1. Diversification in the Stochastic Dominance Efficiency Analysis Timo Kuosmanen University of Copenhagen, Denmark Wageningen University, The Netherlands

2. Definition of SD Risky portfolios j and k, return distributions G j and G k. Portfolio j dominates portfolio k by FSD (SSD, TSD) if and only if FSD: SSD: TSD: with strict inequality for some z.

3. Empirical approach Finite (discrete) sample of return observations of n assets from m time periods represented by matrix Y=(Y ij ) nxm Rearrange each asset (row of Y) in nondecreasing order. Denote the resulting matrix by X=(X ij ) nxm, X i1  X i2  …  X im. Apply the SD criteria to the empirical distribution function (EDF):

4. Equivalence theorem The following equivalence results hold for empirical distributions of all portfolios/assets j and k: FSD: jD 1 k, with the strict inequality for some t. SSD: jD 2 k, with the strict inequality for some t.

5. SD efficiency The set of feasible portfolios is denoted by Definition: Portfolio k is FSD (SSD) efficient in , if and only if, jD 1 k (jD 2 k). 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.

6. SD efficiency Levy, 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.”

7. 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.

8. 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 y 0. Lemma: Portfolio y 0 is l order SD efficient, l = 1,2, if and only if the l order dominating set of y 0 does not include any feasible portfolio, i.e.

9. Example 2 Periods Assets A and B Time series: A: (1,4) or (4,1)? B: (0,3) or (3,0)?

10. .50-.50 portfolio (1,4)&(0,3) or (4,1)&(3,0)(1,4)&(3,0) or (4,1)&(0,3)

11. FSD case Theorem: where P denotes a permutation matrix. Example: Let y 0 = (1,4). FSD dominating set

12. FSD efficiency test Portfolio y 0 is FSD efficient in if and only if

13. SSD case Theorem: Where W denotes a doubly stochastic matrix. Example:

14. Separating hyperplane theorem Since both the portfolio set and the dominating set are convex, if y 0 is SSD efficient, there exists a separating hyperplane which strongly separates  and  0 (y 0 ).

15. SSD test Portfolio y 0 is SSD efficient in if and only if

16. 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.