Generalizations of elliptical distributions, and their portfolio - PowerPoint PPT Presentation

A primer on portfolio separation Market models Distributions Discussion Generalizations of elliptical distributions, and their portfolio separation properties. Ross-type portfolio separation with α -stable, α -symmetric and pseudo-isotropic distributions, with generalizations N.C. Framstad 12 1 Dept. of Economics, University of Oslo 2 also affiliated with (the usual disclaimer applies) The Financial Supervisory Authority of Norway Stochastic analysis seminar, Oslo, April 25th 2012 Friday economics seminar, Oslo, April 27th 2012

A primer on portfolio separation Market models Distributions Discussion A primer on portfolio separation 1 Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies Market models 2 The single-period market Dynamic models Distributions 3 Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α -symmetric X , no riskless opportunity α -stable X Discussion 4 Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion Q: Under what conditions can a large number («market») of investment opportunities, be replaced by a small(er) number of market indices («funds») without welfare loss to the investors? Reduces dimensionality to a small(er) hyperplane. (Models like e.g. CAPM also assume such a property.) When? Answer will depend on the investors’ preferences, and on market: returns distributions and, if applicable, portfolio constraints. It is common to treat separately the case w/o numéraire investment («no risk-free investment») ... which can be formalized as a portfolio constraint (position = 0). Other portfolio constraints are not so frequently discussed in the literature. Original case: preferences as a quadratic utility function, or returns being Gaussian.

A primer on portfolio separation Market models Distributions Discussion Brief history: Mean–variance portfolio optimization: de Finetti (Giornale dell’ Istituto Italiano degli Attuari, 1940) scoops Markowitz (J. Finance 1952) (and Roy (Econometrica 1952) ). Separation: Tobin (RES 1958) , quadratic utility or Gaussians. Conjectured generalization to any two-parameter distribution family; this disproved by Samuelson (J. Finan. Quantit. 1967) , and Feldstein (RES 1969) , NHH’s Karl Borch (ditto) . iid symmetric α -stable returns + risk-free opportunity admit separation: Fama (Management Sci. 1965) . Cass/Stiglitz (JET 1970) : characterizes the utility functions which exhibit separation across wealth. In view of their predecessors, conjecture that α -stability is necessary for the returns distribution version of the theorem ... ... conjecture disproved by Agnew (RES, 1971) . Ross (JET 1978) characterizes (in stochastic dominance terms) the separation-admitting returns distributions. This presentation is about distributions and Ross’ criteria. (Further references follow.)

A primer on portfolio separation Market models Distributions Discussion

A primer on portfolio separation Market models Distributions Discussion Cass–Stiglitz (JET 1970): Utility functions A primer on portfolio separation 1 Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies Market models 2 The single-period market Dynamic models Distributions 3 Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α -symmetric X , no riskless opportunity α -stable X Discussion 4 Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion Cass–Stiglitz (JET 1970): Utility functions The preferences side of it: For the single period model, Cass and Stiglitz characterize the utility functions which exhibit two-fund portfolio separation no matter the returns distribution (as long as the utility function is defined for all possible terminal wealth states). The material content: no matter what wealth. Agents with such a utility function but different wealth, choose same two funds (but different allocation between them). Monetary separation (case with riskless investment opportunity; then this can be chosen as one fund) for (modulo linear translations) power and exp. Cass–Stiglitz type theorems are still being explored, e.g. Schachermayer et al. (Finance Stoch. 2009) , cont. time. Also include risk measure-induced choices (NCF (IJTAF 2005) , Giorgi et al., (Finance Research Letters 2011) ).

A primer on portfolio separation Market models Distributions Discussion Cass–Stiglitz (JET 1970): Utility functions

A primer on portfolio separation Market models Distributions Discussion Ross (JET 1978): Returns distributions A primer on portfolio separation 1 Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies Market models 2 The single-period market Dynamic models Distributions 3 Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α -symmetric X , no riskless opportunity α -stable X Discussion 4 Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion Ross (JET 1978): Returns distributions For the single period model, Ross gives a characterization of the returns distributions which exhibit k -fund separation for all preferences obeying first-order stochastic dominance: = Z ∗ + [nonpositive r.v.] d Z � Z ∗ if Z (i.e.: preferences only over total returns distributions only, of expected increasing utility type) Also considered in the literature: second-order stochastic dominance (colloquially, Z d = Z ∗ + [independent zero-mean r.v.] Z � Z ∗ if for the risk-averse subclass of preferences; definition must be refined if non-integrable distributions are considered.) Throughout this presentation, agents will be assumed to order according to first-order stochastic dominance. Distributional assumptions must ensure the ordering is total. Argument may be modified to cover portfolio constraints.

A primer on portfolio separation Market models Distributions Discussion Ross (JET 1978): Returns distributions

A primer on portfolio separation Market models Distributions Discussion Distributions for which Ross 2-fund separation applies A primer on portfolio separation 1 Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies Market models 2 The single-period market Dynamic models Distributions 3 Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α -symmetric X , no riskless opportunity α -stable X Discussion 4 Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion Distributions for which Ross 2-fund separation applies Will cover (generalizations of!) the following: The Gaussian (since Tobin) and more: The class is now known to include all elliptical distributions: Chamberlain (JET 1983) : under square integrability, ellipticity ⇔ any expected utility is f ( mean,variance ) , and in a separation-admitting manner. Owen/Rabinovich (J. Finance 1983) : under integrability, the ellipticals admit (i) separation, (ii) CAPM, and (iii) linear regression. These results: 2-fund separation with or without risk-free opportunity, one fund being the risk-free or the minimum variance portfolio (or analogue if infinite variance.) Fama: 2-fund sep. for risk-free + iid symmetric α -stables. Turns out: the case w/o risk-free opportunity does not follow likewise. Special cases will follow. A vector of iid symmetric α -stables are not elliptical (except the Gaussian), contrary to some misleading terminology. Symmetry: − X d = X . Ellipticity is stronger. And: Dependence structure for ❛ -stables cannot be expressed by matrix multiplication.

A primer on portfolio separation Market models Distributions Discussion Distributions for which Ross 2-fund separation applies

A primer on portfolio separation Market models Distributions Discussion The single-period market A primer on portfolio separation 1 Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies Market models 2 The single-period market Dynamic models Distributions 3 Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α -symmetric X , no riskless opportunity α -stable X Discussion 4 Modeling issues For future research