Diversification in the Stochastic Dominance Efficiency Analysis

Timo Kuosmanen

University of Copenhagen, Denmark

Wageningen University, The Netherlands

Definition of SD• Risky portfolios j and k, return distributions Gj and Gk.

• Portfolio j dominates portfolio k by FSD (SSD, TSD) if and only if

FSD:

SSD:

TSD:

with strict inequality for some z.

( ) ( ) 0k jG z G z

z ( ) ( ) 0z

k jG t G t dt

( ) ( ) 0z u

k jG t G t dtdu

Empirical approach• Finite (discrete) sample of return observations of n assets

from m time periods represented by matrix Y=(Yij)nxm

• Rearrange each asset (row of Y) in nondecreasing order. Denote the resulting matrix by X=(Xij)nxm , Xi1 Xi2 … Xim.

• Apply the SD criteria to the empirical distribution function (EDF):

m

xzTtMaxzH it

i

)(

Equivalence theoremThe following equivalence results hold for empirical

distributions of all portfolios/assets j and k:

FSD: jD1k , with the strict inequality for

some t.

SSD: jD2k , with the strict inequality

for some t.

ktjt xx Tt

Tt

t

iki

t

iji xx

11

SD efficiencyThe set of feasible portfolios is denoted by

Definition: Portfolio k is FSD (SSD) efficient in , if and only if, jD1k (jD2k) . Otherwise k is FSD (SSD) inefficient.

Typical approach is to apply the basic pairwise comparisons to a sample of assets/portfolios. However, there are infinite numbers of alternative diversified portfolios.

;ty y Y

jy

SD efficiencyLevy, H. (1992): Stochastic Dominance and Expected Utility:

Survey and Analysis, Management Science 38(4), 555-593:

“Ironically, the main drawback of the SD framework is found in the area of finance where it is most intensively used, namely, in choosing the efficient diversification strategies. This is because as yet there is no way to find the SD efficient set of diversification strategies as prevailed by the M-V framework. Therefore, the next important contribution in this area will probably be in this direction.”

The source of the problem• In contrast to some opinions, the problem with diversification

is by no means an inherent feature of SD. It arises from the conventional method of application.

• There problem arises in sorting the data in ascending order (translation Y -> X) – loss of the information of the time series structure.

• In general, it is impossible to recover the EDF of a diversified portfolio from the knowledge of EDFs of the underlying assets.

Solution• Preserve the time series structure of the data to keep track of

the diversification possibilities, i.e. work with Y instead of X.

• Instead, re-express the SD criteria in terms of time-series.

Definition: The set , l = 1,2, is the l order dominating set of the evaluated portfolio y0.

Lemma: Portfolio y0 is l order SD efficient, l = 1,2, if and only if the l order dominating set of y0 does not include any feasible portfolio, i.e.

0 0( ) Dml ky y y y

0( )l y

Example• 2 Periods• Assets A and B

• Time series:A: (1,4) or (4,1)?B: (0,3) or (3,0)?

0

0.5

1

-2 -1 0 1 2 3 4 5

4for 1

41for

1for 0

)( 21

z

z

z

zH A

3for 1

30for

0for 0

)( 21

z

z

z

zH B

.50-.50 portfolio(1,4)&(0,3) or (4,1)&(3,0) (1,4)&(3,0) or (4,1)&(0,3)

0

0.5

1

0 1 2 3 4 5

0

0.5

1

0 1 2 3 4 5

5.3for 1

5.30.5for

5.0for 0

)( 21

z

z

z

zH

2for 1

2for 0)('

z

zzH

(4,1)

(4,4)(1,4)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

FSD case• Theorem:

where P denotes a permutation matrix.

Example: Let y0 = (1,4).

1 0 0 0( ) : ;my y P y y P y y P

FSD dominating set

FSD efficiency testPortfolio y0 is FSD efficient in if and only if

0)( 01 y

1,0

11

11

0

11

..

1)( )(

0

0,

01

ij

TT

T

T

P

P

P

P

PyY

ts

PyYMaxy

TYyY 0

SSD case• Theorem:

Where W denotes a doubly stochastic matrix.

Example:

  2 0 0 0( ) : ; my y W y y W y y P P

(4,1)

(4,4)(1,4)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

Separating hyperplane theorem• Since both the portfolio set and the dominating set are

convex, if y0 is SSD efficient, there exists a separating hyperplane

which strongly separates and 0(y0).

 

0 0 0: ; ,t mj k j kH z w zw y w y y w w j k T

SSD test

 Portfolio y0 is SSD efficient in if and only if 2 0( ) 1y

2 0 0

0 0

( )

. .

1

+ , :

1

w

j

l k l k

T

y Max y w

s t

Y w j N

w w l k T y y

w

Accounting for diversification...• can improve the power of the SD as ex post evaluation

criteria.• might bring forth interesting diversification strategies

when applied as decision-aid instrument in portfolio selection.

• enables one to truly compare the SD to other criteria like MV which do account for diversification.

• can immediately accommodate additional features like chance constraints or additional constraints.

• can be immediately supported by additional statistical/computational techniques like bootsrapping.