Modeling asset allocation strategies and a new portfolio performance score
MModeling asset allocation strategies and a newportfolio performance score
Apostolos Chalkis and Ioannis Z. Emiris
Department of Informatics & Telecommunications, National &Kapodistrian University of Athens, Greece ATHENA Research & Innovation Center, Greece
Abstract
We discuss a powerful, geometric representation of financial portfolios andstock markets, which identifies the space of portfolios with the points lying in asimplex convex polytope. The ambient space has dimension equal to the numberof stocks, or assets. Although our statistical tools are quite general, in this paperwe focus on the problem of portfolio scoring. Our contribution is to introduce anoriginal computational framework to model portfolio allocation strategies, whichis of independent interest for computational finance. To model asset allocationstrategies, we employ log-concave distributions centered on portfolio benchmarks.Our approach addresses the crucial question of evaluating portfolio management,and is relevant to the individual private investors as well as financial organizations.We evaluate the performance of an allocation, in a certain time period, by providinga new portfolio score, based on the aforementioned framework and concepts. Inparticular, it relies on the expected proportion of actually invested portfolios that itoutperforms when a certain set of strategies take place in that time period. We alsodiscuss how this set of strategies –and the knowledge one may have about them–could vary in our framework, and we provide additional versions of our score inorder to obtain a more complete picture of its performance. In all cases, we showthat the score computations can be performed efficiently. Last but not least, weexpect this framework to be useful in portfolio optimization and in automaticallyidentifying extreme phenomena in a stock market. a r X i v : . [ q -f i n . P M ] D ec Introduction
Modern finance has been pioneered by Markowitz who set a framework to study choice inportfolio allocation under uncertainty, see [14], and which earned him the Nobel Prizein economics, 1990. Within this framework, Markowitz characterized portfolios by theirreturn and their risk; the latter is formally defined as the variance of the portfolios’ returns.An investor would build a portfolio that will maximize its expected return for a chosenlevel of risk; it has since become common for asset managers to optimize their portfoliowithin this framework. This approach has led a large part of the empirical finance researchto focus on the so-called efficient frontier which is defined as the set of portfolios presentingthe lowest risk for a given expected return. Figure 1 presents such an efficient frontier.The efficient frontier is associated with a well-known family of convex functions, studiedby Markowitz in [15]. However, building a portfolio in that (or any other) framework doesnot always guarantees superior performance in practice comparing to other allocationchoices. Thus, evaluating the performance of a certain allocation is a challenging task ofspecial interest. We discuss previous work on portfolio scoring and then we present ourmain contributions.
The fast growth of asset management industry during the past few decades has highlightedthe analysis of portfolio allocation performance as an important aspect of modern finance.Research in this area is axed on Sharpe-like ratios proposed in the 1960’s [11, 19, 21]. Inpractice, the performance of a portfolio manager, over a given period, is usually measuredas the ratio of his ”excess” return with respect to a benchmark portfolio over a riskmeasure [8]. Managers are then ranked according to these ratios, and the one achievingthe highest and steadiest returns receives the best score. The major drawback of thesetechniques is the identification of benchmark portfolios, while the formation of suchportfolios remains controversial. Thus, we assume that the best score corresponds to a“good” portfolio allocation, but without having a universal measure of goodness for thisallocation. Moreover, they suffer from significant estimation errors [12], which preventany performance comparison to be significant.In [16] -and independently in [9, 1]- they use the geometric representation of a stockmarket, presented also in this paper, to define a cross-sectional score of a portfolio given avector of assets’ returns. In particular the score of a portfolio is defined as the proportion ofallocations that the portfolio outperforms. The aim is to measure the relative performance-in terms of return- of an asset allocation with respect to all possible alternative allocationsoffered to the manager. The term cross section is used to underline that the score takesinto account portfolios that are diversified over all sections of assets, without studying-separately- the performance on specific sections of stocks. In [16, 17, 10], the relativeperformance of value-weighted indices with respect to long-only portfolios is assessed inthe Dutch, Spanish and German markets; they considered the MSCI Netherlands 24,IBEX 35, and DAX 30 components, respectively. Interestingly, in [1], they follow thesame approach by defining what they call naive investor’s strategy . A naive investor’sstrategy selects uniformly a portfolio from the set of all portfolios, as it is agnostic aboutthe assets’ returns generating process, and hence does not use any such information.In [16, Thm 4.2.2] they compute the score by sophisticated geometrical algorithms.However, this computation is not valid when some asset returns are equal and it presents2igure 1: An illustration of the efficient frontier.floating point errors limiting its use to around 20 assets. As a consequence, in [16] andin related studies [1], the score is estimated by a quasi-Monte Carlo sampling of theportfolios; one may refer to [18] for uniform sampling methods over a simplex of generaldimension. Finally, in [2] they show that an algorithm in [22] computes this score veryefficiently and robustly (a few milliseconds, in stock markets with thousands of assets).Moreover, in [3] they characterize statistically the distribution of portfolios’ returns, wherethe aforementioned portfolio score corresponds to its Cumulative Density Function (CDF),and they rely on powerful techniques in computational geometry to compute exactlythe CDF and Probability Density Function, as well as the moment of portfolios’ returnsdistribution of any order, with several applications [20, 7].
At first we employ a geometric representation of the set of portfolios in a stock market(Section 2) also appeared in [2, 3]. In particular, we focus on the long-only strategiesand thus, we represent the set of portfolios with the canonical simplex, which is a convexpolytope. However, the computational framework we provide can be generalized for anyconvex set. In the sequel, our aim is twofold:(a) to introduce original models of portfolio allocation strategies, which should be criticalfor other problems in finance.(b) to employ the latter framework in evaluating portfolio’s performance in a certain timeperiod by introducing a new score.We introduce a new mathematical model of portfolio allocation strategies in a stockmarket. It is of independent interest and may be used to address several questions infintech besides those in Section 5. We consider the concept where portfolio managerscompute and propose asset allocations, which we call formal allocation proposals . Then,an investor first decides which allocation proposal to select and second how much tomodify this proposal to create his final investment / portfolio. Thus, we expect that theportfolios of the investors, that choose the proposal of a certain portfolio manager, will be3concentrated around” that proposal. To model this procedure we employ multivariatedistributions. The support of the Probability Density Function (i.e. the subset of R n which are not mapped to zero) of each distribution is the set of all portfolios. In particular,we say that a portfolio allocation strategy F π is induced from a distribution π as follows:to create a portfolio with strategy F π sample a point/portfolio from π . According to theprevious observations, the most intuitive choice for π is a unimodal distribution. Then,we call the mode of π formal allocation proposal of the allocation strategy F π .We focus on Markowitz’s framework to leverage log-concave distributions inducedby the family of convex functions of Equation (3) in Section 3.1. We discuss how weparameterize the allocation strategies by the level of risk that a certain group of investorsselect. Similarly, for a given level of risk, we use the variance to parameterize how stickaround the formal allocation proposal a subgroup of investors may decide to be. In otherwords, when we say “the investors that create their portfolio according to strategy F π ”we denote the proportion of the investors, in a certain stock market and time period,that select risk according to the mode of π and they stick around the formal allocationproposal of F π according to the variance of π . Finally, as in a stock market appear plentyof strategies followed by group of investors, we define the mixed strategy induced by aconvex combination of distributions, i.e. a mixture distribution.We evaluate the performance of a portfolio for a given time period and compare theportfolio against a mixed strategy F π , when a certain set of strategies take place in thattime period. Thus, we define the score of a portfolio as the expected number of actuallyinvested portfolios that the first outperforms, when the portfolios have been investedaccording to the mixed strategy F π . We provide an efficient algorithm, based on MarkovChain Monte Carlo integration, to estimate the new score within arbitrarily small error (cid:15) (Section 3). Furthermore, in extreme cases our new score becomes equal to that of[16, 9, 1]. Thus, it can also be seen as a generalization of the latter score.Lastly, one may have limited knowledge about a certain stock market and how theinvestors behave in it, or her/his knowledge may vary from a time period to another.We extend our framework to handle these issues (Section 4). We also provide differentversions of our score. Each version provides a different information about the portfolioallocation we would like to evaluate.We expect that the frameworks and the computational tools we present in the sequelcan be generalized and used to handle further problems in fintech. For example, they couldbe combined with various asset-pricing models and methods to predict assets’ returns byMachine Learning and AI methods [6]. Additionally, we believe that the new score canbe used to define new performance measures and optimal portfolios according to thesemeasures. Finally, despite the fact that in this paper we focus on the long-only strategies,the tools we present can be easily extend to any set of portfolios.The work presented here is also appears in [4]. Paper structure.
The next section presents the geometric representation of portfolioswe use. Section 3 introduces our new framework for modeling allocation strategies, andevaluating portfolio performance by defining a new score of a portfolio. In Section 4we discuss how one can parameterize our framework to have further in-depth study ofportfolio performance. In Section 5 we briefly discuss conclusions and future work.4
Geometric representation of the set of portfolios
In this section we formalize the geometric representation of sets of portfolios with anarbitrary large number of assets n . We handle the case of long-only strategies. Thus, theset of all portfolios becomes a specific convex set.In particular, let a portfolio x investing in n assets, whose weights are x = ( x , . . . , x n ) ∈ R n . The portfolios in which a long-only asset manager can invest are subject to n (cid:80) i =1 x i = 1and x i ≥ , ∀ i . Thus, the set of portfolios available to this asset manager is the unit( n − − dimensional canonical simplex, denoted by ∆ n − and defined as∆ n − := (cid:40) ( x , . . . , x n ) ∈ R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) i =1 x i = 1 , and x i ≥ , ∀ i ∈ { , . . . , n } (cid:41) ⊂ R n . (1)The simplex ∆ n − is the smallest convex polytope with nonzero volume in a givendimension. For instance, in the plane any triangle is a simplex, while a triangular pyramid,or tetrahedron, is the simplex in 3d space.Here the space dimension n represents the number of assets. Each point in the interiorof the simplex represents a portfolio since its coordinate vector is a convex combination ofthe vertex coordinates: if we use all vertices, this combination is unique and is known asbarycentric coordinates of the point. The vertices represent portfolios composed entirelyof a single asset. This is the most common investment set —of long-only strategies—in practice today, as portfolio managers are typically forbidden from short-selling orleveraging. We now discuss an original method for modeling allocation choices and for evaluatingportfolio performance by a new portfolio score. We define the new score of a portfolio asthe expected value of the proportion of actually invested portfolios that it outperforms,when the portfolios have been built according to, what we call, a mixed strategy .Here, we assume that in a stock market the portfolio managers make allocationproposals and then the investors choose which proposal to follow and how much to modifyit before they create their final portfolio. We model allocation strategies in Markowitz’framework using multivariate log-concave distributions with ∆ n − being the support ofeach Probability Density Function (PDF). A proper choice of log-concave distributionsallows us to parameterize a strategy by the level of risk and the level of dispersion aroundthe formal allocation proposal of the strategy. However, the framework presented in thisSection allow us to use any unimodal distribution centered at any benchmark portfolio. Definition 1.
Let π be a unimodal distribution truncated in ∆ n − with PDF π ( x ) . Then,a portfolio allocation strategy F : π → ∆ n − is said to be induced by the distribution π ,and we write F π . More precisely, F π is induced by the following state:“To build a portfolio with strategy F π sample a point/portfolio from π ”. The mode of π can be seen as the allocation proposal that a portfolio manager hasbeen made. Then, we expect that the invested portfolios of the investors who have chosenthat proposal will be concentrated around that proposal/mode as the mass of π implies.5 efinition 2. Let strategy F π induced by the unimodal distribution π . We call the modeof π formal allocation proposal or formal proposal of the portfolio allocation strategy F π . In the sequel, we assume that in a stock market the set of actually invested portfoliosare created by a combination of different strategies used by the investors (mixed strategy).First, we consider a sequence of log-concave distributions π , . . . , π M truncated in ∆ n − .Then, each distribution induces a portfolio allocation strategy, i.e. F π , . . . , F π M . Then,the mixed strategy is induced by a convex combination of π i , i.e. by a mixture distribution,as the following definition states. Definition 3.
Let π , . . . , π M be a sequence of unimodal distributions, and let the mixturedensity be π ( x ) = (cid:80) Mi =1 w i π i ( x ) , where w i ≥ , (cid:80) Mi =1 w i = 1 . We call F π the mixedstrategy induced by the mixture density π . In Definition 3 each weight w i corresponds to the proportion of investors that build theirportfolios according to the allocation strategy F π i . Thus the vector of weights w ∈ R M implies how the investors in a certain stock market and time period tends to behave. Nowwe are ready to define the cross-sectional score of an allocation versus a mixed strategy. Definition 4.
Let a stock market with n assets and F π a mixed strategy induced by themixture density π . For given asset returns R ∈ R n over a single period of time, the scoreof a portfolio, providing a value of return R ∗ , is s = (cid:90) ∆ n − g ( x ) π ( x ) dx, g ( x ) = (cid:26) . if R T x ≤ R ∗ , , otherwise. (2)Notice that the Definition 4 can be generalized for any set of portfolios. The value ofthe integral in Equation (2) corresponds to the expected proportion of portfolios that anallocation outperforms when the portfolios are invested according to the mixed strategy F π . In this Section, we consider the Markowitz’ framework and we discuss the selection ofa proper log-concave distribution so that we could fix a sequence π , . . . , π M . In thisframework the assets’ returns are random variables distributed normally, with mean µ and covariance matrix Σ.In general, using Markowitz’ framework one can define, under certain assumptions,the optimal portfolio ¯ x as the maximum of a concave function h ( x ) , x ∈ ∆ n − . Then thelog-concave distribution with PDF π ( x ) ∝ e αh ( x ) has its mode equal to ¯ x and its variance σ = 1 /α . We again call the mode of π formal allocation proposal of the induced strategy F π as we do in Section 3.Notice that as the variance grows, π converges to the uniform distribution and asthe variance diminishes, the mass of π concentrates around the mode of π ( x ). Thus, weuse the variance to parameterize the sequence π i ∝ e α i h ( x ) . Small variances correspondto allocation strategies that are used by investors who stick around the formal proposal.Thus, the created portfolios with such a strategy F π would be highly concentrated aroundthe formal allocation proposal of F π (or mode of π ) as the mass of π implies. Largevariances correspond to allocation strategies that are used by investors who may modifythe formal proposal a lot. The portfolios created with such a strategy F π , would be highly6igure 2: Left: illustration of PDFs π q ∝ e − αφ q ( x ) , where α = 1 and from left to right q = 0 . , q = 1 , q = 1 .
5. Right: 3 illustrations of the mixture density of Equation (5),where M = 3 , M = 2. In both plots the black point corresponds to the formal allocationproposal of each strategy. From yellow to blue: high to low density regions.dispersed around the mode of π . In the extreme case of very large variance, π is close tothe uniform distribution and the induced allocation strategy becomes the naive strategyas defined in [1]. We employ the distance between π i and the uniform distribution tocharacterize how dispersed the portfolios created with F π are, around the formal allocationproposal. Definition 5.
Let π ∝ e αh ( x ) be any log-concave distribution and let F π be the inducedportfolio allocation strategy. We say that F π is − D )% -dispersed, where D is thedistance between π and the uniform distribution, in terms of total variation distance. Our main approach is to leverage the family of convex functions which is widely usedby investors to compute the efficient frontier (EF). In particular, in Markowitz’s frameworkthe assets’ returns are assumed to be normally distributed following N ( µ, Σ). Then, theparameterized function φ q ( x ) = x T Σ x − qµ T x, x ∈ K, q ∈ [0 , + ∞ ] , (3)where K is the set of portfolios, is used to compute the efficient frontier and optimalportfolios. The x T Σ x is called risk term, the µ T x is called return term and the parameter q controls the trade-off between return and risk. To make an efficient portfolio allocation,in modern finance, a portfolio manager typically compute the EF. In particular, themanager selects a value q —which determines the level of risk of his allocation— andthen, according to [15] he solves the following optimization problem:min φ q ( x ) = x T Σ x − q µ T x, subject to x ∈ ∆ n − . We call the portfolio ¯ x = min x ∈ K φ q ( x ) as the optimal mean-variance portfolio for the riskimplied by q . Thus, the efficient frontier can be seen as a parametric curve on q (seeFigure 1).Let the log-concave distribution, π α,q ∝ e − αφ q ( x ) . (4)7he left plot in Figure 2 illustrates some examples of the density function π α,q where µ and Σ are randomly sampled once. Notice that for different q , the mode (or the formalallocation proposal of the strategy F π α,q ) is shifted.We can use parameter q to denote the level of risk of a portfolio allocation strategy F π α,q . Small values of q correspond to low risk strategies whereas large values of q to highrisk strategies. Thus a sequence of such densities can be parameterized by both q (risk)and α (dispersion). In particular, a mixed strategy F π can be induced by the followingmixture density: π ( x ) = M (cid:88) i =1 M (cid:88) j =1 w ij e − a ij φ i ( x ) , where φ i = x T Σ x − q i µ T x, (5)where each q i denotes the level of risk and for each q i the parameters α ij imply the levelof dispersion of F π ij . Notice that for each level of risk q i there are M different levelsof dispersion that different groups of investors’ portfolios may appear around the sameformal allocation proposal. The right plot of Figure 2 illustrates some examples of thismixture density.A definitely important question is how one could set the risk and dispersion parameters q i , α ij and the weight w ij of each allocation strategy F π qi,αij in a certain stock market.The issue is that our knowledge about the stock market and the behavior of the investorsin it might be weak or vary from a time period to another. In Section 4 we extend ourframework to address these issues. We also provide different versions of the score thanthose given in Section 3. Each version provides a different information about the portfolioallocation we would like to evaluate for given assets returns. This section discusses Markov Chain Monte Carlo (MCMC) integration to guarantee fastand robust approximation within arbitrarily small error for the computation of the scorein Section 3. Let the density π ( x ) = (cid:80) Mi =1 w i π i ( x ) in Equation (2) to be the probabilitydensity function of a mixture of log-concave distributions. Furthermore, let the vector ofassets’ returns R ∈ R n , the halfspace H ( R ∗ ) := { x ∈ R n | R T x ≤ R ∗ } and the indicatorfunction g ( x ) = (cid:26) . if x ∈ H ( R ∗ ) , , otherwise. . Then the score of Equation (2) can be written, s = (cid:90) ∆ n − g ( x ) M (cid:88) i =1 w i π i ( x ) dx = M (cid:88) i =1 w i (cid:90) ∆ n − g ( x ) π i ( x ) dx = M (cid:88) i =1 w i (cid:90) ∆ n − ∩ H ( R ∗ ) π i ( x ) dx = M (cid:88) i =1 w i (cid:90) S π i ( x ) dx, (6)where S := ∆ n − ∩ H ( R ∗ ) is the intersection of the canonical simplex with a halfspace.It is clear that the computation of the score s is reduced to integrate M log-concavefunctions over a convex set S , i.e. to compute each (cid:82) S π i ( x ) dx, i = [ M ]. For each oneof these M integrals we use the algorithm presented in [13] to approximate it within anarbitrarily small error after a polynomial in dimension (number of assets) n number ofoperations. First, we use an alternative representation of the volume of S , employing a8og-concave density π ( x ),vol( S ) = (cid:90) S π ( x ) dx (cid:82) K π β ( x ) dx (cid:82) S π ( x ) dx (cid:82) S π β ( x ) dx (cid:82) S π ( x ) β dx · · · (cid:82) S dx (cid:82) S π ( x ) β k dx ⇒ (cid:90) S π ( x ) dx = vol( S ) (cid:82) S π ( x ) β k dx (cid:82) S dx · · · (cid:82) S π ( x ) dx (cid:82) S π ( x ) β dx , (7)where the sequence β j , j = [ k ] are factors applied on the variance of π ( x ).Since S is the intersection of a halfspace with the canonical simplex ∆ n − we useVarsi’s algorithm to compute the exact value of vol( S ) after n operations at most. Thus,the computation of (cid:82) S π ( x ) dx is reduced to compute k ratios of integrals. This problemseems intractable at first glance. However, for each ratio we have, r j = (cid:82) S π ( x ) β j − dx (cid:82) S π ( x ) β j dx = 1 (cid:82) S π ( x ) β j dx (cid:90) S π ( x ) β j − π ( x ) β j ( x ) π ( x ) β j ( x ) dx = (cid:90) S π ( x ) β j − π ( x ) β j π ( x ) β j (cid:82) S π ( x ) β j dx dx. (8)Thus, to estimate r j we just have to sample N points from the distribution proportionalto π ( x ) β j and truncated to S . Then, r j ≈ N N (cid:88) i =1 π ( x i ) β j − π ( x i ) β j (9)as N grows. The key for an efficient approximation of r j using Monte Carlo integra-tion is to set β j , β j +1 such that the variance of r j is as small as possible (ideally aconstant) for N as small as possible. To estimate the score in Equation (6) suffices toestimate each (cid:82) S π i ( x ) dx, i = 1 , . . . , M as the Equation (7) implies. Then the score s = (cid:80) Mi =1 w i (cid:82) S π i ( x ) dx can be easily derived. The following Lemma provides the totalnumber of operations required to approximate the score s in Equation (2) within arbitrarilysmall error, employing MCMC integration and the algorithm in [13]. Lemma 6.
Let the density π ( x ) in the Definition 4 be a mixture of M log-concave densities.Then the portfolio score in Equation (2) can be estimated after O ∗ ( M n ) operations, where O ∗ ( · ) suppresses polylogarithmic factors and dependence on error e .Proof. In [13], they prove that the sequence of β , . . . , β k can be fixed such that the varianceof each r j , j = [ k ] is bounded by a constant. Moreover, N = O ∗ ( √ n ) points per integralratio r j and k = O ∗ ( √ n ) ratios in total suffices to approximate each (cid:82) S π i ( x ) dx, i = [ M ]within error e . Thus, O ∗ ( n ) points suffices to estimate each (cid:82) S π i ( x ) dx .To sample from each target distribution proportional to π ( x ) β j and truncated to S in[23] they use the Hit-and-Run random walk [23]. This implies a total number of O ∗ ( n )arithmetic operations per generated point. Thus the total number of arithmetic operationsto estimate the score s is O ∗ ( M n ).Considering practical computations, a plenty of random walks for sampling fromlog-concave densities in high dimensions are implemented in the software package volesti [5]. For an extended introduction to geometric random walks we suggest [23].9 Determine a mixed strategy
In this Section, we discuss how we set the parameters of a sequence of log-concavedistributions π ij = e − a ij φ i ( x ) , where φ i = x T Σ x − q i µ T x, i = [ M ] and j = [ M ]which induce a mixed strategy as in Equation (5). Let q i ∈ [0 , Q U ] , Q U < ∞ , i = [ M ].When q i = Q U the term of risk x T Σ x is negligible in φ i ( x ) with respect to the term ofreturn µ T x . Thus, q = Q U corresponds to the optimal mean-variance portfolio with highestexpected return. We recall that q = 0 corresponds to the allocation strategy of zero risk.Let for each q i , the parameters α L i < α ij < α U i , j = [ M ]. The variance 1 /α L i correspondsto a 100(1 − e )%-dispersed allocation strategy and the variance 1 /α U i corresponds tothe log-concave density π α Ui ,q i ( x ), whose mass is almost entirely concentrated around theformal allocation proposal of the induced strategy. The bounds on the parameters α ij and q i can be easily extracted from the observations in [13].Now we select equidistant values in both intervals above to set the sequences of q i and α ij . The aim is to represent allocation strategies with various levels of risk and dispersionin a certain stock market. It is clear that as both M , M grow, the representativeness ofstrategies improves. Set the sequence of q i and α ij
1. Select M equidistant values q < · · · < q M from [0 , Q U ].2. For each q i , select M equidistant values α i < · · · < α iM from [ α L i , α U i ].The construction of both sequences of q i and α ij allow to specify the sequence oflog-concave distributions π ij = e − α ij φ qi ( x ) . However, to determine a mixed strategy onehas to determine the weights w ij in the corresponding mixture distribution. We recallthat each w ij implies the proportion of investors that create their portfolios accordingthe allocation strategy induced by π ij . Setting w ij forms the mixed strategy F π while thescore of Section 3 becomes, s = M (cid:88) i =1 M (cid:88) j − w ij (cid:90) S π ij ( x ) dx, S := ∆ n − ∩ H ( R ∗ ) , (10)as also denoted by Equation (6) in Section 3.2. However, one may have a weak knowledgeon how the investors behave in a certain stock market, in order to determine explicitlythe weights w ij . First we allow to set further bounds on w ij . For example, one wouldprovide an upper bound on the proportion of the investors who chose a specific allocationstrategy. We allow these degrees of freedom as follows and we additionally provide threedifferent versions of our score.In particular, let us assume that we estimate the M = M M integrals of Equation(10) as described in Section 3.2, where M is the number of allocation strategies in acertain stock market. Then, let the M values to form a vector c ∈ R M and also let thecorresponding weights w ij in Equation (10) to be given as a vector w ∈ R M . Then thescore, s = (cid:104) c, w (cid:105) , (11)10here (cid:104)· , ·(cid:105) denotes the inner product between two vectors. Given a matrix A ∈ R N × M and a vector b ∈ R N which express N further constraints on the weights (e.g. specifylower, upper bounds or any linear constraint on w ij ), let Q ⊂ R M the following feasibleregion of weights, Aw ≤ bw i ≥ M (cid:88) i w i = 1 (12)Notice that if no further constraints are given on the weights, then the feasible region Q is the canonical simplex ∆ M − . Now let us define three new versions of score s . Eachnew score provides a different information about the allocation we evaluate.Let the weights w ∈ Q , where Q ⊂ R M the feasible region in Equation (12).1. min score , s := min (cid:104) c, w (cid:105) , subject to Q .2. max score , s := max (cid:104) c, w (cid:105) , subject to Q .3. mean score , s := vol ( Q ) (cid:82) Q (cid:104) c, w (cid:105) dw .For the scores s and s one has to solve a linear program for each one of them. Thescore s requires the computation of an integral which can be computed with MCMCintegration employing uniform sampling from Q ; otherwise it can be reduced to thecomputation of the volume of a convex polytope P ⊆ R M since (cid:104) c, w (cid:105) is a linear functionof w with the domain being the set Q . For the latter computation there are severalrandomized approximations algorithms and efficient C++ software provided by package volesti [5].Let w ∈ Q such that the min score s = (cid:104) c, w (cid:105) . The weights denoted by the vector w imply the proportions of the investors that follow each allocation strategy such thatthe portfolio score s takes its minimum value. Similarly, the vector of weights w ∈ Q suchthat the max score s = (cid:104) c, w (cid:105) , implies the proportions of the investors that follow eachallocation strategy such that the portfolio score s takes its maximum value. Moreover,it is easy to prove that the mean score s = (cid:104) c, ¯ w (cid:105) , where the vector of weights ¯ w is thecenter of mass of Q . For example, if Q = ∆ M − (i.e. the case where no further constraintsare given on the weights) the vector ¯ w is the equally weighted vector.However, one may have an additional knowledge on how the investors tend to behavein a certain stock market, i.e. which allocation strategies they tend to prefer. We alsoallow for these degrees of freedom by providing the notion of behavioural functions . In this Section we assume that we are given a set of functions which represents theknowledge, that one may have, related to which allocation strategies the investors tend toprefer in a certain stock market and time period. We assume that we are given M + 1functions f q , f α i with the domain being [0 , Q U ] and [ α L i , α U i ] , i = [ M ] respectively. Wecall these functions behavioural functions and we use them to create a vector of weights11igure 3: Examples of behavioral functions. w ∈ R M , that emphasizes specific strategies, where M = M M the number of allocationstrategies that take place in the stock market.The plots in Figure 3 demonstrate 4 possible choices of such functions. For example, ifplot C is f q then the investors tend to prefer low risk investments; the value of f q is high forsmall values of q (low risk) and low for high values of q (high risk). If in addition the plot D is f α i then the investor tends to be highly sticked around the formal allocation proposalthat corresponds to q i ; the value of f α i is large for large values of α (low dispersion) andsmall for small values of α (high dispersion). The following pseudo-code describes how wecompute such a weight vector w when M + 1 behavioural functions are given. Construct the vector weight w Input : risk and dispersion parameters q i and α ij , i = [ M ] , j = [ M ]computed as in Section 4 and M + 1 behavioural functions f q , f α i .1. For each pair of ( i, j ) set r ( i − M + j ← f q ( q i ) f α i ( α ij )2. Normalize the vector r j ← r j / (cid:80) Mi =1 r i , j = [ M ] and M = M M
3. Set the weight vector w ← r .Note that for each q i we request a behavioural function f α i to emphasize strategies withlevel of risk q i and level of dispersion denoted by f α i . Given the behavioral functions, onecould use the vector of weights —determined as in the above pseudo-code— to computethe portfolio score s = (cid:104) c, w (cid:105) , while c is again the vector that contains the values of theintegrals of Equation (10) in Section 4. 12igure 4: In both plots: Probability density functions p T i ( w ) ∝ e rw/T i where (from left toright) T = 2 , T = 1 , T = 2 /
3. The bias vector r ∈ R is given in each title. In this Section we allow a weaker knowledge than in Section 4.1 that we might have abouthow the investors tends to behave. Thus, we do not explicitly determine the vector ofweights w ∈ R M — M is the number of allocation strategies in a certain stock market—as in Section 4.1. In particular, let the coordinates of the vector r ∈ R M as in Section 4.1, r ( i − M + j ← f q ( q i ) f α i ( α ij ) , i = [ M ] , j = [ M ]where f q , f α i the M + 1 behavioral functions. Then, we use the vector r to denote a bias on the behavior of the investors. First, we again allow further bounds and linearconstraints on the weights. Thus, we assume —as in Section 4— that the feasible regionof the weights is the set Q of Equation (12). To denote the bias on the behavior of theinvestors we employ the exponential distribution p T ( w ) ∝ e rw/T , T > , with the support of p T ( w ) being the set Q . The distribution p T ( w ) ∝ e rw/T is usually calledBoltzmann distribution and the vector r bias vector . In general, Boltzmann distributiongives the probability that a system will be in a certain state as a function of that state’senergy and the temperature of the system. The bias vector r determines how the masstends to distribute in Q and the (temperature) parameter T how strong the bias denotedby r is. The plots in Figure 4 illustrate some examples of the density function of p T inthe simple case of Q = ∆ and two different choices of the bias vector r . Notice that themass tends to concentrate around the vertices which correspond to the coordinates of r with larger values than the other coordinates. Moreover, as the temperature T → r . As T → ∞ , p T converges tothe uniform distribution and the bias denoted by r disappears.It is clear that our intention is to use the temperature T to parameterize how strongthe tendency on the investors’ behavior, that the behavioral functions and the bias vector13 imply, is. Then the parametric score is given as, s ( T ) := (cid:90) S (cid:104) c, w (cid:105) p T ( w ) dw, where p T ( w ) ∝ e rw/T , T > r ( i − M + j = f q ( q i ) f α i ( α ij ) , i = [ M ] , j = [ M ] (13)Let the center of mass ¯ w T in Q when the mass is distributed according to p T ( w ).Notice that ¯ w T can be seen as a parametric curve on T . Furthermore, it is easy to provethat, for fixed T , the parametric score s ( T ) = (cid:104) c, ¯ w T (cid:105) . Thus, the score s ( T ) is evaluatedon that parametric curve. Following these observations we are ready to state the followingLemma. Lemma 7.
Let a stock market with M allocation strategies. Assume that we are given theparameters q i , α ij of Section 4 and any behavioral functions f q , f α i , i = [ M ] , j = [ M ] and M = M M the number of allocation strategies that take place in the stock market.Let the feasible set Q ⊂ R M of the weights as in Equation (12), the min score s , the maxscore s and the mean score s of Section 4 and the parametric score in Equation (13).Then, the followings hold, s ≤ s ( T ) ≤ s , ∀ T > ,s = lim T →∞ s ( T ) (14)Notice that the Equation (14) holds for any set of behavioral functions. Thus, thescores s , s always bound the parametric score. In particular, for given M allocationstrategies, the parametric score when T → s when all the investorsselect the allocation strategy denoted by the largest coordinate of the bias vector r .Furthermore, when T → ∞ the distribution p T ( w ) converges to the uniform distributionover the feasible region of the weights Q and thus the parametric score is equal to themean score s . On the other hand, let the weights w , w that correspond to scores s , s as in Section 4. The w , w imply how the investors have to be distributed among theallocations strategies such that the score s takes the smallest possible and the largestpossible value respectively. Thus, they can be seen as lower and upper bound on theparametric score respectively. A future direction would be to employ the present computational framework for theproblem of detecting financial crisis. In particular, we could compute copulae as in [2]but instead of uniform sampling to employ sampling from a mixture distribution as inEquation (5). Moreover, we could introduce parametric copulae following the notion ofparametric score in Section 4.2.We also believe that it would be of special interest to use the new score to define newperformance measures and thus, compute the optimal portfolios with respect to thosemeasures. In particular, for a given portfolio one could estimate its score distribution.Then, the problem reduces to compute a portfolio with a “good” score distribution.From an implementation point of view, the latter two applications require to samplefrom various log-concave distributions truncated to convex sets and perform MCMCintegration multiple times. Thus, new, practical sampling methods leveraging modernrandom walks (e.g. Hamiltonian Monte Carlo) will be required. We plan to develop thecorresponding methods based on package volesti [5].14 eferences [1] A. Banerjee and C.-H. Hung. Informed momentum trading versus uninformed “naive”investors strategies.
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