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

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• 1. About OLZ AG

OLZ AG was founded in 2001 by

• C. Orlacchio
• Prof. C. Loderer (University of Bern)
• P. Zgraggen

Headquartered in Bern with offjces and subsidiaries in Zurich Liechtenstein Singapore The investment philosophy is

Effjcient Investing

asset management without confmicts of interest scientifjcally sound investment concept The main focus is on

Risk Based Strategies

OLZ AG provides investment solutions for Institutional Investors Private Clients

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• 1. About OLZ AG

OLZ AG was founded in 2001 by

• C. Orlacchio
• Prof. C. Loderer (University of Bern)
• P. Zgraggen

Headquartered in Bern with offjces and subsidiaries in Zurich Liechtenstein Singapore The investment philosophy is

Effjcient Investing

asset management without confmicts of interest scientifjcally sound investment concept The main focus is on

Risk Based Strategies

OLZ AG provides investment solutions for Institutional Investors Private Clients

4

• 1. About OLZ AG

OLZ AG was founded in 2001 by

• C. Orlacchio
• Prof. C. Loderer (University of Bern)
• P. Zgraggen

Headquartered in Bern with offjces and subsidiaries in Zurich Liechtenstein Singapore The investment philosophy is

Effjcient Investing

asset management without confmicts of interest scientifjcally sound investment concept The main focus is on

Risk Based Strategies

OLZ AG provides investment solutions for Institutional Investors Private Clients

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?

In Portfolio Optimization we search for the weights ˆ w such that: ˆ w = argmax

w∈C

f (w) The objective function can be Linear and Quadratic Ex.: Mean-Variance (Markowitz) w w w w where X X X Non-linear Ex.: Prospect Theory (Kahneman-Tversky) w w X where .

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• 2. Dear Investor, What’s Your Objective?

In Portfolio Optimization we search for the weights ˆ w such that: ˆ w = argmax

w∈C

f (w) The objective function f can be Linear and Quadratic Ex.: Mean-Variance (Markowitz) f (w) = µT w − λ 2 wT Σw where µ = E [X] Σ = E (X − µ) (X − µ)T Non-linear Ex.: Prospect Theory (Kahneman-Tversky) w w X where .

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• 2. Dear Investor, What’s Your Objective?

In Portfolio Optimization we search for the weights ˆ w such that: ˆ w = argmax

w∈C

f (w) The objective function f can be Linear and Quadratic Ex.: Mean-Variance (Markowitz) f (w) = µT w − λ 2 wT Σw where µ = E [X] Σ = E (X − µ) (X − µ)T Non-linear Ex.: Prospect Theory (Kahneman-Tversky) f (w) = E g wT X where g(x) =

(x − θ)a

x ≥ θ −b(−(x − θ))a x < θ a ∈ [0, 1] , b > 1.

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• 2. Dear Investor, What’s Your Objective?

The constraints C can be Budget 1T w = 1 No Short-Selling (Lower Bounds) Liquidity (Upper Bounds) bl ≤ w ≤ bu Country Sector Asset Classes Aw ≤ b, with A binary Tracking Error Variance w w w w Turnover w w Tail Risk g w

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• 2. Dear Investor, What’s Your Objective?

The constraints C can be Budget 1T w = 1 No Short-Selling (Lower Bounds) Liquidity (Upper Bounds) bl ≤ w ≤ bu Country Sector Asset Classes Aw ≤ b, with A binary Tracking Error Variance (w − ¯ w)T Σ (w − ¯ w) ≤ ¯ c Turnover w w Tail Risk g w

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• 2. Dear Investor, What’s Your Objective?

The constraints C can be Budget 1T w = 1 No Short-Selling (Lower Bounds) Liquidity (Upper Bounds) bl ≤ w ≤ bu Country Sector Asset Classes Aw ≤ b, with A binary Tracking Error Variance (w − ¯ w)T Σ (w − ¯ w) ≤ ¯ c Turnover w − ¯ w1 ≤ ¯ c Tail Risk g (w) ≤ 0

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• 2. Dear Investor, What’s Your Objective?

The constraints C can be Linear: Aw ≤ b Mixed Integer: w Quadratic: w w Non-Linear: w

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• 2. Dear Investor, What’s Your Objective?

The constraints C can be Linear: Aw ≤ b Mixed Integer: w Quadratic: wT Qw ≤ r Non-Linear: w

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• 2. Dear Investor, What’s Your Objective?

The constraints C can be Linear: Aw ≤ b Mixed Integer: w ∈ Zn Quadratic: wT Qw ≤ r Non-Linear: w

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• 2. Dear Investor, What’s Your Objective?

The constraints C can be Linear: Aw ≤ b Mixed Integer: w ∈ Zn Quadratic: wT Qw ≤ r Non-Linear: g(w) ≤ 0

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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)

A set C ⊆ Rn is convex if for all x, y ∈ C and for all α ∈ [0, 1], αx + (1 − α) y ∈ C.

Defjnition (Convex Function)

A function f : Rn → R is convex if for all x, y ∈ C and for all α ∈ [0, 1], f (αx + (1 − α) y) ≤ αf (x) + (1 − α) f (y) .

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• 3. Dear Audience, Remember Geometry?

Defjnition (Lp-norm)

For p ∈ R, p ≥ 1, the Lp-norm of w ∈ Rn is xp = n

k=1 |xk|p 1

p .

Defjnition (Polyhedron)

An n-dimensional (convex) polyhedron is the intersection of I n-dimensional half-spaces Si = w ∈ Rn

aT

i w ≤ bi

• for i = 1, . . . , I

P =

I

• i=1

Si = {w ∈ Rn |Aw ≤ b} where the i-th row of A is ai for i = 1, . . . , I. This is the H-representation of a polyhedron.

Defjnition (Polytope)

A (convex) polytope P is a bounded (convex) polyhedron.

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• 3. Dear Audience, Remember Geometry?

Defjnition (Standard Simplex)

A standard n-simplex is S = w ∈ Rn+1

1T w = 1, w ≥ 0 Defjnition (Ellopsoid)

An ellipsoid centred in c with shape matrix is

c

w w c w c

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• 3. Dear Audience, Remember Geometry?

Defjnition (Standard Simplex)

A standard n-simplex is S = w ∈ Rn+1

1T w = 1, w ≥ 0 Defjnition (Ellopsoid)

An ellipsoid centred in c with shape matrix Σ 0 is EΣ,c = w ∈ Rn

(w − c)T Σ (w − c) ≤ 1

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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.

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• 4. Dear Manager, Are You Reliable Enough?

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• 4. Dear Manager, Are You Reliable Enough?

Bias-Variance Tradeofg

The total squared error of an estimator can be decomposed as E

ˆ

Σ − Σ2 = Varˆ Σ + Biasˆ Σ, Σ2 Variance An estimator ˆ Σ computed on real data

• f the quantity Σ is afgected by noise.

Varˆ Σ = Eˆ Σ2 − Eˆ Σ2 is high. Bias A matrix ¯ Σ chosen for it’s structure has a bias w.r.t. Σ. Bias¯ Σ, Σ = E¯ Σ − Σ is high. Shrinkage Real Covariance Matrix Estimated Covariance Matrix Shrinkage Target Shrinkage Intensity

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• 4. Dear Manager, Are You Reliable Enough?

Bias-Variance Tradeofg

The total squared error of an estimator can be decomposed as E

ˆ

Σ − Σ2 = Varˆ Σ + Biasˆ Σ, Σ2 Variance An estimator ˆ Σ computed on real data

• f the quantity Σ is afgected by noise.

Varˆ Σ = Eˆ Σ2 − Eˆ Σ2 is high. Bias A matrix ¯ Σ chosen for it’s structure has a bias w.r.t. Σ. Bias¯ Σ, Σ = E¯ Σ − Σ is high. ˆ Σ ¯ Σ Σ (1 − α)ˆ Σ + α¯ Σ Shrinkage Σ Real Covariance Matrix ˆ Σ Estimated Covariance Matrix ¯ Σ Shrinkage Target α Shrinkage Intensity

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• 4. Dear Manager, Are You Reliable Enough?
• 1.08

−0.2 0.8 −0.2 0.92 −0.3 0.8 −0.3 1.12

• = ˆ

Σ ¯ Σ =

• 1

1 1

• Σ =
• 1.06

−0.21 0.78 −0.21 0.95 −0.28 0.78 −0.28 1.1

• Σα

Σα = (1 − α)ˆ Σ + α¯ Σ =

• 1.06

−0.2 0.8 −0.2 0.94 −0.3 0.8 −0.3 1.09

• with α = 0.25

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• 4. Dear Manager, Are You Reliable Enough?

Theorem (Shrinkage Interpretation of Constraints)

Given a covariance matrix ˆ Σ and a set of constraints C there exist ΣC such that argmin

w∈C

wT ˆ Σw = ˆ w = argmin

w∈Rn wT ΣCw

In addition ΣC = ˆ Σ + α∆ = (1 − α) ˆ Σ + αˆ Σ + ∆ so ΣC is a shrinked version of ˆ Σ. ˆ Σ ¯ Σ = ˆ Σ + ∆ Σ Σα = ΣC

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• 4. Dear Manager, Are You Reliable Enough?

Variance-Turnover Tradeofg

When reallocating a portfolio, it can be interesting to have a turnover constraint in order to limit the impact of transaction costs w − ¯ w1 ≤ ¯ c. The marginal benefjt (risk reduction) decreases substantially when turnover increases.

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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?

• 5. Dear Solver, Can You Find It?

Objective Function Linear Quadratic Non-Linear Constraints Linear LP QP NLP Quadratic QCLP QCQP NLP Mixed-Integers MILP MIQP MINLP Non-Linear NLP NLP GNLP

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• 5. Dear Solver, Can You Find It?

Objective Function Linear Quadratic Non-Linear Constraints Linear LP QP NLP Quadratic QCLP QCQP NLP Mixed-Integers MILP MIQP MINLP Non-Linear NLP NLP GNLP

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• 5. Dear Solver, Can You Find It?

Objective Function Linear Quadratic Non-Linear Constraints Linear LP QP NLP Quadratic QCLP QCQP NLP Mixed-Integers MILP MIQP MINLP Non-Linear NLP NLP GNLP

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• 5. Dear Solver, Can You Find It?

Objective Function Linear Quadratic Non-Linear Constraints Linear LP QP NLP Quadratic QCLP QCQP NLP Mixed-Integers MILP MIQP MINLP Non-Linear NLP NLP GNLP

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• 5. Dear Solver, Can You Find It?

Objective Function Linear Quadratic Non-Linear Constraints Linear LP QP NLP Quadratic QCLP QCQP NLP Mixed-Integers MILP MIQP MINLP Non-Linear NLP NLP GNLP

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• 5. Dear Solver, Can You Find It?

Objective Function Linear Quadratic Non-Linear Constraints Linear LP QP NLP Quadratic QCLP QCQP NLP Mixed-Integers MILP MIQP MINLP Non-Linear NLP NLP

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• 6. Appendix

References

– Ledoit O., Wolf M. «Improved estimation of the covariance matrix of stock returns with an application to portfolio selection» Journal of Empirical Finance, Vol. 10, No. 5 (December 2003) pp. 603–621. – Ledoit O., Wolf M. «A well-conditioned estimator for large-dimensional covariance matrices» Journal of Multivariate Analysis, Vol. 88, No. 2 (February 2004) pp. 365–411. – Ledoit O., Wolf M. «Honey, I shrunk the sample covariance matrix» The Journal of Portfolio Management, Vol. 30,

• No. 4 (Summer 2004) pp. 110–119.

– Markowitz H. «Portfolio Selection» Journal of Finance, Vol. 7, No. 1 (1952) pp. 77-91. – Roncalli T. «Understanding the Impact of Weights Constraints in Portfolio Theory» (January 31, 2011). Available at SSRN: https://ssrn.com/abstract=1761625

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