Incorporating Stochastic Dominance and Progressive CVaR Levels in Portfolio Models

Incorporating Stochastic Dominance and Progressive CVaR

Levels in Portfolio Models

Gautam Mitra

Co-authors: Diana RomanCsaba Fabian

Victor Zviarovich

LQG Investment Technology Day

Outline

•The problem of portfolio construction

•Models of Choice

•Second order stochastic dominance

•Index tracking and outperforming

•Using SSD for enhanced indexation

•Numerical results

•Summary and conclusions

The portfolio selection problem

Models for choice

Proposed approach

Second Order

Stochastic Dominance

Numerical results

Conclusions

Index tracking /

outperforming

3

Three leading problemsThree leading problems

• Valuation or pricing of assetscash flows and returns are random; pricing theory

has been developed mainly for derivative assets.

• Ex-ante decision of asset allocation… optimum risk decisions

portfolio planning or portfolio rebalancing decisions..?

• Timing of the decisions

when to execute portfolio rebalancing decisions..?

Research Problems in Research Problems in FinanceFinance


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

4

The messageThe message

• The investment community follows classical{=modern} portfolio

theory based on (symmetric) risk measure.. variance

• Computational and applicable models have been enhanced

through capital asset pricing model (CAPM) and arbitrage pricing theory (APT)

• In contrast to investment community… regulators are concerned with downside (tail) risk of portfolios

• The real decision problem is to limit downside risk and improve upside potential

The main focus of the talkThe main focus of the talk


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

5

A historical perspectiveA historical perspective

• Markowitz ..mean-variance 1952,1959

• Hanoch and Levy 1969, valid efficiency criteria individual’s utility function

• Kallberg and Ziemba’s study.. alternative utility functions

• Sharpe ..single index market model 1963

• Arrow- Pratt.. absolute risk aversion


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

6

A historical A historical perspective..contperspective..cont

• Sharpe 64, Lintner 65, Mossin 66…CAPM model

• Rosenberg 1974 multifactor model

• Ross.. Arbitrage Pricing Theory(APT) multifactor equilibrium model

• Text Books: Elton & Gruber, Grinold & Kahn, Sortino & Satchell

• LP formulation 1980s.. computational tractability

• Konno MAD model.. also weighted goal program

• Perold 1984 survey…


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

7

Evolution of Portfolio Evolution of Portfolio ModelsModels

Current practice and R&D focus: Mean variance

Factor model

Rebalancing with turnover limits

Index Tracking (+enhanced indexation)[Style input and goal oriented model]

Cardinality of stock held: threshold constraints

Cardinality of trades: threshold constraints


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

8

Target return and risk measuresTarget return and risk measures Symmetric risk measures a critique.

0.0 0.5 1.0 1.5 2.0

Return

Relative Frequency (Density Function) Portfolio Y Portfolio X

Distribution properties of a portfolio

…shaping the distribution


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

9


•An amount of capital to invest now

•n assets

•Decision: how much to invest in each asset

•Purpose: the highest return after a specified time T

•Each asset’s return at time T is a random variable -> decision making under risk

Notations:

•n = the number of assets

•Rj = the return of asset j at time T

•x=(x1,…,xn) portfolio: decision variables; xj = the fraction of wealth invested in asset j

•X: the set of feasible portfolios


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

10

3 major problems:

• the distribution of (R1,…,Rn) ( -> scenario generation)

• the model of choice used

• the timing / rebalancing

•Portfolio x=(x1,…,xn). Its return: RX=x1R1+…+xnRn

•Portfolio y=(y1,…,yn). Its return: RY=y1R1+…+ynRn

•RX and RY - random variables

•How do we choose between them?


Models for choosing between random variables!


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming


•S scenarios: rij=the return of asset j under scenario i; j in 1…n, i in 1..S. (pi=probability of scenario i occurring)

•The (continuous) distribution of (R1,…,Rn) is replaced with a discrete one, with a finite number of outcomesasset1 asset2 … asset n probability

scenario 1 r11 r12 … r1n p 1

scenario 2 r21 r22 … r2n …

… … … … … …

scenario S rS1 rS2 … rSn p S


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

12

Models for choice under risk

-Mean-risk models

-Stochastic dominance / Expected utility maximisation

“Max” Rx

Subject to: x X

-Index-tracking modelsThe index’s return distribution is available: The index’s return distribution is available: RRII

“Min” |Rx – RI |

Subject to: x X

-Enhanced indexation modelsThe index’s return distribution is available as a reference; The index’s return distribution is available as a reference; this distribution should be improved .this distribution should be improved .

(1)

(2)


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

13

Models for choice under risk: Mean-risk models

•2 scalars attached to a r.v.: the mean and the value of a risk measure.

•Let be a risk measure: a function mapping random variables into real numbers.

•In the mean-risk approach with risk measure given by , RX is preferred to r.v. RY if and only if: E(RX)E(RY) and

(RX) (RY) with at least one strict inequality.


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

14

Expected Utility Maximisation

- A utility function: a real valued function defined on real numbers (representing possible wealth levels).

- Each random return is associated a number: its “expected utility”.

- Expected utilities are compared (larger values preferred)

- Q: How should utility functions be chosen?


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

15

Expected Utility Maximisation: Risk aversion behaviour

wealth

U(w)

U

Risk-aversion: the observed economic behaviour

A surplus of wealth is more valuable at lower wealth levels concave utility function


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

16

Models for choice under risk: Stochastic dominance (SD)

SD ranks choices (random variables) under assumptions about general characteristics of utility functions.

It eliminates the need to explicitly specify a utility function.

• First order stochastic dominance (FSD);

• Second order stochastic dominance (SSD);

• Higher orders.


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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First order Stochastic dominance (FSD)

The “stochastically larger” r.v. has a smaller distribution function: F FSDG

Strong requirement!

outcome

probability

1

x

F(x)

G(x)

F G


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

18

Second order Stochastic dominance (SSD)

A weaker requirement: concerns the “cumulatives” of the distribution functions.

Typical example: F starts lower (meaning smaller probability of low outcomes); F SSD G.

outcome

probability

1

F G


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

19

Second Order Stochastic dominance (SSD)

Particularly important in investment!

Several equivalent definitions:

•The economist’s definition: RXSSDRY E[U(RX)] E[U(RY)], U non-decreasing and concave utility function.

(Meaning: RX is preferred to RY by all rational and risk-averse investors).

•The intuitive definition: RXSSDRY E[t- RX]+ E[t- RY]+, tR

[t- RX]+= t- RX if t- RX 0

[t- RX]+= 0 if t- RX < 0


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Second Order Stochastic dominance (SSD)

Thus SSD describes the preference of rational and risk-averse investors: observed economic behaviour.

Unfortunately, very demanding from a computational point of view.


Models for choice

Proposed approach

Second Order


Numerical results

Index tracking /

outperforming

Conclusions

Index Tracking and Enhanced Indexation

21

• Over the last two to three decades, index funds have gained tremendous popularity among both retail and institutional equity investors. This is due to

(i) disillusionment with the performance of active funds, also (ii) predominantly it reflects attempts by fund managers to minimize their costs. Managers adopt strategies that allocate capital to both passive index and active management funds.

• The funds are therefore run at a reduced cost of passive funds, and managers concentrate on a few active components.

As Dan DiBartolomeo says “Enhanced index funds generally involve a quantitatively defined strategy that ‘tilts’ the portfolio composition away from strict adherence to some popular market index to a slightly different composition that is expected to produce more return for similar levels of risk”.


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

22

Index tracking models

Traditionally, minimisation of “tracking error”: the standard deviation of the difference between the portfolio and index returns.

Other approaches:

•Based on minimisation of other risk measures for the difference between the portfolio and index returns: MAD, semivariance, etc.

•Regression of the tracking portfolio’s returns against the returns of the index


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

23

Models for choice under risk

-Mean-risk models

-Stochastic dominance / Expected utility maximisation

“Max” Rx

Subject to: x X

-Index-tracking modelsThe index’s return distribution is available: The index’s return distribution is available: RRII

“Min” |Rx – RI |

Subject to: x X

-Enhanced indexation modelsThe index’s return distribution is available as a reference; The index’s return distribution is available as a reference; this distribution should be improved .this distribution should be improved .

(1)

(2)


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

24

Index tracking models

A few models have been proposed: concerned with overcoming the computational difficulty (less focus on the actual fund performance).

Issues raised: large number of stocks in the portfolio’s composition, low weights for some stocks.

Thus: Threshold constraints... cardinality constraints, to reduce transaction costs are imposed -> requires use of binary variables-> leads to computational difficulty.


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Enhanced indexation models

• Aim to outperform the index: generate “excess” return.

• The computational difficulty is a major issue.

• Relatively new area; no generally accepted approach.

• Regression of the tracking portfolio’s returns against the returns of the index; the resulting gap between the intercepts is the excess ‘alpha’ which is to be maximsed


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

26

SD under equi-probable scenarios

Let RX, RY r.v. with equally probably outcomesOrdered outcomes of RX: 1 … S Ordered outcomes of RY: 1 … S

RX SSDRY 1+…+ i 1+…+ i , i = 1…S

Taili(RX) Taili(RY)

RX FSDRY i i , i = 1…S


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

27

Proposed approach

Purpose: to determine a portfolio whose return distribution• is non-dominated w.r. to SSD. • tracks (enhances) a “target” known return distribution (e.g. an index)

Assumption: equi-probable scenarios (not restrictive!)

the SD relations greatly simplified!


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

SSD under equi-probable scenarios: an example

Consider the case of 4 equi-probable scenarios and two random variables X, Y whose outcomes are:

X: 0 2 -1 3

Y: 1 0 0 3

Rearrange their outcomes in ascending order:

X: -1 0 2 3

Y: 0 0 1 3

None of them dominates the other with respect to FSD.Cumulate their outcomes:X: -1 -1 1 4

Y: 0 0 1 4

Y dominates X w.r.t. SSD. Intuitively: it has better outcomes under worst-case scenarios.


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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SSD under equi-probable scenarios

Equivalent formulation using Conditional Value-at-Risk

Confidence level (0,1). =A%.

CVaR(RX) = - the mean of its worst A% outcomes

1

1( ) ( ... )i X i

S

CVaR Ri

Thus:

( ) ( ), 1...X SSD Y i X i Y

S S

R R CVaR R CVaR R i S


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

Conditional Value-at-Risk: an example

Consider a random return with 100 equally probable outcomes.We order its outcomes; suppose that its worst 10 outcomes are:

1

100

( ) ( 0.2) 20%XCVaR R

-0.2

-0.18

-0.15

-0.13

-0.1

-0.1

-0.08

-0.05

-0.05

-0.03

Confidence level =0.01=1/100:

The average loss under the worst 1% of scenarios is 20%.


The average loss under the worst 5% of scenarios is 15.2%.


The average loss under the worst 10% of scenarios is 10.7%.

CVaR5/100(Rx)=-1/5[(-0.2)+(-0.18)+…+(-0.1)]=0.152

CVaR10/100(Rx)=-1/10[(-0.2)+(-0.18)+…+(-0.03)]=0.107


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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A multi-objective model

The SSD efficient solutions: solutions of a multi-objective model:

Or:

1max( ( ),..., ( ))X S XV Tail R Tail RSuch that:

(1)

x X1/ /min( ( ),..., ( ))S X S S XV CVaR R CVaR R

Such that:

(2)

Worst outcome Sum of all outcomes


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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The reference point method

How do we choose a specific solution?

Specify a target (goal) in the objective space and try to come close (or better) to it:

If the target is not efficient, outperform it “quasi-satisficing”decisions (Wierzbicki 1983)

Target = the tails (or scaled tails) of an index.


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

33

The reference point method

Consider the “worst achievement”:

Let z* =(z1*,…,zS*) be the target

zi*= the Taili of the index (sum of i worst outcomes)

*1

( ) min( ( ) * )z i x ii S

x Tail R z

The problem we solve:

*max( ( ))zx X

x

• Basically, it optimises the “worst achievement”.

Alternatively, zi*= the “scaled” Taili of the index (mean of the worst i outcomes)


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

34

Expressing tails

Cutting plane representation of CVaR / tails (Künzi-Bay and Mayer 2006)

( )j T

j J

R x

Such that:

Taili(RX) = Min

• Similar representation for the “scaled” tails.

{1,..., }, | |J S J i

= realisation of RX under scenario j( )j TR xwhere


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Model formulation

( ) *, i

j Ti

j J

R x z

Such that:

{1,..., }, | |i iJ S J i

Max

, R x X

for each

1,...,i S• Similar formulation when “scaled” tails are

considered; different results obtained.

• Both formulations lead to SSD efficient portfolios that track and improve on the return distribution of the index.


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Computational behaviour and…

• Very good computational time; problems with tens of thousands of scenarios solved in seconds. ( Pentium 4 , 3.00 GHz, 2 Gbytes Ram. )

• Portfolios computed by this model possess good return distributions (in-sample).


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Computational study

• FTSE100: 101 stocks, 115 scenarios• Nikkei: 225 stocks, 162 scenarios• S&P 100: 97 stocks, 227 scenarios

3 data sets: past weekly returns considered as equally probable scenarios.

The corresponding indices, the same time periods.


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Computational study

• We construct portfolios based on our proposed models (i)scaled tails (ii) unscaled tails and (iii) tracking error minimisation. No cardinality constraints imposed.

• The actual returns are computed for the next time period and compared to the historical return of the index.

• Rebalancing frame (weekly): back-testing over the period 5 Jan – 15 March 2009 (10 weeks).

• Practicality of the resulting solutions: number of stocks in the composition, necessary rebalancing.


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

39

Computational study: FTSE 100

Back-testing: Ex-post returns, 5 Jan – 15 Mar 2009

-0.12-0.1

-0.08-0.06-0.04-0.02

00.020.040.060.08

1 2 3 4 5 6 7 8 9 10

time period

retu

rnSSD Index TrackError


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Computational study: FTSE 100

Back-testing: Ex-post compounded returns,5 Jan – 15 Mar 2009

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1 2 3 4 5 6 7 8 9 10

time

cum

ula

tive

ret

urn

SSD Index TrackError


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Computational study: Nikkei 225


-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

1 2 3 4 5 6 7 8 9 10

time period

retu

rnSSD index TrackError


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Computational study: Nikkei 225

Back-testing: Ex-post compounded returns, Jan – 15 Mar 2009

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1 2 3 4 5 6 7 8 9 10

time period

cum

ula

tive

ret

urn



Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Computational study: S&P100


-0.16

-0.12

-0.08

-0.04

0

0.04

0.08

1 2 3 4 5 6 7 8 9 10

time period

retu

rn



Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Computational study: S&P100

Backtesting: Ex-post compounded returns, Jan – 15 Mar 2009

0.6

0.7

0.8

0.9

1

1.1

1 2 3 4 5 6 7 8 9 10

cum

ula

tive

ret

urn

time period

SSD Index TracKError


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Computational study: composition of portfolios

No of stocks (on average)

SSD_scaled SSD_unscaled TrackError

FTSE 100 9 11 58

Nikkei 225 12 3 118

S&P 100 14 17 73

No need to impose cardinality constraints in the SSD based models.


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

46


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Computational study: composition of portfolios

• Composition of SSD portfolios: very stable, only little rebalancing necessary.

• Particularly, the case of “unscaled” SSD model: rebalancing is only needed when the new scenarios taken into

account make the previous optimum change (lead to a higher difference between worst outcome of the portfolio and the worst outcome of the index).

• Case of Nikkei 225 and FTSE100, unscaled SSD model: NO rebalancing was necessary for the 10 time periods of backtesting.


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Summary and conclusions

• SSD represents the preference of risk-averse investors;

• The proposed model selects a portfolio that is efficient w.r.t. SSD, and…

• Tracks (improves) a desirable, “target”, “reference” distribution, e.g. that of an index;

• Use in the context of enhanced indexation;

• The resulting model is solved within seconds for very large data sets;


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Summary and conclusions

• Back-testing: considerably and consistently realised improved performance over the indices and the index tracking strategies (trackers).

• Good strategy in a rebalancing frame:

o Naturally few stocks are selected (no need of cardinality constraints);

o Little (or no) rebalancing necessary: use as a rebalancing signal strategy.


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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References

• Canakgoz, N.A. and Beasley, J.E. (2008): Mixed-Integer Programming Approaches for Index Tracking and Enhanced Indexation, European Journal of Operational Research 196, 384-399

• Fabian, C., Mitra, G. and Roman, D. (2009): Processing Second Order Stochastic Dominance Models Using Cutting Plane Representations, Mathematical Programming, to appear.

• Kunzi-Bay, A. and J. Mayer (2006): Computational aspects of minimizing conditional value-at-risk, Computational Management Science 3, 3-27.

• Ogryczak, W. (2002): Multiple Criteria Optimization and Decisions under Risk, Control and Cybernetics, 31, no 4

• Roman, D., Darby-Dowman, K. and G. Mitra: Portfolio Construction Based on Stochastic Dominance and Target Return Distributions, Mathematical Programming Series B 108 (2-3), 541-569.

• Wierzbicki, A.P. (1983): A Mathematical Basis for Satisficing Decision Making, Mathematical Modeling, 3, 391-405.


Models for choice

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Second Order


Numerical results

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Index tracking /

outperforming

THANK YOU

• CONTACT US : [email protected]• [email protected]• [email protected]

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Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Evolution of Portfolio Evolution of Portfolio ModelsModels

• Tracking error as a constraint…[discuss ]

• Nonlinear transaction cost /market impact[discuss ]

• Trade scheduling =algorithmic trading.. [discuss ]

• Resampled efficient frontier

• Risk attribution and risk budgeting


Models for choice

Proposed approach

Second Order


Numerical results

Conclusions

Index tracking /

outperforming

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Incorporating Stochastic Dominance and Progressive CVaR Levels in Portfolio Models

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