Beating the Market with Generalized Generating Portfolios
BBeating the Market with Generalized GeneratingPortfolios ∗ Patrick Mijatovic † Abstract
Stochastic portfolio theory aims at finding relative arbitrages, i.e. trading strategieswhich outperform the market with probability one. Functionally generated portfolios,which are deterministic functions of the market weights, are an invaluable tool indoing so. Driven by a practitioner point of view, where investment decisions are basedupon consideration of various financial variables, we generalize functionally generatedportfolios and allow them to depend on continuous-path semimartingales, in additionto the market weights. By means of examples we demonstrate how the inclusionof additional processes can reduce time horizons beyond which relative arbitrage ispossible, boost performance of generated portfolios, and how investor preferences andspecific investment views can be included in the context of stochastic portfolio theory.Striking is also the construction of a relative arbitrage opportunity which is generatedby the volatility of the additional semimartingale. An in-depth empirical analysisof the performance of the proposed strategies confirms our theoretical findings anddemonstrates that our portfolios represent profitable investment opportunities even inthe presence of transaction costs.
A large part of the research in investment mathematics relies on normative assump-tions on the underlying assets in the financial market. This trend was initiated by HarryMarkowitz in his famous work [M52]. Even though normative assumptions lead to a richresearch landscape and can help in order to arrive at closed form solutions, one should tryto model observed properties instead of imposing unrealistic ones, such that the discrep-ancy between mathematical theory and financial market realizations is kept small. Thisis the path Robert Fernholz went by introducing stochastic portfolio theory (SPT) in hisseminal work [F02]. An essential role in SPT is played by functionally generated portfolios,which represent a very robust asset allocation tool as estimation of drifts and volatilitiesis absent in order to construct portfolios with controllable characteristics.The setting of stochastic portfolio theory as introduced by Robert Fernholz only usesthe market weights as inputs to the generating function. As a consequence, functionallygenerated portfolios are deterministic functions of the market weights. However, investmentdecisions are generally made by taking into consideration also other stock characteristicsbeside the size of a company in the market. Hence, it is desirable to have a theory in whichgenerating functions can take as inputs additional information which could potentially take ∗ Acknowledgement: The author greatly acknowledges Prof. Dr. Josef Teichmann from ETH Zurich andProf. Dr. Martin Larsson from Carnegie Mellon University for many constructive talks, fruitful discussionsand helpful suggestions. † Swiss Federal Institute of Technology (ETH Zurich), [email protected] a r X i v : . [ q -f i n . M F ] J a n dvantage of market inefficiencies. A first step into this direction was made by Strong in[S13]. In this manuscript, the author proves a master equation for generating functionswhich in addition to the market weights depend on a process of finite variation. Additionalrelated research was done by Schied, Speiser and Voloshchenko in [SSV18]. In the namedpaper, the authors prove similar results to those in [S13], but in a model-free setting, bymeans of pathwise Ito calculus. Moreover, Ruf and Xie extend in their paper [RX19] theframework of “additive functional generation” introduced by Karatzas and Ruf in [KR17],and allow portfolio generating functions do depend on a supplementary process of finitevariation. Let us also mention the recent work [KK20] by Karatzas and Kim, in whichthe authors further expand concepts similar to those from [RX19], in a probability-freecontext.The value of this work is twofold. We first generalize the master equation for gener-ating functions which take continuous-path semimartingale arguments beyond the marketweights. Second, we explicitly demonstrate how the inclusion of additional processes intogenerating functions can be beneficial. We come up with an example which demonstratesthat this generalization can be advantageous for reduction of times beyond which relativearbitrage is possible. Furthermore, we show that it can increase the performance of gener-ated portfolios and how one is able to implement quantitative investment preferences in theframework of stochastic portfolio theory. We also present a relative arbitrage opportunityversus the market which is generated by the volatility of the additional stock characteristicssemimartingale. To the best of our knowledge these are the first such examples across SPTliterature.This paper is organized as follows. In Section 2 we summarize the main notions fromstochastic portfolio theory. Specifically, we discuss stocks and portfolios in continuous time,the excess growth rate of a portfolio, the market portfolio, relative arbitrage and functionalportfolio generation. In Section 3 we extend the notion of functionally generated portfoliosto include continuous semimartingales as inputs, in addition to the market weights. Weargue how such a generalization can be advantageous from an investor point of view. This isfollowed by propositions which help establish the construction of relative arbitrages in thissetting. Moreover, four new examples of relative arbitrages versus the market are presentedin order to illustrate our points. In Section 4 we perform a detailed empirical analysis ofthe proposed strategies, where we show their ability to outperform the market and SPTalternatives, not only pathwise, but also in terms of a number of performance measures.Afterwards we perform regression analysis on the three Fama French risk factors in orderto understand the sources driving the returns of our portfolios. We finish this paper inSection 5, where we present a discussion of our main findings and listen the current openresearch problems in the field of SPT. We also suggest several new questions which wereraised during the development of this work. In this section we will introduce stochastic portfolio theory. We first set the stage andgive the basic definitions for stocks and portfolios in continuous time. We also discuss theexcess growth rate of a portfolio, which is a central quantity in SPT. Afterwards we devotea subsection to the market portfolio and its main properties. We end this section with abrief summary on functional portfolio generation.2 .1 Stocks and Portfolios in Continuous Time
Let (Ω , F , F , P ) be a probability space with F = ( F t ) t ≥ a continuous time filtrationwhich we assume to be right-continuous and P -complete. Throughout this work, we shallwork with this probability space. Furthermore, let n ≥ B = ( B t , ..., B nt ) t ≥ we denote a standard n -dimensional Brownian motion defined on our probability space. Hence we have (cid:10) B i , B j (cid:11) t = δ ij t , i, j = 1 , ..., n , for all t ≥
0, a.s., where δ ij = 1 if i = j and δ ij = 0 otherwise.By (cid:104) X, Y (cid:105) = ( (cid:104)
X, Y (cid:105) t ) t ≥ we represent the quadratic covariation process between thestochastic processes X = ( X t ) t ≥ and Y = ( Y t ) t ≥ . Compactly, we shall denote (cid:104) X (cid:105) = (cid:104) X, X (cid:105) . Let us write also F B for the filtration generated by B . Note that we do not require F = F B , but assume only that F contains F B .We make the assumption that shares of companies are infinitely divisible. Because ofthis, there is no loss in generality by assuming that each firm has a single share outstand-ing. Thus, in this setting the stock price is equal to the total market capitalization of acompany for all times. A financial market is a family M = { X , ..., X n } of stocks, where X i = ( X it ) t ≥ denotes the price process of stock i ∈ { , ..., n } , and we postulate that itsatisfies the stochastic differential equation (SDE) d log( X it ) := γ it dt + n (cid:88) j =1 ξ ijt dB jt , t ≥ , i = 1 , ..., n, (2.1.1)where the growth rate γ i and volatilities ξ i , ..., ξ in are F -progressively measurable andsatisfy (cid:90) T | γ it | dt + n (cid:88) j =1 (cid:90) T ( ξ ijt ) dt < ∞ , T ≥ , a.s. (2.1.2)for each i = 1 , ..., n .Let us fix an n ≥ M = { X , ..., X n } . We work with thismarket throughout this paper, unless stated otherwise. Moreover, we define the R n × n -valued covariance process σ = ( σ ijt ) ≤ i,j ≤ nt ≥ componentwise as σ ijt := ddt (cid:10) log( X i ) , log( X j ) (cid:11) t = n (cid:88) k =1 ξ ikt ξ jkt , t ≥ , i, j = 1 , ..., n. (2.1.3)At this point we want to outline the importance of the covariance process σ which describesthe volatility structure of a financial market. Appropriate assumptions on the latter leadto existence of portfolios which “outperform the market” almost surely after a finite timehorizon. The two standard assumptions on the covariance process σ under which theexistence of relative arbitrage opportunities can be proven are strict nondegeneracy andsufficient intrinsic volatility.We say that M is strictly nondegenerate if there exists an ε > x · σ t x ≥ εx · x ,for all x ∈ R n and t ≥
0, almost surely, where we use the notation x · y = x y + ... + x n y n , x = (1 , ..., x n ), y = ( y , ..., y n ) ∈ R n . Strict nondegeneracy is a fairly strong conditionwhich states that the smallest eigenvalue of the random matrix σ should be bounded awayfrom 0 for all times with probability one. This condition can not be expected to hold inrealistic financial markets. 3he sufficient intrinsic volatility is a much weaker condition. However, we shall in-troduce it when we discuss the market portfolio. Next, we introduce the definitions of a portfolio and the portfolio value process. A portfolio π = ( π t , ..., π nt ) t ≥ in the financial market M is a bounded, F -progressively measurable process which fulfils n (cid:88) i =1 π it = 1 , t ≥ , a.s. (2.1.4)The components of π are called portfolio weights . We say that π is long in stock i at time t if π it > π it < π is short in stock i at time t . We refer to aportfolio which is long in all stocks for all times as a long-only portfolio . Let π be a portfolio in M . The portfolio value process Z π = ( Z πt ) t ≥ isthe process which satisfies following stochastic differential equation dZ πt Z πt := n (cid:88) i =1 π it dX it X it , t > , Z π = Z , (2.1.5)where Z > π .We also define the portfolio variance process σ ππ = ( σ ππt ) t ≥ as σ ππt := n (cid:88) i,j =1 π it σ ijt π jt , t ≥ . (2.1.6)The next proposition gives the logarithmic representation of the value process of a portfolio. Let π be a portfolio in M . Then the portfolio value process Z π satisfies d log( Z πt ) = n (cid:88) i =1 π it d log( X it ) + γ π, ∗ t dt, t ≥ , a.s. , (2.1.7)where γ π, ∗ = (cid:0) γ π, ∗ t (cid:1) t ≥ is the excess growth rate of a portfolio , given by γ π, ∗ t := 12 (cid:32) n (cid:88) i =1 π it σ iit − σ ππt (cid:33) , t ≥ . (2.1.8) Proof.
See proof of Proposition 1.1.5 and Corollary 1.1.6 in [F02]. (cid:3)
From Proposition 2.1.4 we infer that the excess growth rate of a portfolio is a centralquantity as it influences directly the portfolio value process. It will appear again when wecharacterize the performance of a portfolio versus the market portfolio (see Proposition2.2.3). In the following we aim to show that the excess growth rate can be expressed con-veniently by means of the relative covariance process . Let π be a portfolio in M . We define the R n × n -valued relative co-variance process τ π = ( τ π,ijt ) ≤ i,j ≤ nt ≥ componentwise as τ π,ijt := σ ijt − σ iπt − σ jπt + σ ππt , t ≥ , i, j = 1 , ..., n, (2.1.9)4here the process σ iπt is defined by σ iπt := n (cid:88) j =1 π jt σ ijt , t ≥ , i = 1 , ..., n. (2.1.10) Let π be a portfolio in M . Then τ πt is positive semidefinite for all t ≥
0, a.s. Moreover, the kernel of τ πt is spanned by π t . Proof.
See proof of Lemma 1.2.2 in [F02]. (cid:3)
The value of the relative covariance process lies in the fact that it can be used veryconveniently to describe the properties of the excess growth rate. This is illustrated by thefollowing proposition, respectively corollary.
Let π and ζ be any two port-folios in M . Then the excess growth rate fulfils γ π, ∗ t = 12 n (cid:88) i =1 π it τ ζ,iit − n (cid:88) i,j =1 π it τ ζ,ijt π jt , (2.1.11)for all t ≥
0, almost surely.
Proof.
See proof of Lemma 1.3.4 in [F02]. (cid:3)
The numeraire invariance property can now be used to express the excess growth rateof a portfolio in a very compact form.
Let π be a portfolio in M . Then for all t ≥
0, it holds a.s. that γ π, ∗ t = 12 n (cid:88) i =1 π it τ π,iit . (2.1.12) Proof.
See proof of Corollary 1.3.6 in [F02]. (cid:3)
This subsection is devoted to discuss the single most important portfolio in SPT, namelythe market portfolio. We provide the definition of the market portfolio, relative arbitrageand discuss how performance of a portfolio with respect to the market can be measured.Moreover, we state the definition of market diversity and the sufficient intrinsic volatility.
For the financial market M , the market portfolio µ = ( µ t , ..., µ nt ) t ≥ isdefined by µ it := X it X t + ...X nt , t ≥ , i = 1 , ..., n. (2.2.1)The components of the market portfolio µ , ..., µ n are called the market weights .It is clear that the market portfolio satisfies Definition 2.1.2. Furthermore, the marketweights are determined by the ratio between the company’s market capitalization and the5arket capitalization of all stocks in the market. Hence, the market weight of a stocktells us the proportion of the market that a firm constitutes in terms of capitalization.Throughout this work we shall assume that the market portfolio process is (cid:52) n + -valuedwith probability one, where (cid:52) n + := (cid:40) x ∈ (0 , n : n (cid:88) i =1 x i = 1 (cid:41) . (2.2.2)Hence, we exclude the possibility of a default of any company in M . It can be easilychecked that the value process of the market portfolio satisfies dZ µt = dX t + ... + dX nt , t ≥ , a.s. (2.2.3)All through this paper we shall use µ to denote the market portfolio and Z µ to denote itsvalue process. Also, we use the simpler notation τ for τ µ .The goal of SPT is to find portfolios which outperform the market portfolio over a cer-tain time horizon with probability one. This is captured in the notion of relative arbitrage. Let π and ζ portfolios in M and T ∈ (0 , ∞ ). We say that π is arelative arbitrage opportunity versus ζ over [0 , T ] if P ( Z πT ≥ Z ζT ) = 1 and P ( Z πT > Z ζT ) > Z π = Z ζ . Weshall refer to π as a strong relative arbitrage opportunity versus ζ if P ( Z πT > Z ζT ) = 1 holdstrue.If π is a relative arbitrage versus ζ over [0 , T ], then log( Z πT /Z ζT ) ≥
0, almost surely.Inspired by this, let us define the relative return process of a portfolio π versus ζ aslog( Z πT /Z ζT ), for T ≥
0. The relative return process is a very convenient way of describingthe performance of one portfolio versus another. The next proposition gives insight intothe dynamics of the relative return process in the case ζ = µ . Let π be any portfolio in M . Then the following holds a.s., for t ≥ d log( Z πt /Z µt ) = n (cid:88) i =1 π it d log( µ it ) + γ π, ∗ t dt. (2.2.5) Proof.
See proof of Proposition 1.2.5 in [F02]. (cid:3)
Next, we introduce the notion of market diversity. Informally, a stock market is diverseif no single company is allowed to dominate the entire market in terms of market capital-ization. This means that all the market weights are bounded away from 1 for all times.Market diversity is a very weak condition which is empirically observed and holds in anyreasonable financial market model.
We call the financial market M diverse if there exists a δ ∈ (0 , i = 1 , ..., n µ it ≤ − δ, t ≥ , a.s. (2.2.6)6ven though market diversity seems like an innocent assumption on a financial market,it has strong implications and consequences. There are examples in which it helps toestablish relative arbitrage or even yield certain performance benefits (see [FKK05] andExample 3.2.5).Now, we present the definition of sufficient intrinsic volatility, which is an assumptionon the volatility structure of the market and leads to the existence of relative arbitrageopportunities. We say that the financial market M is weakly sufficiently volatile ifthere exists a strictly increasing continuous function Υ : [0 , ∞ ) → [0 , ∞ ) with Υ(0) = 0and Υ( ∞ ) = ∞ such that ∞ > (cid:90) T n (cid:88) i =1 µ it τ iit dt = (cid:90) T γ µ, ∗ t dt ≥ Υ( T ) , (2.2.7)for all T ≥
0, a.s. Furthermore, we say that M is sufficiently volatile if there exists an ε >
0, such that for the market excess growth rate the following holds for all t ≥ γ µ, ∗ t ≥ ε. (2.2.8)Sufficient intrinsic volatility is a much weaker condition than strict nondegeneracy andis argued to hold in real financial markets (see [FK05]). We shall see a couple of instanceswhere sufficient intrinsic volatility leads to the existence of relative arbitrage opportunities(see Example 2.3.3, Example 3.2.5, Example 3.2.6 and Example 3.3.1). In this subsection we present functional generation of portfolios and an example whichshows how it can be used for construction of relative arbitrage opportunities. First, weintroduce convenient notation. Afterwards, we state the definition of a functionally gener-ated portfolio and present the key theorem of SPT, which gives the desired master equation.Let X ⊂ R (cid:96) , Y ⊂ R be open. By C ( X, Y ), X ⊂ R (cid:96) , Y ⊂ R we denote the space offunctions f : X → Y for which ( ∂ ) β ... ( ∂ (cid:96) ) β (cid:96) f ( x ) (2.3.1)is continuous for all β + ... + β (cid:96) ≤ x ∈ X . Here, ∂ i denotes the partial derivativewith respect to the i -th variable. Moreover, we shall use ∂ ij and ∂ i,j interchangeably todenote the second partial derivative with respect to the i -th and j -th variable. We use also R ++ to refer to the set of strictly positive real numbers. Furthermore, we introduce the set of generating functions G n as the set of all functions f for which there exists an openneighbourhood U of (cid:52) n + , such that f ∈ C ( U, R ++ ) and x i ∂ i log( f ( x )) is bounded for all x ∈ (cid:52) n + and i = 1 , ..., n . Let S : (cid:52) n + → R ++ be a continuous function and π a portfolio in M . We say that π is functionally generated by S if there exists an F -adapted, continuous-path, finite variation process Θ = (Θ t ) t ≥ such thatlog( Z πt /Z µt ) = log( S ( µ t ) /S ( µ )) + Θ t , (2.3.2)7or all t ≥
0, a.s. We refer to the portfolio π as generating portfolio and the process Θ asthe drift process corresponding to the generating function S . Let S ∈ G n . Then S generates the portfolio π with weights givenby π it = µ it ( ∂ i log( S ( µ t )) + 1 − n (cid:88) j =1 µ jt ∂ j log( S ( µ t ))) , (2.3.3)for t ≥ i = 1 , ..., n . The drift process satisfies d Θ t = − S ( µ t ) n (cid:88) i,j =1 ∂ ij S ( µ t ) µ it µ jt τ ijt dt, (2.3.4)for t ≥
0, almost surely.We call expression (2.3.2) with the drift process given as in Theorem 2.3.2 the masterequation . We shall prove the latter theorem in the context of a more general theory ofportfolio generating functions (see the proof of Theorem 3.1.2 in the appendix). In thesubsequent instance we replicate an example from [FK05] in order to show how Theorem2.3.2 can be used for construction of relative arbitrage opportunities.
This examplewas originally proposed in [FK05]. Let us assume that M is weakly sufficiently volatile.We start by defining the generalized entropy function S c ( x ) := c − n (cid:88) i =1 x i log( x i ) = c + S ( x ) , x = ( x , ..., x n ) ∈ (cid:52) n + , (2.3.5)for c >
0, and S ( x ) = − (cid:80) ni =1 x i log( x i ) is the standard entropy function . Since S c ∈ G n ,we can take advantage of Theorem 2.3.2. The function S c generates the portfolio π it = cµ it − µ it log( µ it ) c − (cid:80) nj =1 µ jt log( µ jt ) , t ≥ , i = 1 , ..., n, (2.3.6)to which we refer as the entropy weighted portfolio . One can also check that the driftprocess is given by d Θ t = 12 n (cid:88) i =1 τ iit µ it S c ( µ t ) dt, t ≥ , a.s. (2.3.7)The claim is that the portfolio (2.3.6) does the job, i.e. it is a relative arbitrage opportunityversus the market portfolio. In order to show this, we will need c ≤ S c ( µ t ) ≤ c + log( n ) , t ≥ , a.s. , (2.3.8)which follows from 0 ≤ S ( x ) ≤ log( n ), x ∈ (cid:52) n + , and P ( µ t ∈ (cid:52) n + ) = 1, t ≥
0. The masterequation yields log( Z πT /Z µT ) = log( S c ( µ T ) /S c ( µ )) + (cid:90) T γ µ, ∗ t S c ( µ t ) dt ≥ log (cid:18) cc + S ( µ ) (cid:19) + Υ( T ) c + log( n ) , a.s. , (2.3.9)8or T ≥
0, where we have used the bounds on S c and the assumption that the market isweakly sufficiently volatile. The right hand side of (2.3.9) is strictly positive if T > T ∗ ( c ) := Υ − (cid:18) ( c + log( n )) log (cid:18) S ( µ ) c (cid:19)(cid:19) , (2.3.10)which is well defined since Υ is strictly increasing and continuous, and hence invertible.Thus, the entropy weighted portfolio (2.3.6) is a strong relative arbitrage opportunityversus market over all time horizons [0 , T ] with T > T ∗ ( c ). Moreover, from T ∗ := lim c →∞ T ∗ ( c ) = Υ − ( S ( µ )) , (2.3.11)we can conclude that for each T > T ∗ we can choose a c sufficiently large, such that π outperforms the market over [0 , T ]. (cid:3) In this section we present the main results of this paper. We first introduce generalizedgenerating functions and state the generalized master equation. Afterwards we prove acouple of propositions which indicate what functions are appropriate in order to obtainbenefits in this generalized framework. We then apply these results in order to constructfour novel trading strategies which outperform the market almost surely.
We first adapt the definition of generating portfolios such that the corresponding gener-ating functions take as inputs continuous semimartingales, beyond the market weights. Weimpose that the additional semimartingale is F -progressively measurable, has continuouspaths almost surely and is not equal to a deterministic function of the market weights. Ifthis is the case, this additional process is called the stock characteristics process. Through-out this paper we denote by K a subset of R k , for some k ∈ N , and we make additionalspecifications on K only when needed. Let P be a stock characteristics process which is valued in K , let S : (cid:52) n + × K → R ++ be a continuous function and π a portfolio in M . We say that S gen-erates the portfolio π with stock characteristics P if there exists an F -adapted, continuous-path, finite variation process Θ, such that the following holdslog( Z πt /Z µt ) = log( S ( µ t , P t ) /S ( µ , P )) − k (cid:88) i =1 (cid:90) t ∂ n + i log( S ( µ s , P s )) dP is + Θ t , (3.1.1)for all t ≥
0, almost surely. We shall refer to such a portfolio as a generalized generatingportfolio and the process Θ = (Θ t ) t ≥ is called the drift process corresponding to the gen-erating function S .Comparing Definition 3.1.1 with Definition 2.3.1 it is evident that the generalized ver-sion has an additional integral. If the stock characteristics process P is a finite variationprocess, then the integral is a Lebesgue-Stieltjes integral which is also of finite variation.In this case we define the extended drift process (cid:101) Θ by (cid:101) Θ t := Θ t − k (cid:88) i =1 (cid:90) t ∂ n + i log( S ( µ s , P s )) dP is , (3.1.2)9or t ≥
0. Hence, the formal structure of Definition 2.3.1 remains for finite variation stockcharacteristics processes. If P is a semimartingale with a local martingale part which hasinfinite first variation, then the integral is a stochastic integral.We further extend the set of generating functions introduced in the previous section. Letthe stock characteristics process be K -valued. By G Kn we denote the space of all functions f for which there exists an open neighbourhood U of (cid:52) n + × K such that f ∈ C ( U, R ++ ),and for i = 1 , ..., n , the quantity x i ∂ i log( f ( x, y )) is bounded for all ( x, y ) ∈ (cid:52) n + × K . Assume that P is a stock characteristics process which is valued in K and let S ∈ G Kn . Then S generates the portfolio π with weights given by π it = µ it ( ∂ i log( S ( µ t , P t )) + 1 − n (cid:88) j =1 µ jt ∂ j log( S ( µ t , P t ))) , (3.1.3)for t ≥ i = 1 , ..., n . The drift process satisfies d Θ t = − S ( µ t , P t ) n (cid:88) i,j =1 ∂ ij S ( µ t , P t ) µ it µ jt τ ijt dt − k (cid:88) i,j =1 ∂ n + i,n + j log( S ( µ t , P t )) d (cid:10) P i , P j (cid:11) t − n (cid:88) i =1 k (cid:88) j =1 ∂ i,n + j log( S ( µ t , P t )) d (cid:10) µ i , P j (cid:11) t , (3.1.4)for all t ≥
0, almost surely.We provide a proof of Theorem 3.1.2 in the appendix. Expression (3.1.1) along withthe drift process given by Theorem 3.1.2 is referred to as the generalized master equation . Remarks.
From the generalized master equation one is able to obtain many advantagescompared to the classical one. We already have two benefits: • For a fixed time horizon [0 , T ], for some
T >
0, an appropriately chosen generatingfunction S and stock characteristics process P can increase the outperfomance relativeto the market via the additional terms in the drift process. • For a fixed goal of outperformance, i.e. log( Z πT /Z µT ) ≥ c , for some c >
0, by anappropriate choice of S and P one can reduce the time T by which this goal isachieved.The generalized master equation is an ideal framework for comparing the performanceof functionally generated portfolios to the market portfolio. However, comparison to otherbenchmark portfolios is possible as well. Usually these benchmark portfolios are very sim-ple and do not depend on additional stock characteristics beside the market weights. Thenext proposition gives insight how this can be done. Let the stock characteristics process P be valued in K and assumethat S ζ ∈ G n and S π ∈ G Kn . Let ζ, π be the portfolios generated by S ζ , S π respectively.Then the following holds true for t ≥
0, almost surely d log( Z πt /Z ζt ) = d log( S π ( µ t , P t ) /S ζ ( µ t )) − k (cid:88) i =1 ∂ n + i log( S π ( µ t , P t )) dP it + d Θ πt − d Θ ζt , (3.1.5)10here Θ x denotes the drift process of portfolio x ∈ { π, ζ } . Proof.
Equation (3.1.5) follows from d log( Z πT /Z ζT ) = d log( Z πT /Z µT ) − d log( Z ζT /Z µT ) andDefinition 3.1.1. (cid:3) We use here thenotation from Proposition 3.1.3. The equally weighted portfolio (EWP) is a prominentbenchmark in the financial industry. It is defined as ζ it := 1 n , (3.1.6)for i = 1 , ..., n and all t ≥
0. The equally weighted portfolio is functionally generated by S ( x ) = ( x · ... · x n ) /n , x = ( x , ..., x n ) ∈ (cid:52) n + . (3.1.7)Furthermore, the drift process corresponding to S satisfies for t ≥
0, a.s. d Θ ζt = 12 n n (cid:88) i =1 τ iit − n n (cid:88) i,j =1 τ ijt dt = γ ζ, ∗ t dt, (3.1.8)where we have used the numeraire invariance property in the last equality. Since the EWPis a long-only portfolio, we have γ ζ, ∗ t > t ≥ T >
0, that Θ T = (cid:82) T γ ζ, ∗ t dt >
0, a.s., which explains thesurprisingly good performance of the EWP.Let now the stock characteristics process be valued in K , and choose an S π ∈ G Kn .Proposition 3.1.3 states that for the value process of the portfolio π generated by S π andthe portfolio ζ the following holds truelog( Z πT /Z ζT ) = log (cid:32) S πT ( µ · ... · µ n ) /n S π ( µ T · ... · µ nT ) /n (cid:33) − k (cid:88) i =1 (cid:90) T ∂ n + i log( S ( µ t , P t )) dP it + Θ πT − (cid:90) T γ ζ, ∗ t dt, (3.1.9)for all T ≥
0, almost surely. Note the strictly negative contribution of the last term in(3.1.9). It is for this reason, why it is a difficult task to find portfolios in practice whichbeat the EWP. Note also that (3.1.9) is well defined due to the standing assumption thatno company in M defaults. (cid:3) In this subsection we want to give examples and applications of the generalized mas-ter equation. We are mainly interested to motivate why the inclusion of additional stockcharacteristics processes into generating functions can be beneficial. Before doing so, weprove two lemmas which give insight into which functions are appropriate for our discussion.
Let S : U (cid:52) n + × U K → R ++ be a continuous function, where U (cid:52) n + is an open neighbourhood of (cid:52) n + and U K is an open neighbourhood of K . We say that S is multiplicative if there exist continuous functions f : U (cid:52) n + → R ++ and g : U K → R ++ such that S ( x, y ) = f ( x ) g ( y ) for all ( x, y ) ∈ U (cid:52) n + × U K .11 .2.2 Lemma. Assume that the stock characteristics process P is K -valued almost surely.Let S be a multiplicative function with S ∈ G Kn . Then neither the portfolio weights π gen-erated by S , nor the return of π relative to the market depend on P . Proof.
Let the stock characteristics process P be K -valued almost surely and let S bemultiplicative and S ∈ G Kn . By definition there exist functions f ∈ C ( U (cid:52) n + , R ++ ) and g ∈ C ( U K , R ++ ) with U (cid:52) n + ⊃ (cid:52) n + and U K ⊃ K open such that S ( x, y ) = f ( x ) g ( y ), for all( x, y ) ∈ (cid:52) n + × K . Using this it is easy to see that the identities ∂ i log( S ( µ t , P t )) = ∂ i log( f ( µ t )) , i = 1 , ..., n, (3.2.1) ∂ ij S ( µ t , P t ) S ( µ t ) = ∂ ij f ( µ t , P t ) f ( µ t ) , i, j = 1 , ..., n, (3.2.2) ∂ n + i log( S ( µ t , P t )) = ∂ i log( g ( P t )) , i = 1 , ..., k, (3.2.3) ∂ i,n + j log( S ( µ t , P t )) = 0 , i = 1 , ..., n, j = 1 , ..., k, (3.2.4)hold true for t ≥
0, a.s. Since S is assumed to be a member of G Kn , we can apply Theorem3.1.2. Using identity (3.2.1) we can deduce that the portfolio weights generated by S aregiven by π it = µ it ( ∂ i log( f ( µ t )) + 1 − n (cid:88) j =1 µ jt log( f ( µ t ))) , t ≥ , (3.2.5)for i = 1 , ..., n , which is independent of P . The generalized master equation reads d log( Z πt /Z µt ) = d log( f ( µ t )) − f ( µ t ) n (cid:88) i,j =1 ∂ ij f ( µ t ) τ ijt µ it µ jt dt + d log( g ( P t )) − k (cid:88) i =1 ∂ i log( g ( P t )) dP it − k (cid:88) i,j =1 ∂ ij log( g ( P t )) d (cid:10) P i , P j (cid:11) t = d log( f ( µ t )) − f ( µ t ) n (cid:88) i,j =1 ∂ ij f ( µ t ) τ ijt µ it µ jt dt, t ≥ , a.s. , (3.2.6)where the identities (3.2.2), (3.2.3) and (3.2.4) helped us in the first equality. In order toarrive at the second equality, we have taken advantage of Ito’s lemma applied on log( g ( P t )).This gives the desired claim. (cid:3) Let the stock characteristics process P be valued in K and S ∈ G Kn .Choose any F ∈ C ( U K , R ++ ), where U K is an open neighbourhood of K . Let π and ζ denote the portfolios generated by S and S · F , respectively. It holds true that π it = ζ it , t ≥ , i = 1 , ..., n, , a.s. , (3.2.7)and hence also Z πt = Z ζt , t ≥ , a.s. Remarks.
Once we have proven (3.2.5), namely that for multiplicative generating func-tions the portfolio weights do not depend on the stock characteristics process, an intuitivereasoning suggests that the relative returns versus the market can not depend on the ad-ditional characteristics either. This is then confirmed also rigorously by (3.2.6).12emma 3.2.2 and Corollary 3.2.3 play an important role in the discussion to comesince they restrict the type of functions for which we can obtain the benefits of includingadditional processes as arguments of generating functions. We conclude that only functionswhich are non-multiplicative are able to give additional performance benefits compared tothe classical SPT approach where portfolios are constructed solely based on market weights.In general, relative arbitrages versus the market can be generated by functions forwhich the drift process is increasing. Propostion 3.1.15 and Proposition 3.4.2 in [F02]state that the drift process is increasing for concave generating functions. Hence, thesefunctions are good candidates to generate portfolios which outpeform the market. How-ever, this statement is only valid in the absence of stock characteristics processes. The nextlemma generalizes Proposition 3.4.2 from [F02] in the case of generating functions whichadditionally take increasing or decreasing stock characteristics processes as inputs. For afunction S ∈ G Kn , we say that S is concave on (cid:52) n + if for every fixed y ∈ K we have that S ( λx + (1 − λ ) x , y ) ≥ λS ( x , y ) + (1 − λ ) S ( x , y ), for all λ ∈ (0 ,
1) and any x , x ∈ (cid:52) n + .Moreover, we say that S is increasing (decreasing) on K if for every fixed x ∈ (cid:52) n + it holdsthat ∂ n + i S ( x, y ) ≥ ∂ n + i S ( x, y ) ≤ i = 1 , ..., k and y ∈ K . Assume that the stock characteristics process P is either increasing ordecreasing and K -valued. Let S ∈ G Kn be a non-multiplicative function which is concaveon (cid:52) n + and has the opposite monotonicity of P on K . Then the extended drift process (cid:101) Θis increasing almost surely.
Proof.
Assume that P is an increasing K -valued stock characteristics process and let S ∈ G Kn be non-multiplicative, concave on (cid:52) n + and decreasing on K . Recall that for afinite variation stock characteristics process P we define the extended drift process by (cid:101) Θ T = − n (cid:88) i,j =1 (cid:90) T ∂ ij S ( µ t , P t ) S ( µ t , P t ) µ it µ jt τ ijt dt − k (cid:88) i =1 (cid:90) t ∂ n + i log( S ( µ t , P t )) dP it , T ≥ . (3.2.8)We will show that both integrals in (3.2.8) are increasing in T . Regarding the first in-tegral in (3.2.8), we follow the proof of Proposition 3.1.15 in [F02]. Let us denote by HS y ( x ) := ( ∂ ij S ( x, y )) ≤ i,j ≤ n the Hessian matrix of S with respect to the first n variablesof S evaluated at x ∈ (cid:52) n + for a fixed y ∈ K . Let now t ≥ HS P t ( µ t ) is diagonalizable with eigenvalues λ , ..., λ n and corresponding normalizedeigenvectors e , ..., e n . Hence we may write( HS y ( x )) ij = ∂ ij S ( µ t , P t ) = n (cid:88) (cid:96) =1 λ (cid:96) e (cid:96)i e (cid:96)j , (3.2.9)where e (cid:96)i denotes the i -th component of the eigenvector e (cid:96) , for i, (cid:96) = 1 , ..., n . Then itfollows that n (cid:88) i,j =1 ∂ ij S ( µ t , P t ) µ it µ jt τ ijt = n (cid:88) (cid:96) =1 λ (cid:96) n (cid:88) i,j =1 e (cid:96)i e (cid:96)j µ it µ jt τ ijt ≤ , (3.2.10)where the inequality follows from the fact that λ , ..., λ n ≤
0, since S is concave on (cid:52) n + ,and from Proposition 2.1.6 which states that the R n × n -valued relative covariance process τ is positive semidefinite for all times. Hence, since t ≥ − n (cid:88) i,j =1 (cid:90) T ∂ ij S ( µ t , P t ) S ( µ t , P t ) µ it µ jt τ ijt dt ≥ T , almost surely. It remains to prove that the secondintegral in (3.2.8) is increasing as well. Observe that for any t ≥ i = 1 , ..., k , ∂ n + i S ( µ t , P t ) ≤
0, almost surely, since S is assumed to be decreasing on K . This howevermeans that − k (cid:88) i =1 ∂ n + i log( S ( µ t , P t )) = k (cid:88) i =1 ∂ n + i S ( µ t , P t ) S ( µ t , P t ) ≥ . (3.2.12)Combining this with the fact that P is increasing and that t ≥ − k (cid:88) i =1 (cid:90) T ∂ n + i log( S ( µ t , P t )) dP it ≥ , (3.2.13)and is increasing as a function of T , almost surely. Hence, the extended drift process (cid:101) Θis positive and increasing as wished. The proof for the case of P being decreasing and S increasing on K is analogous. (cid:3) In the sequel we examine two examples to illustrate the benefits arising from the gen-eralized master equation.Recall Example 2.3.3, where it is shown that in a market which is weakly sufficientlyvolatile, relative arbitrage versus the market exists for sufficiently long time horizons. Inthe example below we show that the lower bound on the time horizon (2.3.10) can beshortened using the generalized master equation. We shall demonstrate how the inclusionof a single stock characteristics process can already generate a strictly positive contributionto the relative return versus the market. This in turn leads to the reduction of the minimaltime horizon beyond which relative arbitrage is possible.
Let us assume the market M is diverse and weakly sufficiently volatile. Recall from Definition 2.2.5 that the latter propo-sition holds if there exists a strictly increasing continuous function Υ : [0 , ∞ ) → [0 , ∞ ),such that for the market excess growth rate γ µ, ∗ the following holds almost surely (cid:90) T γ µ, ∗ t dt ≥ Υ( T ) , T ≥ . We consider time as our stock characteristics process, i.e. in terms of the notation ofDefinition 3.1.1 we have P t = t and K = [0 , ∞ ).Since the market is diverse, it follows from Proposition 2.3.2 in [F02] that there existsan ε > S ( µ t ) ≥ ε, t ≥ , a.s. , (3.2.14)where S ( µ t ) := − (cid:80) ni =1 µ it log( µ it ) is the market entropy at time t , for t ≥
0. Consider nowthe generating function (cid:101) S c defined by (cid:101) S c ( x, y ) := c − n (cid:88) i =1 x i log( x i ) − ε tanh( y ) = S c ( x ) − ε tanh( y ) , (3.2.15)for x = ( x , ..., x n ) ∈ (cid:52) n + and y ∈ [0 , ∞ ). Here, S c ( x ) = c + S ( x ) denotes the generalizedentropy function from Example 2.3.3, c > ε is the market entropylower bound from (3.2.14). It is evident that (cid:101) S c ∈ G [0 , ∞ ) n . By this, we are allowed to takeadvantage of Theorem 3.1.2. The function (cid:101) S c generates the portfolio π it = µ it ( c − ε tanh( t ) − log( µ it )) c − ε tanh( t ) − (cid:80) nj =1 µ jt log( µ jt ) , t ≥ , i = 1 , ..., n. (3.2.16)14he fact that our stock characteristics process is of finite variation significantly simplifiesthe drift term (3.1.4), as only the first sum remains. The drift process is determined by d Θ t = 12 n (cid:88) i =1 (cid:101) S c ( µ t , t ) τ iit µ it dt, t ≥ , a.s. (3.2.17)The following bound on (cid:101) S c will prove to be useful c ≤ (cid:101) S c ( µ t , t ) ≤ c + log( n ) , t ≥ , a.s. (3.2.18)This follows from − ε ≤ − ε tanh( y ) ≤ y ∈ [0 , ∞ ), and ε ≤ S ( x ) ≤ log( n ), x ∈ (cid:52) n + .Hence, the drift process satisfies a.s.Θ T ≥ c + log( n ) (cid:90) T γ µ, ∗ t dt ≥ c + log( n ) Υ( T ) , T ≥ , (3.2.19)where in the last inequality we have used the weak sufficient volatility assumption. Themaster equation readslog( Z πT /Z µT ) = log( (cid:101) S c ( µ T , T ) / (cid:101) S c ( µ , T + ε (cid:90) T − tanh ( t ) (cid:101) S c ( µ t , t ) dt, (3.2.20)for T ≥
0, a.s. In order to arrive at (3.2.20), the identity d (tanh( x )) /dx = 1 − tanh ( x ), x ∈ R , has helped us. The integral on the right hand side of (3.2.20) is a Lebesgue Stieltjesintegral and can be estimated thanks to (3.2.18) as follows (cid:90) T (1 − tanh ( t )) (cid:101) S c ( µ t , t ) dt ≥ tanh( T ) c + log( n ) , T ≥ , a.s. , (3.2.21)where we have used tanh(0) = 0. Using (3.2.18), (3.2.19) and (3.2.21) in the masterequation (3.2.20), we conclude that the following holds almost surely for the relative returnof the portfolio π versus the marketlog( Z πT /Z µT ) ≥ log (cid:18) cc + S ( µ ) (cid:19) + Υ( T ) c + log( n ) + εc + log( n ) tanh( T ) , (3.2.22)for all T ≥
0. Let us define the function (cid:101) Υ( T ) = Υ( T ) + ε tanh( T ) , T ≥ , (3.2.23)which is strictly increasing and continuous, and hence it possesses an inverse. From (3.2.22)it follows then that after a time T which satisfies T > (cid:101) T ( c ) := (cid:101) Υ − (cid:18) ( c + log( n )) log (cid:18) S ( µ ) c (cid:19)(cid:19) , (3.2.24)the portfolio (3.2.16) will outperform the market almost surely, for all c >
0. Similar tothe final argument in Example 2.3.3, we can deduce that for all
T > (cid:101) T := (cid:101) Υ − ( S ( µ )), wecan choose a c > π “beats” the market over [0 , T ].Under the assumption that the Υ from Example 2.3.3 and from this example coincide,let us compare the times (cid:101) T ( c ) , (cid:101) T with the times T ∗ ( c ) , T ∗ from Example 2.3.3. Recall that T ∗ ( c ) and T ∗ are given by T ∗ ( c ) = Υ − (cid:18) ( c + log( n )) log (cid:18) S ( µ ) c (cid:19)(cid:19) , ∗ = Υ − ( S ( µ )) . Since (cid:101) Υ( T ) > Υ( T ), it follows that (cid:101) Υ − ( T ) < Υ − ( T ), for all T >
0. Hence, we concludethat (cid:101) T ( c ) < T ∗ ( c ), for all c >
0, and especially (cid:101)
T < T ∗ . Therefore, considering stockcharacteristics processes and the generalized master equation can lead to reduction of timehorizons beyond which a certain goal of outperformance is desired. (cid:3) In the next instance we give an example of a generalized generating portfolio whichoutperforms the entropy weighted portfolio from Example 2.3.3.
Suppose that the market M is weakly sufficiently volatile and let ζ denote the entropy weighted portfolio and S c the generalized entropy function from Example 2.3.3. Moreover, Υ : [0 , ∞ ) → [0 , ∞ ) is thestrictly increasing continuous function which represents the lower bound on the cumulativeexcess growth rate of the market. For a fixed α ∈ (0 , / (cid:101) S c by (cid:101) S c ( x, y ) := c − n (cid:88) i =1 x i log( x i ) + αc (tanh( − y ) − , x = ( x , ..., x n ) ∈ (cid:52) n + , y ∈ [0 , ∞ ) . (3.2.25)The latter generates the portfolio π it = µ it ( c + αc (tanh( − t ) − − log( µ it )) c + αc (tanh( − t ) − − (cid:80) nj =1 µ jt log( µ jt ) , t ≥ , i = 1 , ..., n. (3.2.26)Moreover, the drift process reads d Θ t = 12 n (cid:88) i =1 (cid:101) S c ( µ t , t ) τ iit µ it dt, t ≥ , a.s. (3.2.27)Let us also comment that (cid:101) S c admits the following bounds c − αc ≤ (cid:101) S c ( µ t , t ) ≤ c + log( n ) − αc, t ≥ , a.s. (3.2.28)We will show that π outperforms ζ over a sufficiently long time horizon. From the definitionof S c and (cid:101) S c and the identity − tanh( − x ) = tanh( x ), x ∈ R , it follows that S c ( µ t ) − (cid:101) S c ( µ t , t ) = αc (1 + tanh( t )) ≥ αc , t ≥
0, a.s. Taking advantage of this together withProposition 3.1.3, we can conclude that the following holds for the relative return of π versus ζ log( Z πT /Z ζT ) = log (cid:32) (cid:101) S c ( µ T , T ) S c ( µ ) (cid:101) S c ( µ , S c ( µ T ) (cid:33) + 12 n (cid:88) i =1 (cid:90) T αc (1 + tanh( t )) (cid:101) S c ( µ t , t ) S c ( µ t ) τ iit µ it dt + αc (cid:90) T − tanh ( − t ) (cid:101) S c ( µ t , t ) dt, T ≥ , a.s. , (3.2.29)where we have also used the drift process (2.3.7) from Example 2.3.3. It also holds that (cid:101) S c ( µ T , T ) S c ( µ ) (cid:101) S c ( µ , S c ( µ T ) = S c ( µ T ) + αc (tanh( − T ) − S c ( µ T ) · S c ( µ ) S c ( µ ) − αc> S c ( µ T ) − αcS c ( µ T ) = 1 − αcS c ( µ T ) ≥ − α, T ≥ , a.s. , (3.2.30)16here we have used S c ( µ T ) ≥ c , T ≥ , a.s. , in order to arrive at the last inequality. Notealso that (cid:90) T − tanh ( − t ) (cid:101) S c ( µ t , t ) dt ≥ tanh( T ) c (1 − α ) + log( n ) , T ≥ , a.s. , (3.2.31)similar to (3.2.21). Using these observations in (3.2.29), together with the bounds on S c and (cid:101) S c and the weak sufficient intrinsic volatility assumption yields the followinglog( Z πT /Z ζT ) > log (1 − α ) + αc Υ( T )( c (1 − α ) + log( n ))( c + log( n ))+ αcc (1 − α ) + log( n ) tanh( T ) , T ≥ , a.s. (3.2.32)By defining Υ c ( T ) = Υ( T ) + ( c + log( n )) tanh( T ) , T ≥ , (3.2.33)for all c >
0, we deduce that on a time horizon [0 , T ], where T ≥ (cid:98) T c ( α ) := Υ − c (cid:18) ( c (1 − α ) + log( n ))( c + log( n )) αc log (cid:18) − α (cid:19)(cid:19) , (3.2.34)the portfolio π outperforms the entropy weighted portfolio (2.3.6) for all c >
0, almostsurely. Note that the inverse of Υ c ( T ) with respect to T exists as it is a strictly increasingcontinuous function. Furthermore, from (cid:98) T c = lim α ↓ (cid:98) T c ( α ) = Υ − c (cid:18) c + log( n )) c (cid:19) , c > , (3.2.35)we can infer that for a given c > T > (cid:98) T c , we can choose a small enough α >
0, suchthat the portfolio π outperforms ζ on [0 , T ]. (cid:3) Remarks.
Note that the previous example constructs a generalized generating portfo-lio which outperforms a strong relative arbitrage opportunity versus the market. Hence,we may conclude that considering stock characteristics processes in the context of the mas-ter equation can improve the performance of generating portfolios and that it can increasethe outperformance with respect to the market.Let us emphasize the contribution of the additional stock characteristics process in theprevious two examples. As it is evident from the master equations (3.2.20) and (3.2.29),they generate a strictly positive, strictly increasing contribution to the relative return.Moreover, they improve the drift process, since the generating functions have a compo-nent which is strictly decreasing with time, and this leads to an amplification in (3.2.17)and (3.2.27). However, this benefit comes with a trade-off, namely since t (cid:55)−→ (cid:101) S ( · , t ) isdecreasing, it leads also to a smaller lower bound on log( (cid:101) S ( µ T , T ) / (cid:101) S ( µ , .3 Preference Based Investing in the Framework of SPT In this subsection we want to give two examples in order to illustrate how quantitativepreferences can be included in making investment decisions in our framework in order tooutperform the market.In our first example we construct a portfolio based on market beta. Specifically, thetrading strategy invests a larger amount of cash into stocks which have a smaller absolutecorrelation to the market portfolio. This is an inspiration from [FP14], where a “bettingagainst beta” factor is constructed in order to explain equity returns.
The beta of a stockis a key ingredient of the Capital Asset Pricing Model in order to explain equity returns.It measures the correlation between the returns of a stock and the returns of the marketportfolio. In this paper, we shall define the beta process of a stock as β it := (cid:10) X i , Z µ (cid:11) t , t ≥ , (3.3.1)for i = 1 , ..., n . Applying Ito’s lemma on the process exp(log( X i )) and using definition(2.1.1), we obtain dX it = (cid:32) γ it + 12 n (cid:88) i =1 ( ξ ijt ) (cid:33) X it dt + X it n (cid:88) j =1 ξ ijt dB jt , t ≥ , a.s. , (3.3.2)for i = 1 , ..., n . This observation, together with Z µ = X + ... + X n gives dβ it = d (cid:10) X i , Z µ (cid:11) t = X it n (cid:88) j =1 n (cid:88) k =1 n (cid:88) (cid:96) =1 X kt ξ ijt ξ k(cid:96)t d (cid:68) B j , B (cid:96) (cid:69) t = X it n (cid:88) k =1 σ ikt X kt dt, t ≥ , a.s. , (3.3.3)for i = 1 , ..., n , where we have used the independence of Brownian motions and the defini-tion of the covariance process (2.1.3), in order to arrive at the last equality.We make the following two assumptions: i) The market M is sufficiently volatile, i.e. there exists an ε >
0, such that γ µ, ∗ t ≥ ε, t ≥ , a.s. (3.3.4) ii) The quantities s it := sign (cid:32) X it n (cid:88) k =1 σ ikt X kt (cid:33) , t ≥ , i = 1 , ..., n, (3.3.5)do not change over time, i.e. s it = s i , with s i ∈ { +1 , − } , for t ≥ i = 1 , ..., n ,almost surely. In (3.3.5), sign( x ) = +1, if x ≥
0, and − (cid:101) β i := s i β i , for i = 1 , ..., n , as our stock characteristicsprocess. We define the generating function S A,c,p by S A,c,p ( x, y ) := A + n (cid:88) i =1 ( x i ) p ( c + e − y i ) , x = ( x , ..., x n ) ∈ (cid:52) n + , y = ( y , ..., y n ) ∈ [0 , ∞ ) n , (3.3.6)18or all A ≥ c > p ∈ (0 , S A,c,p ∈ G [0 , ∞ ) n n . By taking advantage ofTheorem 3.1.2, we conclude that the portfolio generated by S A,c reads π it = µ it (cid:32) p ( µ it ) p − ( c + e − (cid:101) β it ) S A,c,p ( µ t , (cid:101) β t ) + 1 − pS A,c,p ( µ t , (cid:101) β t ) n (cid:88) i =1 ( µ it ) p ( c + e − (cid:101) β it ) (cid:33) = p ( µ it ) p ( c + e − (cid:101) β it ) + µ it ((1 − p ) S A,c,p ( µ t , (cid:101) β t ) + pA ) S A,c,p ( µ t , (cid:101) β t ) , t ≥ , i = 1 , ...n. (3.3.7)It is evident that π is a long-only portfolio for any A ≥ c > p ∈ (0 , beta weighted portfolio . The correspondingdrift process reads d Θ t = p (1 − p )2 S A,c,p ( µ t , (cid:101) β t ) n (cid:88) i =1 ( µ it ) p ( c + e − (cid:101) β it ) τ iit dt, t ≥ , a.s. (3.3.8)Let us also note that the generating function S A,c,p admits the following bounds A + c ≤ S A,c,p ( x, y ) ≤ A + (1 + c ) n − p , ( x, y ) ∈ (cid:52) n + × [0 , ∞ ) n . (3.3.9)The lower bound in (3.3.9) follows from (cid:80) ni =1 ( x i ) p ≥ x ∈ (cid:52) n + , p ∈ (0 , S A,c,p , we have used the fact that z (cid:55)−→ z p , z >
0, isconcave for p ∈ (0 , z p ≤ z p + pz p − ( z − z ) holds for all z > z = 1 /n gives for all x ∈ (cid:52) n + n (cid:88) i =1 ( x i ) p ≤ n (cid:88) i =1 (cid:18) n p + pn − p ( x i − /n ) (cid:19) = n − p . (3.3.10)Thanks to Theorem 3.1.2, the relative return of the beta weighted portfolio with respectto the market satisfieslog( Z πT /Z µT ) = log( S A,c,p ( µ T , (cid:101) β T ) /S A,c,p ( µ , (cid:101) β )) + n (cid:88) i =1 (cid:90) T p (1 − p )2 S A,c,p ( µ t , (cid:101) β t ) ( µ it ) p ( c + e − (cid:101) β it ) τ iit dt + n (cid:88) i =1 (cid:90) T ( µ it ) p e − (cid:101) β it S A,c,p ( µ t , (cid:101) β t ) d (cid:101) β it , T ≥ , a.s. (3.3.11)Note that the last sum in (3.3.11) is positive and increasing almost surely, and henceimproves the return of the beta weighted portfolio relative to the market. Using the upperand lower bound on S A,c,p , the fact that a p > a , for a, p ∈ (0 , Z πT /Z µT ) > log (cid:18) A + cA + (1 + c ) n − p (cid:19) + p (1 − p ) cA + (1 + c ) n − p (cid:90) T n (cid:88) i =1 µ it τ iit (cid:124) (cid:123)(cid:122) (cid:125) = γ µ, ∗ t dt ≥ log (cid:18) A + cA + (1 + c ) n − p (cid:19) + p (1 − p ) cA + (1 + c ) n − p εT, T ≥ , a.s. , (3.3.12)where we have made usage assumption ii) in order to arrive at the last inequality. Hence,on a time horizon [0 , T ], where T ≥ A + (1 + c ) n − p p (1 − p ) cε log (cid:18) A + (1 + c ) n − p A + c (cid:19) , (3.3.13)19he beta weighted portfolio outperforms the market almost surely. (cid:3) Remarks.
In the previous example we make the assumption that the market M is suffi-ciently volatile and that the beta process of each company is either increasing or decreas-ing. Let us remark that such market models indeed exist. Consider the volatility stabilizedmodel from [FK05], where the dynamics of stock prices are given by d log( X it ) := α µ it dt + 1 (cid:112) µ it dB it , t ≥ , i = 1 , ..., n, (3.3.14)for an α ≥
0. In terms of the notation used in (2.1.1), we have that γ i = α/ (2 µ i ) and ξ ij = δ ij / (cid:112) µ i , for i, j = 1 , ..., n . From this we can conclude that the covariance process σ fulfils σ ijt = δ ij /µ it , for all t ≥ i, j = 1 , ..., n . A simple calculation shows then thatthe excess growth rate of the market is given by γ µ, ∗ t = n − , t ≥ , a.s. (3.3.15)Hence, by Definition 2.3.5, the market model (3.3.14) is sufficiently volatile. Let us alsoremark that dβ it = X it n (cid:88) k =1 σ ikt X kt = ( X it ) µ it dt, t ≥ , i = 1 , ..., n, a.s. (3.3.16)From (3.3.16) we are able to conclude that s it = +1, t ≥
0, for all i = 1 , ..., n , almost surely.Thus, volatility stabilized models satisfy both assumptions made in the previous example.In the next example we construct a portfolio based on the performance metric returnon assets (ROA). This metric tells how good a company is investing its assets in order togenerate profit. Informally, we would expect companies which possess a smaller ROA tobe less profitable than firms with a larger ROA, which is an idea from quality investing .Inspired by this, we construct a functionally generated portfolio which invests a largeramount of cash into stocks with a smaller ROA, and show that it underperforms the mar-ket portfolio with probability one, after a sufficiently long time horizon. Then using thisobservation, we construct an long-only portfolio which outperforms the market. Let us introduce the F -adapted,continuous-path semimartingales R , ..., R n , which represent the ROA processes of the com-panies considered in M . Specifically, R it is the ROA of company i ∈ { , ..., n } at time t ≥ R , ..., R n , we stay descriptive in our approach. Weassume the following: i) The market weights are non-constant for all times and satisfy the non-failure condi-tion, i.e. there exists a δ ∈ (0 , /n ) such that µ it ≥ δ, t ≥ , i = 1 , ..., n, a.s. (3.3.17)Interpretation: There exists movement in the financial market and no company in M can go bankrupt. ii) There exists an ς > P (0 < R it < ς ) = 1, for i = 1 , ..., n and t ≥
0. Interpretation: the ROA processes are bounded from above and below by adeterministic constant and companies cannot have a negative or arbitrary high ratioof net income to total assets. 20 ii)
The quadratic covariation process between the market weights and the ROA processesvanishes for all companies, i.e. P ( (cid:10) µ i , R i (cid:11) t ≡
0) = 1, for i = 1 , ..., n and t ≥ iv) There exist F -adapted, continuous-path processes (cid:101) R , ..., (cid:101) R n such that d (cid:10) R i (cid:11) t = (cid:101) R it dt, t ≥ , i = 1 , ..., n, a.s. (3.3.18) n (cid:88) i =1 (cid:101) R it ≥ η > , t > , a.s. (3.3.19)Interpretation: the ROA processes are “sufficiently volatile”. v) There exist constants
A, ε ≥
0, with ε < δ e − ς η/
2, such that for all
T < ∞ , thefollowing holds for the market weights and the ROA processes P (cid:32) n (cid:88) i =1 (cid:90) T µ it e − R it dR it < A + εT (cid:33) = 1 . (3.3.20)Interpretation: the market M is “ROA diverse”, in the sense that ROA processes ofdifferent companies are not comonotonic, but rather some move upwards and otherdownwards at same times.We consider the ROA processes of the companies in M as our stock characteristics process.Let us define the generating function S by S ( x, y ) := exp (cid:32) n (cid:88) i =1 x i e − y i (cid:33) , x = ( x , ..., x n ) ∈ (cid:52) n + , y = ( y , ..., y n ) ∈ (0 , ς ) n . (3.3.21)It is easy to verify that S ∈ G (0 ,ς ) n n and that it generates the portfolio π it = µ it e − R it + 1 − n (cid:88) j =1 µ jt e − R jt , t ≥ , i = 1 , ..., n. (3.3.22)The trading strategy (3.3.22) invests a larger amount of wealth in stocks with a smallerROA, and a smaller amount of wealth into stocks with a larger ROA. Hence, we wouldexpect this portfolio to underperform the market on the long run. Note that this portfoliois long-only. Indeed, sinceexp( − ς ) ≤ exp( − R it ) ≤ , i = 1 , ..., n, t ≥ , a.s. , (3.3.23)we have for i = 1 , ..., nπ it ≥ µ it e − R it + 1 − n (cid:88) j =1 µ jt = µ it e − R it > , t ≥ , a.s. (3.3.24)Next, let us look at the drift process generated by S . Recall that it is determined by d Θ t = − S ( µ t , R t ) n (cid:88) i,j =1 ∂ ij S ( µ t , R t ) µ it µ jt τ ijt dt − k (cid:88) i,j =1 ∂ n + i,n + j log( S ( µ t , R t )) d (cid:10) R i , R j (cid:11) t − n (cid:88) i =1 k (cid:88) j =1 ∂ i,n + j log( S ( µ t , R t )) d (cid:10) µ i , R j (cid:11) t , t ≥ , a.s.21hanks to the expression for the generating function and assumption iii ) on the ROAprocess, the expression for the drift process readsΘ T = − n (cid:88) i,j =1 (cid:90) T e − R it e − R jt µ it µ jt τ ijt dt − (cid:90) T n (cid:88) i =1 µ it e − R it d (cid:10) R i (cid:11) t , t ≥ , a.s. (3.3.25)Defining the R n -valued process z = ( z t , ..., z nt ) t ≥ componentwise as z it = e − R it µ it , t ≥ i = 1 , ..., n , yields then for all T ≥
0, almost surelyΘ T = − (cid:90) T z t · τ t z t dt − (cid:90) T n (cid:88) i =1 µ it e − R it d (cid:10) R i (cid:11) t ≤ − (cid:90) T n (cid:88) i =1 µ it e − R it d (cid:10) R i (cid:11) t , (3.3.26)where we have used Proposition 2.1.6, namely that the matrix-valued relative covarianceprocess τ t is positive semidefinite for all t ≥
0. Furthermore, using assumption i ) and theestimate (3.3.23) in the first inequality, and assumption iv ) in the second inequality we getthe following upper bound on the drift process generated by S − n (cid:88) i =1 (cid:90) T µ it e − R it d (cid:10) R i (cid:11) t ≤ − δ e − ς (cid:90) T n (cid:88) i =1 d (cid:10) R i (cid:11) t ≤ − δ e − ς (cid:90) T ηdt = − δ e − ς ηT, T ≥ , a.s. (3.3.27)Now, we want to show that the portfolio (3.3.22) underperforms the market after a suffi-ciently long time horizon. For this purpose we look at the relative return of π with respectto µ , which according to Theorem 3.1.2 readslog( Z πT /Z µT ) = log( S ( µ T , R T ) /S ( µ , R )) − n (cid:88) i =1 (cid:90) T ∂ n + i log( S ( µ t , P t )) dR t + Θ T = n (cid:88) i =1 ( µ iT e − R iT − µ i e − R i ) + n (cid:88) i =1 (cid:90) T µ it e − R it dR it + Θ T , T ≥ , a.s. (3.3.28)Note that (3.3.23) and the fact that P ( µ t ∈ (cid:52) n + , t ≥
0) = 1 can be used to conclude thatan upper bound for the first sum in (3.3.28) is given by 1 − e − ς . With the help of thisremark, assumption v ) and (3.3.26) and (3.3.27) we findlog( Z πT /Z µT ) ≤ − e − ς + A + εT − δ e − ς ηT, T ≥ , a.s. (3.3.29)Hence, if T > T ∗ := 2(1 + A − e − ς ) δη e − ς − ε , (3.3.30)the market overperforms the ROA weighted portolio over [0 , T ] almost surely. (cid:3) In the next instance, we make usage of the result of Example 3.3.2 in order to constructa long-only portfolio based on ROA, which outperforms the market almost surely.
Let the financial market M and the stock char-acteristics process be as in Example 3.3.2. Moreover, let π denote the portfolio (3.3.22).22e consider a portfolio η which initially invests 1 + a dollars in µ and sells a dollars ofportfolio π . Here a denotes the quantity a := e − b − e − b − e − ς , (3.3.31)where we have defined b := 1 + A >
0, and A is the constant from (3.3.20), whereas ς > a >
0. The value of the portfolio η is given by Z ηt = (1 + a ) Z µt − aZ πt , t ≥ , (3.3.32)with Z η = 1. First, we show that Z ηt > , for t ≥
0, a.s. Indeed, from (3.3.29) it followsthat Z µt ≥ Z πt exp( − b ) , t ≥ , a.s. (3.3.33)Using this inequality, we get Z ηt ≥ Z πt (e − b (1 + a ) − a ) = Z πt e − b − e − ς − b − e − b − e ς > , t ≥ , a.s. (3.3.34)The portfolio weights of η read η it = 1 Z ηt (cid:0) (1 + a ) Z µt µ it − aZ πt π it (cid:1) , t ≥ , i = 1 , ..., n. (3.3.35)We shall refer to the trading strategy (3.3.35) as the ROA weighted portfolio . In the sequelwe proceed to show that η is a long-only portfolio which outperforms the market over asufficiently long time horizon. Note that for arbitrary t ≥ n (cid:88) i =1 η it = (1 + a ) Z µt Z ηt n (cid:88) i =1 µ it − aZ πt Z ηt n (cid:88) i =1 π it = (1 + a ) Z µt − aZ πt Z ηt = 1 . (3.3.36)Thus, η defines a portfolio. In order to show that η is long-only, it is enough to prove that κ it := (1 + a ) Z µt µ it − aZ πt π it = (2 − e − ς ) Z µt µ it − e − b Z πt π it − e − b − e − ς ≥ , (3.3.37)for all t ≥ i = 1 , ..., n , almost surely. By taking advantage of (3.3.33) along with thedefinition κ it , we get κ it ≥ Z πt µ it e − b − e − b − e − ς (cid:18) − e − ς − π it µ it (cid:19) , t ≥ , i = 1 , ..., n, a.s. (3.3.38)Moreover, the definition of the portfolio π and (3.3.23) imply π it ≤ µ it (2 − e − ς ) , t ≥ , i = 1 , ..., n, a.s. (3.3.39)Using this finding in (3.3.38) finally yields that κ it ≥ t ≥ i = 1 , ..., n , almostsurely. Hence, η is a long-only portfolio.Next, we show that η represents a strong relative arbitrage opportunity versus themarket over [0 , T ], for all T > T ∗ , where T ∗ is given by (3.3.30). Indeed, in Example 3.3.2we have shown Z µT > Z πT , T > T ∗ , a.s. (3.3.40)23hus, we deduce that Z ηT = (1 + a ) Z µT − aZ πT > (1 + a ) Z µT − aZ µT = Z µT , (3.3.41)for all T > T ∗ , almost surely. (cid:3) Remarks.
Note the significance of Example 3.3.3 as we end up with a strong relativearbitrage opportunity versus the market, which is generated by the volatility of the addi-tional stock characteristics process. To the best of our knowledge it is the first such exampleacross SPT literature, and it illustrates again the strength of the generalized setting weintroduced in this section.
In this section we report an empirical analysis of the trading strategies proposed inSection 3.3 and compare their performance to the market portfolio, the entropy weightedportfolio and the equally weighted portfolio. First, we describe our dataset along withthe methodology of implementing the mentioned portfolios. Afterwards we present andvisualize their performance statistics. Finally, we end this section with a regression analysisin order to understand the origins of the returns of our trading strategies.
The universe of stocks we consider consists out of n = 40 firms. In particular wehave considered the companies AAPL, AIG, AMGN, AXP, BA, BAC, C, CAT, CRM, CSCO,CVX, DD, DIS, GE, GS, HD, HON, HPQ, IBM, INTC, JNJ, JPM, KO, MCD, MDLZ, MMM,MO, MRK, MSFT, NKE, PFE, PG, RTX, T, TRV, UNH, VZ, WBA, WMT and
XOM . Our test-ing period ranges from the 3rd of January 2006 till the 31st of December, 2020. We weregiven the access to the daily prices, historical shares outstanding and ROA data for theabove stocks. Note that the latter two quantities are not reported on a daily basis, henceat each day in the time range we simply use the last reported value. Accounting data isusually published with a lag of several months, which is the reason why the dates in ourshares outstanding and ROA dataset do not correspond to the dates the data was actuallypublished. Let us also remark that we did not have access to historical data of delistedcompanies. The previous two remarks lead us to the conclusion that survivorship bias and look-ahead bias are present in our analysis. Due to these unavoidable biases, we can deducethat the quality of our datasets is rather poor. Finally, aiming to understand the sourcesdriving the returns of our portfolios, we have used the daily historical returns of the threeFama French risk factors. These are available together with the historical risk-free rate onthe website of Kenneth French .We rebalance all portfolios on a daily basis. The first portfolio for each strategy isformed on January 3rd, 2006, and the last portfolio is implemented on December 30th,2020. This corresponds to a discrete time horizon { , , ..., T } with T = 3775. Eachtrade causes proportional transaction costs and an additional “overnight fee” for portfolioswhich short stocks. We describe in detail the used model in the next subsection. All ourcalculations and analysis have been performed in the programming language Python . https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html c = 10 − . In order to implement the beta weighted portfolio from Example 3.3.1, wehave set A = 10 − , c = 10 − and p = 0 .
7. All these choices are motivated by the masterequations of the corresponding portfolios, aiming to amplify the drift processes. Moreover,we have estimated the “betas” as β it = t (cid:88) s =1 ( X is − X is − )( Z µs − Z µs − ) , t = 1 , ..., T, (4.1.1)with β i = 0, for all i = 1 , ..., n . We have also assumed s i = +1, for i = 1 , ..., n , regarding thebeta weighted portfolio. This is indeed observed empirically in our dataset for a majorityof time points. In order to implement the ROA weighted portfolio from Example 3.3.3 thevalue of a was set to 2 .
5. Note that the choice of a is rather greedy and indeed results withthe ROA weighted portfolio occasionally selling stocks. Let us remark that arbitrarily highvalues of a would not result in a desirable portfolio since its value process would depletequickly to 0 due to our transaction costs model which accounts for “overnight fees” whena portfolio is short in a stock. Let us also comment that the ROA of a stock is a numberrelatively close to 0, and hence the portfolio (3.3.22) will approximately be equal to themarket portfolio. In order to prevent this, we scaled all ROA values by multiplying themwith a factor of 10. We also want to stress that at no point, any optimization of the aboveparameters was performed. This is in order to reduce the likelihood of a backtest overfit .However, a rigorous search of parameters may be done on a dataset which possesses alarger time range. One could split the dataset into two subsamples, where one is used forchoosing the optimal parameters (e.g. by a simple grid search algorithm) and the otherfor performing an out-of-sample test. In Figure 1 we present the portfolio value processes of the market portfolio (Market),the entropy weighted portfolio from Example 2.3.3 (Entropy), the equally weighted port-folio from Example 3.1.4 (EWP), the beta weighted portfolio from Example 3.3.1 (Beta)and the ROA weighted portfolio from Example 3.3.3 (ROA). In Table 1, we listen theperformance statistics of these portfolios and their corresponding value processes. In par-ticular, for each portfolio we report the annualized return, the annualized Sharpe ratio, theannualized information ratio, terminal value of the portfolio, annualized turnover, as wellas the annualized “alpha” and “beta”. In the sequel, we summarize how we have computedthese performance measures and characteristics.Let us fix some ε , ε ≥
0. The discrete portfolio value process Z π = ( Z πt ) t =0 ,...,T isdetermined by Z πt := Z t (cid:89) s =1 (1 + R πs ) , t = 1 , ..., T, (4.2.1)and Z π := Z , for any portfolio π = ( π t ) t =0 ,...,T − . The quantity Z represents the ini-tial fortune of an investor, which we have normalized to 1 for all portfolios and R π =( R πt ) t =1 ,...,T is the portfolio return process given as R πt := n (cid:88) i =1 π it − (cid:18) X it X it − − (cid:19) − ε n (cid:88) i =1 | π it − − ˆ π it − | − ε n (cid:88) i =1 π ↓ ,it − , t = 1 , ..., T. (4.2.2)25n (4.2.2), the second sum captures the effect of proportional transaction costs and ˆ π =(ˆ π t , ..., ˆ π nt ) t =0 ,...,T − are the readjusted portfolio weights given byˆ π it := π it − X it X it − Z πt − Z πt , t = 1 , ..., T − , (4.2.3)and ˆ π i = 0 for i = 1 , ..., n . The third sum represents the additional costs caused byportfolios which short stocks. Specifically, π ↓ = ( π ↓ , t , ..., π ↓ ,nt ) t =0 ,...,T − is the “ short leg ”of π determined by π ↓ ,it := max(0 , − π it ) , t = 0 , ..., T − , i = 1 , ..., n. (4.2.4)We have set ε = 0 .
3% and ε = 0 .
5% in our numerical experiments. Furthermore, the annualized return of a portfolio is calculated according to R π := (cid:18) Z πT Z (cid:19) /T − . (4.2.5)For realizations X , ..., X m of a random variable X , we denote by (cid:98) E [ X ] and ˆ σ ( X ) itsempirical mean and standard deviation, respectively. In particular (cid:98) E [ X ] := 1 m m (cid:88) i =1 X m , (4.2.6) (cid:98) σ ( X ) := 1 m − m (cid:88) i =1 ( X i − (cid:98) E [ X ]) . (4.2.7)We define the annualized Sharpe ratio of a portfolio π as SR ( π ) := √ (cid:98) E [ R π ] (cid:98) σ ( R π ) . (4.2.8)We also define the annualized information ratio of a portfolio π with respect to the marketportfolio as IR µ ( π ) := √ (cid:98) E [ R π − R µ ] (cid:98) σ ( R π − R µ ) . (4.2.9)We use the following equation to determine the annualized turnover of π ∆ π := 252 T T − (cid:88) t =0 n (cid:88) i =1 | π it − ˆ π it | . (4.2.10)Finally, we calculate the “ alpha ” α π and “ beta ” β π of a portfolio π as the regressioncoefficients in the model R πt − R ft := α π + β π ( R µt − R ft ) + ζ πt , t = 1 , ..., T, (4.2.11)where R ft is the risk-free rate and ζ π , ..., ζ πT are i.i.d. with mean 0. The annualized alpha α π is then determined by α π := 252 · α π . (4.2.12)Note that the α π reported in Table 1 are expressed in percentages. In addition, we alsolisten the coefficient of determination R associated to the regression (4.2.11), for each26ortfolio. For the ROA weighted portfolio we have also computed the mean amount shorted π ↓ with help of the expression π ↓ = 1 T T − (cid:88) t =0 n (cid:88) i =1 π ↓ ,it , (4.2.13)which resulted in π ↓ = 0 . R π SR ( π ) IR µ ( π ) Z πT ∆ π α π β π R Market 10.30% 0.596 - 4.345 0.126 0 1 1Entropy 9.92% 0.577 -0.218 4.124 0.706 -0.371 1.000 0.995EWP 10.88% 0.600 0.167 4.703 2.338 0.253 1.031 0.958ROA 12.52% 0.724 0.420 5.856 0.414 2.686 0.927 0.955Beta 14.33% 0.787 0.612 7.442 0.813 4.124 0.941 0.915Table 1: Performance statistics and portfolio characteristics of the tested strategies fromFigure 1. For a portfolio π , the quantity R π denotes the annualized return, SR ( π ) is theannualized Sharpe ratio, IR µ ( π ) is the annualized information ratio with respect to themarket portfolio, Z πT represents the portfolio value on terminal time (31st of December,2020), ∆ π is the annualized turnover, α π and β π are the annualized alpha and market betarespectively, and R is the coefficient of determination associated to the regression model(4.2.11).From Figure 1 and Table 1 we can observe that the ROA weighted and beta weighted27ortfolios indeed have the potential to be desirable investment strategies. Not only do theyoutperform the considered SPT alternatives pathwise, they also admit a higher Sharperatio, which means that they deliver a higher return for the same level of risk undertaken.Moreover, the generalized generating portfolios record high information ratios, larger alphasand smaller betas compared to the entropy weighted portfolio and the EWP. This allowsus to conclude that they are able to outperform the market significantly on a consistentbasis, which is also confirmed visually in Figure 1. In order to understand the factors driving the returns of our portfolios, we have per-formed regressions on the three Fama French risk factors (see [FF92] and [FF93]). Specif-ically, we have estimated the regression coefficients α π , β π , s π and h π in the model R πt − R ft := α π + β π ( R µt − R ft ) + s π R s t + h π R h t + ζ πt , t = 1 , ..., T. (4.3.1)In (4.3.1), R s t is the return of the “Small Minus Big” (SML) size factor and R h t is the returnof the “High Minus Low” (HML) value factor. In Table 2 we report the annualized alpha α π , the “factor loadings” β π , s π , h π , as well as the R for the regression model (4.3.1), forall portfolios.Portfolio α π β π s π h π R Market 0.422 0.963 -0.256 0.006 0.970Entropy 0.408 0.954 -0.230 0.074 0.974EWP 1.669 0.969 -0.170 0.200 0.967ROA 1.710 0.935 -0.261 -0.238 0.936Beta 4.072 0.914 -0.204 -0.088 0.870Table 2: The loadings on the three Fama French risk factors of the returns of our portfolios.Specifically, for portfolio π , α π is the annualized alpha, β π is the loading on the marketrisk factor, s π is the loading on SMB and h π is the exposure on HML. The quantity R isthe coefficient of determination associated to the regression model (4.3.1).From Table 2 it is evident that all our portfolios load positively on the market riskfactor and negatively on the SMB risk factor. This was indeed expected in beforehand asour investment universe consists mainly of large capitalization stocks which tend to havehigh weights in the overall market portfolio. Moreover, we observe that the returns of themarket portfolio, entropy weighted portfolio and EWP load positively on the value factor,whereas the returns of the ROA weighted portfolio and the beta weighted portfolio have anegative exposure on HML. In addition, the ROA weighted portfolio and the beta weightedportfolio show the largest annualized alpha. However, the regressions related to the lattertwo portfolios also possess the lowest R .Even though the model (4.3.1) manages to explain the returns of our portfolios ac-curately, as all R are higher than 0.87, it would be interesting to examine the factorexposures of our portfolios in other models as well. In particular, we have in mind theCarhart four factor model, which is an extension of (4.3.1) by an additional term whichdescribes momentum (see [C97]). 28 Discussion
In this paper we have further generalized the master equation originally introduced in[F02]. While there have been already attempts in doing so, by means of adding a process offinite variation as the argument of the generating function, our approach goes a step furtheras we allow generating functions to depend on continuous-path semimartingales, in additionto the market weights. To the best of our knowledge, this is the first paper which makesa step in this direction and in addition explicitly demonstrates the value of a generalizedmaster equation. We have shown that it is possible to shorten time horizons beyondwhich relative arbitrage is possible, increase the outperformance of generating portfolioswith respect to the market and include preference based investing in the framework ofSPT. Semimartingales represent a more interesting class of stock characteristics than finitevariation processes, as the former can enhance the drift process significantly. Furthermore,they allow for a greater modelling flexibility, as one is able to drop potentially unrealisticassumptions on the volatility of the market weights, and replace them by more realisticassumptions on the stock characteristics process, in order to generate relative arbitrages.However, this comes then with a cost of having a stochastic integral in the master equation.Stochastic portfolio theory still offers a rich amount of research possibilities. Themain open problem is the incorporation of transaction costs within the master equation.Transaction costs can affect the returns of a portfolio significantly and it would be ofgreat value to incorporate them in the context of SPT. An empirical study of the impactof proportional transaction costs on some functionally generated portfolios is given in[RX20]. From a practitioner’s point of view, it would be interesting to rigorously examinethe validity of SPT results in discrete time. This was pointed out in [V15]. From thestandpoint of our work it would be appealing to question the existence of short-term relativearbitrage opportunities, possibly generated by the additional stock characteristics process.Moreover, it would be interesting to further examine stock characteristics processes andgenerating functions which amplify the drift process in the generalized master equation,and possibly look for portfolios which beat the market on average, instead of almost surely.To search for models which fulfil E [log( Z πT Z µT )] ≥ E [ · ] denotes the expected valueunder the probability measure P ) would allow flexibility in handling the stochastic integralfrom the master equation, while maintaining the benefits of the improved drift process. Appendix: Proof of Theorem 3.1.2
The weights given by (3.1.3) sum to 1. Furthermore, they are bounded and F -progressivelymeasurable. This implies that π is a portfolio. The process Θ given by (3.1.4) is clearly offinite variation.Let P be a K -valued stock characteristics process and S ∈ G Kn . We start by applying29to’s lemma on log( S ( µ, P )) d log( S ( µ t , P t )) = n (cid:88) i =1 ∂ i log( S ( µ t , P t )) dµ it + k (cid:88) i =1 ∂ n + i log( S ( µ t , P t )) dP it + 12 n (cid:88) i,j =1 ∂ ij log( S ( µ t , P t )) d (cid:10) µ i , µ j (cid:11) t + 12 k (cid:88) i,j =1 ∂ n + i,n + j log( S ( µ t , P t )) d (cid:10) P i , P j (cid:11) t + n (cid:88) i =1 k (cid:88) j =1 ∂ i,n + j log( S ( µ t , P t )) d (cid:10) µ i , P j (cid:11) t , t ≥ , a.s. , (A.1)where we take advantage of Schwarz’s theorem and the symmetry of (cid:104)· , ·(cid:105) to deduce thatthe last term in (A.1) lacks a factor 1 /
2. Since S generates a portfolio π with stockcharacteristics P , we have from Definition 3.1.1 d log( Z πt /Z µt ) = d log( S ( µ t , P t )) − k (cid:88) i =1 ∂ i log( S ( µ t , P t )) dP it + d Θ t , t ≥ , a.s. (A.2)By inserting (A.1) into (A.2) we get d log( Z πt /Z µt ) = n (cid:88) i =1 ∂ i log( S ( µ t , P t )) dµ it + 12 n (cid:88) i,j =1 ∂ ij log( S ( µ t , P t )) d (cid:10) µ i , µ j (cid:11) t + 12 k (cid:88) i,j =1 ∂ n + i,n + j log( S ( µ t , P t )) d (cid:10) P i , P j (cid:11) t + n (cid:88) i =1 k (cid:88) j =1 ∂ i,n + j log( S ( µ t , P t )) d (cid:10) µ i , P j (cid:11) t + d Θ t , t ≥ , a.s. (A.3)Proposition 2.2.3 states that the relative return process of a portfolio π versus the marketsatisfies for all t ≥
0, almost surely d log( Z πt /Z µt ) = n (cid:88) i =1 π it d log( µ it ) + 12 n (cid:88) i =1 π it τ iit dt − n (cid:88) i,j =1 π it τ ijt π jt dt, (A.4)where we have taken advantage of the numeraire invariance property (Proposition 2.1.7)to express the excess growth rate γ π, ∗ in terms of the relative covariance process of themarket τ . By definition of the market weights, respectively the relative covariance processwe have for t ≥
0, a.s. d (cid:10) log( µ i ) , log( µ j ) (cid:11) t = d (cid:10) log( X i /Z µ ) , log( X j /Z µ ) (cid:11) t = τ ijt dt. (A.5)Furthermore, an application of Ito’s lemma on µ it = exp(log( µ it )) yields dµ it = µ it d log( µ it ) + 12 µ it τ iit dt, t ≥ , i = 1 , ..., n, a.s. , (A.6)where (A.5) was used. From (A.6) it also follows that d (cid:10) µ i , µ j (cid:11) t = µ it µ jt τ ijt dt, i, j = 1 , ..., n, (A.7)30or all t ≥
0, a.s. Expressing d log( µ it ) by means of (A.6) and inserting it into (A.4) gives d log( Z πt /Z µt ) = n (cid:88) i =1 π it µ it dµ it − n (cid:88) i,j =1 π it τ ijt π jt dt, t ≥ , a.s. (A.8)In order for (A.2) to hold, the local martingale parts of (A.3) and (A.8) have to be equal.This is indeed satisfied if π it = µ it ( ∂ i log( S ( µ t , P t )) + ϑ t ) , i = 1 , ..., n, t ≥ , (A.9)for any R -valued stochastic process ϑ = ( ϑ t ) t ≥ . To see the above statement we remarkthat n (cid:88) i =1 π it µ it dµ it = n (cid:88) i =1 ( ∂ i log( S ( µ t , P t )) + ϑ t ) dµ it = n (cid:88) i =1 ∂ i log( S ( µ t , P t )) dµ it (A.10)holds for t ≥
0, a.s., since (cid:80) ni =1 dµ it = 0. Hence, the local martingale parts of (A.3) and(A.8) are equal. The process ϑ is determined in such a way that (A.9) defines a portfolio.It is easy to see that this is the case if ϑ t is given by ϑ t = 1 − n (cid:88) j =1 µ jt ∂ j log( S ( µ t , P t )) , t ≥ , a.s. (A.11)This proves (3.1.3). Now that we have equality of the local martingale parts of (A.3)and (A.8), we also want the finite variation parts of the two equations to be equal. Thisrequirement determines the drift process. In particular, by comparing (A.3) and (A.8), wehave for any t ≥
0, almost surely d Θ t = − n (cid:88) i,j =1 π it τ ijt π jt dt − n (cid:88) i,j =1 ∂ ij log( S ( µ t , P t )) µ it µ jt τ ijt dt − k (cid:88) i,j =1 ∂ n + i,n + j log( S ( µ t , P t )) d (cid:10) P i , P j (cid:11) t − n (cid:88) i =1 k (cid:88) j =1 ∂ i,n + j log( S ( µ t , P t )) d (cid:10) µ i , P j (cid:11) t , (A.12)where we have taken advantage of identity (A.7) in the first line. Using the obtainedexpression for π it it follows that n (cid:88) i,j =1 π it τ ijt π jt = n (cid:88) i,j =1 (cid:0) µ it ( ∂ i log( S ( µ t , P t )) + ϑ t ) (cid:1)(cid:0) µ jt ( ∂ j log( S ( µ t , P t )) + ϑ t ) (cid:1) τ ijt = n (cid:88) i,j =1 ∂ i log( S ( µ t , P t )) ∂ j log( S ( µ t , P t )) µ it µ jt τ ijt , t ≥ , a.s. , (A.13)where we have used Proposition 2.1.6, which states that µ t spans the kernel of τ t for all t ≥ x, y ∈ (cid:52) n + × K , i, j = 1 , ..., n∂ ij log( S ( x, y )) = ∂ ij S ( x, y ) /S ( x, y ) − ∂ i log( S ( x, y )) ∂ j log( S ( x, y )) . (A.14)31sing (A.13) in the first equality, along with (A.14) in the second equality, we see that thefirst two terms on the right hand side of (A.12) satisfy − n (cid:88) i,j =1 π it τ ijt π jt dt − n (cid:88) i,j =1 ∂ ij log( S ( µ t , P t )) µ it µ jt τ ijt dt = − n (cid:88) i,j =1 (cid:0) ∂ i log( S ( µ t , P t )) ∂ j log( S ( µ t , P t )) + ∂ ij log( S ( µ t , P t )) (cid:1) µ it µ jt τ ijt dt = − S ( µ t , P t ) n (cid:88) i,j =1 ∂ ij S ( µ t , P t ) µ it µ jt τ ijt dt, t ≥ , a.s. , (A.15)which finally gives the desired expression (3.1.4) for the drift process. (cid:3) References [C97] Mark Carhart,
On Persistence in Mutual Fund Performance , The Journal of Finance,52(1):57–82, 1997.[CE15] Samuel Cohen and Robert Elliott,
Stochastic Calculus and Applications ,Birkh¨auser Verlag, 2015.[CEH16] Lawrence Cunningham, Torkell Eide and Patrick Hargreaves,
Quality Investing:Owning the best companies for the long term , Harriman House, 2016.[EK19] Ernst Eberlein and Jan Kallsen,
Mathematical Finance , Springer, 2019.[F02] Robert Fernholz,
Stochastic Portfolio Theory , Springer, 2002.[FF92] Eugene Fama and Kenneth French,
The cross-section of expected stock returns , TheJournal of Finance, 47(2):427–465, 1992.[FF93] Eugene Fama and Kenneth French,
Common risk factors in the returns on stocksand bonds , Journal of Financial Economics, 33(1):3–56, 1993.[FKK05] Robert Fernholz, Ioannis Karatzas and Constantinos Kardaras,
Diversity andrelative arbitrage in equity markets , Finance Stoch., 9(1):1–27, 2005.[FK05] Robert Fernholz and Ioannis Karatzas,
Relative arbitrage in volatility-stabilizedmarkets , Ann. Finance, 1:149–177, 2005.[FP14] Andrea Frazzini and Lasse Heje Pedersen,
Betting against beta , Journal of FinancialEconomics, 111(1):1–25, 2014.[KK20] Ioannis Karatzas and Donghan Kim,
Trading strategies generated pathwise by func-tions of market weights , Finance Stoch., 24:423–463, 2020.[KR17] Ioannis Karatzas and Johannes Ruf,
Trading strategies generated by Lyapunovfunctions , Finance Stoch., 21(3):753-787, 2017.[KS98] Ioannis Karatzas and Steven Shreve,
Brownian Motion and Stochastic Calculus ,Springer, 1998. 32KV15] Ioannis Karatzas and Alexander Vervuurt,
Diversity-weighted portfolios with neg-ative parameter , Ann. Finance, 11:411–432, 2015.[L65] John Lintner,
Security prices, risk and maximal gains from diversification , Journalof Finance, 20:587–615, 1965.[M52] Harry Markowitz,
Portfolio selection , The Journal of Finance, 7(1):77–91, 1952.[M65] Jan Mossin,
Equilibrium in a capital asset market , Econometrica, 35:768-783, 1965.[RX19] Johannes Ruf and Kangjianan Xie,
Generalised Lyapunov functions and function-ally generated trading strategies , Appl. Math. Finance, 26(4):293-327, 2019.[RX20] Johannes Ruf and Kangjianan Xie,
The impact of proportional transaction costson systematically generated portfolios , SIAM J. Financ. Math., 11(3):881-896, 2020.[S13] Winslow Strong,
Generalizations of Functionally Generated Portfolios with Applica-tions to Statistical Arbitrage , SIAM J. Financ. Math., 5(1):472-492, 2014.[S64] William Sharpe,
Capital asset prices: A theory of market equilibrium under condi-tions of risk , The Journal of Finance, 19(3):425–442, 1964.[SSV18] Alexander Schied, Leo Speiser and Iryna Voloshchenko,
Model-free portfolio theoryand its functional master formula , SIAM J. Financ. Math., 9(3):1074-1101, 2018.[V15] Alexander Vervuurt,