Incorporating Stochastic
Dominance and Progressive CVaR
Levels in Portfolio Models
Gautam Mitra
Co

authors:
Diana Roman
Csaba 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 problems
•
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..?
Research Problems in Finance
The portfolio
selection
problem
Models for
choice
Proposed
approach
Second Order
Stochastic
Dominance
Numerical
results
Conclusions
Index
tracking /
outperforming
4
The 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 talk
The portfolio
selection
problem
Models for
choice
Proposed
approach
Second Order
Stochastic
Dominance
Numerical
results
Conclusions
Index
tracking /
outperforming
5
A 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 perspective..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 Models
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 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
•
R
j
= the return of asset
j
at time
T
•
x=(x
1
,…,x
n
)
portfolio: decision variables;
x
j
= 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
(R
1
,…,R
n
)
(

> scenario generation)
•
the model of choice used
•
the timing / rebalancing
•
Portfolio
x=(x
1
,…,x
n
).
Its return:
R
X
=x
1
R
1
+…+x
n
R
n
•
Portfolio
y=(y
1
,…,y
n
).
Its return:
R
Y
=y
1
R
1
+…+y
n
R
n
•
R
X
and
R
Y

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
: r
ij
=the return of asset j under scenario i; j in
1…n, i in 1..S. (
p
i
=probability of scenario i occurring)
•
The (continuous) distribution of (R
1
,…,R
n
) is replaced with a
discrete
one, with a
finite number of outcomes
asset1
asset2
…
asset n
probability
scenario 1
r
11
r
12
…
r
1n
p
1
scenario 2
r
21
r
22
…
r
2n
…
…
…
…
…
…
…
scenario S
r
S1
r
S2
…
r
Sn
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”
R
x
Subject to:
x
X

Index

tracking models
The index’s return distribution is available:
R
I
“Min” 
R
x
–
R
I

Subject to:
x
X

Enhanced indexation models
The index’s return distribution is available as a reference;
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
,
R
X
is preferred to r.v. R
Y
if and only if: E(
R
X
)
E(
R
Y
) and
(
R
X
)
(
R
Y
) 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
捯湣慶c
畴楬楴礠晵湣f楯i
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
FSD
G
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:
R
X
SSD
R
Y
E[U(R
X
)]
E[U(R
Y
)]
,
U
non

decreasing and concave utility function.
(Meaning:
R
X
is preferred to
R
Y
by all rational and risk

averse
investors).
•
The intuitive definition:
R
X
SSD
R
Y
E[t

R
X
]
+
E[t

R
Y
]
+
,
t
R
[t

R
X
]
+
= t

R
X
if
t

R
X
0
[t

R
X
]
+
= 0
if
t

R
X
<
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”
R
x
Subject to:
x
X

Index

tracking models
The index’s return distribution is available:
R
I
“Min” 
R
x
–
R
I

Subject to:
x
X

Enhanced indexation models
The index’s return distribution is available as a reference;
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
R
X
,
R
Y
r.v. with equally probably outcomes
Ordered outcomes of R
X
:
1
…
S
Ordered outcomes of
R
Y
:
1
…
S
R
X
SSD
R
Y
1
+…+
i
1
+…+
i
,
i = 1…S
Tail
i
(
R
X
)
Tail
i
(
R
Y
)
R
X
FSD
R
Y
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
(R
X
) =

the mean of its worst A% outcomes
1
1
( ) (...)
i X i
S
CVaR R
i
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%
X
CVaR 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%.
CVaR
5/100
(Rx)=

1/5[(

0.2)+(

0.18)+…+(

0.1)]=0.152
CVaR
10/100
(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:
1
max( ( ),...,( ))
X S X
V Tail R Tail R
Such that
:
(1)
x X
1//
min( ( ),...,( ))
S X S S X
V 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* =(z
1
*,…,z
S
*)
be the target
z
i
*= the
Tail
i
of the index (
sum
of
i
worst outcomes)
*
1
( ) min( ( ) * )
z i x i
i S
x Tail R z
The problem we solve:
*
max( ( ))
z
x X
x
•
Basically, it optimises the “worst achievement”.
Alternatively,
z
i
*= the “scaled”
Tail
i
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
:
Tail
i
(
R
X
) = Min
•
Similar representation for the “
scaled”
tails.
{1,...,},  
J S J i
= realisation of
R
X
under scenario
j
( )
j T
R x
where
The portfolio
selection
problem
Models for
choice
Proposed
approach
Second Order
Stochastic
Dominance
Numerical
results
Conclusions
Index
tracking /
outperforming
35
Model formulation
( )
*,
i
j T
i
j J
R x z
Such that
:
{1,...,},  
i i
J 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
0
0.02
0.04
0.06
0.08
1
2
3
4
5
6
7
8
9
10
time period
return
SSD
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
cumulative return
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
return
SSD
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
cumulative return
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
return
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
cumulative return
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
N
o 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
•
CONTACT US :
gautam@optirisk

systems.com
•
gautam.mitra@brunel.
ac.uk
•
diana.roman@brunel.ac.uk
»
51
The portfolio
selection
problem
Models for
choice
Proposed
approach
Second Order
Stochastic
Dominance
Numerical
results
Conclusions
Index
tracking /
outperforming
52
Evolution of Portfolio
Models
•
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
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