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. Framstad12

• 1Dept. of Economics, University of Oslo

2also 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

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A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

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Market models The single-period market Dynamic models

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Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

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Discussion 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

• f 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

• n 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

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A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

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Market models The single-period market Dynamic models

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Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

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Discussion 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

• pportunity; 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

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A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

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Market models The single-period market Dynamic models

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Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

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Discussion 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 Z ∗ if Z

d

= Z ∗ + [nonpositive r.v.] (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 Z ∗ if Z d = Z ∗ + [independent zero-mean r.v.] 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

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A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

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Market models The single-period market Dynamic models

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Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

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Discussion 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

• pportunity, 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

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A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

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Market models The single-period market Dynamic models

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Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

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Discussion Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion The single-period market

n «risky» investment opportunities. One numéraire investment opportunity (usually referred to as «riskless»), enumerated with coordinate no. 0.

Absence of such will be modeled by a constraint that the risky investment positions equal total wealth.

Returns distribution in terms of excess return over numéraire.

Numéraire return: ˆ X «Risky» opportunities return vector: ˆ X1 + b ˆ R + XR where the ˆ R and R are there in order to fit into the language of elliptical/sub-Gaussian/sub-stable distributions

ˆ R and R independent, and both ≥ 0. We can specify X conditionally on ( ˆ X, ˆ R, R). It will turn out that can work as if this triplet is independent of X ... ... or even constant, fitting the usual terminology of the numéraire as «risk-free». (Hardly any generality gained by the ˆ

• R. Think of it as 1.)

Market assumed free of arbitrage opportunities and of redundant investment opportunities.

A primer on portfolio separation Market models Distributions Discussion The single-period market

If w is initial wealth and u is the portfolio vector over the riskies, then terminal wealth is: w ˆ X + u⊤ b ˆ R + XR

• Whenever needed, we will rule out negative-wealth agents from

the market! (Yes, this is a potential restriction.)

Key observation: If u∗⊤X|( ˆ X, ˆ R, R)

d

= u⊤X|( ˆ X, ˆ R, R) then u∗⊤XR

d

= u⊤XR.

If in addition, u∗⊤b ≥ u⊤b then the portfolio return using u∗ first-order stochastically dominates the one using u.

This tacitly assumes that an optimal portfolio exists. However, with possibly risk-seeking behaviour, that is easily violated.

Rather than repeatedly stating the existence assumption (and approximation ramifications), we shall always assume existence

• f optimal portfolio choice.

A primer on portfolio separation Market models Distributions Discussion The single-period market

A primer on portfolio separation Market models Distributions Discussion Dynamic models

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A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

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Market models The single-period market Dynamic models

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Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

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Discussion Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion Dynamic models

Finite-period models: recursive application of single-period case. Continuous time: Consider the following construction: n deflated (with the numéraire) prices evolving according to a geometric Lévy process dSi = Si(bi dt + dZi). Forming a portfolio, self-financed except financing consumption (also deflated) yields a wealth dynamics dY = u⊤(b dt + dZ) − dC where ui equals the value held in opportunity i at the time.

If the investment opportunities are not tradeables, but e.g. insurance premiums/losses, we arrive at the same dynamics without invoking the self-financedness concept.

Now assume Z = Z(t) a Lévy process with Z(1) distributed as the single-period «X».

This is possible if and only if X is infinitely divisible.

A primer on portfolio separation Market models Distributions Discussion Dynamic models

Wealth dynamics dY = u⊤(b dt + dZ) − dC Copy the stochastic dominance argument of the discrete time setup for a small increment:

If (u∗)⊤ dZ

d

= u⊤ dZ but yields higher drift, then consume the excess: Consider the strategy (C(history), u). Then the strategy (C∗, u∗) with dC∗ = dC + (u∗ − u)⊤b dt ( ≥ dC!) has wealth process with the same law, but first-order dominating consumption.

This approach due to Khanna and Kulldorff (Finance Stoch. 1999), Z assumed Brownian, but works equally well for any Lévy process).

Essential property: the simple functions way of defining the Itô integral.

A primer on portfolio separation Market models Distributions Discussion Dynamic models

The single-period case is the more general model the Lévy process model is defined if we have infinite divisibility, ... in which case the results follow from considering simple integrands and considering the discrete-time model ... which follows as a recursive application of the single-period model. No more stochastic calculus required than the Khanna/Kulldorff construction. The remaining part: multivariate distributional theory. Back to the single-period model: w ˆ X + u⊤ b ˆ R + XR

• ,

What distributions for X admit separation?

A primer on portfolio separation Market models Distributions Discussion Dynamic models

A primer on portfolio separation Market models Distributions Discussion Elliptical X

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A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

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Market models The single-period market Dynamic models

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Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

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Discussion Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion Elliptical X

Concept originating with Schoenberg (Annals of Math., 1938). An(y) elliptical distribution can be constructed as follows: Start with U being uniform on the unit sphere, multiply with an invertible matrix: X = KU, scale X with an independent R ≥ 0, translate (i.e., adding the b). Notice: We may scale (with R) before transforming (with K): An UR is called spherical. E.g., R = |N(0, 1)| std. multinormal. In the language of elliptical distributions, it is custom to call M = K K ⊤ the «covariance matrix» and b «mean», regardless

• f what R is (... even if E[R] = +∞!).

Properties: Provided absolute continuity: Elliptical level curves for the density, often referred to as the ’elliptical contours’. What makes separation, is the analogue property for the chf!

E[exp(iϑX)] depends on ϑ only through b⊤ϑ and ϑ⊤Mϑ (positive-definite!).

A primer on portfolio separation Market models Distributions Discussion Elliptical X

The linear regression property: Let X be a finite-mean random vector, put d = # of linear

• dependents. Suppose d ≥ 2 (otherwise empty theory!)

Def.: X has the linear regression property if:

for any linear combinations L0 and L = (L1, . . . , Lk) of the Xi, there is a vector a = (a1, . . . , ak) s.t. E[L0|L] = a⊤L.

Hardin (ZWvG/PTRF 1982): (assuming merely finite-mean, which I simplify to the EX = 0 case) Any elliptical has the linear regression property (any d ≥ 2). Conversely, if X has the linear regression property, then

X is elliptical, or if not: d = 2 and E[X⊤X] = ∞

There are indeed non-elliptical cases: For every p ∈ (1, 2) there exists a non-elliptical (X1, X2) with finite p-moment, possessing the linear regression property. If d > 2 and E[X⊤X] < ∞, the «any k» part of the LRT definition can be weakened to k = 1 and still prove ellipticity. Note: Under ellipticity, L0|L is iself elliptical.

A primer on portfolio separation Market models Distributions Discussion Elliptical X

The portfolio separation property: Textbook approach: minimize variance given mean (and subject to u⊤1 = w if no risk-free opportunity)

Assumes risk-aversion; If no risk-free opportunity and (b, 1) lin.dep., risk-seekers may need a «pure volatility fund» ⊥ b. Concave problem (no constraint qualification), and pedagogically nice – Lagrange multipliers as portfolio weights.

Ross (and Khanna/Kulldorff in continuous time): Max mean given dispersion, without assuming risk-aversion. Under some differences in integrability assumptions: Proposition #1 (Chamberlain and Owen/Rabinovich) An elliptical X admits two-fund separation with or without risk-free

• pportunity. (And monetary separation in the «with» case.)

The funds are the following: (i) the minimum variance portfolio (i.e. riskless or M−11), and (ii) M−1b – or, if linear dependence, a pure volatility fund.

A primer on portfolio separation Market models Distributions Discussion Elliptical X

Proof: put c(u) = √ u⊤Mu; then c > 0 except for u = 0. Unconstrained case:

Assume the agent chooses ˜ u∗. Put Q = c(˜ u∗). By first-order stochastic dominance, ˜ u∗ solves: max

u

u⊤b subject to c(u) = Q By homogeneity, a solution u∗ for Q = 1 will yield solutions ˜ u∗ = Qu∗ for arbitrary Q ≥ 0. Thus any agent has optimum using the same two funds: u∗ and the riskless numéraire.

Case w/o numéraire investment opportunity:

Agent’s choice must solve max

u

u⊤b subject to (c(u))2 = Q2 and u⊤1 = w Lagrange: b = νMu + λ1, funds M−1b and M−11. Degeneracies: if b, 1 lin.dep., augment with some vector ⊥ b.

(which risk-averse agents will not need).

A primer on portfolio separation Market models Distributions Discussion Elliptical X

The proofs are different, and admit different generalizations: The case with risk-free opportunity merely uses homogeneity;

Portfolio constraints to a (closed) radial set – i.e., a family H

• f half-lines from the origin – poses no problem.

Therefore, if short sale is forbidden for some (or all) of the risky investment opportunities, the result with risk-free

• pportunity, carries over. Only the risky fund is modified.

The case without risk-free opportunity uses Lagrange. Constraining to a radial set destroys the setup. In return, it behaves nicely under linear constraints:

Three-fund separation under «no borrowing» ↔ constraint u⊤1 ≤ w. (Just like the no risk-free case, except that the risk-free position does not zero out ;-) .) Even, three-fund separation under interest-rates being different for lending/borrowing or for leverage degree. Another (lin.indep.) linear constraint another fund.

A primer on portfolio separation Market models Distributions Discussion Elliptical X

The Lagrange case also provides for non-traded income. Suppose that e.g. opportunity # n is non-tradeable, agent j has ℓj of it: Pretend it is tradeable, introduce it to the market (possibly amending the budget constraint). Impose the linear constraint un = ℓj. Get another fund M−1en where e⊤

n = (0, . . . , 0, 1).

Observe that this fund does not depend on ℓj, so agents can have different shares of it.

It isn’t necessary that all hold a share of the same random variable, as long as they all have the same correlation with the tradeables. («Correlation» can be defined much the same as «covariance» is, even without integrability assumption.)

Slogan: one non-tradeable another fund.

A primer on portfolio separation Market models Distributions Discussion Skew-elliptical X and generalizations

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A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

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Market models The single-period market Dynamic models

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Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

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Discussion Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion Skew-elliptical X and generalizations

The Lagrange approach applies to (generalized) skew-elliptical distributions in the Azzalini (Scand. J. Statist. 1985) sense: Let E⊤ = (E1, . . . , En, . . . , En+q), be elliptical, and ...

... such that the «covariance» M has form

• Σ

∆

• , Σ is n × n.

Following Díaz-García and González-Farías (J. Korean Statist. Soc.

2008), we form the vector V =

In×n Π Iq×q

• E.

If X is (V1, . . . , Vn)⊤ = (E1, . . . , En)⊤ but conditioned upon (Vn+1, . . . , Vn+q) ∈ some given (translated!) orthant, then X has a (possibly singular extended) skew-elliptical distribution.

Generalization: condition upon (Vn+1, . . . , Vn+q) ∈ Ξ (arbitrary set).

Notice that if Ξ is a singleton, then X is elliptical. So the only substantial generalization is where we have partial, but not full information of these (untradeable) risk factors.

A primer on portfolio separation Market models Distributions Discussion Skew-elliptical X and generalizations

Proposition #2 (NCF, Stat. Prob. Letters 2011) (2 + rank Π) fund separation with or without riskless investment

• pportunity.

Proof by LQ programming, observing that the distribution of u⊤X only depends on u through u⊤Σu and ΠΣu: max

u

u⊤b subject to u⊤Σu = Q and ΠΣu = q Apply Lagrange (and check degeneracies). Extra (lin.indep.) linear constraint? Extra fund!

but: ruling out the risk-free opportunity also removes a fund.

What it says? Each (independent, untradeable, partial) piece

• f information yields one (linear) portfolio adjustment

Just like non-tradeable income does!

A primer on portfolio separation Market models Distributions Discussion Skew-elliptical X and generalizations

A digression: insider traders. The agent with no information on 1Ξ, will choose the fund Σ−1b and the riskless (or if none such, Σ−11). The agent who knows 1Ξ, may also («will also» except degenerate cases) invest in the rows of Π. We cannot tell how much in each. It may require full knowledge of preferences. In which case: if the model is distorted by e.g. correlated individual income from outside the financial market, it might be impossible to tell whether a position orthogonal to the uninformed agent’s two funds, is due to insider information or to other distortions.

A primer on portfolio separation Market models Distributions Discussion Skew-elliptical X and generalizations

A primer on portfolio separation Market models Distributions Discussion Pseudo-isotropic X

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A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

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Market models The single-period market Dynamic models

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Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

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Discussion Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion Pseudo-isotropic X

Pseudo-isotropic (multivar.) distributions generalize the spherical: X is pseudo-isotropic if chf = E[exp(iϑX)] = can be written as a function g(c(ϑ)), where c ≥ 0 is positively homogeneous (a quasi-norm).

For simplicity, assume all coordinates a.s. = 0. Alternative defining property: ϑ⊤X d = h(ϑ)X1, any ϑ ∈ Rn. («All linear combinations are of the same type.») By a linear transformation by some invertible K, we can assume all coordinates identically distributed

(but not independent – the dependence structure is more complex than that, in contrast to the Gaussian)

We can then write portfolio return as w ˆ X + u⊤ b ˆ R + XR = w ˆ X + u⊤K −1 Kb ˆ R + KXR

• Solving for u⊤K −1 instead: Same number of funds.

Assume K = I without loss of generality.

A primer on portfolio separation Market models Distributions Discussion Pseudo-isotropic X

What pseudoisotropics are there? Only known cases are the «α-symmetric» (Cambanis et al.,

• J. Multivariate Analysis 1983): c is (up to linear transformation) the

Lα quasi-norm, some α ∈ (0, 2].

Again, we assume this linear transformation done, i.e., c is the standard Lα quasi-norm. α = 2 are the sphericals. α ∈ (0, 2) behave different: infinite α-moment, and positive density everywhere.

Conjecture: there are no others. (Misiewicz (Soobslich Akad. Nauk

Gruzin, 1988 and Dissertationes Math. 1996), with further support by

e.g. Koldobsky (High Dim. Prob. vol V, 2009)).

If there are others, they must be quite strange – in particular, they must have E[|Xi|ǫ] = ∞, all ǫ ∈ (0, 1).

Symmetric α-stable (SαS) if any two coord’s independent.

A primer on portfolio separation Market models Distributions Discussion Pseudo-isotropic X

Like for the ellipticals, the homogeneity argument immediately yields: Proposition #3 (NCF, preprint 2011) Pseudo-isotropic X with risk-free opportunity ⇒ 2-fund monetary separation. Proof as in the elliptical case, except the form of c.

Admits portfolio constraint to radial set H (e.g. no short sale).

And if so, only the positive homogeneity for ϑ ∈ H is used. Admits generalizations in distributions. Little is known (to me) about such a distribution class, not even any terminology («positively pseudo-isotropic»?) There are well-established examples: certain skew α-stable.

Two-fund separation fails without risk-free opportunity! (Special cases can be will be recovered below.)

A primer on portfolio separation Market models Distributions Discussion Pseudo-isotropic X

A primer on portfolio separation Market models Distributions Discussion «Positively pseudo-isotropic» (... terminology ?) X

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A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

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Market models The single-period market Dynamic models

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Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

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Discussion Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion «Positively pseudo-isotropic» (... terminology ?) X

If the portfolio is constrained to a radial set H, then the chf need only assume the g(c(ϑ)) form on H, for Prop. 3 to carry

• ver.

There are such distributions which are not pseudo-isotropic, e.g. certain skew α-stables (to be covered later), but to my knowledge, there is not even an established name for this class

• f distributions.

What about «positively pseudo-isotropic»?

Arias and Koldobsky (2003?, manuscript) consider the positive-valued subclass – they find that c cannot be a norm.

Again, such distributions exist: let X be a vector of iid completely right-skewed α-stables, α < 1. If −X has such a distribution, then most agents will be «risk averse» (careful to define this!) – the investment opportunities have expected yield of negative infinity. So if we consider the risk-averse agents, who will minimize volatility for given value

• f u⊤b = d:

A primer on portfolio separation Market models Distributions Discussion «Positively pseudo-isotropic» (... terminology ?) X

Proposition #4 (in an ad-hoc setup only sketched) Consider a reinsurance market where volatility-minimizing agents are exposed to Arias–Koldobsky-distributed losses (i.e. per unit premium incomes b, loss X), and where they trade arbitrary units

• f the insurance policies (no derivative securities!).

Then there is no Walrasian equilibrium except in special cases. Proofsketch: Preferences are nowhere convex, so agents want corner solutions. Preference structure leads to lexicographical

• rdering of corners, and any price ratio will make agents rush

to the same positions (except degenerate cases). Result first obtained for −X being first-orthant α-stable, iid. components, α < 1 (NCF, dr. thesis 2002), but works when the c-level curve is nowhere concave in the first orthant. ... well arguably: if the agents are volatility minimizers, why would they expose themselves to these risks in the first place?

A primer on portfolio separation Market models Distributions Discussion «Positively pseudo-isotropic» (... terminology ?) X

A primer on portfolio separation Market models Distributions Discussion ❛-symmetric X, no riskless opportunity

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A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

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Market models The single-period market Dynamic models

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Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

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Discussion Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion ❛-symmetric X, no riskless opportunity

For the case without riskless investment opportunity, we have: Proposition #5 (NCF, preprint 2011) Let X be α-symmetric for α = 1 + 1/k, some odd natural k. Then the case of no riskless opportunity, admits k + 1 fund separation. Except special cases, k funds are insufficient even when n = k + 1. Proofsketch: The Lagrange conditions associated to max

u

u⊤ ˆ b subject to

∑ |ui|α = Qα,

u⊤ ˆ a = w (where ˆ b = M−1b, ˆ a = M−11), lead to |ui|1/ksign(ui) = 1 να(ˆ bi + λˆ ai) which if k is odd, yields ui = (να)−k(ˆ bi + λˆ ai)k Expand the power, collect terms (& check degeneracies)

A primer on portfolio separation Market models Distributions Discussion ❛-symmetric X, no riskless opportunity

k + 1 agents with different λ’s, typically require k + 1 funds: Essentially because λ → (ˆ bi + λˆ ai)k is a kth order polynomial, uniquely determined by k + 1 distinct points.

More formally, if U gathers k + 1 agents’ choices into U = (u1, . . . , uk+1), then U = α−kAL where

L⊤ is Vandermonde: row #j is (λ0

j , . . . , λk j ), determinant

equals ∏1≤i<j≤k(λj − λi) Row #i of A is (k

0)ρk i , . . . , (k k)ρ0 i

• ˆ

ak

i , where ρi = ˆ

bi/ˆ

• ai. Its

largest minors expressible as a Vandermonde determinant × lots of binomial coefficients × product of the ˆ ak

i .

If there are k + 1 distinct ρi’s then U has full rank.

Other remarks:

What if k is even? |a + b|ksign(a + b) does not expand to a

• polynomial. Conditions for separation unknown (to me).

(There are some (quirky!) results for α ≤ 1.)

A primer on portfolio separation Market models Distributions Discussion ❛-symmetric X, no riskless opportunity

A primer on portfolio separation Market models Distributions Discussion ❛-stable X

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A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

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Market models The single-period market Dynamic models

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Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

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Discussion Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion ❛-stable X

Recall that pseudo-isotropy is defined by: all linear combinations of the coordinates are of the same type. α-stability defined by: all convex combinations of independent copies, are of same type:

X stable if for any two X(1) d = X(2) d = X (all iid), any positive numbers a1, a2, there ∃a ≥ 0, d ∈ Rn (non-random!) such that a1X(1) + a2X(2) d = aX + d.

In which case, ∃α ∈ (0, 2] s.t. a = (aα

1 + aα 2)1/α, hence the

term «❛-stable». The 2-stable is Gaussian, the symmetric univariate 1-stable is the Cauchy.

CLT: If X is infinite-dim., iid coord’s, then ∃α ∈ (0, 2] s.t. n−1/α(

n

1, . . . , 1, 0, 0, . . . )(X − m)

d

− − − →

n→∞

α-stable (for a location m which = mean iff X is integrable iff α > 1).

For the α-stables themselves, this holds also for all finite n. The classical CLT: finite (co)variance ⇒ α = 2 (Gaussian).

Pseudo-isotropy: allow for dependent coordinates. If X is α-stable and ˆ α-symmetric, then ˆ α ∈ [α, 2] (if dim.> 2).

A primer on portfolio separation Market models Distributions Discussion ❛-stable X

Suppose X is an α-stable vector. Then u⊤X is univariate α-stable: The parametrization comedy: The probably most common parametrization yields for the univariate marginals, a chf = exp

• iµϑ − σα|ϑ|α(1 − iβ tan πα

2 signϑ)

• except for α = 1 where the tan πα

2 is replaced by − 2 π ln |ϑ|.

Iff α > 1 then mean is finite and equals µ. Since we have b in the model, we can and will assume µ = 0. (This parametrization discontinuous in distribution at α = 1.)

There exists a unit sphere-supported spectral measure Γ s.t. the linear combination u⊤X, has the following parameters: (Scale)α: σα = σα

u =

• |u⊤s|α Γ(ds)

Skewness: β = βu =

• |u⊤s|α sign(u⊤s) Γ(ds) · σ−α

u

Location: µ = µu = −χα=1 · 2

π

• (u⊤s) ln |u⊤s| Γ(ds)

A primer on portfolio separation Market models Distributions Discussion ❛-stable X

Proposition #6 (NCF (preprint 2002, extending Fama (1965)) Suppose that the portfolio is constrained to a radial set H on which βu is constant. Then we have 2-fund separation. Straightforward proof (some separate calculations for α = 1). Each level set for βu is radial. This is fairly ad hoc, but there is a more useful-looking corollary: Corollary (NCF) Suppose that Γ is supported by only the negative orthant and H := the positive orthant, and if Γ(H) > 0 assume that ∀ Borel A ⊆ H we have Γ(−A) = γΓ(A) or Γ(H) = 0. Then the case without short sale, admits 2-fund separation. Proof: βu is constant on H. (Indeed, βu = −1 if Γ(H) = 0 and (1 − γ)/(1 + γ) otherwise.)

A primer on portfolio separation Market models Distributions Discussion ❛-stable X

The continuous-time Lévy process corresponding to the result

• f the Corollary, would be the difference between two

independent processes, where for each, all coordinates have

• nly negative jumps.

IOW, no negatively correlated jump signs. This does not imply comonotonicity! In fact, the continuous «upward» movement is of infinite variation on every open set (and so is the downward jump part!)

α = 1 is an oddball case with an oddball result: Proposition #7 (NCF) Suppose α = 1 and that for some γ = 0 we have s Γ(ds) = γ1. Then there is 2-fund separation for the case without numéraire investment (1-fund separation (!) if in addition the portfolio is constrained to a H as in Prop. 6.)

Why? The assumption removes one degree of freedom: σuβu = wγ.

A primer on portfolio separation Market models Distributions Discussion

A primer on portfolio separation Market models Distributions Discussion Modeling issues

1

A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

2

Market models The single-period market Dynamic models

3

Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

4

Discussion Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion Modeling issues

The literature on portfolio separation has traditionally not considered applicability. For example, the prominence of the Gaussian, despite its unbounded downside making it less suitable for limited liability asset prices.

Elliptical returns may however be bounded – the prototypical elliptical distribution is the uniform on the (unit) sphere. Non-elliptical pseudo-isotropic distributions are unbounded. The first-orthant Arias & Koldobsky example has no downside, but infinite mean.

For the continuous-time problem, we want to know: which distributions are infinitely divisible?

The α-stables are, regardless of skewing. The skew-ellipticals are usually not (the half-normal isn’t).

The Azzalini skewing approach has – on the surface – nothing to do with the skewed ❛-stables!

The ’arrival of partial information’ interpretation can make a useful continuous-time model out of the skew-ellipticals, regardless of infinite divisibility.

A primer on portfolio separation Market models Distributions Discussion Modeling issues

The infinite variance / infinite mean discussion: One might argue that infinite-mean losses or infinite-variance losses, constitute unrealistic model behaviour. Even more so, than the unboundedness exhibited by the Gaussian (might itself be questionable). Empirical analyses might yield very heavy tails.

Ex.: Various operational loss datasets in the banking industry (Moscadelli, Banca d’Italia wp, 2004): estimates tail index both > 1 and < 1, somtimes 1 ∈ relevant confidence intervals

The emergence of e.g. quantile measures. Models determine or characterize behaviour?

A primer on portfolio separation Market models Distributions Discussion Modeling issues

Suppose we have infinite divisibility. Will the geometric Lévy process changes sign? The geometric Brownian does not, despite the driving noise having full support.

Bm with drift → geometric process = exp(Bm with drift). That is exceptional!

If the Lévy measure is supported by the unit ball, then the price process will not change sign.

Leading to the topic of pseudo-isotropic Lévy measures. By the Bochner theorem, a pseudo-isotropic non-elliptical finite Lévy measure, must have full support. What about the infinite Lévy measures?

OTOH, the value of an insurance policy should with positive probability change sign. But is symmetric driving noise desirable? Arguably, you could want heavier-tailed losses than gains.

The completely skewed α-stables possess this property!

A primer on portfolio separation Market models Distributions Discussion For future research

1

A primer on portfolio separation Cass–Stiglitz (JET 1970): Utility functions Ross (JET 1978): Returns distributions Distributions for which Ross 2-fund separation applies

2

Market models The single-period market Dynamic models

3

Distributions Elliptical X Skew-elliptical X and generalizations Pseudo-isotropic X «Positively pseudo-isotropic» (... terminology ?) X α-symmetric X, no riskless opportunity α-stable X

4

Discussion Modeling issues For future research

A primer on portfolio separation Market models Distributions Discussion For future research

For future research: The «positively pseudo-isotropic» property.

What is the class like? Which distributions (that are neither pseudo-isotropics nor substables)? Which ones lead to returns a.s. ≥ −100%? Which ones have full measure? Which ones have one light and one heavy tail? (Cf. again Arias and Koldobsky (2003?)). Which ones are infinitely divisible? How do their Lévy measures behave? Which ones do not have full measure?

More generally, extend these questions to the class of X which admit the representation (with or w/o restriction to u ∈ Rn

+)

u⊤X

d

= c1(u) ˜ X1 + · · · + cℓ(u) ˜ Xℓ (ℓ < n given), where each ci is positive homogeneous.

Example: the skew-elliptical, with ℓ = q + 1, q being the dimension of the conditioned vector.

Is there any useful unification of Azzalini skewing with e.g. skew-stables?