LQG Investment Technology Day
Incorporating Stochastic
Dominance and Progressive CVaR
Levels in Portfolio Models
Gautam Mitra
Co-authors:
Diana Roman
Csaba Fabian
Victor Zviarovich
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
Research Problems in Finance
Three leading problems
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
• Valuation or pricing of assets
cash 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..?
3
The main focus of the talk
The message
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
• 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
4
A historical perspective
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
• 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
5
A historical perspective..cont
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
• 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…
6
Evolution of Portfolio Models
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
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
Proposed
approach
Numerical
results
Conclusions
7
Target return and risk measures
Symmetric risk measures a critique.
The portfolio
selection
problem
Distribution properties of a portfolio
…shaping the distribution
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Portfolio Y
0.0
Relative Frequency
(Density Function)
0.5
Portfolio X
1.0
1.5
2.0
Proposed
approach
Numerical
results
Conclusions
8
Return
The portfolio selection problem
• An amount of capital to invest now
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
• 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:
Proposed
approach
•n = the number of assets
Numerical
results
•Rj = the return of asset j at time T
Conclusions
•x=(x1,…,xn) portfolio: decision variables; xj = the fraction of
wealth invested in asset j
•X: the set of feasible portfolios
9
The portfolio selection problem
3 major problems:
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
• 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?
Conclusions
Models for choosing between random variables!
10
The portfolio selection problem
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
•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 outcomes
…
asset1
asset2
asset n
probability
scenario 1
r11
r12
…
r1n
p1
scenario 2
…
scenario S
r21
…
rS1
r22
…
rS2
…
…
…
r2n
…
rSn
…
…
pS
Models for choice under risk
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
-Mean-risk models
-Stochastic dominance / Expected utility maximisation
“Max” Rx
Subject to: x  X
(1)
-Index-tracking models
The index’s return distribution is available: RI
“Min” |Rx – RI |
Subject to: x  X
(2)
-Enhanced indexation models
The index’s return distribution is available as a reference;
this distribution should be improved .
12
Models for choice under risk: Mean-risk models
The portfolio
selection
problem
•2 scalars attached to a r.v.: the mean and the value of a risk
measure.
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
•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.
13
Expected Utility Maximisation
The portfolio
selection
problem
-
A utility function: a real valued function defined on
real numbers (representing possible wealth levels).
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
-
Each random return is associated a number: its
“expected utility”.
-
Expected utilities are compared (larger values
preferred)
-
Q: How should utility functions be chosen?
Numerical
results
Conclusions
14
Expected Utility Maximisation:
Risk aversion behaviour
The portfolio
selection
problem
U
U(w)
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
wealth
Proposed
approach
Numerical
results
Conclusions
Risk-aversion: the observed economic behaviour
A surplus of wealth is more valuable at lower wealth
levels  concave utility function
15
Models for choice under risk:
Stochastic dominance (SD)
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
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.
16
First order Stochastic dominance (FSD)
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
probability
1
G(x)
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
G
F
F(x)
outcome
x
The “stochastically larger” r.v. has a smaller distribution
function: F FSDG
Strong requirement!
17
Second order Stochastic dominance (SSD)
The portfolio
selection
problem
Models for
choice
probability
1
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
G
F
outcome
A weaker requirement: concerns the “cumulatives” of the
distribution functions.
Typical example: F starts lower (meaning smaller
probability of low outcomes); F SSD G.
18
Second Order Stochastic dominance (SSD)
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Particularly important in investment!
Several equivalent definitions:
•The economist’s definition: RXSSDRY  E[U(RX)]  E[U(RY)],
U non-decreasing and concave utility function.
Index
tracking / (Meaning: R is preferred to R by all rational and risk-averse
X
Y
outperforming
investors).
Proposed
approach
Numerical
results
Conclusions
•The intuitive definition: RXSSDRY  E[t- RX]+ E[t- RY]+, tR
[t- RX]+= t- RX if t- RX  0
[t- RX]+= 0 if t- RX < 0
19
Second Order Stochastic dominance (SSD)
The portfolio
selection
problem
Models for
choice
Thus SSD describes the preference of rational and riskaverse investors: observed economic behaviour.
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Unfortunately, very demanding from a computational
point of view.
Proposed
approach
Numerical
results
Conclusions
20
Index Tracking and Enhanced
Indexation
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
•
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”.
21
Index tracking models
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
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
22
Models for choice under risk
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
-Mean-risk models
-Stochastic dominance / Expected utility maximisation
“Max” Rx
Subject to: x  X
(1)
-Index-tracking models
The index’s return distribution is available: RI
“Min” |Rx – RI |
Subject to: x  X
(2)
-Enhanced indexation models
The index’s return distribution is available as a reference;
this distribution should be improved .
23
Index tracking models
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
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.
Proposed
approach
Numerical
results
Conclusions
A few models have been proposed: concerned with
overcoming the computational difficulty (less focus on
the actual fund performance).
24
Enhanced indexation models
The portfolio
selection
problem
• Aim to outperform the index: generate “excess” return.
Models for
choice
Second Order
Stochastic
Dominance
• Relatively new area; no generally accepted approach.
Index
tracking /
outperforming
• 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
Proposed
approach
Numerical
results
Conclusions
• The computational difficulty is a major issue.
25
SD under equi-probable scenarios
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Let RX, RY r.v. with equally probably outcomes
Ordered outcomes of RX: 1 …  S
Ordered outcomes of RY: 1 …  S
RX FSDRY  i  i , i = 1…S
RX SSDRY  1+…+ i  1+…+ i , i = 1…S
Conclusions
Taili(RX)
Taili(RY)
26
Proposed approach
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
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)
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Assumption: equi-probable scenarios (not restrictive!)
 the SD relations greatly simplified!
Conclusions
27
SSD under equi-probable scenarios:
an example
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
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.
SSD under equi-probable scenarios
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Equivalent formulation using Conditional Value-at-Risk
Confidence level (0,1). =A%.
CVaR(RX) = - the mean of its worst A% outcomes
Proposed
approach
1
CVaR i ( RX )   (1  ...  i )
i
S
Numerical
results
Thus:
Conclusions
RX  SSD RY  CVaR i ( RX )  CVaR i ( RY ), i  1...S
S
S
29
Conditional Value-at-Risk: an example
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Consider a random return with 100 equally probable outcomes.
We order its outcomes; suppose that its worst 10 outcomes are:
-0.2
-0.18
-0.15
-0.13
-0.1
Confidence level =0.01=1/100:
CVaR 1 ( RX )  (0.2)  20%
100
The average loss under the worst 1% of scenarios is 20%.
-0.1
-0.08
Confidence level =0.05=5/100:
-0.05
-0.05
CVaR5/100(Rx)=-1/5[(-0.2)+(-0.18)+…+(-0.1)]=0.152
-0.03
Proposed
approach
Numerical
results
Conclusions
The average loss under the worst 5% of scenarios is 15.2%.
Confidence level =0.1=10/100:
CVaR10/100(Rx)=-1/10[(-0.2)+(-0.18)+…+(-0.03)]=0.107
The average loss under the worst 10% of scenarios is 10.7%.
A multi-objective model
The portfolio
selection
problem
Models for
choice
The SSD efficient solutions: solutions of a multi-objective
model:
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
V max(Tail1 ( RX ),..., TailS ( RX ))
(1)
Such that:
Worst outcome
Sum of all outcomes
Or:
Numerical
results
Conclusions
V min(CVaR1/ S ( RX ),..., CVaRS / S ( RX ))
Such that:
x X
31
(2)
The reference point method
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
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  “quasisatisficing”decisions (Wierzbicki 1983)
Proposed
approach
Numerical
results
Target = the tails (or scaled tails) of an index.
Conclusions
32
The reference point method
Let z* =(z1*,…,zS*) be the target
The portfolio
selection
problem
zi*= the Taili of the index (sum of i worst outcomes)
Models for
choice
Alternatively, zi*= the “scaled” Taili of the index (mean
of the worst i outcomes)
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Consider the “worst achievement”:
 z* ( x)  min(Taili ( Rx )  z *i )
Proposed
approach
1i  S
Numerical
results
The problem we solve:
max( z* ( x))
Conclusions
x X
•
Basically, it optimises the “worst achievement”.
33
Expressing tails
The portfolio
selection
problem
Cutting plane representation of CVaR / tails (Künzi-Bay
and Mayer 2006)
Models for
choice
Taili(RX) = Min
Second Order
Stochastic
Dominance
jJ
Such that:
Index
tracking /
outperforming
Proposed
approach
Numerical
results
R
( j )T
x
J  {1,..., S}, | J | i
where R ( j )T x = realisation of RX under scenario j
Conclusions
•
Similar representation for the “scaled” tails.
34
Model formulation
Max 
The portfolio
selection
problem
Such that:
   R( j )T x  zi *,
Models for
choice
jJi
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
  R, x  X
for each
J i  {1,..., S}, | J i | i
i  1,..., 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.
35
Computational behaviour and…
The portfolio
selection
problem
•
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
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
36
Computational study
3 data sets: past weekly returns considered as equally
probable scenarios.
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
•
•
•
FTSE100: 101 stocks, 115 scenarios
Nikkei: 225 stocks, 162 scenarios
S&P 100: 97 stocks, 227 scenarios
The corresponding indices, the same time periods.
Numerical
results
Conclusions
37
Computational study
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
•
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.
Proposed
approach
Numerical
results
Conclusions
38
Computational study: FTSE 100
The portfolio
selection
problem
Back-testing: Ex-post returns, 5 Jan – 15 Mar 2009
SSD
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
return
Models for
choice
Index
TrackError
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
1
2
3
4
5
6
7
8
time period
39
9
10
Computational study: FTSE 100
The portfolio
selection
problem
Back-testing: Ex-post compounded returns,5 Jan – 15 Mar 2009
Proposed
approach
Numerical
results
1.15
cumulative return
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
TrackError
Index
SSD
Models for
choice
1.1
1.05
1
0.95
0.9
0.85
0.8
1
2
3
4
6
5
7
Conclusions
time
40
8
9
10
Computational study: Nikkei 225
The portfolio
selection
problem
Back-testing: Ex-post returns, 5 Jan – 15 Mar 2009
Models for
choice
Proposed
approach
Numerical
results
Conclusions
0.06
0.04
0.02
return
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
TrackError
index
SSD
0
-0.02
-0.04
-0.06
-0.08
-0.1
1
2
3
4
5
6
8
7
time period
41
9
10
Computational study: Nikkei 225
The portfolio
selection
problem
Back-testing: Ex-post compounded returns, Jan – 15 Mar 2009
Proposed
approach
Numerical
results
Conclusions
1.1
cumulative return
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
TrackError
Index
SSD
Models for
choice
1.05
1
0.95
0.9
0.85
0.8
0.75
1
2
3
4
5
6
8
7
time period
42
9
10
Computational study: S&P100
The portfolio
selection
problem
Back-testing: Ex-post returns, 5 Jan – 15 Mar 2009
Models for
choice
0.08
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
0.04
Proposed
approach
Numerical
results
Conclusions
TrackError
Index
SSD
return
0
-0.04
-0.08
-0.12
-0.16
1
2
3
4
5
6
7
8
time period
43
9
10
Computational study: S&P100
The portfolio
selection
problem
Backtesting: Ex-post compounded returns, Jan – 15 Mar 2009
Models for
choice
Proposed
approach
Numerical
results
Conclusions
Index
TracKError
1.1
cumulative return
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
SSD
1
0.9
0.8
0.7
0.6
1
2
3
4
5
6
7
8
time period
44
9
10
Computational study: composition of portfolios
The portfolio
selection
problem
No of stocks (on average)
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
SSD_scaled
SSD_unscaled
TrackError
FTSE 100
9
11
58
Nikkei 225
12
3
118
S&P 100
14
17
73
Proposed
approach
Numerical
results
Conclusions
No need to impose cardinality constraints in the SSD
based models.
45
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
46
Computational study: composition of portfolios
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
•
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).
Proposed
approach
Numerical
results
Conclusions
•
Case of Nikkei 225 and FTSE100, unscaled SSD
model: NO rebalancing was necessary for the 10
time periods of backtesting.
47
Summary and conclusions
The portfolio
selection
problem
•
SSD represents the preference of risk-averse investors;
Models for
choice
•
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;
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
48
Summary and conclusions
The portfolio
selection
problem
•
Back-testing: considerably and consistently realised
improved performance over the indices and the index
tracking strategies (trackers).
•
Good strategy in a rebalancing frame:
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
o Naturally few stocks are selected (no need of
cardinality constraints);
o Little (or no) rebalancing necessary: use as a
rebalancing signal strategy.
49
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.
•
Index
tracking /
outperforming
Kunzi-Bay, A. and J. Mayer (2006): Computational aspects of
minimizing conditional value-at-risk, Computational Management
Science 3, 3-27.
•
Proposed
approach
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), 541569.
•
Wierzbicki, A.P. (1983): A Mathematical Basis for Satisficing
Decision Making, Mathematical Modeling, 3, 391-405.
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Numerical
results
Conclusions
50
The portfolio
selection
problem
Models for
choice
THANK YOU
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
• CONTACT US : [email protected]
• [email protected]
• [email protected]
Conclusions
»
51
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Evolution of Portfolio
Models
• Tracking error as a constraint…[discuss ]
• Nonlinear transaction cost /market
impact[discuss ]
• Trade scheduling =algorithmic trading..
[discuss ]
• Resampled efficient frontier
Conclusions
• Risk attribution and risk budgeting
52
The portfolio
selection
problem
Models for
choice
Second Order
Stochastic
Dominance
Index
tracking /
outperforming
Proposed
approach
Numerical
results
Conclusions
53