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30 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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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