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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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…
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
9
The portfolio selection problem
•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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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?
The portfolio selection problem
Models for choosing between random variables!
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
The portfolio selection problem
•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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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)
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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?
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
17
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
20
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.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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”.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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)
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
25
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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!
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
29
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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%.
Confidence level =0.05=5/100:
The average loss under the worst 5% of scenarios is 15.2%.
Confidence level =0.1=10/100:
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
31
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
32
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.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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)
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
35
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.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
36
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).
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
37
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.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
38
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.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
40
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
41
Computational study: Nikkei 225
Back-testing: Ex-post returns, 5 Jan – 15 Mar 2009
-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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
42
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
SSD Index TrackError
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
43
Computational study: S&P100
Back-testing: Ex-post returns, 5 Jan – 15 Mar 2009
-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
SSD Index TrackError
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
44
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
45
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.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
46
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
47
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.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
48
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;
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
49
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.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
50
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.
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
THANK YOU
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The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
52
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
The portfolio selection problem
Models for choice
Proposed approach
Second Order
Stochastic Dominance
Numerical results
Conclusions
Index tracking /
outperforming
53