IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
Opinion pooling and portfolio optimization
Francisco Gochez, Mango Solutions,[email protected]
July 2, 2009
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
BLCOP - Implementation of opinion pooling methods (BlackLitterman and Copula Opinion Pooling).
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
Black-Litterman: Market distribution and the mean
I Market M ∼ N(µ,Σ).
I Mean µ ∼ N(π, τΣ)
I π are “equilibrium means”
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
Black-Litterman: Market distribution and the mean
I Market M ∼ N(µ,Σ).
I Mean µ ∼ N(π, τΣ)
I π are “equilibrium means”
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
Black-Litterman: Market distribution and the mean
I Market M ∼ N(µ,Σ).
I Mean µ ∼ N(π, τΣ)
I π are “equilibrium means”
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
Black-Litterman: Specification of views
I Views: E [pi ,1A1 + pi ,2A2 + ...+ pi ,nAn] = qi + ε1
I P = (pij),q = (qi )
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
Black-Litterman: Specification of views
I Views: E [pi ,1A1 + pi ,2A2 + ...+ pi ,nAn] = qi + ε1
I P = (pij),q = (qi )
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
Black-Litterman: Specification of views (cont)
I Confidences: εi ∼ N(0, ω2i )
I “Automatic confidences” : Ω = κdiag(PΣPT )
I “Overall confidence” : 0 < τ
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
Black-Litterman: Specification of views (cont)
I Confidences: εi ∼ N(0, ω2i )
I “Automatic confidences” : Ω = κdiag(PΣPT )
I “Overall confidence” : 0 < τ
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
Black-Litterman: Specification of views (cont)
I Confidences: εi ∼ N(0, ω2i )
I “Automatic confidences” : Ω = κdiag(PΣPT )
I “Overall confidence” : 0 < τ
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
Black-Litterman: Posterior distribution calculation
I Mean : π + τΣPT (τPΣPT + Ω)−1(q − Pπ)
I Covariance : (1 + τ)Σ− τ2ΣPT (τPΣPT + Ω)−1PΣ
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
Black-Litterman: Posterior distribution calculation
I Mean : π + τΣPT (τPΣPT + Ω)−1(q − Pπ)
I Covariance : (1 + τ)Σ− τ2ΣPT (τPΣPT + Ω)−1PΣ
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
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
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
COP: Specification of views
I Market A has arbitrary distribution
I pi ,1A1 + pi ,2A2 + ...+ pi ,nAn ∼ θi (·), θi arbitrary, pdf fi
I P = (pij) is the “pick” matrix.
I View i has confidence ci , 0 < ci < 1
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
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
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
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
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
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
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
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
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
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
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
COP: Obtaining the posterior distribution
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
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
COP: Obtaining the posterior distribution
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
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
COP: Obtaining the posterior distribution
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
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
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
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization
IntroductionBlack-Litterman:Theory
Copula Opinion Pooling: TheoryReferences
See also the vignette in the BLCOP package, vignette(”BLCOP”)
Francisco Gochez, Mango Solutions, [email protected] pooling and portfolio optimization