Hindawi Publishing Corporation Journal of Probability and Statistics Volume 2015, Article ID 235452, 11 pages http://dx.doi.org/10.1155/2015/235452

Research Article Portfolio Theory for 𝛼-Symmetric and Pseudoisotropic Distributions: 𝑘-Fund Separation and the CAPM

Nils Chr. Framstad

DepartmentofEconomics,UniversityofOslo,P.O.Box1095,Blindern,0317Oslo,Norway

Correspondence should be addressed to Nils Chr. Framstad; [email protected]

Received 30 June 2015; Revised 6 October 2015; Accepted 12 October 2015

Academic Editor: Chunsheng Ma

Copyright © 2015 Nils Chr. Framstad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The shifted pseudoisotropic multivariate distributions are shown to satisfy Ross’ stochastic dominance criterion for two-fund monetary separation in the case with risk-free investment opportunity and furthermore to admit the Capital Asset Pricing Model 𝛼 under an embedding in L condition if 1<𝛼≤2, with the betas given in an explicit form. For the 𝛼-symmetric subclass, the market without risk-free investment opportunity admits 2𝑑-fund separation if 𝛼 = 1+1/(2𝑑−1), 𝑑∈N, generalizing the classical elliptical case 𝑑=1, and we also give the precise number of funds needed, from which it follows that we cannot, except degenerate cases, have a CAPM without risk-free opportunity. For the symmetric stable subclass, the index of stability is only of secondary interest, and several common restrictions in terms of that index can be weakened by replacing it by the (no smaller) indices of symmetry/of embedding. Finally, dynamic models with intermediate consumption inherit the separation properties of the static models.

1. Introduction the portfolio returns distributions (univariate) can be ordered by their mean once a single dispersion parameter is given, Portfolio separation, that is, the property of reducing the and for the second-order case: by their dispersion once the dimension of a portfolio optimization problem to a low mean is given. Subsequently, Owen and Rabinovitch [7] and number of vectors (“funds”) without welfare loss to the Chamberlain [8] establish that the elliptical (also frequently agents in question, has been treated extensively since Tobin referred to as “elliptically contoured”) distributions satisfy [1]. There are two main directions: the one which is the Ross’ conditions for two-fund separation. Their setting is a subject of this paper is the characterization of those returns mean-variance tradeoff, tying the knot back to the Markowitz probability distributions for which those funds will do for all [9] approach as employed by Tobin [1]. Over these decades, agents. The other is the characterization of preferences which thedevelopmenthasofferedsurprisestoquiteafewofthe admit the property for all suitable returns distributions (the giants who bear today’s theory on their shoulders: Markowitz standard work being Cass and Stiglitz [2], but see even the turned out predated by more than a decade by de Finetti modern probabilistic approach of Schachermayer et al. [3]); [10](seeMarkowitz’account[11]wherehealsocreditsRoy there are also other routes to the separation property, for [12]). Tobin conjectured that any two-parameter portfolio example, risk measures, falling somewhat in between beliefs returns distribution family would admit two-fund separation; and preferences (contributions include this author [4] and counterexamples were given by Samuelson [13], Borch [14], independently Giorgi et al. [5]). and Feldstein [15]. Fama’s discovery ([16], can also be read This paper concerns the distributional side of the theory, out of Samuelson [17]) that vectors of iid symmetric 𝛼-stables where the standard literature reference is Ross [6]. Ross con- admitted two-fund separation led Cass and Stiglitz to suggest siders preferences compatible with second-order stochastic that 𝛼-stability was even necessary, making a reservation dominance (and in footnotes, preferences merely assumed for independence, which quickly turned out to be a most compatible with first-order dominance). The core of his result appropriate one, as Agnew [18] provided a nonstable example. is the property that the returns distribution vector be such that However,thepropertiesthatenabledOwen,Rabinovitch,and 2 Journal of Probability and Statistics

Chamberlain to verify the Ross [6] criterion for the ellipticals vectors if bold) are either nonrandom or choice variables; were to be found as far back as Schoenberg [19, 20] in 1938, a 𝐷-vector 𝜉 is called a portfolio ifittakesvaluesina before modern portfolio theory. given set to be denoted by 𝐻 or 𝐿 (notation to depend The classical (elliptical) 2-fund separation result holds on shape; “unrestricted” if no such set restriction is given). irrespective of whether a “risk-free” numeraire´ opportunity 1 isthevectorofones,and0 is the null vector. Vectors ⊤ exists,andoneofthefundscanbechosentobethe are columns by default, unless indicated by superscript “ ” safest available (the “minimum variance” portfolio, so the (transposition) or given as a gradient. We apply the signed ⟨𝑝⟩ 𝑝 risk-free case admits so-called “monetary separation”). This power notation 𝑥 fl |𝑥| sign(𝑥) even to vectors, element- paper considers the generalization to the (shifted) so-called ⟨𝑝⟩ ⟨𝑝⟩ ⟨𝑝⟩ ⊤ ⟨𝑝⟩ wise: x =(𝑥 ,...,𝑥 ) .(Noticethatx 󳨃→ x is pseudoisotropic distributions,amultivariateclassofsymmet- 1 𝐷 invertible.) Matrices are Greek uppercase boldfaced letters. ric random variables such that all linear combinations of Aset𝐻⫌{0} is radial if it is composed as a union of half- thecoordinatesareofthesametype.Thepseudoisotropic x ∈𝐻⇔𝑞x ∈𝐻∀𝑞>0 distributions admit a dispersion quasinorm 𝜍 (often called the lines from the origin: . Constraining “standard”) which is symmetric and positively homogeneous the portfolio to the closed first orthant models a “no short and which, together with the excess returns entering via a sale” constraint, and we will use that terminology as well. No location shift, characterizes the portfolio return distribution shortsaleonsome,butnotall,investmentopportunitieswill completely, which is briefly summarized as follows: also correspond to a radial constraint. As commonplace in the literature, we will frequently refer to the numeraire´ as the (i) With risk-free opportunity, the (shifted) pseudo- “risk-free” investment opportunity and the other investments isotropic distributions admit two-fund monetary sep- as “risky.” aration just like the elliptical subclass, or like the The ∼ symbol denotes equal probability law. A random subclass of iid symmetric 𝛼-stable random variables variable and its distribution are symmetric if X ∼−X;then as established already by Fama [16] (for results in 𝜇 + X is called shifted symmetric for nonrandom 𝜇. continuous time: this author [21] and Ortobelli et al. [22]). Assumption 1. We will allow for constraints to be specified 𝜍 (in the single-period model, we will consider either the con- Furthermore, we have a CAPM if is differentiable straint to a radial set, covering, e.g., no short sale conditions, outside the origin; a fortiori, so is the case for the so- to an affine half-space representing no borrowing or limited called 𝛼-symmetric distributions with 𝛼>1,andfor >1 degree of leverage, or to the affine hyperplane of no risk- the symmetric stables with index of stability ,but free opportunity). After having restricted the opportunity set also for certain nonintegrables. according to these constraints, we will assume the market (ii) Without risk-freeopportunity,separationwillonlybe to be free from arbitrage opportunities and from redundant admitted by a few special cases leading to 2𝑑-fund investment opportunities.(Ifthereisaredundantopportunity, separation if the index of symmetry is one of the then we can leave it out and rebuild the model without it.) In values 𝛼 = 1+1/(2𝑑−1), 𝑑∈N,thatis,one- particular, the independent radial scalings 𝑅0 and 𝑅 are never and-an-oddth, where 𝑑=1subsumes the elliptical Dirac at zero; if any of these is constant, it is without loss of distributions 𝛼=2=1+1/1. generality = 1.

AlsotheCAPMbreaksdownexceptinellipticalor Note that, in line with the literature on portfolio separa- degenerate cases. Fama [23, section VI.B] remarks tion, we do not assume limited liability, which in fact holds that the presence of risk-free opportunity “greatly only in a few well-known cases, all elliptical. simplifies determination of the efficient set of port- 2𝑑 folios,” and indeed, for the -fund separation cases 3. The Single-Period Market and just mentioned, the efficient set is no longer convex. the Preferences The paper will first introduce terminology and then in 𝑤 Section 3 state the single-period market model and review Consider a single-period investment allocating wealth 𝜉 𝐷∈N stochastic dominance. Section 4 will introduce pseudoiso- between in “risky” investment opportunities and 𝑤−1⊤𝜉 tropic random variables, and Sections 5–7 will point out how/ the remaining in a numeraire´ (enumerated as the when they admit, respectively, monetary separation, sepa- 0th coordinate) that returns 𝑋0 per monetary unit invested. ration without risk-free opportunity, and CAPM. Section 8 Writing the risky returns vector as 𝑋01 + 𝜇𝑅0 + X𝑅 for some sketches how a dynamic model inherits the separation prop- nonrandom location parameter 𝜇 (resembling a representa- erties from the static model. tion common for elliptical distributions, e.g., Cambanis et al. [24]), the portfolio return then becomes our model ansatz 2. Notation, Terminology, and ⊤ 𝑤𝑋0 + 𝜉 (𝜇𝑅0 + X𝑅) . (1) Standing Assumptions (𝑅0,𝑅,X) will be specified conditional on 𝑋0,with(𝑅0,𝑅) 𝐷 We work in R for arbitrary finite 𝐷≥2;someresultswillbe conditionally independent of X. We will later assume X to vacuous for low 𝐷. Random quantities are denoted by Latin be symmetric (but not that it is integrable!). It will represent letters (boldfaced if vector-valued). Minuscles (Greek/Latin, no loss of generality to interpret, or even formally assume, Journal of Probability and Statistics 3

𝑋0 as “risk-free”; we say that arisk-freeopportunityexists be invoked only in a few instances, where they can do with ⊤ unless all agents are constrained to 1 𝜉 =𝑤. fewer funds: As is well-known, all risk-averse agents can do We first define agents and then separation (to hold over with the fund 1 if there is no risk-free opportunity and, for all agents); note that “𝑘-fund separation” implies 𝑘+1-fund example, all returns are iid Gaussian, but a non-risk-averse separation; Theorem 11 will specify when a result cannot be agent could need another fund to boost variance. Theorem 11 improved upon. will touch this issue.

∗ Definition 2. To compare 𝑋 and 𝑋,suppose𝑋≥0̂ a.s. and Taking the well-known Gaussian as example, monetary consider the formula two-fundseparationisduetothefollowingfeatures,assum- ing for simplicity 𝑋0 =0: the set of all possible portfolio ∗ 𝑋 + 𝑋∼𝑋+̆ 𝑋.̂ (2) returns is a family wherein each distribution is fully charac- terized by location (which is a good to every agent!) and scale. (i) By an agent we mean a pair of initial wealth 𝑤∈R and Both these functionals are homogeneous of degree one, so apartialpreference ordering over random variables, 𝜑 ∗ if maximizes location given scale (standard deviation) of where preferences are such that 𝑋 is always (weakly) 1,then𝑄𝜑 maximizes location given a scale of 𝑄,andfor ̆ preferred to 𝑋 whenever (2) holds with 𝑋=0a.s. for every 𝜉 the return is first-order stochastically dominated by ̂ ∗ some 𝑋≥0a.s., in which case we say that 𝑋 (weakly) 𝑄𝜑 for the appropriate 𝑄. The next section will introduce the first-order stochastically dominates 𝑋. more general class of pseudoisotropic distributions which, ∗ (ii) An agent is risk-averse if 𝑋 is always (weakly) when shifted by a location, share these features. What will not preferred to 𝑋 even whenever we admit any 𝑋̆ which carry over, except in a much weaker result valid in exceptional is independent of the three others and symmetric. cases, is separation under restriction to an affine subspace not containing 0. (iii) Suppose that there exist 𝑘≥1vectors (“funds”) 𝜑1,...,𝜑𝑘 such that, for any given portfolio 𝜉,there 𝑞 ,...,𝑞 4. Pseudoisotropic and exist 1 𝑘 sothatreturn(1)isweaklyfirst-order 𝛼 stochastically dominated by the return obtained using -Symmetric Distributions in place of 𝜉 in (1) the portfolio The pseudoisotropic random vectors form a multivariate ∗ distribution class which contains, among others, the symmet- 𝜉 =𝑞1𝜑1 +⋅⋅⋅+𝑞𝑘𝜑𝑘. (3) ric ellipticals (and no other square-integrable distributions!) 𝛼 𝑘+1 and symmetric -stables. The following will give a primer Then we say that the returns distribution admits - on the theory assuming the basics of these subclasses are fund monetary separation if a risk-free opportunity known (see, e.g., Cambanis et al. [24] and the beginning of exists (fund 𝑘+1being the numeraire)´ and 𝑘-fund ⊤ ∗ Samorodnitsky and Taqqu [26]). For the idea of finding 𝐷- separation if numeraire´ holdings 𝑤−1 𝜉 vanish dimensional versions of univariate distributions, see Eaton identicallyforallagents. [27]; the term pseudoisotropic does refer to the multivariate X 3 Remark 3. Notice first that we do not assume that each agent ; see Jasiulis and Misiewicz [28, Definition ]. has an “optimal” (finite) portfolio; rather, the property says R𝐷 that for any given portfolio there is one which is at least Definition 4. A symmetric distribution in is called as good and which uses only the funds (implying that the pseudoisotropic if, for some order 1 positive-homogeneous 𝜍:R𝐷 →[0,∞) restriction to the funds is without welfare loss). standard and some (complex) function ℎ 𝑞𝜃 󳨃→ Item (i) is the so-called mass-transfer criterion for first- , the characteristic function can be represented as 𝑖𝑞𝜃⊤X order stochastic dominance. It is equivalent to either of the E[𝑒 ] = ℎ(|𝑞|𝜍(𝜃)). following; see, for example, Østerdal [25] for more on the ∗ 𝜃⊤X ∼𝜍(𝜃)𝑋̃ 𝑋̃ various definitions: CDF𝑋∗ ≤ CDF𝑋,orE[𝑢(𝑋 )] ≥ E[𝑢(𝑋)] Thus we have for some marginal for every bounded nondecreasing (i.e., “utility”) function 𝑢. (any non-Dirac marginal 𝑋𝑖 will do!), a property obviously ∗ Wewillfrequentlyusethatifneither𝑋 first-order dominates preserved under matrix transformations. Pseudoisotropy 2 𝑋 nor vice versa, then there are two agents which disagree generalizes ellipticity located at zero (then, 𝜍 is a quadratic over preference between them; indeed, the utility function form), but ellipticity admits some special properties. For ∗ 1𝑥≥𝑥̂ prefers 𝑋 to 𝑋 iff CDF𝑋∗ (𝑥)̂ ≤ CDF𝑋(𝑥)̂ .Noticethat example, nonelliptical pseudoisotropic distributions cannot there is no first-order dominance between 𝑋 and 𝜇+𝜎𝑋if have finite second-order moments and must be absolutely 𝑋 is real and symmetric and has full support and 𝜎>1and continuous with respect to 𝐷-dimensional Lebesgue measure 𝜇≥0; if one can increase 𝜎 without decreasing 𝜇, then there except a possible point mass at the origin, or a marginal is some agent who will prefer it and some who will not. being Dirac. Those exceptions can be done away with the Second-order stochastic dominance corresponds to risk latter by the assumption of no arbitrage or no second risk-free aversion, but in contrast to the common literature, which opportunity and the former by incorporating it in 𝑅.Thereis assumes sufficient integrability for E𝑋=0̆ and 𝑢 concave, we thus no loss of generality in the following restriction to what merely ask if an agent will reject any independent symmetric Misiewicz [29, Remark II.2.1] calls “pure” pseudoisotropic noise.Riskaversionisnotamainpointofthispaperandwill measures. 4 Journal of Probability and Statistics

⊤ ⊤ 1/2 Assumption 5. For the pseudoisotropic distributions consid- (𝜃 ∫ xx 𝑑𝜘 𝜃) . This exhibits a very special feature of the ered in this paper, no coordinate has any point mass at the ellipticals; namely, that matrix transformation (along with the origin, and 𝜍(𝜃)=0only iff 𝜃 = 0. radial 𝑅) suffices to characterize dependence. Further prop- erties unique to the ellipticals are that the 2-spectral measure Pseudoisotropy also generalizes symmetric 𝛼-stability; need not be unique (for 𝛼<2,however,allthe𝛼-spectral indeed, if any two coordinates of a pseudoisotropic variable measures of a spherical distribution must be uniform on the are independent, we do have symmetric stability. Like for unit sphere); furthermore, only for the ellipticals we have that symmetric stable distributions, there are some geometric the probability measure exhibits the same elliptical symmetry properties to observe. We introduce some terminology, com- (affinely transformed isotropy) as the characteristic function; pare, for example, Koldobsky [30, 31] (wherein the reader also and as mentioned, only for the elliptical class there are can find why the restriction to 𝑝≤2does not rule out any distributions integrable at the order of the embedding index. interesting cases for our purposes) and Kalton et al. [32]. A special subclass of the pseudoisotropy, generalizing 𝛼 𝐷 the sphericals, is the so-called -symmetric distributions, Definition 6. An origin-symmetric star body 𝐾 in R is introduced by Cambanis et al. [24], which exist for 𝛼 ∈ (0, 2]. an origin-symmetric compact with a continuous boundary crossed precisely twice by each line through 0,required Definition 8. A pseudoisotropic Z is called 𝛼-symmetric or interior to 𝐾.Letthe𝐾-quasinorm ‖⋅‖𝐾 be the (well-defined!) standard 𝛼-symmetric if one can take 𝜍 as the standard 𝛼- ‖𝜃‖ = {𝑎 > 0; 𝜃/𝑎 ∈ 𝛼 1/𝛼 associated Minkowski functional 𝐾 min norm ‖𝜃‖𝛼 =(∑𝑖 |𝜃𝑖| ) (by slight abuse of notation). one 𝐾}.Fix𝑝 ∈ [0, 2]; we say that the quasinormed space then calls X = ΣZ transformed 𝛼-symmetric,orΣ-transformed 𝐷 𝑝 (R ,‖⋅‖𝐾) embeds in L if ‖⋅‖𝐾 admits a so-called Blaschke- 𝛼-symmetric.onecalls𝛼 (coinciding with the embedding Levy´ representation index!) the index of symmetry.

󵄨 󵄨𝑝 1/𝑝 𝛼 𝛼 {(∫ 󵄨𝜃⊤x󵄨 𝜘 (𝑑x)) ,𝑝∈(0, 2 , Thus a transformed -symmetric has -spectral measure { 󵄨 󵄨 ] (1) (𝐷) ‖𝜃‖ = supported by only 2𝐷 unit vectors ±x ,...,±x which 𝐾 { 󵄨 󵄨 (4) { 󵄨 ⊤ 󵄨 R𝐷 𝛼 x(𝑖) = e exp ∫ ln 󵄨𝐴𝜃 x󵄨 𝜘 (𝑑x) ,𝑝=0 span ; for the standard -symmetrics, we have 𝑖. { Apartfromtheellipticals,thearguablybestknown 𝛼 for some finite spectral measure 𝜘 supported by the unit examples are the vectors of iid symmetric -stables. Such a distribution has, if normalized to unit scale, characteristic sphere (necessarily symmetric, and for 𝑝=0it integrates to 𝛼 𝑝=0 𝐴>0 function exp(−‖𝜃‖𝛼). More generally, it is known since Paul one) and for some (explicitly computed in terms 𝛼 𝛼 of 𝜘 in [32, p. 3-4]). The supremum over those 𝑝∈[0,2]such Levy´ that there exist -symmetric -stable distributions iff 2≥𝛼≥𝛼>0 (−‖𝜃‖𝛼) that (4) holds seems not to have an established term: we call (with characteristic function form exp 𝛼 ; suchonecanbegeneratedbyscalingan𝛼-symmetric 𝛼-stable it the embedding index. 𝛼 X by an independent radial 𝑅 such that 𝑅 is 𝛼-stable). The Notice that if 𝑝>1,then‖𝜃‖𝐾 is strictly quasiconvex, reader should beware the confusion in the literature, where ⊤ ⟨𝑝−1⟩ ⊤ with (by bounded convergence) gradient =∫(𝜃 x) x 𝑑𝜘 the notion of symmetry most often in the modern literature continuous for 𝜃 ≠ 0. meansantipodalsymmetry(asinthispaper),althoughitis The connection to pseudoisotropic variables is the fol- used by other authors in the past for rotational invariance lowing fact, conjectured by Lisitski˘ı[33]andlaterprovenby (isotropy, implying ellipticity). This translates to a confusion Koldobsky [31, Corollorary 1], that for our purposes we do as to whether the canonical choice for a multidimensional L0 version of a symmetric 𝛼-stableistheonewithiidcoordinates have embedding in (and thus the embedding index is a ⊤ 𝛼/2 well-defined number ∈ [0, 2]). (𝛼-symmetric), or the elliptical one (chf = exp(−(𝜃 𝜃) )), for example, in Owen and Rabinovitch [7, footnote 4]. 𝐷 Theorem 7. For any pseudoisotropic X in R , 𝜍 is the Generally, when it comes to stable laws, the reader should Minkowski functional ‖⋅‖𝐾 of some origin-symmetric star body be warned against the literature’s inconsistent language and 𝐷 0 𝐾,suchthat(R ,‖⋅‖𝐾) embeds in L . notation, dubbed by Hall [34] as a “comedy of errors.” For some 𝛼-symmetric distributions not generated from 𝑝 Embedding in L for 𝑝 ∈ (0, 2] implies embedding in stables, see Gneiting [35]. 𝛼 L for all 𝛼∈[0,𝑝], each with its own spectral measure, 𝛼 henceforth the -spectral measure if needed to distinguish; a 5. Portfolio Separation with Risk-Free stronger assertion than Theorem 7 is therefore the Misiewicz conjecture that the embedding index is >0.Thisquestion Investment Opportunity remains open, though potential counterexamples must satisfy The symmetry and positive homogeneity of the 𝜍 functional restrictive conditions (see [30]). In particular, they must be immediately yield two-fund monetary separation for the extremely tail-heavy, as it is well-known that nonembeddabil- pseudoisotropics, in much the same way as the elliptical case L𝑝 𝑝 ∈ (0, 2] 𝑝 ity in ( )impliesinfinite th-order moment. or the case of linearly transformed iid 𝛼-stable components The elliptical distributions located at the origin are pre- treated already by Fama [16]. It is already known that the 2 cisely the ones which are pseudoisotropic and embed in L . independence of (linearly transformed) coordinates is not 𝛼 The Blaschke-Levy´ representation (4) then takes the form essential, for example, [36–38]. For 𝛼≤1,theL unit ball is Journal of Probability and Statistics 5

2 not only a nonconvex set; indeed, its complement intersected know. A sketch in R with 𝜇2 >𝜇1 >0with the “unit sphere” with any orthant is a convex set (the first-orthant part of the beinganellipsearoundtheoriginwillinterpretanalogues epigraph defining any component as a convex function of the of “correlation” and “hedging” graphically, but the geometric others). This motivates the nondiversification final part of the arguments work for nonelliptical smooth 𝜍-unit spheres too; following result. if the 𝜍-sphere through (0, 1) does not fall too steeply there, we must have adaptation in the first quadrant. However, if we Theorem 9. Consider market (1) with the restriction that the have a corner at (0, 1), the situation is different, in particular, if 𝐻 portfoliosarerestrictedtosomeclosedradialset (possibly 𝜍 is the 1-norm |𝑥1|+|𝑥2|. But even for nonintegrable cases we 𝐷 𝑝 = the entire R ). Suppose that conditionally on 𝑋0, X is have differentiability if they admit embedding in L for some pseudoisotropic with standard 𝜍. 𝑝>1. Section 7 will utilize these arguments for the CAPM. Then there is two-fund monetary separation: for any given 𝜉, the return is first-order stochastically dominated by the ∗ 6. Some Special Results under return using portfolio 𝜉 =𝜍(𝜉)𝜑,where𝜑 solves Constrained Leverage or No Risk-Free ⊤ max 𝜉 𝜇 𝜉∈𝐻 Investment Opportunity (5) 𝛼 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜𝜍 (𝜉) =1. This section assumes transformed -symmetry. By Assumption 1, we can take the linear transformation Σ Suppose in the following that 𝐻 is a convex set such that of Definition 8 to be invertible even under constraint of type 𝜉⊤𝜇 =0̸ 𝜉 ∈𝐻 𝜑 𝜍 (7) below; should the constraint remove an unrestricted for some .Then is unique if the -unit ball ⊤ is a strictly convex set, in particular if the embedding index is arbitrage, with Σ 𝜃 = 0 for some 0 ≠ 𝜃 ⊥ 𝜂,thisrisk-free >1.Ontheotherhand,ifthereisanextremumonanaxisthen investment opportunity would violate Assumption 1. it is optimal to only invest in one opportunity, either a positive X Σ position in the one with highest excess return/dispersion ratio, Assumption 10. Throughout this section, is -transformed 𝛼 X = ΣZ 𝛼 Z or shorting the one with the largest negative such; this in -symmetric: for some -symmetric and some Σ particular occurs with 𝛼-symmetric distributions when 𝛼≤ invertible .Introducethenotation 1. Under the additional assumption of iid coordinates and ⊤ ⊤ 𝐷 𝜁 = 𝜉 Σ, 𝐻=R , holding only one risky opportunity implies either 𝛼- 𝛼 𝜇 −1 symmetric -stability or that all but one 𝑖 vanish. ^ = Σ 𝜇, (6) Proof. 𝐾 has continuous boundary, so 𝜑 exists by the extreme 𝜂 = Σ−11 ∗ value theorem. Consider 𝜉 and 𝜉 =𝜍(𝜉)𝜑.Wehave ∗⊤ ⊤ ⊤ ⊤ 𝜉 X ∼ 𝜉 X and thus (by independence and nonnegativity) so that 𝜉 (𝜇 + X)=𝜁 (^ + Z) and the total invested ∗⊤ ⊤ ∗ ⊤ 𝜉 (𝜇𝑅0 + X𝑅) ∼ 𝜉 (𝜇𝑅0 + X𝑅) +𝜉 ( − 𝜉)𝜇𝑅0, identifying risky amount becomes 𝜁 𝜂. The only portfolio constraints ̂ the latter a.s. nonnegative (since 𝑅0 ≥0)termas𝑋 in (2). considered in this section are the following leverage constraint Suppose 𝐻 is convex. If the embedding index is >1the type, where 𝐿 = {𝑤} means no risk-free opportunity: 𝜍-unit sphere is smooth, yielding unique maximum unless 𝜁⊤𝜂 ∈𝐿 ( ). (contrary to assumption) 𝐻 is orthogonal to 𝜇.Otherwise, possibly agent-dependent (7) we can have corner solutions; in particular for the standard 𝛼- 𝛼≤1 In particular we have no further restriction to arbitrary radial quasinorms with , the convex hull of the unit ball is the 𝐻. standard 1-norm, and pushing a plane as far as possible in one direction while intersecting this leads to a corner. Finally, as The ellipticals admit 2-fund separation without risk-free independent coordinates of a pseudoisotropic imply stability, opportunity, and we will see that this generalizes, at the cost iid coordinates imply 𝛼-symmetric 𝛼-stability, which is a of additional funds, to the special 𝛼-values 𝛼=1+1/odd. well-known case. Theorem 11. Consider the market under Assumption 10. Put A comment is appropriate. From a first course in finance, oneobservesthatagentswilldiversifyandthatifweintroduce 1 𝛼 𝑑= ⋅ (8) a new investment opportunity which offers return exceeding 2 𝛼−1 the risk-free, one will buy a positive amount of it as long as the 𝛼 ∈ (1, 2] ⇔𝑑≥1 hedging benefit of shorting (from a positive correlation) is not and assume in parts (a)–(d) that ( ), while 𝛼 ∈ (0, 1] too large. Of course, the argument is based on some degree in part (e) assume . of integrability, and it is long known that nonintegrability (a) For 𝛼 ∈ (1, 2], the minimum-dispersion portfolio for no may lead to plunging all eggs into one basket (and from risk-free opportunity, that is, the one which minimizes ⊤ −1 the literature’s focus on the iid coordinate case, e.g., Fama the dispersion 𝜍 subject to the constraint, is 𝜉 =(Σ ) 𝜁 [16], this behaviour often shows up). It is generally not with straightforward to describe the dependence structure outside 2 1 𝑤 L (nor excess return outside L ) but in the pseudoisotropic 𝜁 = 𝜂⟨2𝑑−1⟩. 𝜇 𝜍 󵄩 󵄩2𝑑 (9) case, the location and the -ballwillrevealwhatweneedto 󵄩𝜂󵄩2𝑑 6 Journal of Probability and Statistics

(b) Suppose 𝑑∈N (i.e., 𝛼=1+1/𝑜𝑑𝑑). Then we have Proof. In order not to be first-order dominated, any agent ⊤ 2𝑑 + 1-fund monetary separation under constrained who chooses the level 𝑤∈𝐿̂ for 𝜁 𝜂 (where 𝑤=𝑤̂ is leverage and 2𝑑-fund separation if there is no risk- mandatory if there is no risk-free opportunity) and the level free investment opportunity. In both cases, the 2𝑑 risky 𝜍 for dispersion must choose a solution of the problem 0 funds are (with the convention 0 =1) ⊤ max 𝜁 ^ −1 𝑗−1 2𝑑−𝑗 ⊤ 𝜁 𝜑 =(Σ⊤) (𝜂 ] ,...,𝜂𝑗−1]2𝑑−𝑗) . (10) 𝑗 1 1 𝑛 𝑛 𝛼 𝛼 subject to ‖𝜁‖𝛼 = 𝜍 , (12) Call a portfolio “efficient” if it is a linear combination 𝜁⊤𝜂 = 𝑤̂ of these 𝜑𝑗 and satisfies leverage constraint (7). When 𝑑>1 𝛼 ∈ {4/3, 6/5, 8/7, ..} (i.e., ), the set of efficient with associated Lagrange condition to be used in what to portfolios is not convex except in degenerate cases (this follow: in contrast with the case 𝛼=2=𝑑+1). ^ −𝜆𝜂 =𝛿𝛼𝜁⟨𝛼−1⟩ (c) Lettheassumptionsofpart(b)hold,andconsiderthe (13) number (mnemoniac: 𝜆 for the leverage constraint, 𝛿 for the disper- ̃ sion constraint). 𝑘 fl min {2𝑑, 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑖𝑠𝑡𝑖𝑛𝑐𝑡𝑟𝑒𝑎𝑙 (a) For the minimum-dispersion portfolio, consider the (11) 𝜂 problem 𝑖 V𝑎𝑙𝑢𝑒𝑠 𝑝𝑙𝑢𝑠𝑖; 1𝑖𝑓∃ 𝜂 =0≠ ] }. ] 𝚤 𝚤 𝑖 min ‖𝜁‖𝛼 𝜁 (14) ̃𝑘 ⊤ If there is no risk-free opportunity, then there are subject to 𝜁 𝜂 =𝑤 linearly independent funds such that all risk-averse agents can choose optimally among them, while any and in case 𝛼>1this is a concave problem with proper subset of these funds will fail to satisfy some risk- solution uniquely given by (9) (which is a limiting averse agent. case of (13)). ̃ If 𝑘>1, still assuming no risk-free opportunity exists, (b) To cover the last part first, (13) has only two param- thesefundsalsosufficeforallagents,notnecessarily eters, so the possibly optimal portfolios will, at least ̃ risk-averse. If 𝑘=1,thenallrisk-averseagentswill piecewise, form at most 2-dimensional surface, not choose minimum-dispersion portfolio (9) (recovering convex unless subset of a plane, which, for the 2𝑑- the classical degeneracy of one-fund separation over fund cases, 𝑑>1,requiresthenumberoffundsto risk-averse agents), while other agents require an arbi- degenerate to at most 3. The possible degeneracies are trary (nonnull) free portfolio in addition. addressed in item (c). ̃ Under constrained leverage, we have min{3, 𝑘+1}fund Now for the separation result itself, odd signed pow- monetary separation; the above funds together with the ers are just ordinary powers, so (13) yields risk-free are sufficient to satisfy any agent. 2𝑑−1 𝜁 ⋅ (𝛿𝛼)2𝑑−1 = (] −𝜆𝜂) . (15) (d) Part (b) does not generalize to the case where 𝑑−1/2∈ 𝑖 𝑖 𝑖 N (i.e., 𝛼=1+1/𝑒V𝑒𝑛); if we formally consider the If 𝛿 =0̸ , expand the power and collect terms to get the fundsof(10)withanoddnumberfor2𝑑,therearecases 2𝑑 risky funds given by (10), and in addition there is where some agent cannot be satisfied by these funds. the risk-free, unless it vanishes identically. To address 𝛼 ∈ (0, 1] (e) Suppose in this part that .Thenanyagent degeneracies, the constraint qualification could fail, holds the zero position in all but at most two opportu- butonlyattheminimum-dispersionportfolio,which nities (where in contrast to separation results, different is the fund 𝜑2𝑑.Andtheremainingcase𝛿=0implies agents may require different pairs). The minimum- ^ =𝜆𝜂 ⊤ , which is subsumed in the next item. dispersion portfolio for the equality constraint 𝜁 𝜂 =𝑤 (c) Let us first cover the case when ^ and 𝜂 are propor- canbechosenononeaxis(possiblynonunique).This tional. Then the left-hand side of (13) collapses toone portfolioischosenbyallrisk-averseagentsinthespecial vector, a scaling of (9). In addition there is the risk- case where ^ is proportional to 𝜂. free, if one such exists, but if it does not, then by Before proceeding to the proof, notice that the case, where proportionality the excess return is uniquely given by 𝑤,sothat𝛿=0. (11) yields 1, is the only where an opportunity with 𝜂𝑖 = ]𝑖 =0 is not redundant. Indeed, if Σ istheidentity,then(11)counts If 𝛿=0, there has to be an additional fund ⊥ 𝜂,at the number of different marginal distributions of nonzero zero price, but also not contributing to excess return, excess returns; then if there are at least two, one with zero to satisfy agents who want higher than minimum excess return (possibly desired by a non-risk-averse agent) dispersion. (Risk-averse agents will choose the zero can be generated as a linar combination. position in this fund.) Journal of Probability and Statistics 7

To establish the number of funds needed, that is, the whichdoesnotexpandtoapolynomial.Supposefora number of linearly independent vectors in expansion counterexample that ]𝑛/𝜂𝑛 >⋅⋅⋅>]1/𝜂1 >0,withall (10) (cf., (15)), assume 𝛿 =0̸ (otherwise we have the 𝜂𝑖 >0.Let𝜍 grow from minimum dispersion (which is proportionality just covered) and 𝜆 such that of the form of expansion (10)). At the point where the optimum falls outside the appropriate simplex (e.g., 2𝑑 2𝑑 − 1 2𝑑−1 𝑗−1 the unit simplex if 𝜂 = 1 and 𝑤=1), opportunity #1 (𝛿𝛼) 𝜁 = ∑ ( ) (−𝜆) 𝜑𝑗. (16) 𝑗=1 𝑗−1 is shorted, requiring one more fund. (e) Finally, assume 𝛼 ∈ (0, 1]. Then the intersection of 𝛼 We wish to pick 2𝑑 agents with distinct 𝜆 values. That each orthant with the exterior of the L unit sphere is convex. Except in the proportional case, and as long is possible (cf., Remark 3) as two distributions with ⊤ different dispersions are never ordered by first-order as dimension exceeds 2, maximizing 𝜁 ^ subject to ⊤ 𝛼 dominance; dot each side of (13) with 𝜂 to eliminate 𝛿 being in the plane 𝜁 𝜂 and on the L (quasi-norm) and get, for 𝑤>0, sphereistomovealineinparalleltothisplaneuntil 𝛼 it no longer intersects the interior of the L ball; then 𝑤 ⟨2𝑑−1⟩ 𝜁 = (^ −𝜆𝜂) . some coordinate becomes zero. Remove that coordi- ⊤ ⟨2𝑑−1⟩ (17) 𝜂 (^ −𝜆𝜂) nate from the model and repeat the argument until there are only two left (in which case the constraints Scaling the problem by 𝑤 by replacing 𝜍 by 𝑤𝜍̂,we form a discrete set and the process cannot be iterated). have a static maximization problem where different Notice that the only way an agent can obtain disper- choicesofdispersionleadtodifferent𝜆’s. sion as low as 𝑐(𝜁) = |𝑤|/max𝑖|𝜂𝑖| is to choose all Gather the 2𝑑 agents’ portfolios in a matrix Ξ.Then coordinates of 𝜁 as zero except for a (not necessarily we can write unique) 𝑖 with highest |𝜂𝑖| (nonzero, as the Σ matrix is 𝑘 ⊤ assumed invertible), in which the position should be 𝛼 ΞΔ = ΓΠΛ , (18) 𝑤/𝜂𝑖; note that in case of nonuniqueness, the minimum dispersion is not attained by mixing two opportuni- Δ 2𝑑 × 2𝑑 where is the diagonal invertible matrix ties, except in the case 𝛼=1.Thisresolvesthespecial with the agents’ 𝛿 multipliers on the main diago- Π 2𝑑 × 2𝑑 case. Obviously, a minimum-dispersion portfolio is nal, is the diagonal invertible matrix indispensable, as some agent would choose minimum ( 2𝑑−1 ) with the binomial coefficients 𝑗−1 on the main dispersion. However, an agent choosing higher dis- Γ (𝜑 ,...,𝜑 ) diagonal, is the matrix of the funds 1 2𝑑 , persioncouldverywellchoosetwodifferentopportu- Λ (−𝜆) and is the Vandermonde matrix of the ’s, nities, as the minimum-dispersion portfolio may not that is, with row 𝑗 being the geometric sequence ] 2 2𝑑−1 ⊤ payoffverywellintermsof 𝑖 (say, it could be zero). (1, −𝜆𝑗,(−𝜆𝑗) ,...,(−𝜆𝑗) ) ,andinvertibleasthe 𝜆’s are distinct. It remains to find the rank of Γ,anditfollowsby Remark 12. In item (c) the first part asserts that all funds properties of Vandermonde determinants and their are needed in order to satisfy all agents, even all risk-averse, ℓ≤2𝑑 ] minors. Pick rows each with 𝑖 nonzero; but the last sufficiency claim does not; although any level of ]2𝑑−1 these rows are then 𝑖 times a geometric sequence dispersion will be chosen by some agent, it is not necessary so 2𝑑−1 (1, 𝜂𝑖/]𝑖,...,(𝜂𝑖/]𝑖) ),andwehavefullrankwhen- that any 𝑤̂ will be chosen. Assume, with no claim to realism, ever these rows have 𝜂𝑖/]𝑖 distinct but not if two such that all ]𝑖 <0; then the opportunities will be shorted, and any ⊤ ratios coincide. Let ℓ be the maximum number of (positive) upper bound on 𝜁 𝜂 would be inactive. linearly independent ]𝑖 =0̸ rows, and form a matrix of these rows and an arbitrary nonnull row of the Using the leverage constraint, we can extend the separa- 2𝑑 form (0,...,0,𝜂𝚤 ) (equivalent to ]𝚤 =0=𝜂̸ 𝚤), if tion result to agent-specific leverage-dependent interest rates as follows. Suppose that agent number 𝑎 has interest spread of there is one. If such a nonnull row does exist and ⊤ ⊤ ℓ<2𝑑= 𝑟𝑎 =𝑟𝑎(𝜉 1)=𝑟𝑎(𝜁 𝜂) relative to the risk-free opportunity; ( thenumberofcolumns),itisanother ⊤ linearly independent row. intuitively it makes sense that 𝑟𝑎 has the same sign as 𝜁 𝜂 −𝑤 (if it is interest paid). Then the agent’s excess return at leverage The last statement follows as the unconstrained opti- ⊤ 𝑤̂ is not anymore 𝜁 ^,but mumisspannedby(10),namely,thesinglefund𝜑1. 𝜁⊤^ −𝑟 𝜁⊤𝜂. (d) This part will implicitly use Remark 3 so that a 𝑎 (20) continuum of multiplier pairs will actually be chosen The following property then easily carries over from the by different agents. Observe that, in the even-power classical case. case, (13) does not yield (15), but Corollary 13. Theorem 11 applies to the case of individual 𝑘 ]𝑖 −𝜆𝜂𝑖 ]𝑖 −𝜆𝜂𝑖 leverage-dependent interest rate just as for constrained lever- 𝜁𝑖 =( ) sign ( ) (19) 𝛿𝛼 𝛿𝛼 age. Also, it admits 𝐿=𝐿𝑎 individual. 8 Journal of Probability and Statistics

Proof. For whatever choice of 𝜍, 𝑤̂ agent 𝑎 considers, the Theorem 15 (pseudoisotropic CAPM when the embedding ⊤ −𝑟𝑎𝜁 𝜂 =−𝑟𝑎(𝑤)̂ 𝑤̂ term goes outside the maximization, index exceeds 1). Consider a market of agents trading a given and the problem reduces to the problem for an agent with supply of risky opportunities with excess returns 𝜇 + X and one wealth 𝑤=𝑤̂,choice𝜍, and no risk-free opportunity, except risk-free opportunity, where X is pseudoisotropic with standard 1 𝐷 that agent 𝑎’s position in the risk-free opportunity does not 𝜍∈C (R \{0}) (in particular: if the embedding index is >1). vanish. Then the hypothesis of Proposition 14 applies, with 𝑆 being asingleton.IfX is furthermore Σ-transformed 𝛼-symmetric for 𝛼>1, then the betas are (uniquely) given as 7. When Do We Have a Capital Asset Pricing Model? Σ⊤𝜉∗ ⟨𝛼−1⟩ 𝛽 = Σ ( ) . 󵄩 ⊤ ∗󵄩 (21) 󵄩Σ 𝜉 󵄩 This section establishes a Capital Asset Pricing Model for the 𝛼 pseudoisotropic distributions provided the embedding index Observe that we cannot obtain the so-called zero-beta exceeds 1, and there is a risk-free opportunity. We will in this CAPM where no risk-free opportunity exists, as we do case obtain some elements of the elliptical CAPM: there is not have two-fund separation; we cannot then claim that a the Markowitz bullet (namely, a (strictly!) convex risk/return market aggregate of (agents’ individual) efficient portfolios is set for the risky opportunities), a pricing characterization efficient. formula in the form of risk free return plus beta times market excess return times, and a securities market line where the Remark 16. CAPM versions valid for integrable symmetric agents will adapt. stable X are recovered as corollaries: first, the symmetric- Σ ACAPMcanbededucedassuming tradeoff between stable CAPM of Fama [23], who assumed -transformed iid’s, 𝜉⊤𝜇 is precisely Theorem 15 with the additional assumption that excess return ( , desired) and some dispersion functional, 𝛼>1 and so there is nothing novel to the following derivation save is the index of stability and of symmetry. for the fact that (shifted) pseudoisotropy makes the location- Also we obtain and generalize even within the class of dispersion ansatz valid for all agents. Apart from that, the symmetric-stables the CAPM of Belkacem et al. [37] and of Gamrowski and Rachev [38]. Their approach employs the argument mimics a textbook approach and we only sketch 2.7 it: starting from a position in a location-dispersion efficient covariation (see [26, Section ]) which unlike covariance ∗ portfolio 𝜉 ≠ 0 (same for all agents, up to scaling) the agent is not symmetric: we speak of the covariation of a security’s return on (not “and”!) the market portfolio’s return; dividing can then consider buying a (sufficiently small) portfolio 𝛿 ∗ ∗ this quantity by the dispersion as quantified by the standard and scale the risky portfolio 𝜉 by a factor 1−𝑏(𝛿)/𝜍(𝜉 ) 𝜍(⋅), we get a nonsymmetric “correlation coefficient” which as to maintain the portfolio returns dispersion 𝜍(𝛿 +(1− ∗ ∗ ∗ becomes the security’s beta. Indeed, formulating Theorem 15 𝑏(𝛿)/𝜍(𝜉 ))𝜉 ) fixed at level 𝜍(𝜉 ), this implicitly defining 𝑏. 𝜍 ∇𝑏(0) = ∇𝜍(𝜉∗)=:𝛽⊤ in terms of the embedding index and the shape of extends Formal differentiation yields .Bythe not only the CAPM of [37, 38] but also the covariation func- assumed efficiency, 𝛿 = 0 must maximize location given ⊤ tion itself both beyond symmetric stability and to many non- dispersion, yielding the formal first-order condition 𝜇 = ⊤ ∗ ∗ integrable stable cases. The latter is not surprising though: the (𝜇 𝜉 /𝜍(𝜉 ))∇𝑏(0) 𝛼 . Without rigorously defining “CAPM,” result for an 𝛼-symmetric 𝛼-stable X and 𝑅 being 𝛼-stable 󸀠 󸀠 we give the following stylized fact. (𝛼≤1≤𝛼) should not change if we instead consider X 𝑅 󸀠 󸀠 where 𝑅 ≡1and X fl X𝑅,whichis𝛼-symmetric 𝛼-stable. Proposition 14 (location-dispersion CAPM). Suppose that the excess returns are 𝜇 + X and a risk-free opportunity exists and that each agent chooses portfolio (unrestricted) as to trade 8. Outline: Dynamic Models ⊤ off the value of 𝜉 𝜇 (of which more is preferred) against only Inheriting the Separation Properties of adispersionmeasure𝜍(𝜉) which is positive for 𝜉 ≠ 0 and the Static Model homogeneous of degree one and has a subdifferential 𝑆 at ∗ the market portfolio 𝜉 (defined as the total risky investment The results generalize to dynamic models where the price made in the economy, or, by homogeneity, an arbitrary positive processes have the appropriately distributed increments. For scaling). a motivating example, reconsider the single-period market 𝜇 = treated this far as a two-stage decision with preferences Then we have a CAPM with excess returns satisfying over initial consumption, terminal consumption, and bequest (𝜉∗⊤𝜇/𝜍(𝜉∗))𝛽 𝛽 ∈𝑆 ,forsome . (= whatever remains), with an initial investment decision. Should a dominance result like Theorem 9 apply, the agent The hypothesis of this assertion is however a theorem can improve a strategy that uses portfolio 𝜉 the same way: under the assumption of pseudoisotropy and embedding 𝜉 𝜉∗ =𝜍(𝜉)𝜑, 1 keep initial consumption, replace by and, index above 1; then 𝜍 is C outside 0,andeachagent(bythe rather than keeping the excess wealth, increase terminal ∗ ⊤ definition of “agent” in this paper) will choose a nondomi- consumption by (𝜉 −𝜉) 𝜇𝑅0, leaving the bequest unchanged nated portfolio return, which by Theorem 9 is unique up to in distribution. This should be preferable to an agent who scaling. We summarize the following. prefers more to less, and we will adapt the preference Journal of Probability and Statistics 9

assumption to the dynamic setting by modifying the mass- possiblyanullset.Supposethatdiscountedwealthattime𝑡𝑚, transfer criterion, for brevity omitting risk aversion in this 𝑚∈N,isgivenby section. The approach is based on [21], which in turn is based on an approach of Khanna and Kulldorff [39] which makes 𝑌(𝑡𝑚)=𝑦0 −𝛾(𝑡𝑚) a somewhat less rigorous (but very neat!) argument for a 𝑚−1 (22) geometric Brownian Black-Scholes-type model. ⊤ + ∑ 𝜉 (𝑡𝑛) [𝜇 (𝑡𝑛)+Σ (𝑡𝑛) Z (𝑡𝑛+1)] . Wetakeasgivenasequence 𝑡0 <𝑡1 <𝑡2 <⋅⋅⋅,andaprob- 𝑛=0 ability space (Ω, F, P) equipped with a filtration generated 𝑡 𝜇(𝑡 ) 𝜇 by a sequence of independent Z(𝑡𝑛); F0 is generated by Assume that at time 𝑛 the following hold with 𝑛 for X fl Σ(𝑡 )Z(𝑡 ) the null sets of F0 and inductively F𝑛 by F𝑛−1 and Z(𝑡𝑛). and 𝑛 𝑛+1 : Fixing initial cumulative consumptiontozerowithoutlossof (a) If there is a risk-free opportunity, assume the hypothesis generality and initial (but postconsumption!) wealth at given of Theorem 9 is satisfied with 𝐻 fl the (radial) portfolio 𝑦 arbitrary 0, we extend the mass-transfer concept of stochas- restriction at time 𝑡𝑛. Then for every agent in the tic dominance over the probability laws of consumption- dynamicmarketandeachstrategy(𝛾, 𝜉),thereisone (𝑌,𝛾) wealth process pairs as follows. strategy which at time 𝑡𝑛 uses the portfolio 𝜍(𝜉(𝑡𝑛))𝜑, and which leads to a preferred wealth-consumption (𝑦 ,⪰) Definition 17. By an agent we mean a pair 0 of wealth process. 𝜑 (given in Theorem 9) is common to all agents. 𝑦0 ∈ R at time 𝑡0 and a partial ordering ⪰ over adapted ∗ ∗ (b) Ifthereisnorisk-freeopportunityassumethehypothe- process pairs such that (𝑌 ,𝛾 ) is weakly preferred over (𝑌,𝛾) sis of some part (a) to (e) of Theorem 11 holds at time 𝑡𝑛. whenever there is a.s. nondecreasing adapted process 𝛾̂ with ∗ ∗ Then analogously, the conclusion of the respective part 𝛾(𝑡̂ 0)=0such that (𝑌 ,𝛾 ) ∼ (𝑌,𝛾 + 𝛾)̂ . of Theorem 11 holds at time 𝑡𝑛 (with the same funds, as By a strategy we mean an adapted process pair therein). {(𝛾(𝑡𝑛), 𝜉(𝑡𝑛))}𝑛=0,1,... with 𝛾(𝑡0)=0, where we assume 𝜉(𝑡𝑛) to take values in a set of precisely one of the two following forms In words, this means that the dynamic model inherits, (the choice of which is predetermined and nonrandom): time-by-time, the separation properties that the distribution either a given closed radial set 𝐻𝑛 (if a risk-free opportunity would infer in a single-period model. ⊤ exists) or to the set {𝜉(𝑡𝑛) 1 =𝑌(𝑡𝑛)} for each agent (if it does A proof can easily be constructed from the proof of not exist). the single-period model by following [21], which shows the elliptical or stable case in the more complicated continuous Remark that we do not want the agent to care about pre- time model, based on Khanna and Kulldorff [39] for the consumption wealth other than through consumption and Gaussian case. The essence is that we can simply consume the postconsumption wealth, and Theorem 18 will be formulated excess at each time, and the (strong) Markov property will accordingly, defining 𝑌(𝑡𝑛+1) netoftheconsumptionattime leave us with the same opportunity set for all future. To see 𝑡𝑛+1.Notealsothatasweonlycareaboutthelaw,thatis,the this,considerasingletime𝑡𝑛 for which the hypothesis holds finite-distributional distributions, we need only to show sep- true. Whatever portfolio 𝜉 the strategy yields, we can replace ∗ ∗ aration for every natural number of periods. It would be nat- it by some 𝜉 ofthesamescale𝜍(𝜉 )=𝜍(𝜉) which uses the ural to restrict the strategies further, for example, requiring fund. Imagine for the moment that we simply dispose of the ∗ ⊤ insolvent agents to close out their positions and stay on a fixed excess (𝜉 − 𝜉) 𝜇(𝑡𝑛) (≥0); then we have merely replaced consumption per period (note that agents will become insol- next period’s wealth and consumption by one of the same vent in this model); however, as we only compare strategies (conditional) distribution, and thus the opportunity set, the {(𝑌∗,𝛾∗)} {(𝑌∗,𝛾∗)} pairwise, we can show separation without any such admissi- set of possible laws of 𝑡>𝑡𝑛 (thus of 𝑡≥𝑡0 ), bility restriction, and then discard any nonadmissible strate- is the same; by assumption this replacement is (weakly) gies, as long as the opportunity set only depends on past preferred by every agent. Now drop the fictitious disposal and ∗ ⊤ throughtheagent’shistoryandisnotrestrictedbyincreasing 𝑡𝑛+1 (𝜉 − 𝜉) 𝜇(𝑡𝑛) ∗ increase the consumption at time by .By consumption from 𝛾 to 𝛾 . assumption,thisincreaseis(weakly)preferredbyeveryagent. As we have seen from the single-period model, the independent radial scalings in (1) play no part in the result, 9. Concluding Remarks and neither does the distribution of the risk-free opportunity; we do not lose any generality by normalizing the radials to the It was natural to develop portfolio theory for shifted sym- constant 1 and the risk-free return to the constant 0. metric stable returns, from the defining property of stability, We then have the dynamic model and the separation as long as one did not realize that matrix multiplication was theorem as follows, which in the interest of brevity is not sufficient to capture the dependence structure. Indeed, formulated a bit loosely especially in part (b). with preferences only over portfolio return, not of the returns of the individual opportunities, the defining property of Theorem 18. Suppose that each Z(𝑡𝑛+1) is pseudoisotropic with pseudoisotropic distributions almost begs the question of (nonrandom) standard 𝜍𝑛. Assume given nonrandom 𝜇(𝑡𝑛) and portfolio separation, and this paper has extended the classical Σ(𝑡𝑛), the latter satisfying for each agent and each strategy portfolio theory to those distributions. Doing so, we are able ⊤ 𝜉(𝑡𝑛) Σ(𝑡𝑛)Z(𝑡𝑛+1)=0 only on the event {𝜉(𝑡𝑛)=0} and to recast the theory for symmetric stables as well, in a way 10 Journal of Probability and Statistics that not only is much less restrictive but shows that the Acknowledgments basic properties associated with symmetric stables are not the crucial ones; the essential properties are in the geometry of The author has been affiliated with The Financial Supervisory the standard 𝜍. Authority of Norway; the content does not reflect their views. Within the class of pseudoisotropic distributions, the Preliminary special cases of Theorem 11 parts (b) and (e) possible diversification properties that follow from suitable that appeared in [36] were obtained under funding from the integrability are not integrability properties; they are geomet- Research Council of Norway. ric embeddability properties. For 𝛼-stables, the assumption that 𝛼>1is sufficient (but not necessary) that the index References of embedding is >1,whichagainissufficientthatthe𝜍- spheres are smooth, from which the qualitative properties [1] J. Tobin, “Liquidity preference as behavior toward risk,” The of the classical cases are recovered. Thus for the symmetric Review of Economic Studies,vol.25,no.2,pp.65–86,1958. stables, it is not the index of stability that is crucial. [2] D. Cass and J. E. Stiglitz, “The structure of investor preferences The class of (transformed) 𝛼-symmetric distributions and asset returns, and separability in portfolio allocation: a highlights this: the 𝛼-symmetric stable distributions behave contribution to the pure theory of mutual funds,” Journal of in some sense, the same for given 𝛼, no matter which index Economic Theory,vol.2,no.2,pp.122–160,1970. 𝛼 of stability. Inspecting Fama’s characterization [16, formulae [3] W. Schachermayer, M. Sˆırbu, and E. Taflin, “In which financial (14)–(17)] we note that they hold true if we merely replace his markets do mutual fund theorems hold true?” Finance and assumption of 𝛼-stability by an assumption of 𝛼-symmetry. Stochastics,vol.13,no.1,pp.49–77,2009. This is well understood for the elliptical case (Owen and [4] N. C. Framstad, “Coherent portfolio separation—inherent sys- Rabinovitch [7]), and we have established a direct extension, temic risk?” International Journal of Theoretical and Applied under the assumption that a risk-free opportunity exists. Finance,vol.7,no.7,pp.909–917,2004. For the case without risk-free opportunity, the elliptical [5]E.D.Giorgi,T.Hens,andJ.Mayer,“Anoteonreward-risk distributions have unique properties, and it is directly con- portfolio selection and two-fund separation,” Finance Research nected to the shape of the 𝜍-spheres. For the (transformed) Letters,vol.8,no.2,pp.52–58,2011. 𝛼-symmetric distributions, the geometry enables us to make [6] S. A. Ross, “Mutual fund separation in financial theory—the a sequence of symmetry indices that admit weaker separation separating distributions,” Journal of Economic Theory,vol.17,no. results, namely, 𝑘-fundseparationresultfor1 + 1/(𝑘 − 2,pp.254–286,1978. 1)-norm-symmetric variables when 𝑘 is even, generalizing [7] J. Owen and R. Rabinovitch, “On the class of elliptical distribu- the elliptical case 𝑘=2. Thus within the class of 𝛼- tions and their applications to the theory of portfolio choice,” symmetric distributions (𝛼>1), separation now looks like an The Journal of Finance,vol.38,no.3,pp.745–752,1983. exceptional property, one of a sequence inside a continuum, [8] G. Chamberlain, “A characterization of the distributions that although it is a generalization of a property of a family widely imply mean-variance utility functions,” Journal of Economic considered an adequate approximation of reality (at least Theory,vol.29,no.1,pp.185–201,1983. implicitly, ellipticity is necessary for linearity of regression in [9] H. Markowitz, “Portfolio selection,” The Journal of Finance,vol. dimension >2, Hardin Jr. [40]). 7, no. 1, pp. 77–91, 1952. Althoughportfolioseparationisatheoreticalresultwhich [10] B. de Finetti, “Il problema dei ‘pieni’,” Giornale dell Istituto has historically not concerned fit to real data, we make a Italiano degli Attuari,vol.11,pp.1–88,1940. remark on applicability. Various heavy-tailed models have [11] H. Markowitz, “De Finetti scoops Markowitz,” Journal of Invest- been introduced to find a better fit to data, though one can ment Management,vol.4,no.3,pp.5–18,2006. question whether a low order of integrability is in line with [12] A. D. Roy, “Safety first and the holding of assets,” Econometrica, the real world. However, the exact asymptotical tail index vol.20,no.3,pp.431–449,1952. is not necessarily the scope of application for a financial [13] P. A. Samuelson, “General proof that diversification pays,” The model. Indeed, with the emergence of quantile measures (the Journal of Financial and Quantitative Analysis,vol.2,no.1,pp. infamous value-at-risk), financial risk is often measured ina 1–13, 1967. way that totally disregards the order of integrability. Not only [14] K. Borch, “A note on uncertainty and indifference curves,” The does this make the objections less valid, one does not extract Review of Economic Studies,vol.36,no.1,pp.1–4,1969. from the model the properties that are most questionable, [15] M. S. Feldstein, “Mean-variance analysis in the theory of but also the nonsubadditivity of value-at-risk may in certain liquidity preference and portfolio selection,” The Review of cases penalize diversification; behaviour according to this Economic Studies,vol.36,no.1,pp.5–12,1969. does in fact require nonintegrability, and even that is not [16] E. F. Fama, “Portfolio analysis in a stable Paretian market,” sufficient: in the pseudoisotropic model this translates into Management Science, vol. 11, no. 3, pp. 404–419, 1965. 𝜍 the -sphere having corner (nondifferentiability points) and [17] P. A. Samuelson, “Efficient portfolio selection for Pareto- an embedding index of at most one. Levy´ investments,” The Journal of Financial and Quantitative Analysis,vol.2,no.2,pp.107–122,1967. Conflict of Interests [18] R. A. Agnew, “Counter-examples to an assertion concerning the normal distribution and a new stochastic price fluctuation The author declares that there is no conflict of interests model,” The Review of Economic Studies,vol.38,no.3,pp.381– regarding the publication of this paper. 383, 1971. Journal of Probability and Statistics 11

[19] I. J. Schoenberg, “Metric spaces and completely monotone [37] L. Belkacem, J. L. Vehel,´ and C. Walter, “CAPM, risk and functions,” Annals of Mathematics,vol.39,no.4,pp.811–841, portfolio selection in ‘𝛼-stable markets’,” Fractals,vol.8,no.1, 1938. pp. 99–115, 2000. [20] I. J. Schoenberg, “Metric spaces and positive definite functions,” [38] B. Gamrowski and S. T.Rachev, “Atestable version of the Pareto- Transactions of the American Mathematical Society,vol.44,no. stable CAPM,” Mathematical and Computer Modelling,vol.29, 3, pp. 522–536, 1938. no. 10–12, pp. 61–81, 1999. [21] N. C. Framstad, “Ross-type dynamic portfolio separation, [39] A. Khanna and M. Kulldorff, “A generalization of the mutual (almost) without Itoˆ stochastic calculus,” Memorandum fund theorem,” Finance and Stochastics,vol.3,no.2,pp.167– 21/2013, Department of Economics, University of Oslo, Oslo, 185, 1999. Norway, 2013. [40] C. D. Hardin Jr., “On the linearity of regression,” Zeitschriftur f¨ [22] S. Ortobelli, I. Huber, S. T. Rachev, and E. S. Schwartz, “Portfolio Wahrscheinlichkeitstheorie und Verwandte Gebiete,vol.61,no.3, choice theory with non-Gaussian distributed returns,”in Hand- pp. 293–302, 1982. book of Heavy Tailed Distributions in Finance,S.T.Rachev,Ed., pp.547–594,North-Holland,Amsterdam,TheNetherlands, 2003. [23] E. F. Fama, “Risk, return, and equilibrium,” Journal of Political Economy,vol.79,no.1,pp.30–55,1971. [24] S. Cambanis, R. Keener, and G. Simons, “On 𝛼-symmetric multivariate distributions,” Journal of Multivariate Analysis,vol. 13, no. 2, pp. 213–233, 1983. [25] L. P. Østerdal, “The mass transfer approach to multivariate discrete first order stochastic dominance: direct proof and implications,” Journal of Mathematical Economics,vol.46,no. 6, pp. 1222–1228, 2010. [26] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman&Hall,NewYork,NY,USA,1994. [27] M. L. Eaton, “On the projections of isotropic distributions,” The Annals of Statistics,vol.9,no.2,pp.391–400,1981. [28] B. H. Jasiulis and J. K. Misiewicz, “On the connections between weakly stable and pseudo-isotropic distributions,” Statistics and Probability Letters,vol.78,no.16,pp.2751–2755,2008. [29] J. K. Misiewicz, “Substable and pseudo-isotropic processes. 𝛼 Connections with the geometry of subspaces of L -spaces,” Dissertationes Mathematicae (Rozprawy Matematyczne),vol. 358, 1996. [30] A. Koldobsky, “A note on positive definite norm dependent functions,” in High Dimensional Probability V: The Luminy Volume, C. Houdre,V.Koltchinskii,D.M.Mason,andM.´ Peligrad, Eds., vol. 5 of Institute of Mathematical Statistics Lecture Notes—Collections,pp.30–36,InstituteofMathematical Statistics, Beachwood, Ohio, USA, 2009. [31] A. Koldobsky, “Positive definite functions and stable random vectors,” Israel Journal of Mathematics,vol.185,no.1,pp.277– 292, 2011. [32]N.J.Kalton,A.Koldobsky,V.Yaskin,andM.Yaskina,“The geometry of 𝐿0,” Canadian Journal of Mathematics,vol.59,no. 5, pp. 1029–1049, 2007. [33] A. D. Lisitskii, “The Eaton problem and multiplicative proper- ties of multidimensional distributions,” Teoriya Veroyatnoste˘ıiee Primeneniya, vol. 42, no. 4, pp. 696–714, 1997. [34] P. Hall, “A comedy of errors: the canonical form for a stable characteristic function,” The Bulletin of the London Mathemati- cal Society,vol.13,no.1,pp.23–27,1981. [35] T. Gneiting, “Criteria of Polya´ type for radial positive definite functions,” Proceedings of the American Mathematical Society, vol. 129, no. 8, pp. 2309–2318, 2001. [36] N. C. Framstad, Portfolio Separation without Stochastic Calculus (Almost), Preprint Pure Mathematics 10/2001, University of Oslo, Oslo, Norway, 2001. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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