Optimal Portfolio Allocation Algorithms FREDRIK LÖÖW

Transcription

1 Optimal Portfolio Allocation Algorithms FREDRIK LÖÖW Master of Science Thesis Stockholm, Sweden 2009

2 Optimal Portfolio Allocation Algorithms FREDRIK LÖÖW Master s Thesis in Computer Science (30 ECTS credits) at the School of Computer Science and Engineering Royal Institute of Technology year 2009 Supervisor at CSC was Johan Håstad Examiner was Johan Håstad TRITA-CSC-E 2009:093 ISRN-KTH/CSC/E--09/093--SE ISSN Royal Institute of Technology School of Computer Science and Communication KTH CSC SE Stockholm, Sweden URL:

3 Abstract The problem of portfolio allocation is analyzed within the framework of Cover s universal portfolios. The analysis is performed by establishing upper and lower bounds on the performance of the portfolio allocation algorithms. Upper bounds represent how well it is theoretically possible for an algorithm to perform. Here, the existing results of Ordentlich and Cover are extended by considering markets where the possible price moves are limited in order to better simulate an actual financial market. We find that if the initial constraints are relaxed to allow short selling, the upper bounds remain unchanged. Under the original portfolio constraints a new upper bound is established that is a function of the allowed price move. Moreover, in an effort to approximate a continuous market, a model where a fixed time period is divided into a number of time slices and the size of the allowed price move is tied to the length the time slices is examined. In this setting the upper bound is shown to converge to a constant value as the number of time slices tends to infinity and the allowed price move tends to zero in a way inspired by Brownian motion. Lower bounds represent how well en existing algorithm is guaranteed to perform. We extend the analysis of Cover s universal portfolio to allow for the use of any parameter 0 < δ 1 in the Dirichlet(δ) distribution in contrast to the specific values δ = 1/2 and δ = 1 analyzed in existing research. It turns out that by choosing δ slightly smaller than 1/2, the constant factor of the asymptotic lower bound approaches the optimal value, as opposed to coming within a factor 2 for the δ = 1/2 instance previously analyzed.

4 Referat Optimala algoritmer för portföljallokering Portföljallokeringsproblemet analyseras i ramverket för universella portföljer som introducerades av Cover. Analysen utförs genom att etablera övre och undre gränser för hur väl portföljallokeringsalgoritmer kan prestera under förutsättningarna i detta ramverk. De övre gränserna visar hur bra det är teoretiskt möjligt för en algoritm att prestera. Resultat tidigare etablerade av Ordentlich och Cover utökas här genom att betrakta marknader där bara vissa begränsade prisrörelser är tillåtna. Vi visar att om man lättar på de initiala portföljbegränsningarna genom att tillåta blankning så står sig de tidigare etablerade resultaten för generella marknader även under dessa förutsättningar. Under bibehållna begränsningar visas en ny övre gräns som beror av den tillåtna rörelsen. Vidare studeras en modell där en fix tidsperiod delas in i ett antal delperioder, och den tillåtna prisrörelsen kopplas till längden på delperdioden. Under dessa förutsättningar visar vi att den övre gränsen konvergerar mot en konstant när antalet delperioder går mot oändligheten samtidigt som den tillåtna prisrörelsen går mot noll på ett sätt inspirerat av brownsk rörelse. Undre gränser visas genom att analysera specifika algoritmer. Här utökar vi analysen av Covers universella portfölj genom att tillåta parametern δ i Dirichlet(δ)-fördelningen anta alla värden, 0 < δ 1, och inte bara δ = 1/2 och δ = 1 som analyserats i tidigare forskning. Det visar sig då att genom att välja δ strax under 1/2 så kommer den konstanta faktorn i den undre gränsen asymptotiskt gå mot dess optimala värde. Detta skiljer sig från instansen av algoritmen med δ = 1/2 som tidigare analyserats, vilken endast uppnår en faktor 2 av det optimala värdet.

5 Contents 1 Introduction 1 2 Background History and Prior Work Notation and Definitions Constant Rebalanced Portfolios - the Benchmark Some Properties of the Best Constant Rebalanced Portfolio The Wealth Ratio Universal Portfolios The Universal Algorithm Some Properties of the Universal Portfolio Effectiveness of the Algorithm: Lower Bounds on the Wealth Ratio Transaction Costs Efficiency of the Algorithm: Implementation Optimal Investment Strategies Upper Bounds for General Markets Upper Bounds for Markets with Limitations Toward a Continuous Market Concluding Remarks Conclusions Suggestions for Future Research Bibliography 47

6 Appendices 50 A Mathematical Background and Extended Proofs 51 A.1 Determining the optimal weights of a CRP A.2 The Beta and Gamma Functions A.3 Finding the Worst Case A.4 Proof of the Upper Bound in Cover s Minimax Regret Portfolios for Restricted Stock Sequences Acknowledgment I would like to thank my supervisor, Johan Håstad, not only for his curiosity and open mind in agreeing to supervise a Master s project outside his normal research areas but more importantly for being a great source of enthusiasm and inspiration throughout the whole process of writing this thesis.

7 Chapter 1 Introduction What is the best way to invest in a universe of stocks and other financial instruments? The topic of this thesis is algorithms for portfolio allocation, and that is the question we seek to answer. An effective method of allocating capital among different investment opportunities is very valuable, for individuals as well as institutions and society at large, for instance in the case of management of pension liabilities. Imagine that you know how the stock market will develop over a given time period. The problem of allocating capital is then trivial just invest in the portfolio that turned out to perform the best. This, the best portfolio in hindsight, is the benchmark to which the results of the algorithms studied in this thesis are compared. We present an online algorithm first introduced by Cover (1991) that actually comes close to achieving the same performance as this, perhaps best described as clairvoyant, benchmark portfolio. The algorithm is analyzed in the setting of online algorithms. The performance is measured in terms of a competitive ratio which measures how well the online algorithm performs in relation to how well an offline algorithm, in essence the benchmark, would have performed. We start by presenting the original algorithm along with a few modifications and analyzing its competitiveness. Then, we turn to the question of whether the presented algorithm is optimal or not and consider if it is possible to create another algorithm that performs better within the same framework. 1

8 CHAPTER 1. INTRODUCTION The portfolio given by the algorithm is called the universal portfolio. Universality is an intriguing and important distinguishing property of the analyzed approach to portfolio allocation that sets it apart from most other models, especially those of traditional finance. The meaning of universality is that no assumptions are made on the distribution of the input data of the algorithm, in this case prices of financial instruments. Another differentiating property is that no assumptions are made as to the preferences the investor has toward risk. This also lets us avoid the tricky problem of deciding on how to define and measure risk. Hence, we are able to show some quite impressive theoretical guarantees on performance, while avoiding the strong assumptions on the price movements of financial instruments prevalent in the models of classical finance. As a final point, the focus of this thesis is on the theoretical analysis of the problem of portfolio allocation within the framework of Cover (1991) and the performance of algorithms within this framework. For investigations of the empirical performance of these algorithms we refer to the work of, for example, Lundahl (2005) and Agarwal, Hazan, Kale, and Schapire (2006). Outline The structure of the thesis is as follows. Chapter 2 contains a short background of the subjects of portfolio allocation and online algorithms followed by the introduction of notation and basic definitions that are used throughout the analysis. Chapter 3 proceeds by introducing Cover s (1991) universal portfolio and analyzing its performance while allowing for some additional flexibility regarding the choice of a central building block of the algorithm. Chapter 4 is devoted to the analysis of how well it is possible to solve the problem. In essence, an upper bound of the possible performance is established. The general case analyzed by (Ordentlich and Cover 1998) is reproduced and similar analysis is performed under variations of the model framework. Finally, the results are summarized in chapter 5, and some suggestions for future research are made. In addition, the appendix contains some mathematical reference material and proofs that were not deemed suitable for the main text. 2

9 Chapter 2 Background This chapter contains a short overview of the framework within which the work of this report is performed. Since we analyze the problem of portfolio construction using online algorithms, some background is given on both these subjects. This is followed by a presentation of the inputs and outputs of the algorithm, and a description of how we measure the performance of the algorithm on a given input. Finally, we define the wealth ratio, which is what is used to compare the worst case performance of two algorithms on any input. 2.1 History and Prior Work The subjects of portfolio allocation and online algorithms have rich independent histories. Here, we give give a short background that covers the parts that are relevant to our problem statement Portfolio construction The analysis of investments and portfolio allocation from an information theoretic point of view started with a paper by Kelly (1956). He established optimal fractions of available capital to bet on independent wagers assuming that the odds were known and that the investor had a logarithmic utility function. In other words, the 3

10 CHAPTER 2. BACKGROUND strategy he proposed maximizes the expectation of the logarithm of the achieved wealth. Thorpe (1971), and others, later conducted similar analysis for a portfolio of financial instruments. It was still assumed that the investor had a logarithmic utility function. In this case the distribution of the price movements of the instruments was also assumed to be known beforehand. In parallel, what is now known as modern portfolio theory was developed by Markowitz (1952), Sharpe (1964), and others. This theory is built on the assumption that price movements are normally distributed and are thus fully defined by their mean, standard deviation, and pair-wise correlation. Moreover, a quadratic utility function that incorporates the assumption that investors are risk averse is used. Is essence, the portfolio is constructed by maximizing the function u(r, σ) = r dσ 2, where r is the expected value of portfolio return, σ is the standard deviation of the portfolio return and d is a constant that represents the extent to which the investor is risk averse. Such a portfolio is commonly called a Markowitz portfolio. The main focus of this thesis, the universal portfolio introduced by Cover (1991), makes no assumptions about the utility function of the investor or the distribution of price movements. However, it has been shown by Belentepe and Wyner (2005) that the universal portfolio and the Markowitz portfolio, while differing drastically in their theoretical frameworks, asymptotically share important statistical properties Online algorithms An online algorithm deals with the problem of decision making in the absence of perfect information. Such problems arise naturally in cases where information is made available one piece at a time, and action is required upon each new piece of information. Examples of such cases range from dynamically updated data structures such as Sleator and Tarjan s (1985b) self-adjusting binary search tree (the splay tree), packet routing on computer networks, and as is our focus investment management. 4

11 2.1. HISTORY AND PRIOR WORK The analysis of online algorithms is generally performed by comparing the result of the algorithm to the result of the best offline algorithm. This is measured by the ratio of the performance of the online algorithm to the performance of the offline algorithm, and the worst value of that ratio over all allowed series of input data is called the competitive ratio of the online algorithm. The actions of the online algorithm can be seen as a game between an online player and a malicious adversary. The adversary has full knowledge of the online algorithm and is constantly trying to supply the input to the algorithm that causes it to perform the worst relative to the offline algorithm. The objective is then to construct an algorithm such that no matter the input supplied by the adversary, the algorithm remains competitive compared to the offline algorithm. A classical example of the analysis of a problem for online algorithms is the analysis of the list accessing problem. Indeed, it was with the analysis of this problem Sleator and Tarjan (1985a) introduced the general concept of competitive analysis. The list accessing problem consists of providing the operations ACCESS(x), INSERT(x), and DELETE(x) on a list data structure. The algorithm is given an initial list, which may be empty, and is then fed a sequence of operations on this list. Allowed operations on the list by the algorithm are ACCESS(x) and DELETE(x), the cost of which is i if item x is on position i in the list, and INSERT(x) which costs l where l is the number of items already in the list. Moreover, the algorithm may reorganize an existing list by transposing two adjacent items. Each such operation costs one unit. Following the access or insertion of an item, the item may then be moved to any position in the list preceding its current position free of charge. The rationale of this rule is that by performing the initial operation, and paying its cost, the algorithm can store all the information it needs to make the move in constant time. The way the algorithms choose to make use of these operations to reorganize the list in between requests is what differentiates their performance. In this framework, Sleator and Tarjan show that the simple heuristic of moving the last accessed item to the front of list (MTF) after each operation is 2- competitive. This means that it requires no more than twice the number of oper- 5

12 CHAPTER 2. BACKGROUND ations required by the best offline algorithm with full knowledge of the sequence of requests from the start. Irani (1991) later improved this result somewhat and showed that the MTF heuristic is an optimal online algorithm for the list accessing problem. It is easy to frame the portfolio allocation problem as an online problem. The algorithm must decide on the best portfolio allocation knowing only the historic development of the market, and is allowed to update the portfolio each time period as new market data arrives. We use a benchmark that has access to all market data, historic and future, when the allocation is made just as is the case for offline algorithms. Finally, the market is our adversary which we assume constantly supplies price data, and knowing the design of our algorithm, supplies data to decrease our performance as much as possible with regard to the benchmark. Under these circumstances we seek to maximize the competitiveness of the online algorithm. 2.2 Notation and Definitions The first step in formalizing the problem statement is to define the nature of the input and the requirements on the output of the problem Input: Market price data The input data of the algorithms studied in this report is price quotations of financial instruments. These instruments can be stocks, bonds, currencies, commodities, or any other tradable instrument including derivatives such as futures contracts or options. We consider the price quotations of m such instruments over n time periods. The subset of instruments selected is called the market. The typical time period when implementing this kind of algorithms is one day, but no assumptions as to the length of the time period are made. Let the vector p i = (p i1, p i2,..., p im ) T represent the prices at the end of time period i = 0, 1,..., n such that for each instrument j = 1, 2,..., m, p ij is the price of instrument j at the end of time period i. 6

13 2.2. NOTATION AND DEFINITIONS We define x ij = p ij p (i 1)j, (2.1) called the price relative of instrument j for time period i. It is the factor by which the price of the instrument changes from the end of the preceding time period to the end of time period i. Note that r ij = x ij 1 is the percentage return over the same time period. The vector of x i = (x i1, x i2,..., x im ) T fully describes the market development for time period i, and the sequence of vectors X n = x 1, x 2,..., x n fully describes the market development over the entire time period Output: Portfolio weights The output of a portfolio allocation algorithm is the distribution of capital over the market s financial instruments. At time period i, this distribution is represented by the vector of relative portfolio weights b i = (b i1, b i2,..., b im ) T, where b ij 0 and i b ij = 1 for all j. The relative change in portfolio value during time period i is thus the factor m b T i x i = b ij x ij. (2.2) As for the price relatives, we define a sequence of portfolio weight vectors, B n = b 1, b 2,..., b n which fully defines the portfolio configuration over the entire time period. The value of the portfolio at time n can then be calculated as j=1 n n m S n (X n, B n ) = b T i x i = b ij x ij i=1 i=1 j=1 assuming that the initial investment was $1. Taking a closer look at the conditions on the portfolio weights, the implications are that (i) negative exposure is not allowed to any instrument i.e. short selling is not allowed, and (ii) the portfolio must be fully invested at all times and can not invest more than its current value i.e. it can not use leverage. This is the basic setup that is predominately focused on in previous research. However, we also consider the implications of relaxing some of the constraints on the portfolio weights, such as the ban on short selling. 7

14 CHAPTER 2. BACKGROUND 2.3 Constant Rebalanced Portfolios - the Benchmark In order to evaluate the performance of a given algorithm, the benchmark to which the results are to be compared to must be defined. The class of portfolios chosen is that which has constant portfolio weights for all time periods, i.e. b i = b (2.3) for all i = 1, 2,..., n; the constant rebalanced portfolio (CRP). The wealth of a CRP is n n m S n (X n, b) = b T x i = b j x ij. i=1 i=1 j=1 The value of b that maximizes S n (X n, b) for a certain market sequence X n represents the best constant rebalanced portfolio and is defined as and the wealth of this portfolio is b = arg max b S n(x n, b) (2.4) S n(x n ) = S n (X n, b ). (2.5) The performance of the algorithms is evaluated against this, the best possible CRP. Note that b is not known until the full sequence X n is known. Hence, this portfolio can not be decided until after the fact. However, the algorithms must decide on the portfolio allocation for each time period using only the market vectors representing previous time periods. In other words, to calculate the weights b i, only the market vectors x 1,..., x i 1 are available. 2.4 Some Properties of the Best Constant Rebalanced Portfolio Here, we show some elementary properties of the best constant rebalanced portfolio. The objective is to give some motivation as to why it is a suitable choice for a benchmark portfolio. 8

15 2.4. SOME PROPERTIES OF THE BEST CONSTANT REBALANCED PORTFOLIO An investment in a single instrument, j, can be expressed as a CRP by setting b = e j. The change in wealth of this portfolio is S n (X n, e j ) = n i=1 x ij = p nj p 0j. This is equivalent to buying the instrument at the first time period and holding it to the last, i.e. a buy-and-hold strategy. Proposition (Best CRP beats best instrument) Sn(X n ) max S n (X n, e j ) (2.6) j where e j is the j th base vector, and so represents the portfolio where all wealth is placed in one and the same instrument at all time periods. Proof. Since S n(x n ) is the maximization over the full simplex, it must be at least as large as the maximization over the base vectors, which constitute a subset of the simplex. So, the best CRP performs at least as well as the single best instrument. A common investment strategy among private and institutional investors is to allocate funds to low-cost passive mutual funds that replicate well known stock indicies such as the Standard & Poor 500 index of the 500 leading U.S. exchange traded stocks (Standard and Poor s 2008). Such an investment can be expressed as a linear combination of the single instrument CRPs described above: m j=1 α j S n (X n, e j ) where α j 0 and j α j = 1. Proposition (Best CRP beats arithmetic index) Let α j 0 and j α j = 1. Then m Sn(X) α j S n (X, e j ). (2.7) j=1 Proof. From we know that S n(x) max j S n (X, e j ). instrument. Then Let j be the best m m Sn(X) S n (X, e j ) = α j S n (X, e j ) α j S n (X, e j ) (2.8) j=1 j=1 To further illustrate the difference between a CRP and the more commonly known arithmetic indices, let us consider a couple of examples. 9

16 CHAPTER 2. BACKGROUND Example. Consider a two-asset market over three time steps. Let the asset prices develop as ((1, 1), (2, 1), (1, 1)). So, both assets start at a price of one, the first asset then doubles in price while the second remains unchanged, and finally the first asset returns to its initial value. Clearly, an arithmetic index of these two assets would be unchanged after this sequence of prices regardless of its weights on the individual assets, since both assets end up at the same price as they started at. Now consider an equally weighted CRP of the two assets over this time period. We start with a total wealth of $1, and weights of 0.5 on each asset. After the first price change the total wealth is $1.50 since our initial investment of $0.5 in the first asset has increased in value to $1, and the investment in the second asset remains unchanged. This also means that the relative weights of the assets in the portfolio have changed. We now have a weight of 2/3 in the first asset ($1) and a weight of 1/3 in the second asset ($0.5). Since a CRP is rebalanced to its initial weights at the end of each time period, the total wealth is reallocated among the two assets to accomplish this. So we sell $0.25 worth of the first asset and buy $0.25 worth of the second asset, restoring the weights to 0.5. In the next time period, the price of the first asset falls back to $1. So, the value of our investment in that asset falls by a factor of $1/$2 = 0.5 to $0.375 and the total value of the portfolio is $ $0.75 = $ Thus, we have made a profit of $0.125 on this price move, even though the final prices are equal to the starting prices. Example. Now, let us see what happens if the price of one asset decreases instead of increases as in the previous example. Let the setting be as above, but consider the asset price sequence ((1, 1), (0.5, 1), (1, 1)). Then, after the price drop in the second time step, we will rebalance the remaining portfolio wealth of $0.25+$0.5 = $0.75 so that we have $0.375 invested in each asset. When the price of the first asset returns to one in the last time step the total value of portfolio will be $0.75+$0.375 = $1.125, and just as above we will have made a profit of $ What these two examples illustrate is that the dynamic nature of the CRP makes its performance characteristics different from common stock indicies. CRPs will do well not only in rising markets but also where there is movement around a single value, a phenomenon commonly referred to as mean reversion. However, since the 10

17 2.5. THE WEALTH RATIO CRP requires rebalancing, it will in practice also incur transaction costs which will lessen its performance. Finally we consider a somewhat more esoteric index type exemplified by the Value Line Geometric Index; see e.g. (Rothstein 1972) for an overview of financial index types. It is defined as the geometric average of the included instruments, and in our notation the value can be expressed as 1/m m S n (X, e j )). (2.9) j=1 Proposition (Best CRP beats geometric index) m Sn(X) S n (X, e j )) j=1 1/m (2.10) Proof. From we know that Sn(X) max j S n (X, e j ). As above, let j be the best instrument. Then m Sn(X) S n (X, e j ) = S n (X, e j ) 1/m j=1 j=1 m S n (X, e j ) 1/m. (2.11) So, if we have an algorithm that lets us perform on par with the best CRP, we can beat all single instruments and all common financial indicies. That makes it a quite challenging, and therefore useful, benchmark. 2.5 The Wealth Ratio Now that the preliminaries are settled we establish the target that the considered portfolio allocation algorithms are evaluated against. The goal is, obviously, to perform as well as possible in all market conditions, and the performance is measured as the wealth generated by the algorithm compared to the wealth generated by the benchmark portfolio. The ratio we wish to maximize is then min X n Ŝ n (X n ) S n(x n ) (2.12) 11

18 CHAPTER 2. BACKGROUND where Ŝn(X n ) is the wealth generated by the algorithm on market sequence X n and Sn(X n ) is the wealth generated by the benchmark portfolio, the in hindsight best constant rebalanced portfolio (CRP), on the same market sequence. The above ratio is similar to the competitive ratio derived in the classical analysis of online algorithms. However, there is a crucial difference in this case. Normally, the online and the offline algorithms are allowed to perform the same operations. However, in this case the online algorithm may change the allocation at each time step, while the offline algorithm must choose an allocation and keep it constant over the full time period. While this restriction on the offline algorithm severely restricts its performance, we have already noted in section 2.3 that such an offline portfolio can be expected to perform very well nonetheless. An offline algorithm that could change allocation each time period would clearly select the best performing instrument each day. Considering the competitiveness ratio, the adversary would let the best performing stock be the one with the lowest weight in our portfolio, and would let the others go to zero. So, the value of our portfolio would decrease by a factor of at least x/m, and the wealth of the optimal offline algorithm would increase by a factor x over the same time period, where x is the relative change in the best performing asset. So, the relative under-performance would be x m /x = 1/m for each time period, or m n after n time periods. Clearly, the performance of such a benchmark is be too extraordinary to remain a useful target for an online algorithm. 12

19 Chapter 3 Universal Portfolios This section contains the description and analysis of an algorithm introduced by Cover (1991), later improved in (Cover and Ordentlich 1996), and analyzed in greater detail in (Ordentlich and Cover 1998). In particular, we investigate the worst case performance of the algorithm and the type of input data that triggers such performance. Simultaneously, some of the fundamental design choices and their effect on performance are reviewed. 3.1 The Universal Algorithm Let us start be reviewing what the objective of the algorithm is. The goal is, obviously, to generate as large a return as possible on the initial investment by dynamically allocating to the available financial instruments. More precisely, we want to maximize the ratio Ŝ n (X n ) S n(x n ) (3.1) where Ŝn(X n ) is the wealth generated by the algorithm and S n(x n ) is the wealth generated by the, in hindsight, best constant rebalanced portfolio (CRP). There are two major differences in the basic conditions placed on the algorithm and its benchmark. The first is the availability of data. While the best CRP at time t is selected using all the market data available for the full time period up to and including time n, that is x 1, x 2,..., x n, the algorithm is only allowed data up to 13

20 CHAPTER 3. UNIVERSAL PORTFOLIOS the time step immediately preceding that for which the portfolio weights are to be determined. That is, x 1, x 2,..., x t 1 are available to determine the weights at time t. The second difference is the weights. The CRP has, as its name states, portfolio weights that are constant over time. The algorithm, on the other hand, is allowed to update its weights at each time step. The universal portfolio is defined as follows: Definition (The universal portfolio) The weights of the universal portfolio, with respect to the probability distribution µ, at time t + 1 are ˆb t+1 (µ, X t W ) = bs t(x t, b)dµ(b) W S t(x t, b)dµ(b) (3.2) where W is the set of allowed portfolio weight vectors and µ(b) is a distribution function such that W dµ(b) = 1. Initially, the wealth is distributed such that dµ(b) is the amount allocated to the neighborhood db of portfolio b. The initial wealth is defined as one. A natural interpretation of the portfolio weights that follows from this definition is that the portfolio at time t is the performance weighted average over all possible portfolios b W. This is the case since the numerator is the average over the portfolio weights multiplied by the corresponding resultant wealth of that portfolio and the denominator scales this quantity into a portfolio weight between zero and one by normalizing with the average over all portfolios. proposition shows, the wealth of the universal portfolio. The denominator is, as 3.2 Some Properties of the Universal Portfolio Some basic properties of the universal portfolio are established here. Proposition (Wealth of universal portfolio is the average over all CRPs) The wealth of the universal portfolio can be expressed as Ŝ n (µ, X n ) = S n (X n, b)dµ(b). (3.3) W 14

21 3.2. SOME PROPERTIES OF THE UNIVERSAL PORTFOLIO Proof. The following observation allows for an alternative interpretation of the algorithm. Note that: ˆb T t+1(µ, X t )x t+1 = = W bt x t+1 S t (X t, b)dµ(b) W S t(x t, b)dµ(b) W S t+1(x t+1, b)dµ(b) W S t(x t, b)dµ(b) (3.4) By eqn. 2.3 (3.5) Thus, there is a telescoping property in the product of wealth factors ˆb T t (X t 1 )x t which allows us to express the wealth of the portfolio as Ŝ n (µ, X n ) = = n ˆb T t (µ, X t 1 )x t (3.6) t=1 W S n (X n, b)dµ(b). (3.7) Hence, the wealth of the universal portfolio can be interpreted as the µ-weighted average over the wealth of the set of all constant rebalanced portfolios. A possible financial interpretation is that the initial wealth is divided into CRPs in the beginning of the period according to the µ-distribution and that these portfolios grow independently over time. What needs to be examined then is whether a large enough proportion of the initial wealth is allocated to the best CRP and to CRPs sufficiently alike in order to reach a final performance close enough to the best CRP. Proposition (Ŝn(X n ) is independent of the order of the market sequence) The wealth of the universal portfolio, Ŝ n (X n ), is independent of the order of the market vectors in the market sequence X n = x 1, x 2,..., x n. Proof. From proposition we know that Ŝ n (X n ) = S n (X n )dµ(b) (3.8) W ( n ) = b T x t dµ(b). (3.9) W t=1 The integrand of expression 3.9 is clearly independent of the order of the x i vectors, so Ŝn(X n ) is as well. So, proposition shows us that for this class of algorithms, the order of the market events is irrelevant. It does not matter if all assets lose value during the 15

22 CHAPTER 3. UNIVERSAL PORTFOLIOS first half of the considered time period, leaving the portfolio all but bankrupt before any increase in value takes place. The final outcome is not affected compared to the instance where the price changes are evenly distributed over time. A practical consequence of this is that the impact of periods of market turbulence, such as the the period around the market crash of October 1987, will not affect the value of the portfolio any differently than if the days of turbulence would have been spread out evenly over the investment period. 3.3 Effectiveness of the Algorithm: Lower Bounds on the Wealth Ratio Now, we will establish lower bounds on the wealth ratio for the universal portfolio. These bounds are valid at all time steps of the considered time period, so there is no requirement on deciding on an investment time span at the onset. This part of the analysis requires that we specify the distribution function dµ(b) that was left unspecified in the definition of the universal portfolio. The initial algorithm by Cover (1991) used a uniform distribution, while improved performance was shown by Cover and Ordentlich (1996) using the Dirichlet(1/2) distribution. Definition The m-dimensional Dirichlet(δ)-distribution is defined as dµ δ (b) = Γ(mδ) Γ(δ) m bδ 1 1 b δ 1 m db (3.10) where Γ(x) = 0 e t t x 1 dt is the Gamma function. See section A.2 in the appendix for some properties of the Gamma function. Note that Γ(0) = 1 and therefore it follows that Dirichlet(1) is simply the uniform distribution dµ 1 (b) = db. So, the uniform distribution is a special case of the Dirichlet distribution. As δ decreases toward zero the distribution accentuates the portfolios having more focus on single instruments at the expense of equally weighted portfolios. See figure 3.1 for an illustration of the distribution function for m = 2 instruments. To parameterize the algorithm on the δ-parameter of the Dirichlet distribution, the following notation is introduced: 16

23 3.3. EFFECTIVENESS OF THE ALGORITHM: LOWER BOUNDS ON THE WEALTH RATIO dμ b Figure b Dirichlet(δ) distribution function. The Dirichlet(δ) distribution function dµ(b) = Γ(2δ) Γ(δ) 2 (b(1 b)) δ 1 db for δ = 1 (thin solid), δ = 3/4 (dotted), δ = 1/2 (thick solid), and δ = 1/4 (dashed). Definition Let dµ(b) in the definition of the universal portfolio be the Dirichlet(δ) distribution as defined above. Then Ŝδ n(x n ) is the wealth and ˆb δ n(x n 1 ) are the weights of that portfolio after n time steps on the market sequence X n. Using the introduced definition, the wealth of the original universal portfolio using the uniform distribution is denoted Ŝ1 n(x n ). Cover and Ordentlich (1996) show that Ŝn(X 1 n ( ) ) n + m 1 1 Sn(X n ) m 1 1. (3.11) (n + 1) m 1 Similarly, the wealth of the Dirichlet(1/2) version on the universal portfolio is denoted Ŝ1/2 n (X n ) and Cover and Ordentlich (1996) show that Ŝn 1/2 (X n ) Sn(X n ) Γ( m 2 )Γ(n ) Γ( 1 2 )Γ(n + m 2 ) 1. (3.12) 2(n + 1) m 1 2 These are the previously well established results. Now, we consider two additional issues. First, we know now the performance of the universal portfolio algorithm 17

24 CHAPTER 3. UNIVERSAL PORTFOLIOS using the Dirichlet(δ) distribution for two specific values of δ; 1 and 1/2. It remains to see how the performance is for other values of δ, where 0 < δ 1. Second, it remains to see whether the analysis of the algorithm is complete, i.e. if there are sequences of price relatives that realize the established worst-cast bounds. The analysis focuses on the case of m = 2 instruments. This leads us to the main result of this chapter. Let us take a moment and reflect over what we are trying to accomplish. We are comparing the wealth of the universal portfolio, which we showed to be the µ- weighted average over all CRPs in proposition 3.2.1, with the wealth of the benchmark portfolio, the best CRP. In more general terms, we are comparing the weighted average of a distribution with the maximum of the same distribution. What we need to show then is that the weight allocated by µ to the best portfolio, and portfolios sufficiently similar to the best, is large enough for the wealth of the average portfolio to be close to the wealth of the best portfolio. To do this, we argue that extremal market sequences, where all stocks go to zero except one at each time period, are the hardest for the universal portfolio. The resulting wealth of the universal portfolio and of the best CRP is then calculated explicitly for these market sequences and the lower bounds are thus established. It turns out that the exact configuration of the worst-case market sequences depends on the distribution µ δ (b) used and this results in different constants in the lower bounds for different parameter values. The formal proof of the lower bound follows. Proposition (Lower bounds for the Dirichlet(δ) universal portfolio) Let Ŝδ n(x n ) be the wealth of the Dirichlet(δ) universal portfolio and S n(x n ) be the wealth of the best CRP, both on the market sequence X n. Then Ŝ δ n(x n ) S n(x n ) W δ(n) (3.13) for 0 < δ 1, and all market sequences X n, where Γ(2δ) Γ(n+δ) Γ(δ) Γ(n+2δ) for δ 1 2 W δ (n) =, and (3.14) 2 n Γ(2δ) Γ(n+2δ) for δ < 1 2, and for the case δ < 1/2, Γ(δ) 2 Γ( n 2 +δ)2 n 2 1/3 δ2 1/2 δ. (3.15) 18

25 3.3. EFFECTIVENESS OF THE ALGORITHM: LOWER BOUNDS ON THE WEALTH RATIO Furthermore, asymptotically under the same conditions, where 1 W δ (n) k(δ) (n + 2δ) max(δ,1/2) (3.16) k(δ) = Γ(2δ) Γ(δ) Γ(δ+ 1 2 ) Γ(δ) for δ 1 2, and (3.17) 2 for δ < 1 2. Proof. We begin by establishing the following lemma that shows that the wealth ratio can be bounded by the performance on the worst extremal market sequence. An extremal market sequence is one where all instruments except one go to zero at each time step. So X n {(0, 1) T, (1, 0) T } n. Lemma (Worst extremal sequence bounds wealth ratio) For each j n {1, 2} n define the sequence X(j n ) = (x 1 (j 1 ),..., x n (j n )) where (1, 0) T if j i = 1 x i (j i ) = (0, 1) T if j i = 2. (3.18) Also, let K n = {X(j n ) : j n {1, 2} n } (3.19) and b = (b 1, b 2 ) = (b, 1 b). Then, Ŝ n (X n ) 1 ni=i Sn(X n ) min 0 b ji dµ(b) j n ni=1 K n b. (3.20) j i Note that this lemma is independent of the choice of distribution, µ. Proof. Ŝ n (X n ) 1 ni=1 Sn(X n ) = 0 (b 1 x i1 + b 2 x i2 )dµ(b) ni=1 (b 1 x i1 + b 2 x (3.21) i2) 1 ni=1 j = n K n 0 b ji x iji dµ(b) ni=1 j n K n b (3.22) j i x iji 1 ni=1 0 b ji x iji dµ(b) min j n ni=1 K n b (3.23) j i x iji 1 0 = min j n K n ni=1 b ji dµ(b) ni=1 b j i (3.24) The step leading to (3.22) consists of carrying out the multiplication of the product of n factors containing two terms each into the sum of 2 n terms containing 2n factors 19

26 CHAPTER 3. UNIVERSAL PORTFOLIOS each; n weights and n price relatives. To get to (3.23) we observe that if both the numerator and the denominator of (3.22) are divided by K n = 2 n they express the average over all vectors j n K n, and the inequality then follows by noting that the average must be at least as large as the minimum. Finally, we arrive at (3.24) by eliminating the equal price relatives in the numerator and the denominator. If we start by applying lemma and then let k = n i=1 {j i = 1}, and finally use lemma A.1.1 to get the optimal CRP weights, b, we get: Ŝn(X δ n ) 1 ni=1 Sn(X n ) min 0 b ji dµ δ (b) j n ni=1 K n b (3.25) j i 1 0 = min bk (1 b) n k dµ δ (b) ) k k ( ) n k (3.26) n k = min k ( k n n 1 0 bk (1 b) n k Γ(2δ) (b(1 b)) Γ(δ) δ 1 db ( ) 2 k ( ) n k k n k n n (3.27) B(k + δ, n k + δ) k k (n k) n k (3.28) Γ(2δ) Γ(δ) 2 min Γ(k + δ)γ(n k + δ) k k k (n k) n k (3.29) = n n Γ(2δ) Γ(δ) 2 min k = n n Γ(n + 2δ) From lemma A.3 we know that (3.29) is minimized by letting k = n for δ 1/2 and letting k = n/2 for δ < 1/2. For the latter case the lemma requires that that n 2 1/3 δ2 1/2 δ. (3.30) What remains then is to calculate the bound on the wealth ratio for these two cases. Note that expression (3.26) is actually the wealth ratio for extremal sequences, so by calculating these expressions we simultaneously get witness market sequences that show that the bounds on the algorithm s wealth ratio are tight. Let us start with the case of δ 1/2. We then let k = n in (3.29) and proceed by applying the Sterling approximation from equation A.19. Ŝn(X δ n ) Sn(X n ) Γ(2δ) Γ(n + δ)γ(δ) Γ(n + 2δ) Γ(δ) 2 n n (3.31) = Γ(2δ) Γ(n + δ) Γ(δ) Γ(n + 2δ) (3.32) n n 20

27 3.3. EFFECTIVENESS OF THE ALGORITHM: LOWER BOUNDS ON THE WEALTH RATIO Γ(2δ) Γ(δ) = e δ Γ(2δ) Γ(δ) Γ(2δ) Γ(δ) = Γ(2δ) Γ(δ) 2πe n δ (n + δ) n+δ (3.33) 2πe n 2δ (n + 2δ) n+2δ 1 (n + 2δ) δ e δ ( ) n + δ n+δ (3.34) n + 2δ (n + 2δ) δ e δ (3.35) 1 (n + 2δ) δ (3.36) ( ) n+δ To get to (3.35) we used the fact that n+δ n+2δ e δ which can be shown by taking the logarithm of the expression and applying l Hôpital s rule. Moving on to the second case: δ < 1/2, we let k = n/2 in (3.28) to get the worst case ratio and proceed with the Sterling approximation from equation A.20. Ŝn(X δ n ) Γ(2δ) B( n 2 Sn(X n + δ, n 2 + δ) nn ) Γ(δ) 2 n n n 2 2 n 2 2 (3.37) = 2 n Γ(2δ) Γ(δ) 2 B(n 2 + δ, n + δ) (3.38) 2 = 2 n Γ(2δ) Γ( n 2 + δ)2 Γ(δ) 2 Γ(n + 2δ) 2 n Γ(2δ) ( n 2π 2 + δ) n+2δ 1 Γ(δ) 2 (n + 2δ) n+2δ 1 2 = 2 n Γ(2δ) ( ) 2π n 1 Γ(δ) δ 2 = Γ(2δ) Γ(δ) 2 = Γ(δ ) Γ(δ) 2 n+2δ 1 2 (3.39) (3.40) (3.41) 2π 2 2δ 1 (n + 1 2δ) 2 (3.42) 2 (3.43) n + 2δ Specifically for δ = 1 2 we get W 1 (n) = Γ(1) Γ(n Γ( 1 2 ) 2 ) Γ(n + 1) = 1 π Γ(n ) Γ(n + 1) (3.44) (3.45) 21

28 CHAPTER 3. UNIVERSAL PORTFOLIOS which is the same as the result of Cover and Ordentlich (1996) shown in equation Asymptotically, W 1 (n) 1 1. (3.46) 2 π n + 1 As is established in chapter 4, the upper bound on the wealth ratio is V n 2 1 π n. (3.47) Hence, the horizon free universal portfolio with δ = 1 2 is a factor of 2 worse than the optimal algorithm. The remaining question is then: Which value of δ is asymptotically optimal? The exponent of n is clearly the most important determinant and, as equation 3.16 shows, it is minimized at 1 2 for 0 < δ 1 2. For values of δ within that range the determining factor is thus the constant, k(δ). We know that k( 1 2 ) = 1 π and k(δ) = Γ(δ+ 1 2 ) Γ(δ) 2 for δ < 1 2. In fact, numerical evaluation shows that for δ > 0.31, ) and would thus yield better asymptotic worst-case wealth ratios. k(δ) > 1 π = k( 1 2 This is illustrated in figure 3.2. Finally, letting δ approach 1/2 from below, k(δ) approaches lim δ 1 2 Γ(δ ) Γ(1) 2 2 = 2 = Γ(δ) Γ( 1 2 ) π (3.48) which is equal to the constant of the optimal ratio V n and so suggests that the universal portfolio indeed approaches optimality for large n. However, it is important to keep in mind that for lemma A.3 to be applicable, which is a requirement for these results, the smallest n that is required tends to infinity as δ tends to 1/2. Nevertheless, a better constant than that of the original Dirichlet(1/2) strategy can be achieved even for small n for values of δ greater than 0.31 and approaching 1/2. See figure A.1 in the appendix for an illustration of how fast the requirement on n grows as δ tends to 1/2. When establishing the upper bound for general markets in chapter 4 we mention that it is tight. This was shown by Ordentlich and Cover (1998) using a different fixed-horizon algorithm. In essence, if you know the final number of time steps in advance, an instance of the algorithm chosen for that precise number of steps is guaranteed to reach the bound at the final step. The universal portfolio, while 22

29 3.4. TRANSACTION COSTS 0.8 k Figure 3.2. The constant factor. The function y = k(δ) for 0 < δ < 1/2 (solid) shown with the line y = k(1/2) = 1/ π (dashed) to illustrate the constant factor of the wealth ratio for which the exponent of n has the optimal value of 1/2. not being optimal up to the constant factor, does not require a fixed investment horizon. As previously mentioned, the bounds given here are guaranteed to hold at every time step, not only the last. 3.4 Transaction Costs In the analysis so far we have assumed that trading the buying and selling of financial instruments is costless in the sense that we can buy or sell any amount at the given market price without paying an intermediary or affecting the price. In reality this is of course impossible. However, the type of analysis that is carried out here, where the performance relative to a rebalanced benchmark, can be argued to be somewhat forgiving in this regard. The reason for that is that in reality, the benchmark would also be affected by transaction costs incurred by its rebalancing operations. 23

30 CHAPTER 3. UNIVERSAL PORTFOLIOS It is nonetheless worthwhile to extend the setup to include a model of transaction costs and see how the results are affected. We begin by reviewing what transaction costs are in practice and then move on to a reasonable model of them. Harris (2003) defines transaction costs as all costs associated with trading and separates them in three groups; explicit costs, implicit costs, and missed trade opportunity costs. Explicit transaction costs are easily identifiable costs directly related to trading such as commissions paid to brokers, fees paid to exchanges, stamp duty and other transaction based taxes, and also in the case of larger trading organizations the internal costs such as trader salaries, software, and accounting. Implicit transaction costs include such elements as price impact and the bid/ask spread. Price impact is the effect the execution of a sizable order has on the market. While a trader may be able to transact a small order without materially affecting the market price, the larger the order the greater the impact is on the market. This effect is variable in terms of the liquidity conditions of the specific market and depends on factors such as the time-of-day. The bid/ask spread is the difference between the bid price, the price where market participants are willing to buy a financial instrument, and the ask price, the price where market participants are willing to sell a financial instrument. Missed trade opportunity costs are costs that arise when a trader acts in a suboptimal manner and as a consequence forgoes an opportunity to execute a trade, for example by waiting for the market to move in a favorable direction before sending an order to the market, only to see the market move in the opposite direction. In general explicit costs, as a percentage of the volume traded, decrease as volume increases since brokers and exchanges are able to offer more competitive rates to large customers. On the other hand implicit costs, as a percentage of the volume traded, tend to increase as volume increases simply because the impact of a large order on the market is greater than the impact of a small order. When choosing a model for transaction costs the trade off is in general between the complexity of the model and the ability to reliably estimate its parameters. The more complex the model, the more data is required to calibrate it in order to 24

31 3.4. TRANSACTION COSTS generate useful results. In our case, another major consideration is the ability to integrate the model in the existing analysis framework. We consider the simple model used by Blum and Kalai (1999). They use a fixed percentage transaction cost, 0 c 1, which for simplicity and without loss of generality is charged for purchase transactions only. Three fundamental properties of the transaction costs of rebalancing are established: 1. The total costs of rebalancing from a portfolio with weights b 1 to portfolio b 3 are no larger than the total costs of first rebalancing from b 1 to portfolio b 2 and then from b 2 to b 3 for any portfolio b The total cost of rebalancing from portfolio b to portfolio ((1 α)b + αb is at most cα, since at most a fraction α of the initial portfolio is moved. 3. By initially allocating a fraction α to investment strategy I 1 and a fraction (1 α) to investment strategy I 2, the final wealth of the combined portfolio is at least α times the final wealth of I 1 plus (1 α) times the final wealth of I 2. In fact, if at times the strategies perform opposite transaction, for example I 1 sells instrument i at the same time that I 2 buys it, the final wealth is larger since transaction costs are saved by performing the buy and sell operations internally. Using these properties the following proposition is proven for the uniform universal portfolio. Proposition (Lower bound on wealth ratio with transaction costs) Let Ŝδ c,n(x n ) be the wealth of the Dirichlet(δ) universal portfolio on the market sequence X n after n time steps assuming a proportional transaction cost of 0 c 1. Similarly, let S c,n(x n ) be the wealth of the best CRP under the same conditions. Then Ŝc,n(X 1 n ( ) ) (1 + c)n + m 1 1 Sc,n(X n ) m 1 for all markets with m instruments. Proof. See theorem 2 of (Blum and Kalai 1999). 1 ((1 + c)n + 1) m 1 (3.49) So, the loss of competitiveness of the algorithm due to a fixed percentage transaction cost of c is, asymptotically, only the factor 1/(1+c) (m 1). For reasonable transaction 25

32 CHAPTER 3. UNIVERSAL PORTFOLIOS costs and a small number of instruments this factor remains quite close to one. It is somewhat surprising to note that it does not depend on the number of time periods n. It remains an open question how the result of proposition generalizes to Dirichlet(δ) distributions for δ 1. The question if the bound of proposition is tight is also yet unanswered. 3.5 Efficiency of the Algorithm: Implementation Although the mathematical definition of the universal portfolio is quite simple, it is not immediately obvious how it can be implemented in terms of accuracy and efficiency. For a small number of instruments, the integral that gives the portfolio weights can easily be discretisized and approximated but for a large number of instruments this becomes an increasingly difficult problem. In this section a few established methods that seek to overcome this problem are presented. The first method was established by Ordentlich (1996). It is a recursive procedure that can be summarized as follows. For δ = 1 2 where 1 k n 1, and and m = 2, let k 1 2 Q n (k) = x n1 n Q n k 1 2 n 1(k 1) + x n2 Q n 1 (k) (3.50) n Q n (0) = x n2 n 1 2 n Q n 1(0), (3.51) Q n (n) = x n1 n 1 2 n Q n 1(n 1), and (3.52) Q 0 (0) = 1. (3.53) Then, and ˆb 1 2 n = Ŝ 1 2 n (X n ) = 1 Ŝ 1 2 n (X n ) n 1 k=0 n 1 k=0 n Q n (k) (3.54) k=0 k n n k 1 2 n 26 Q n 1 (k). (3.55) Q n 1 (k)