Opinion pooling and portfolio optimization - Rmetrics · Introduction Black-Litterman:Theory Copula...

IntroductionBlack-Litterman:Theory

Copula Opinion Pooling: TheoryReferences

Opinion pooling and portfolio optimization

Francisco Gochez, Mango Solutions,fgochez@mango-solutions.com

July 2, 2009

Francisco Gochez, Mango Solutions, fgochez@mango-solutions.comOpinion pooling and portfolio optimization

BLCOP - Implementation of opinion pooling methods (BlackLitterman and Copula Opinion Pooling).

Black-Litterman: Market distribution and the mean

I Market M ∼ N(µ,Σ).

I Mean µ ∼ N(π, τΣ)

I π are “equilibrium means”

Black-Litterman: Specification of views

I Views: E [pi ,1A1 + pi ,2A2 + ...+ pi ,nAn] = qi + ε1

I P = (pij),q = (qi )

Black-Litterman: Specification of views

I Views: E [pi ,1A1 + pi ,2A2 + ...+ pi ,nAn] = qi + ε1

I P = (pij),q = (qi )

Black-Litterman: Specification of views (cont)

I Confidences: εi ∼ N(0, ω2i )

I “Automatic confidences” : Ω = κdiag(PΣPT )

I “Overall confidence” : 0 < τ

Black-Litterman: Posterior distribution calculation

I Mean : π + τΣPT (τPΣPT + Ω)−1(q − Pπ)

I Covariance : (1 + τ)Σ− τ2ΣPT (τPΣPT + Ω)−1PΣ

Black-Litterman: Posterior distribution calculation

I Mean : π + τΣPT (τPΣPT + Ω)−1(q − Pπ)

I Covariance : (1 + τ)Σ− τ2ΣPT (τPΣPT + Ω)−1PΣ

COP: Specification of views

I Market A has arbitrary distribution

I pi ,1A1 + pi ,2A2 + ...+ pi ,nAn ∼ θi (·), θi arbitrary, pdf fiI P = (pij) is the “pick” matrix.

I View i has confidence ci , 0 < ci < 1

I pi ,1A1 + pi ,2A2 + ...+ pi ,nAn ∼ θi (·), θi arbitrary, pdf fi

I P = (pij) is the “pick” matrix.

COP: Obtaining the posterior distribution

I V = PA

I V inherits a distribution from the the market distribution,vi ∼ θ′i

I θi = ciθi + (1− ci )θ′i

I V = PA

I (C1, ...,CN) = (θ′1(v1), ..., θ′N(vN))

I V = (θ−1(v1), ..., θ−1(vN))

I Rotate V back into “market coordinates” to obtain theposterior

I (C1, ...,CN) = (θ′1(v1), ..., θ′N(vN))

I V = (θ−1(v1), ..., θ−1(vN))

I (C1, ...,CN) = (θ′1(v1), ..., θ′N(vN))

I V = (θ−1(v1), ..., θ−1(vN))

References

Meucci, Attilio. Beyond Black-Litterman: Views onNon-Normal Markets. November 2005, Available at SSRN:http://ssrn.com/abstract=848407

Meucci, Attilio. Beyond Black-Litterman in Practice: AFive-Step Recipe to Input Views on non-Normal Markets. May2006, Available at SSRN:http://papers.ssrn.com/sol3/papers.cfm?abstract id=872577

Meucci, Attilio. The Black-Litterman Approach: OriginalModel and Extensions. April 2008, Available at SSRN:http://ssrn.com/abstract=1117574

See also the vignette in the BLCOP package, vignette(”BLCOP”)

Opinion pooling and portfolio optimization - Rmetrics · Introduction Black-Litterman:Theory Copula...

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