Cyril Bachelard OLZ AG Optimal Portfolios and Where to Find Them - PowerPoint PPT Presentation

Cyril Bachelard OLZ AG Optimal Portfolios and Where to Find Them Hint: Somewhere inside a High-Dimensional Convex Polytope

Agenda 1. Introduction 2. Dear Investor, What’s Your Objective? 3. Dear Audience, Remember Geometry? 4. Dear Manager, Are You Reliable Enough? 5. Dear Solver, Can You Find It?

1. About me MSc in Economics University of Bern Research and Product Development (2011-now) OLZ AG 3

1. About OLZ AG Effjcient Investing Private Clients Institutional Investors OLZ AG provides investment solutions for Risk Based Strategies The main focus is on scientifjcally sound investment concept asset management without confmicts of interest The investment philosophy is OLZ AG was founded in 2001 by Singapore Liechtenstein Zurich Headquartered in Bern with offjces and subsidiaries in P. Z graggen Prof. C. L oderer ( University of Bern ) C. O rlacchio 4

1. About OLZ AG Effjcient Investing Private Clients Institutional Investors OLZ AG provides investment solutions for Risk Based Strategies The main focus is on scientifjcally sound investment concept asset management without confmicts of interest The investment philosophy is OLZ AG was founded in 2001 by Singapore Liechtenstein Zurich Headquartered in Bern with offjces and subsidiaries in P. Z graggen Prof. C. L oderer ( University of Bern ) C. O rlacchio 4

1. About OLZ AG Effjcient Investing Private Clients Institutional Investors OLZ AG provides investment solutions for Risk Based Strategies The main focus is on scientifjcally sound investment concept asset management without confmicts of interest The investment philosophy is OLZ AG was founded in 2001 by Singapore Liechtenstein Zurich Headquartered in Bern with offjces and subsidiaries in P. Z graggen Prof. C. L oderer ( University of Bern ) C. O rlacchio 4

Agenda 1. Introduction 2. Dear Investor, What’s Your Objective? 3. Dear Audience, Remember Geometry? 4. Dear Manager, Are You Reliable Enough? 5. Dear Solver, Can You Find It?

2. Dear Investor, What’s Your Objective? w . where w X w (Kahneman-Tversky) Ex.: Prospect Theory Non-linear X X X where w w w Ex.: Mean-Variance (Markowitz) Linear and Quadratic can be The objective function w such that: In Portfolio Optimization we search for the weights ˆ ˆ w = argmax f ( w ) w ∈C 6

2. Dear Investor, What’s Your Objective? Non-linear . where w X w (Kahneman-Tversky) Ex.: Prospect Theory where Ex.: Mean-Variance (Markowitz) Linear and Quadratic w such that: In Portfolio Optimization we search for the weights ˆ ˆ w = argmax f ( w ) w ∈C The objective function f can be f ( w ) = µ T w − λ 2 w T Σ w µ = E [ X ] Σ = E � ( X − µ ) ( X − µ ) T � 6

2. Dear Investor, What’s Your Objective? Linear and Quadratic where (Kahneman-Tversky) Ex.: Prospect Theory Non-linear Ex.: Mean-Variance (Markowitz) where w such that: In Portfolio Optimization we search for the weights ˆ ˆ w = argmax f ( w ) w ∈C The objective function f can be f ( w ) = µ T w − λ 2 w T Σ w f ( w ) = E � g � w T X �� µ = E [ X ] � ( x − θ ) a x ≥ θ Σ = E � ( X − µ ) ( X − µ ) T � g ( x ) = − b ( − ( x − θ )) a x < θ a ∈ [0 , 1] , b > 1 . 6

2. Dear Investor, What’s Your Objective? Variance g w Tail Risk w w Turnover w w w w Tracking Error Asset Classes Sector Country Liquidity (Upper Bounds) No Short-Selling (Lower Bounds) Budget The constraints C can be 1 T w = 1 b l ≤ w ≤ b u A w ≤ b , with A binary 7

2. Dear Investor, What’s Your Objective? Asset Classes g w Tail Risk w w Turnover Variance Tracking Error Sector Country Liquidity (Upper Bounds) No Short-Selling (Lower Bounds) Budget The constraints C can be 1 T w = 1 b l ≤ w ≤ b u A w ≤ b , with A binary w ) T Σ ( w − ¯ ( w − ¯ w ) ≤ ¯ c 7

2. Dear Investor, What’s Your Objective? Sector Tail Risk Turnover Tracking Error Asset Classes Variance Country Liquidity (Upper Bounds) No Short-Selling (Lower Bounds) Budget The constraints C can be 1 T w = 1 b l ≤ w ≤ b u A w ≤ b , with A binary w ) T Σ ( w − ¯ ( w − ¯ w ) ≤ ¯ c � w − ¯ w � 1 ≤ ¯ c g ( w ) ≤ 0 7

2. Dear Investor, What’s Your Objective? Mixed Integer: w Quadratic: w w Non-Linear: w The constraints C can be Linear: A w ≤ b 8

2. Dear Investor, What’s Your Objective? Mixed Integer: w Non-Linear: w The constraints C can be Quadratic: w T Q w ≤ r Linear: A w ≤ b 8

2. Dear Investor, What’s Your Objective? Non-Linear: w The constraints C can be Quadratic: w T Q w ≤ r Linear: A w ≤ b Mixed Integer: w ∈ Z n 8

2. Dear Investor, What’s Your Objective? The constraints C can be Quadratic: w T Q w ≤ r Linear: A w ≤ b Mixed Integer: w ∈ Z n Non-Linear: g ( w ) ≤ 0 8

Agenda 1. Introduction 2. Dear Investor, What’s Your Objective? 3. Dear Audience, Remember Geometry? 4. Dear Manager, Are You Reliable Enough? 5. Dear Solver, Can You Find It?

3. Dear Audience, Remember Geometry? Defjnition (Convex Set) Defjnition (Convex Function) A set C ⊆ R n is convex if for all x , y ∈ C and for all α ∈ [0 , 1] , α x + (1 − α ) y ∈ C . A function f : R n → R is convex if for all x , y ∈ C and for all α ∈ [0 , 1] , f ( α x + (1 − α ) y ) ≤ αf ( x ) + (1 − α ) f ( y ) . 10

3. Dear Audience, Remember Geometry? Defjnition (Polyhedron) Defjnition (Polytope) Defjnition ( L p -norm) k =1 | x k | p � 1 For p ∈ R , p ≥ 1 , the L p -norm of w ∈ R n is � x � p = �� n p . An n -dimensional (convex) polyhedron is the intersection of I n -dimensional half-spaces w ∈ R n � S i = � � � a T i w ≤ b i for i = 1 , . . . , I I � S i = { w ∈ R n | A w ≤ b } P = i =1 where the i -th row of A is a i for i = 1 , . . . , I . This is the H -representation of a polyhedron. A (convex) polytope P is a bounded (convex) polyhedron. 11

3. Dear Audience, Remember Geometry? Defjnition (Standard Simplex) Defjnition (Ellopsoid) An ellipsoid centred in c with shape matrix is c w w c w c A standard n -simplex is w ∈ R n +1 � S = � � 1 T w = 1 , w ≥ 0 � 12

3. Dear Audience, Remember Geometry? Defjnition (Standard Simplex) Defjnition (Ellopsoid) An ellipsoid centred in c with shape matrix A standard n -simplex is w ∈ R n +1 � S = � � 1 T w = 1 , w ≥ 0 � Σ � 0 is � ( w − c ) T Σ ( w − c ) ≤ 1 � w ∈ R n � E Σ , c = � 12

Agenda 1. Introduction 2. Dear Investor, What’s Your Objective? 3. Dear Audience, Remember Geometry? 4. Dear Manager, Are You Reliable Enough? 5. Dear Solver, Can You Find It?

4. Dear Manager, Are You Reliable Enough? ”THE PROCESS OF SELECTING a portfolio may be divided into two stages. The fjrst stage starts with observation and experience and ends with beliefs about the future performances of available securities. The second stage starts with the relevant beliefs about future performances and ends with the choice of portfolio. This paper is concerned with the second stage.” H. Markowitz (1952). Portfolio Selection. Journal of Finance Vol.7, No.1., pp.77-91. 14

4. Dear Manager, Are You Reliable Enough? 15

4. Dear Manager, Are You Reliable Enough? Variance Real Covariance Matrix Shrinkage is high. Bias Shrinkage Intensity Bias-Variance Tradeofg Shrinkage Target Estimated Covariance Matrix The total squared error of an estimator can be decomposed as �� ˆ Σ − Σ � 2 � Σ , Σ � 2 = Var � ˆ Σ � + Bias � ˆ E An estimator ˆ A matrix ¯ Σ computed on real data Σ chosen for it’s structure has of the quantity Σ is afgected by noise. a bias w.r.t. Σ . Bias � ¯ Σ , Σ � = E � ¯ Σ − Σ � Σ � 2 is high. Var � ˆ Σ � = E � ˆ Σ 2 � − E � ˆ 16

4. Dear Manager, Are You Reliable Enough? Bias Shrinkage Real Covariance Matrix Estimated Covariance Matrix Shrinkage Target Shrinkage Intensity Bias-Variance Tradeofg Variance is high. The total squared error of an estimator can be decomposed as �� ˆ Σ − Σ � 2 � Σ , Σ � 2 = Var � ˆ Σ � + Bias � ˆ E An estimator ˆ A matrix ¯ Σ computed on real data Σ chosen for it’s structure has of the quantity Σ is afgected by noise. a bias w.r.t. Σ . Bias � ¯ Σ , Σ � = E � ¯ Σ − Σ � Σ � 2 is high. Var � ˆ Σ � = E � ˆ Σ 2 � − E � ˆ Σ Σ ˆ Σ ¯ Σ ˆ ¯ Σ Σ α (1 − α )ˆ Σ + α ¯ Σ 16

4. Dear Manager, Are You Reliable Enough? � � 1 . 06 − 0 . 21 0 . 78 Σ = − 0 . 21 0 . 95 − 0 . 28 0 . 78 − 0 . 28 1 . 1 � � � � 1 . 08 − 0 . 2 0 . 8 1 0 0 = ˆ ¯ − 0 . 2 0 . 92 − 0 . 3 Σ Σ = 0 1 0 Σ α 0 . 8 − 0 . 3 1 . 12 0 0 1 � � 1 . 06 − 0 . 2 0 . 8 Σ α = (1 − α )ˆ Σ + α ¯ Σ = − 0 . 2 0 . 94 − 0 . 3 with α = 0 . 25 0 . 8 − 0 . 3 1 . 09 17

4. Dear Manager, Are You Reliable Enough? Theorem (Shrinkage Interpretation of Constraints) In addition Given a covariance matrix ˆ Σ and a set of constraints C there exist Σ C such that w T ˆ w ∈ R n w T Σ C w Σ w = ˆ argmin w = argmin w ∈C Σ C = ˆ Σ + α ∆ = (1 − α ) ˆ Σ + α � ˆ Σ + ∆ � so Σ C is a shrinked version of ˆ Σ . Σ ˆ Σ = ˆ ¯ Σ + ∆ Σ Σ α = Σ C 18