period n (market period n lasts from time n 1to n, time being measured, e.g., in days of trading). To the powerless individ-ual investor described above, the asset prices present themselves as a randomprocess on a common probability space ( , A, P ).The return of an investment of 1 unit of money in asset i at time n 1 yieldsa return

X i,n :=P i,n

P i,n1during the subsequent market period. We collect the returns of the single assetsin a return vector

X n := ( X 1,n ,...,X m,n )T .

We will often work with the log-returns

log X n := (log X 1,n ,..., logX m,n )T .

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1.1 The market model 35

The return process {X n}n=1 and the log-return process {log X n}n=1 arestochastic processes on ( , A, P ).In most of our investigations we will assume that the return process {X n}n isstationary and ergodic in the sense of the following denition (Stout, 1974, Sec.3.5; Shiryayev, 1984, V 3):Denition 1.1.1. Let {X n}n=1 be an IRm -valued stochastic process on a prob-ability space (, A, P ).

1. {X n}n=1 is called stationary , if P (X i ,...,X j ) = P (X i + t ,...,X j + t )

for all integers i , j , t with i j .2. A Ais called an invariant event of {X n}n=1 , if there exists a B

B ((IR m )) such thatA = {X i, X i+1 ,...}1(B)

for all iIN.

3. A stationary process {X n}n=1 is called ergodic , if the probability of any invariant event of {X i}i=1 is either 0 or 1.

Stationarity preserves the stochastic regime over time, ergodicity is the settingwhere time averages along trajectories of the process converge almost surely toexpected values under the process distribution:

Theorem 1.1.2. (Birkhoff Ergodic Theorem, Stout, 1974, Sec. 3.5) Let

{X n}n=1 be an IRm-valued stationary and ergodic stochastic process on a prob-ability space (, A, P ) with E |X 1| < . Then

1n

n

i=1

X i EX 1P -almost surely ( P -a.s.), i.e. for all from a set of probability 1.

Stationarity and ergodicity are the basic assumptions for most statistical inves-tigations. The stationarity of stock returns is a thoroughly investigated eld,


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both by economists (e.g. Francis, 1980, A24-1) and statisticians (e.g. Franke etal., 2001, Sec. 10.6). It is natural to assume that there is short term stationar-ity in most stock returns, some authors (Francis, 1980) even claim that returndata may be treated as stationary if the time horizon comprises at least onecomplete business cycle. There is no conclusive answer that proves or disprovesstationarity for the majority of stock markets, and it seems as though this hasto be decided from case to case. We accept stationarity as a working hypothesis,accounting for the fact that it is common practice to assess and compare theperformance of statistical methods in the stationary setting.

Not much is known about the ergodic properties of stock quotes or stock re-turns, neither from the theoretical economists point of view, nor from empiricalstudies. There are indications that the ergodic properties of a market dependvery much upon the ow of information in the market and on the microeconomicprice generation (Donowitz and El-Gamal, 1997). These are difficult to assess,and so the typical approach has become to derive algorithms under ergodichypotheses and then let the success of the algorithm justify the hypotheses.

Throughout this thesis we consider nonparametric models for {X n}n , i.e.models that do not require a parametrized evolution equation (in contrast toMA, AR, ARMA, ARIMA, ARCH and GARCH models, cf. Brockwell andDavis, 1991, Franke et al., 2001). The nonparametric approach guaranteeshighest exibility in modelling, skipping model parameters which otherwiserequire extensive diagnostic model testing. To be more precise, the followingmodels will be investigated in this thesis:

1. {X n}n is a sequence of independent identically distributed (i.i.d.) randomvariables (e.g. with nitely many outcomes) Chapter 2.1.

2. The conditional distribution of X n+1 given X n ,...,X 1 (which we will de-note by P X n +1 |X n ,...,X 1 ) is log-normally distributed (i.e. P log X n +1 |X n ,...,X 1 hasa normal distribution) Chapter 2.2.

3.

{log X n

}n is a stationary Gaussian time series (i.e. (log X T

n+ k,..., logX T

n))

follows a multivariate normal distribution which depends upon k but notupon n) Chapter 3.

4. {X n}n is a Markov process of order d (i.e., we assume P X n +1 |X n ,...,X 1 =P X n +1 |X n ,...,X n d+1 ) Chapters 4 and 5.


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1.2 Portfolios and investment strategies 37

5. {X n}n is a stationary and ergodic time series Chapter 6.Each of these models has been found useful for describing asset return data inreal nancial markets. Model 1 is the Cox-Ross-Rubinstein model (Cox et al.,1979; Francis, 1980, A24-1 and A24-2; Luenberger, 1998, Ch. 11; Franke et al.,2001, Ch. 7). Models 2 and 3 are models with log-normal returns (Francis, 1980,A24-1; Luenberger, 1998, Ch. 11) which arise, e.g., from a discretisation of theBlack-Scholes model (Luenberger, 1998, Ch. 11; Korn and Korn, 1999, Kap.II). In contrast to the classical Black-Scholes model we allow for autocorrelated

log-returns in Chapter 3 (i.e. Cov (log X n , logX n+ k) = 0 for some k > 0). Inpractice, autocorrelation of the log-returns manifests itself for small time lagsk (Franke et al., 2001, Ch. 10) as well as large k (long range dependence , Dinget al., 1993; Peters, 1997). Many studies have indicated that the logarithmsof stock returns slightly depart from a Gaussian distribution (e.g. by heavy tails , Mittnik and Rachev, 1993; McCulloch, 1996; Franke et al., 2001, Ch. 10and the references there). It is therefore advisable to drop the assumption of log-normality of the stock returns wherever possible. This is done in models 4and 5, model 4 capturing the autocorrelation of stock returns by the Markovproperty.

We will assume that the asset returns correspond to one of the models 1-5.

However, we do not assume the exact form of the true return distribution to beknown to the investor (with the exception of Chapter 4). Hence, the investor hasto apply statistical estimation and forecasting techniques for strategy planning.Clearly, nonparametric models require nonparametric statistical methods andarguments are usually more involved than in the parametric setting (for anintroduction to nonparametric estimation as we will use it see Gy or et al.1989, 2002). Unfortunately, nonparametric methods are not yet common ineconometrics and nancial mathematics (Pagan and Ullah, 1999, and Frankeet al., 2001, being two of the few notable exceptions). In this thesis we aim todemonstrate what powerful impetus nonparamtric statistical estimation maygive to investment strategy planning.

1.2 Portfolios and investment strategiesHaving chosen a market model, we turn to the actions that may be taken by

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the investor. Throughout the investment process, the investor holds varyingportfolios of the m assets. Taking a discrete time trading point of view, weassume that the investor is only allowed to rebalance his portfolio at the be-ginning but not in the course of each market period. The portfolio held at thebeginning of market period n (i.e. from time n 1 to n) can be given by thequantities q 1,n1,...,q m,n 1 of the single assets owned by the investor ( q i,n 1 < 0corresponds to borrowed assets, so-called short positions). The investor thenenters the nth market period with a portfolio value of

W +n1 :=m

i=1P i,n 1q i,n 1.

The remaining value at the end of the market period is

W n :=m

i=1

P i,n q i,n 1 =m

i=1

X i,n P i,n 1q i,n 1.

Hence, if W +n1 = 0, the portfolio achieved a return of

W nW +n1

=m

i=1X i,n bi,n (1.2.1)

withbi,n :=

P i,n1q i,n 1m j =1 P j,n 1q j,n 1

.

Note that mi=1 bi,n = 1, and we will nd it more convenient to denote a portfolioby the portfolio vector

bn := ( b1,n ,...,bm,n )T

rather than listing q 1,n1,...,q m,n 1. If the investor is allowed to consume anamount cn before changing his portfolio bn for bn+1 and entering market periodn + 1, then W +n is given by

W +n = W n cn . (1.2.2)(1.2.1) and (1.2.2) are the equations governing general discrete time investment.

Throughout this thesis we are concerned with an investor who neither consumesnor deposits new money into his portfolio but reinvests his current portfolio


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those portfolios whose acquisition generates no more transaction costs thanthe investors current wealth. Then, roughly speaking, the investor is caughtbetween making as few costly transactions as possible on the one hand andmaking as many transactions as necessary to boost his wealth on the otherhand. No wonder that strategic planning under transaction costs requires muchdeeper arguments and has received considerable attention in literature for both,discrete and continuous time models (see e.g. Blum and Kalai, 1999; Bobrykand Stettner, 1999; Cadenillas, 2000; Bielecki and Pliska, 2000). We shall returnto a typical case of transaction costs in more detail in Chapters 4 and 5.

1.3 Pleading for logarithmic utilityAs can be seen from (1.2.3), invested money grows multiplicatively, as a productof daily returns. Suppose the investor wants to maximize the expected valueof his terminal wealth. If the daily returns {X i}i are stationary, maximizationof the single expected daily returns is not appropriate. It does not captureautocorrelation in the returns, since, in general,

En

i=1 =

n

i=1

E .

The expectation is rather determined by the expectation of the logarithmic dailyreturns, since by Taylor expansion ( hot. denoting terms of order 2 and higher)

En

i=1

 = 1+ E logn

i=1

 + hot. = 1+n

i=1

E log + hot.

It is widely accepted that for returns below 10% and high frequency data (e.g.daily returns) the logarithmic approximation is convincing (Franke et al., 2001,Sec. 10.1). This leads to the notion of log-optimal portfolios , i.e. portfo-lios that maximize the expected logarithmic utility of the investorss growthof wealth. The log-optimal portfolio of a process {X n}n of independent andidentically distributed (i.i.d.) returns X n is dened as

b:= argmaxbS

E(log < b, X 1 > ). (1.3.1)

Log-optimal portfolios have been suggested rst by Kelly (1956), Latane (1959)and Breiman (1961) as diversication strategy for investment in a speculative

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1.3 Pleading for logarithmic utility 41

market given by a process {X n}n=1 of i.i.d. return vectors. Since then, nu-merous investigations, notably by Cover (e.g. Cover, 1980, 1984; Cover andThomas, 1991) and Algoet (e.g. Algoet and Cover, 1988) have explored thetheoretical aspects of this strategy, establishing that investment in log-optimalportfolios yields optimal asymptotic growth rates for the invested wealth. Anintroduction, various results and sources of reference can be found in Cover andThomas (1991, Chapter 15). There, for stationary and ergodic return processes

{X n}n , (1.3.1) is generalized by the conditional log-optimal portfolio (forthe nth investment step)

bn := argmaxbS E[log < b, X n > |X n1,...,X 1]

in stationary ergodic return processes (conditioning being void for n = 1). Theconditional log-optimal portfolio is the log-optimal portfolio under the condi-tional distribution P X n |X n 1 ,...,X 1 and hence a random variable. The log-optimalinvestment strategy b1, b2,... is a member of the class of non-anticipatingstrategies , i.e. sequences of S -valued random variables b1, b2,... with the prop-erty that each bn is measurable w.r.t. the -algebra generated by X 1,...,X n1(hence the strategy requires no more information than available at time n).The technical aspects of conditional log-optimal portfolios (we will often dropconditionalfor brevity) are well explored:

Existence and uniqueness of the log-optimal portfolio has been investigated inOsterreicher and Vajda (1993) and Vajda and Osterreicher (1994), correcting awrong criterion used in Algoet and Cover (1988). The main result is

Theorem 1.3.1. (Vajda and Osterreicher, 1994) Let X = ( X 1,...,X m) be a stock market return vector with distribution P X . Then there exists a log-optimal portfolio bS with |E log < b, X > | < if and only if

E logm

i=1

X i < .

bis unique if P X is not conned to a hyperplane in IRm containing the diagonal {(d,...,d)IRm|dIR}.

A good algorithm for the calculation of a log-optimal portfolio from the (known)distribution P X of the return vector X was given by Cover (1984).


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the log-optimal strategy by an amount that grows exponentially fast (i.e.an amount that couldnt be compensated for by investment in a xedinterest rate bank account).

It proves that the log-optimal portfolio will do at least as well as any othernon-anticipating strategy to rst order in the exponent of capital growth,i.e. it guarantees S n = exp( nW + o(n)) with highest possible rate W .

From the rst part of (1.3.2) Bell and Cover (1988) conclude that

there is no essential conict between good short-term and long-run per-formance. Both are achieved by maximizing the conditional expectedlog-return.

The log-optimality criterion has not been undisputed, however. In his criticism,Samuelson (1971, also discussed in Markowitz, 1976) considers a market withi.i.d. returns X 1, X 2,... and compares the expected wealth ES n from a series of log-optimal investments with the expected wealth ES n from investment in thexed portfolio

b:= argmaxbS

E < b, X 1 >

(maximization of expected return). Using the independence and the identicaldistribution of the returns he nds that

ES nES n

=E ni=1 < b, X i >E ni=1 < bi , X i >

=max bS E < b, X 1 >

E < b1, X 1 >

n

(n ).Hence there are strategies that outperform log-optimal strategies in terms of expected terminal wealth for long run investment. However, when comparingthe investors strategy with a competing strategy we think that the ratio of wealths considered (1.3.2) is more instructive than two seperate expectationsfor the investors strategy and the competing strategy. This is a typical exampleof criticism offered by classical economists who favour the Markowitz mean-variance approach to portfolio optimization (Markowitz, 1959; Luenberger,1998, Ch.6.4ff.). There, the investor seeks to maximize the portfolio perfor-mance E < b, X > under the constraint of not exceeding a certain thresholdfor the risk Var < b, X > (or for the value-at-risk, i.e., quantiles of the returndistribution, in more modern versions of the mean-variance approach).

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We would like to emphasize that it is not a question of taste whether or notto use the log-optimal approach. We strongly plead for investment under loga-rithmic utility because of the following facts:

In their spirited defence of the log-optimal criterion Algoet and Cover(1988) come to the conclusion that the mean-variance approach lacksgenerality (e.g. for non-log-normally distributed returns, see Samuelson,1967, and for multiperiod investment, see Luenberger, 1998, Ch. 8.8).

It is doubtful whether investment analysis should be founded on expecta-tions (where typically S n deviates much more from ES n than log S n fromE log S n , stabilizing effect of the log-transform). Pathwise results as thesecond part of (1.3.2) are more instructive than results on averages.

Realistically, the true distribution of market returns and hence the log-optimalstrategy is not revealed to the investor. Then the key problem is (as Algoet,1992, put it):

Find a non-anticipating portfolio selection scheme {bn}n (a so-called universal portfolio selection scheme) such that for any stationary ergodic market process

{X n}n , the compounded capital S n := ni=1 < bi, X i > will grow exponentially fast almost surely (i.e. with probability 1) with the same maximum rate as under

the log-optimum strategy {bn}n , that is, limnlog S n /n = lim nlog S n /nalmost surely.To obtain a universal portfolio selection scheme, under weak conditions on themarket one may choose the log-optimal portfolio with respect to some appro-priately consistent estimate of P X n |X 1 ,...,X n 1 in the nth investment step (moreprecisely, distribution estimates that almost surely exhibit weak convergence tothe true distribution). This was demonstrated by Algoet (1992, Theorem 7). Healso provides an appropriate, yet complicated estimation scheme (Algoet, 1992,Theorem 9). Instead, we can also use the more transparent scheme of Morvai etal. (1996). Algoet points out that there are universal portfolio selection schemesthat do not require an explicit distribution estimation scheme as a subroutine(Algoet, 1992, Sec. 4.3). But still, all existing algorithms seem to require anenormous amount of past data, making their feasibility in practical situationsdoubtful (as noted, e.g., in Yakowitz, Gy or et al., 1999). More practicableresults have been obtained in the case of independent, identically distributedreturn vectors. For instance, Morvai (1991, 1992) and Osterreicher and Vajda


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MARKET MODEL

INVESTMENT GOALINVESTMENT

ACTIONS

assets 1 stationary, ergodic stochastic

return process { }

i= , ..., m

X n na +m

portfolio process{ }

transaction costs (occasionally) no consumption no short positions

b S n na log-utility: maximisationof expected log-returns good for both short term

and long run investment

investmentstrategy

Figure 1.2: Our approach to investment strategy planning.

(1993) propose portfolio strategies which are based on selecting the log-optimalportfolio with respect to the empirical distribution of the data (the so-calledempirical log-optimal portfolio , more on that in Chapter 2). Those esti-mators can be computed with reasonable effort. Repeated investment followingtheir strategies asymptotically yields the optimal growth rate of wealth withprobability one. However, in merely stationary and ergodic return processesthey produce suboptimal results.

This thesis aims to provide simplied, yet efficient portfolio selection algorithmsif the log-returns follow a Gaussian process (Chapters 2 and 3), a Markov pro-cess (Chapters 4 and 5) or, more general, a stationary and ergodic process(Chapter 6). Our approach is summarized in Figures 1.1 and 1.2.

For the sake of completeness, it should be noted that in recent years, the log-optimality criterion has been generalized in several ways. In particular, re-searchers tried

to make the log-optimality criterion risk sensitive, i.e. to introduce de-vices which allow the investor to adjust the log-optimal strategy to hisindividual risk aversion level. This may be achieved in two different ways:Either, as in the Markowitz mean-variance model, the investor seeks tomaximize the expected log-return under variance constraints (Ye and Li,1999), or the log-utility is extended by the variance, e.g. when maximiz-ing (2/ )log E exp((/ 2)log S n) = E log S n (/ 4)Var log S n + O(2),where > 0 is a risk aversion parameter (Bielecki and Pliska, 1999,


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47

CHAPTER 2

Portfolio benchmarking: rates and di-

mensionalityBased on market observations, the investor can follow many different empiricalportfolio selection rules (empirical being synonymous with based on histor-ical return data). Not all of these necessarily turn out to be a good choicein view of the investors goal. Discriminating between good and bad port-folios requires the investor to compare the performance of his portfolio withthe given investment goal. Naturally, good empirical portfolio selection rulesshould approach the investment goal. It is of serious interest to determine howfast the investor approaches his goal as more and more information about themarket is gathered. This is the primary task of what we might call portfolio

benchmarking . Portfolio benchmarking analyses how well a given portfo-lio b or a given portfolio selection rule b = b(X 1,...,X n) of the past returnsX 1,...,X n performs with respect to a xed benchmark, in our case with re-spect to the expected logarithmic portfolio return in the next market period,E log bX n+1 (to keep notation simple we write log bX n+1 rather than log bT X n+1or log < b, X n+1 > in the sequel). This, of course, requires a standardized wayto assess to what extent an empirical portfolio selection rule underperforms incomparison with the log-optimal rule.

In this chapter we analyse how seriously a log-optimal portfolio selection rulebased on an estimate for the true return distribution may underperform. To thisend, we propose a specic measure of underperformance (cf. (2.1.2)). Establish-ing a lower bound result on this measure, it will be seen that underperformancecannot vanish at arbitrarily high rate as the investor gathers more and moreknowledge about the market (Theorem 2.1.1). All investors are subject to auniversally limited rate at which investment rules can succeed in exploitinghistorical market data.


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48 Chapter 2. Portfolio benchmarking: rates and dimensionality

In fact, the empirical log-optimal portfolio of Chapter 1 turns out to be a se-lection rule that achieves the optimal rate (Theorem 2.1.3). It is particularlystriking that this rate does not depend upon the numbers of stocks includedin the portfolio selection process (Theorem 2.1.4). One is tempted to thinkthat arbitrarily large portfolios can be handled successfully without extra pre-cautions. Reasons will be given why this is fallacious and does not obviate thenecessity of trying to keep the dimension of the portfolio at reasonably low levelby pre-selecting a good subset of all possible stocks.

However, the pre-selection of stocks is far from being an easy going thing: As weshall see, there is no way of pre-selecting the stocks on the basis of the perfor-mance of the single stocks only (Theorem 2.2.1). To nd the optimal portfolioconguration, the investor has to evaluate the log-optimal portfolios of all pos-sible subsets of stocks and compare the resulting expected logarithmic portfolioreturns, a huge though necessary computational effort in high dimensions.

2.1 Rates of convergence in i.i.d. modelsSuppose the m-dimensional stock return vectors X 1, X 2,... constitute a sequenceof independent, identically distributed (i.i.d.) random variables with distribu-tion Q := P X 1 . Q is not disclosed to the investor, who, after n market periods,may exploit the observations X 1,...,X n to obtain a distribution estimate Q n of Q. Let F and F n = F n(, X 1,...,X n) denote the cumulative distribution func-tions associated with Q and Q n , respectively. We shall restrict our analysis toestimators Q n whose sensitivity to outliers is such that

F n (x, X 1,...,X i1, X i, X i+1 ,...,X n)

F n (x, X 1,...,X i1, X i , X i+1 ,...,X n) c(x, X i, X i )

n(2.1.1)

for some function c : IR3m+

IR+ , whatever i

IN,x ,X 1,...,X n , X 1,...X n

IRm+may be. Most of the standard distribution estimates share this property, suchas the empirical distribution

Q n(A) =1n

n

i=1

1A(X i) (AB (IRm ))


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2.1 Rates of convergence in i.i.d. models 49

and kernel estimates

Q n(A) = A1

nh dn

n

i=1

K x X i

hndx (AB (IRm )),

hn being a sequence of nonnegative bandwidths and K : IRm IR+0 a kernelfunction.

Having thus learned a picture of the market, Q n , the investor allocates hiswealth according to the corresponding log-optimal portfolio

bn+1 = bn+1 (X 1,...,X n) = argmaxbS E logbY,

where the expectation is calculated for Y Q n . This choice yields the randomreturn Rn = bn+1 X n+1 during the next market period. In order to determinehow well bn reproduces the true log-optimal portfolio b= argmax bS E log bX 1with return Rn = bX n+1 , we rst observe that

E log Rn E log Rn = E log bX n+1 E E[logbn(X 1,...,X n)X n+1 |X 1,...,X n]

E logbX n+1 E E[logbX n+1 |X 1,...,X n]= E log bX n+1

E log bX n+1 = 0 ,

using the independence of X 1,...,X n+1 . Hence

( Q n , Q) := E logRnRn 0 (2.1.2)

with equality if Q n = Q. On the other hand, from Theorem 1.3.4,

liminf n

1n

n

i=1

logRiR i 0.

Taking expectations and using Fatous lemma we obtain

0 E liminf n1n

n

i=1 logRiR i liminf n

1n

n

i=1 E logRiR i .

Therefore, {bn}n is a good portfolio selection rule if ( Q n , Q) = E log

RnRn 0


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50 Chapter 2. Portfolio benchmarking: rates and dimensionality

with high rate as n tends to .( Q n , Q) measures underperformance of bn w.r.t. the benchmark portfolio b.In the sequel we will derive asymptotic properties of ( Q n , Q). The followingtheorem shows that the limit cannot be achieved at arbitrarily high speed of convergence:

Theorem 2.1.1. For any sequence of distribution estimates Q n satisfying (2.1.1), there exists a market distribution Q and a market constant c for which

(Q n , Q)

c

n (2.1.3)for innitely many n.

As will be seen in the proof, (2.1.1) is not needed when considering unbiasedestimators Q n , i.e., estimators for which EQ n(A) = Q(A) for all AB (IRm+ ).Proof. Consider a 2 stock market with return vector ( X (1) , X (2))IR

2+ and

portfolios (b,1 b), b[0, 1]. We can expandE log(bX (1) + (1 b)X (2)) = E log((Z 1)b + 1) + E logX (2) (2.1.4)

with the return ratio Z := X (1) /X (2) . Thus, in a 2 stock market, the log-optimal portfolio only depends upon the distribution of the return ratio Z . Forsimplicity, let Z be of the form

Z =A with probability p,B with probability 1 p

(2.1.5)

with p(0, 1), A, B > 0 to be chosen later.

We rst consider the classical parameter estimation problem of estimating p,which will be linked with the portfolio selection problem at a later stage. Q nallows the investor to derive an estimate of p,

pn = pn (z n) := Q n (x(1) , x (2)) : x(1) /x (2) = A ,

z n

{A, B

}n being the observed realisations of the i.i.d. return ratios Z 1,...,Z n

(independent of Z ). If k(z n) denotes the number of As in z n and B ni ( p) =ni p

i(1 p)ni denotes the ith Bernstein polynomial of order n, we can identify

f n ( p) := E pn (Z 1,...,Z n) =n

i=0

ni

1

zn :k(zn )= i

pn(z n) B ni ( p) =:n

i=0bi,n B ni ( p)


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2.1 Rates of convergence in i.i.d. models 51

as a Bezier curve. For reasons to become clear later it is important to study itsderivative

f n ( p) =n1

i=0

n(bi+1 ,n bi,n )B n1i ( p).Combinatorial arguments given at the end of this proof and relation (2.1.1)yield

n|bi+1 ,n bi,n | const. (2.1.6)independently of i and n. Using

n1i=0

B n1i ( p) = 1 we obtain

|f n ( p)| const. (2.1.7)for all n and p.

We now choose the true parameter p of the model (

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Non Parametric Estimation for Financial Investment Under Log-utility