Introduction CDRM Optimization Case Studies Conclusions and Future Directions Coherent Distortion Risk Measures in Portfolio Selection (Joint work with Dr Ken Seng Tan) Ming Bin Feng University of Waterloo The 46th Actuarial Research Conference August 11, 2011 Ming Bin Feng 1/ 37
Introduction CDRM Optimization Case Studies Conclusions and Future Directions Abstract The theme of this presentation relates to solving portfolio selection problems using linear and fractional programming. Two key contributions: Generalization of the CVaR linear optimization framework (see Rockafellar and Uryasev [3, 4]). Equivalences among four formulations of CDRM optimization problems. Ming Bin Feng 2/ 37
Introduction CDRM Optimization Motivations Case Studies Goals Conclusions and Future Directions Outline 1 Introduction Motivations Goals CDRM Optimization 2 Case Studies 3 Conclusions and Future Directions 4 Ming Bin Feng 3/ 37
Introduction CDRM Optimization Motivations Case Studies Goals Conclusions and Future Directions Motivations Practical portfolio selection problems Good risk measures Well-studied programming models Question Can we connect this together? We want to solve practical portfolio optimization problems with sophisticated risk measures using a programming model that can be solved efficiently. Ming Bin Feng 4/ 37
Introduction CDRM Optimization Motivations Case Studies Goals Conclusions and Future Directions We wish to.. Incorporate a general class of risk measure into a well-studied programming model Study equivalences among different formulations of portfolio selection problems Solve portfolio selection problems of interest efficiently Ming Bin Feng 5/ 37
Introduction CVaR Optimization CDRM Optimization CDRM Representation Theorem Case Studies CDRM Optimization Conclusions and Future Directions Formulation Equivalences Outline 1 Introduction 2 CDRM Optimization CVaR Optimization CDRM Representation Theorem CDRM Optimization Formulation Equivalences Case Studies 3 Conclusions and Future Directions 4 Ming Bin Feng 6/ 37
Introduction CVaR Optimization CDRM Optimization CDRM Representation Theorem Case Studies CDRM Optimization Conclusions and Future Directions Formulation Equivalences Scenario Generation Loss Matrix p 1 → L 11 L 12 · · · L 1 n → l 1 = l ( x , p 1 ) p 2 → L 21 L 22 · · · L 2 n → l 2 = l ( x , p 2 ) L = . . . . . . ... . . . . . . . . . · · · . . . p m → L m 1 L m 2 · · · L mn → l m = l ( x , p m ) Let l ( 1 ) ≤ · · · ≤ l ( m ) be the ordered losses, p ( i ) , i = 1 , · · · , m be the corresponding probability masses. Return/Price/Premium/Profit Vector c = [ c 1 , · · · , c m ] ′ Ming Bin Feng 7/ 37
Introduction CVaR Optimization CDRM Optimization CDRM Representation Theorem Case Studies CDRM Optimization Conclusions and Future Directions Formulation Equivalences CVaR Optimization Background Consider the special function m 1 � p j ( l j − ζ ) + F ( x , ζ ) = ζ + 1 − α j = 1 Rockafellar and Uryasev [3, 4] showed that CVaR α ( x ) = min ζ ∈ R F ( x , ζ ) 1 min x ∈ X CVaR α ( x ) = min ( x ,ζ ) ∈ X × R F ( x , ζ ) 2 Ming Bin Feng 8/ 37
Introduction CVaR Optimization CDRM Optimization CDRM Representation Theorem Case Studies CDRM Optimization Conclusions and Future Directions Formulation Equivalences CVaR Optimization CVaR portfolio selection problems can be formulated as LPs. Suppose X is the set of all feasible portfolios. CVaR minimization subject to a return constraint m 1 � minimize ζ + p j z j 1 − α j = 1 c ′ x ≥ subject to µ l ( x , p j ) − ζ ≤ z j j = 1 , · · · , m ≤ j = 1 , · · · , m 0 z j ( x , ζ ) ∈ X × R Ming Bin Feng 9/ 37
Introduction CVaR Optimization CDRM Optimization CDRM Representation Theorem Case Studies CDRM Optimization Conclusions and Future Directions Formulation Equivalences CVaR Optimization Return maximization subject to CVaR constraint(s) maximize c ′ x m 1 � subject to ζ i + p j z ij ≤ η i i = 1 , · · · , k 1 − α j = 1 l ( x , p j ) − ζ i ≤ z ij ∀ i , j 0 ≤ z ij ∀ i , j X × R k ( x , ζ ) ∈ Ming Bin Feng 10/ 37
Introduction CVaR Optimization CDRM Optimization CDRM Representation Theorem Case Studies CDRM Optimization Conclusions and Future Directions Formulation Equivalences Definition and Representation Theorem Two Equivalent Definitions A risk measure ρ ( x ) is a CDRM if it is A comonotone law-invariant coherent risk measure A distortion risk measure with a concave distortion function Representation Theorem for CDRM A risk measure ρ ( x ) is a CDRM if and only if there exists a 1 � function w : [ 0 , 1 ] �→ [ 0 , 1 ] , satisfying w α d α = 1, such that α = 0 1 � ρ ( x ) = CVaR α ( x ) w α d α α = 0 Ming Bin Feng 11/ 37
Introduction CVaR Optimization CDRM Optimization CDRM Representation Theorem Case Studies CDRM Optimization Conclusions and Future Directions Formulation Equivalences Representation Theorem in Discrete Case Finite Generation Theorem for CDRM m Given a concave distortion function g , ρ ( x ) = � q i l ( i ) , i = 1 moreover m � ρ ( x ) = w i CVaR i − 1 m ( x ) , where i = 1 q 1 if i = 1 p ( 1 ) m w i = � p ( j ) p ( i ) j = i ( q i − p ( i − 1 ) q i − 1 ) if i = 2 , · · · , m p ( i ) Ming Bin Feng 12/ 37
Introduction CVaR Optimization CDRM Optimization CDRM Representation Theorem Case Studies CDRM Optimization Conclusions and Future Directions Formulation Equivalences CDRM Optimization CDRM minimization subject to a return constraint m m 1 minimize � w i ( ζ i + � p j z ij ) 1 − α i = 1 j = 1 c ′ x subject to ≥ µ l ( x , p j ) − ζ i ≤ z ij ∀ i , j 0 ≤ z ij ∀ i , j X × R m ( x , ζ ) ∈ Ming Bin Feng 13/ 37
Introduction CVaR Optimization CDRM Optimization CDRM Representation Theorem Case Studies CDRM Optimization Conclusions and Future Directions Formulation Equivalences CDRM Optimization Return maximization subject to one CDRM constraint maximize c ′ x m m 1 � � subject to w i ( ζ i + p j z ij ) ≤ η 1 − α i = 1 j = 1 l ( x , p j ) − ζ i ≤ z ij ∀ i , j 0 ≤ z ij ∀ i , j X × R m ( x , ζ ) ∈ Ming Bin Feng 14/ 37
Introduction CVaR Optimization CDRM Optimization CDRM Representation Theorem Case Studies CDRM Optimization Conclusions and Future Directions Formulation Equivalences CDRM Optimization Return-CDRM utility maximization m m 1 c ′ x − τ � w i ( ζ i + � p j z ij ) maximize 1 − α i = 1 j = 1 subject to l ( x , p j ) − ζ i ≤ z ij ∀ i , j ≤ ∀ i , j 0 z ij X × R m ( x , ζ ) ∈ This formulation is very similar to a return maximization problem with m CVaR constraints. Yet we converted m CVaR constraints into the objective function. Ming Bin Feng 15/ 37
Introduction CVaR Optimization CDRM Optimization CDRM Representation Theorem Case Studies CDRM Optimization Conclusions and Future Directions Formulation Equivalences CDRM Optimization CDRM-based Sharpe ratio maximization c ′ x − ν maximize m m 1 � w i ( ζ i + � p j z ij ) 1 − α i = 1 j = 1 subject to l ( x , p j ) − ζ i ≤ z ij ∀ i , j 0 ≤ z ij ∀ i , j X × R m ( x , ζ ) ∈ This is an LFP , but we can solve it by solving at most two related LPs using a variable transformation method studied by Charnes and Cooper [1]. Ming Bin Feng 16/ 37
Introduction CVaR Optimization CDRM Optimization CDRM Representation Theorem Case Studies CDRM Optimization Conclusions and Future Directions Formulation Equivalences Formulation Equivalences Equivalences among four formulations, part 1 Problem Max-Return Min-CDRM Preset η µ Parameter Implied Parameters η = N/A ρ ( x ∗ ) µ = c ′ x ∗ N/A u 1 1 τ = u 2 c ′ x ∗ − u 1 ρ ( x ∗ ) R ( x ∗ ) − 1 u 2 ρ ( x ∗ ) ν = If the return and CDRM constraints are binding at respective optimal solutions, the preset parameter for Max-Return equals to the implied parameter for Min-CDRM and vice versa. Ming Bin Feng 17/ 37
Introduction CVaR Optimization CDRM Optimization CDRM Representation Theorem Case Studies CDRM Optimization Conclusions and Future Directions Formulation Equivalences Formulation Equivalences Equivalences among four formulations, part 1 Problem Max-Utility Max-Sharpe Preset τ ν Parameter Implied Parameters η = ρ ( x ∗ ) ρ ( x ∗ ) µ = c ′ x ∗ c ′ x ∗ c ′ x ∗ − ν τ = N/A ρ ( x ∗ ) c ′ x ∗ − τρ ( x ∗ ) ν = N/A We will see that the preset parameter for Max-Return equals to the implied parameter for Min-CDRM and vice versa. Ming Bin Feng 18/ 37
Introduction CDRM Optimization Case 1: Reinsurance portfolio selection with simulated data Case Studies Case 2: Investment portfolio selection with historical data Conclusions and Future Directions Outline 1 Introduction 2 CDRM Optimization Case Studies 3 Case 1: Reinsurance portfolio selection with simulated data Case 2: Investment portfolio selection with historical data Conclusions and Future Directions 4 Ming Bin Feng 19/ 37
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