Coherent Distortion Risk Measures in Portfolio Selection · August 11, 2011 Ming Bin Feng 1/ 37....

IntroductionCDRM Optimization

Case StudiesConclusions and Future Directions

Coherent Distortion Risk Measuresin Portfolio Selection

(Joint work with Dr Ken Seng Tan)

Ming Bin Feng

University of Waterloo

The 46th Actuarial Research Conference

August 11, 2011Ming Bin Feng 1/ 37



Abstract

The theme of this presentation relates to solving portfolioselection 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 CDRMoptimization problems.

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MotivationsGoals

Outline

1 IntroductionMotivationsGoals

2 CDRM Optimization

3 Case Studies

4 Conclusions and Future Directions

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MotivationsGoals

Motivations

Practical portfolio selection problemsGood risk measuresWell-studied programming models

QuestionCan we connect this together? We want to solve practicalportfolio optimization problems with sophisticated riskmeasures using a programming model that can be solvedefficiently.

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MotivationsGoals

We wish to..

Incorporate a general class of risk measure into awell-studied programming modelStudy equivalences among different formulations ofportfolio selection problemsSolve portfolio selection problems of interest efficiently

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CVaR OptimizationCDRM Representation TheoremCDRM OptimizationFormulation Equivalences

Outline

1 Introduction

2 CDRM OptimizationCVaR OptimizationCDRM Representation TheoremCDRM OptimizationFormulation Equivalences

3 Case Studies


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Scenario Generation

Loss Matrix

p1 →p2 →...

...pm →

L =

L11 L12 · · · L1nL21 L22 · · · L2n

... · · · . . ....

Lm1 Lm2 · · · Lmn

→ l1 = l(x ,p1)→ l2 = l(x ,p2)...

...→ lm = l(x ,pm)

Let l(1) ≤ · · · ≤ l(m) be the ordered losses, p(i), i = 1, · · · ,m bethe corresponding probability masses.

Return/Price/Premium/Profit Vector

c = [c1, · · · , cm]′

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CVaR Optimization

Background

Consider the special function

F (x , ζ) = ζ +1

1− α

m∑j=1

pj(lj − ζ)+

Rockafellar and Uryasev [3, 4] showed that

1 CVaRα(x) = minζ∈R F (x , ζ)

2 minx∈X CVaRα(x) = min(x ,ζ)∈X×R F (x , ζ)

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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

minimize ζ + 11−α

m∑j=1

pjzj

subject to c′x ≥ µl(x ,pj)− ζ ≤ zj j = 1, · · · ,m

0 ≤ zj j = 1, · · · ,m(x , ζ) ∈ X × R

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CVaR Optimization

Return maximization subject to CVaR constraint(s)

maximize c′x

subject to ζi + 11−α

m∑j=1

pjzij ≤ ηi i = 1, · · · , k

l(x ,pj)− ζi ≤ zij ∀i , j0 ≤ zij ∀i , j

(x , ζ) ∈ X × Rk

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Definition and Representation Theorem

Two Equivalent Definitions

A risk measure ρ(x) is a CDRM if it isA comonotone law-invariant coherent risk measureA 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

function w : [0,1] 7→ [0,1], satisfying1∫

α=0wαdα = 1, such that

ρ(x) =

1∫α=0

CVaRα(x)wαdα





Representation Theorem in Discrete Case

Finite Generation Theorem for CDRM

Given a concave distortion function g, ρ(x) =m∑

i=1qi l(i),

moreover

ρ(x) =m∑

i=1

wiCVaR i−1m

(x), where

wi =

q1

p(1)if i = 1

(qi −p(i)

p(i−1)qi−1)

m∑j=i

p(j)

p(i)if i = 2, · · · ,m





CDRM Optimization

CDRM minimization subject to a return constraint

minimizem∑

i=1wi(ζi + 1

1−α

m∑j=1

pjzij)

subject to c′x ≥ µl(x ,pj)− ζi ≤ zij ∀i , j

0 ≤ zij ∀i , j(x , ζ) ∈ X × Rm





CDRM Optimization

Return maximization subject to one CDRM constraint

maximize c′x

subject tom∑

i=1wi(ζi + 1

1−α

m∑j=1

pjzij) ≤ η

l(x ,pj)− ζi ≤ zij ∀i , j0 ≤ zij ∀i , j

(x , ζ) ∈ X × Rm





CDRM Optimization

Return-CDRM utility maximization

maximize c′x − τm∑

i=1wi(ζi + 1

1−α

m∑j=1

pjzij)

subject to l(x ,pj)− ζi ≤ zij ∀i , j0 ≤ zij ∀i , j

(x , ζ) ∈ X × Rm

This formulation is very similar to a return maximizationproblem with m CVaR constraints. Yet we converted m CVaRconstraints into the objective function.





CDRM Optimization

CDRM-based Sharpe ratio maximization

maximize c′x−νm∑

i=1wi (ζi+

11−α

m∑j=1

pj zij )

subject to l(x ,pj)− ζi ≤ zij ∀i , j0 ≤ zij ∀i , j

(x , ζ) ∈ X × Rm

This is an LFP, but we can solve it by solving at most tworelated LPs using a variable transformation method studied byCharnes and Cooper [1].





Formulation Equivalences

Equivalences among four formulations, part 1

Problem Max-Return Min-CDRMPresetParameter

η µ

ImpliedParametersη = N/A ρ(x∗)µ = c′x∗ N/Aτ = u1 1

u2

ν = c′x∗ − u1ρ(x∗) R(x∗)− 1u2 ρ(x∗)

If the return and CDRM constraints are binding at respectiveoptimal solutions, the preset parameter for Max-Return equalsto the implied parameter for Min-CDRM and vice versa.





Formulation Equivalences

Equivalences among four formulations, part 1

Problem Max-Utility Max-SharpePresetParameter

τ ν

ImpliedParametersη = ρ(x∗) ρ(x∗)µ = c′x∗ c′x∗

τ = N/A c′x∗−νρ(x∗)

ν = c′x∗ − τρ(x∗) N/A

We will see that the preset parameter for Max-Return equals tothe implied parameter for Min-CDRM and vice versa.




Case 1: Reinsurance portfolio selection with simulated dataCase 2: Investment portfolio selection with historical data

Outline

1 Introduction

2 CDRM Optimization

3 Case StudiesCase 1: Reinsurance portfolio selection with simulated dataCase 2: Investment portfolio selection with historical data






Case Study 1: Constructing Reinsurance Portfolios

We wish to construct profit-CVaR0.95(L) efficient portfolios fromthe following 10 risk contracts. Simulations are done for 10,000scenarios.

Contract Premium LossesMean STD 95%VaR 95%CVaR

1 554271 311388 1377843 2613161 58854422 364272 222117 1172497 588329 43382143 91763 55953 739026 0 11190654 867176 437968 1806626 3845685 79376105 798005 438464 2913258 0 87692846 107585 43381 263019 0 8676247 878525 375438 1375166 3160679 59740878 3081188 1283828 2199151 5661191 84426349 65162 29352 324061 0 587044

10 885897 385173 1047454 1506500 3693435





Case Study 1: Constructing Reinsurance Portfolios

Balanced portfolio consisting of 0.1 unit of each risk.

Summary of balanced portfolio

Premium Losses ExpectedProfitMean STD 95%VaR 95%CVaR

769384 358306 667647 1716458 2656764 40578

Profit-95%CVaR utility maximization with τ = 0.2

Summary of target portfolio

Premium Losses ExpectedProfitMean STD 95%VaR 95%CVaR

769384 305689 492425 1313074 1815641 463695





Profit-CVaR Efficient Frontier (Enlarged)

95%-CVaR

Pro

fit

420000

440000

460000

480000

500000

520000

τ=0.2

x-intercept = 1815641

y -intercept = 463695

1800000 2000000 2200000 2400000 2600000





Profit-CVaR Efficient Frontier

95%-CVaR

Pro

fit

0e+00

1e+05

2e+05

3e+05

4e+05

5e+05

Intercept = 100567, Slope = 0.2

0 500000 1000000 1500000 2000000 2500000





Data decriptions

2 stocks from each of the 10 sectors defined in GlobalIndustry Classification Standard(GICS).Weekly prices from Jan-02-2001 to May-31-2011Adjusted closing prices obtained from finance.yahoo.com





Sum of these 20 stocks’ prices can be viewed as the “market”

Market Portfolio Value from 2001 to 2011

Year

Mar

ketP

ortfo

lioVa

lue

500

1000

1500

2000

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011





Optimization Settings

Replace scenario generation by historical dataConstant “sample” size of 100.c = expected sample returns, L = negative returns matrix.Weekly rebalancing via CDRM-minimization.x ≥ 0, x ≤ 0.2, budget constraint, and return constraint.





Efficient Frontier, beginning of 2003

95%-CVaR of Negative Returns (in %)

Ret

urn

(in%

)

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 1 2 3 4 5 6 7



Ret

urn

(in%

)

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5







Ret

urn

(in%

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4



Ret

urn

(in%

)

-0.3

-0.2

-0.1

0.0

0.1

0.2

0 2 4 6 8 10





Portfolio Selection over Different CDRMs

Well-known CDRMsCVaRα distortion:gCVaR(x , α) = min{ x

1−α ,1}Wang Transform(WT) distortion:gWT (x , β) = Φ[Φ−1(x)− Φ−1(β)]

Proportional hazard(PH) distortion:gPH(x , γ) = xγ with γ ∈ (0,1]

Lookback(LB) distortion:gLB(x , δ) = xδ(1− δ ln x) with δ ∈ (0,1]





Portfolio Values with Out-of-Sample Returns

Year

Port

folio

Valu

es

100

120

140

160

180

200

2003 2004 2005 2006 2007 2008 2009 2010 2011

Risk MeasuresCVaR0.9

CVaR0.95

CVaR0.99


Year

Port

folio

Valu

es

100

120

140

160

180

200

2003 2004 2005 2006 2007 2008 2009 2010 2011

Risk MeasuresWT0.75

WT0.85

WT0.95





Summary statistics of optimal out-of-sample returns

Mean STD Skew Kurt SharpeCVaR0.9 0.00148 0.01891 -0.93697 6.08202 0.07833CVaR0.95 0.00117 0.02050 -0.56738 4.96513 0.05718CVaR0.99 0.00139 0.02243 -0.20805 4.47107 0.06219WT0.75 0.00164 0.01919 -1.00243 7.06069 0.08560WT0.85 0.00261 0.01915 -0.77534 5.88635 0.07477WT0.95 0.00232 0.02107 -0.30517 5.46812 0.06628






Year

Port

folio

Valu

es

100

150

200

250

300

2003 2004 2005 2006 2007 2008 2009 2010 2011

Risk MeasuresPH0.1

PH0.5

PH0.9


Year

Port

folio

Valu

es

100

120

140

160

180

200

2003 2004 2005 2006 2007 2008 2009 2010 2011

Risk MeasuresLB0.1

LB0.5

LB0.9






Mean STD Skew Kurt SharpePH0.1 0.00130 0.02218 -0.26156 5.14293 0.05844PH0.5 0.00148 0.02091 -0.83421 8.50931 0.07091PH0.9 0.00277 0.02622 -0.95739 6.78000 0.10574LB0.1 0.00134 0.02230 -0.22880 4.59996 0.05995LB0.5 0.00137 0.02130 -0.34008 5.15387 0.06439LB0.9 0.00145 0.01893 -0.80400 6.04230 0.07645






Year

Portfo

lioVa

lues

100

150

200

250

300

2003 2004 2005 2006 2007 2008 2009 2010 2011

LegendCVaR(0.9)WT(0.75)PH(0.9)LB(0.9)1n Portfolio






Mean STD Skew Kurt Sharpe1n -portfolio 0.00208 0.03038 0.25175 13.73943 0.06845CVaR0.9 0.00148 0.01891 -0.93697 6.08202 0.07833WT0.75 0.00164 0.01919 -1.00243 7.06069 0.08560PH0.9 0.00277 0.02622 -0.95739 6.78000 0.10574LB0.9 0.00145 0.01893 -0.80400 6.04230 0.07645




Concluding remarksFuture Directions

Outline

1 Introduction

2 CDRM Optimization

3 Case Studies

4 Conclusions and Future DirectionsConcluding remarksFuture Directions





Linear optimization for CDRM portfolio selection

CDRM portfolio optimization with LPS and LFPsCDRM includes CVaR, WT, PH, and LBChoose CDRM that suits specific risk appetitesFour different CDRM formulations are equivalentEquivalences are helpful for interpretation of parameters,verification of consistencies, and estimation of impliedinformation





Empirical results

Simple portfolio construction rules can be very inefficient,active management is important.Despite the inefficiency of the 1

n -portfolio, its terminalwealth (based on out-of sample returns) can be highWe have found CDRM efficient portfolios with higherSharpe ratio than the 1

n -portfolio’s





Future Directions

Apply various decomposition methods to solve CDRMproblems more efficientlyApply stochastic programming techniques to solve CDRmproblemsApply CDRM approach in multi-period modelsExplore/identify other members of CDRM (Higher momentcoherent risk measure)





References I

A. Charnes and W.W. Cooper.Programming with linear fractional functionals.Naval Research Logistics Quarterly, 9(3-4):181–186, 1962.

P.A. Krokhmal, J. Palmquist, and S. Uryasev.Portfolio optimization with conditional value-at-risk objectiveand constraints.Journal of Risk, 4:43–68, 2002.

R.T. Rockafellar and S. Uryasev.Optimization of conditional value-at-risk.Journal of risk, 2:21–42, 2000.





References II

R.T. Rockafellar and S. Uryasev.Conditional value-at-risk for general loss distributions.Journal of Banking & Finance, 26(7):1443–1471, 2002.


Coherent Distortion Risk Measures in Portfolio Selection · August 11, 2011 Ming Bin Feng 1/ 37....

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