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Symmetric and Pseudoisotropic Distributions: -Fund Separation 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 -Symmetric and Pseudoisotropic Distributions: -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 under an embedding in L condition if 1<≤2, with the betas given in an explicit form. For the -symmetric subclass, the market without risk-free investment opportunity admits 2-fund separation if = 1+1/(2−1), ∈N, generalizing the classical elliptical case =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 -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 -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 -vector is called a portfolio ifittakesvaluesina before modern portfolio theory. given set to be denoted by or (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 ⟨⟩ risk-free case admits so-called “monetary separation”). This power notation fl || sign() even to vectors, element- paper considers the generalization to the (shifted) so-called ⟨⟩ ⟨⟩ ⟨⟩ ⊤ ⟨⟩ wise: x =( ,..., ) .(Noticethatx → x is pseudoisotropic distributions,amultivariateclassofsymmet- 1 invertible.) Matrices are Greek uppercase boldfaced letters. ric random variables such that all linear combinations of Aset⫌{0} is radial if it is composed as a union of half- thecoordinatesareofthesametype.Thepseudoisotropic x ∈⇔x ∈∀>0 distributions admit a dispersion quasinorm (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 -stable random variables variable and its distribution are symmetric if X ∼−X;then as established already by Fama [16] (for results in + X is called shifted symmetric for nonrandom . continuous time: this author [21] and Ortobelli et al. [22]). Assumption 1. We will allow for constraints to be specified (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 -symmetric distributions with >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 2-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 = 1+1/(2−1), ∈N,thatis,one- particular, the independent radial scalings 0 and are never and-an-oddth, where =1subsumes the elliptical Dirac at zero; if any of these is constant, it is without loss of distributions =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- 2 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 Section 3 state the single-period market model and review Consider a single-period investment allocating wealth ∈N stochastic dominance. Section 4 will introduce pseudoiso- between in “risky” investment opportunities and −1⊤ 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 0 per monetary unit invested. ration without risk-free opportunity, and CAPM. Section 8 Writing the risky returns vector as 01 + 0 + X for some sketches how a dynamic model inherits the separation prop- nonrandom location parameter (resembling a representa- erties from the static model. tion common for elliptical distributions, e.g., Cambanis et al. [24]), the portfolio
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