ACADEMY OF ECONOMIC STUDIES DOCTORAL SCHOOL OF FINANCE-BANKING ORDERED MEAN DIFFERENCE AND STOCHASTIC DOMINANCE AS PORTFOLIO PERFORMANCE MEASURES with an approach to cointegration Superviser: Professor Moisă Altăr MSc Student: George Popescu

Scheme THE EQUIVALENT MARGIN THE OMD. UTILITY FUNCTION AND POVERTY GAP FUNCTION. STOCHASTIC DOMINANCE THE ECONOMETRIC MODEL EMPIRICAL APPLICATION

3 THE EQUIVALENT MARGIN r: fund return R: benchmark return t: penalty levied on the fund return x: investment in fund investor’s decision problem:

The equivalent margin:

5 The OMD The OMD (Bowden, 2000) = the special case when, in the equivalent margin formula, the utility function has the form of a put pay-off P

6 Motivation for this kind of utility function Investor is interested in obtaining a target return P, being indifferent to values of R in excess of P and negatively exposed if the return falls below the target P- established according to his appetite for risk exactly the converse of the poverty gap function (Davidson and Duclos, 2000) idea from Merton (1981) and Henriksson and Merton (1981)

7 A and B: two random variables A second order stochastically dominates (SSD) B up to a poverty line z if:

Davidson and Duclos (2000) demonstrate that the SSD condition can be written as: The Poverty gap function:

Interpretation of SSD condition in terms of poverty gap function: The average poverty gap in B (the dominated distribution) is greater than in A (the dominant distribution) for all poverty lines less than or equal to z. There is a longer way from the actual level of income B to the poverty threshold than from the actual level of A to the same poverty threshold.

The put payoff - like utility function: The poverty gap function: So: this kind of utility function shows how far we are from the poverty threshold, after we surpassed the threshold

The OMD Introducing the utility function in the equivalent margin formula gives: OMD = the average area between the regression curve of the fund return on the benchmark return and the benchmark return itself, taken on the Ox axis If t(P)>0 for all P, then the fund was superior to the benchmark

The equivalent margin can be written as a weighted average of OMD’s (for all P)

Interpretation: - each investor can be seen as a spectrum of elementary investors (“gnomes” as named by Bowden), each having a put option profile utility function, but differing by the “strike price” (P), which represents the degree of aversion to risk (P moves to the right as the aversion to risk decreases) t U : independent of the degree of aversion to risk

Testing for SSD or, in terms of the poverty gap function:

OMD for r with R as benchmark: OMD for R with r as benchmark:

16 THE ECONOMETRIC MODEL Using the Forsyhte polynomials, transform the initial regression of the fund return on the benchmark return into a regression of the fund return on a set of regressors whose matrix is orthogonal the benchmark: divided into several indexes insures of non-multicollinearity between independent variables

17 The Forsythe polynomials:

The estimated equation: The estimated values for OMD [t(P)]

19 EMPIRICAL APPLICATION Data: –r: Capital Plus return (VUAN series) –R: mutual fund index return (IFM series) Period: –3 January April 2002 Frequency: –weekly Number of observations: –118

Initial (gross) regression equation (34 regressors): Only the significant regressors maintained in terms of t-Statistic (p-values <0.05):

21 Verifying the OLS presumptions:

Independence of explanatory variables of residuals:

23 Stationarity of residuals

COMPUTATION OF OMD - series sorted in ascending order after the IFM values

25

26 Interpretation: OMD positive for every realisation of the benchamark the fund was superior (OMD dominant) to the benchmark and preferred by every risk averse investor, no matter his degree of aversion to risk (because if OMD is positive, then the equivalent margin, which is a weighted average of OMD’s, is also positive)

27 Preferred by both less and more risk averse investors a downward trend the more risk averse investors prefer more than the less risk averse investors the fund the fund added utility to both less and more risk averse investors, but the more risk averse ones appreciate more the utility given by the fund than the less risk averse investors.

28 OMD: the average area between the regression of the fund return on the benchmark return

29 The area is always positive the fund was OMD dominant over the market, though there were points where the fund return was less than the benchmark return Inconvenient: the first values for OMD are computed using few values Remedy: Baysian approach; I tried implement the exponentially weighted OMD (EWOMD), which gives less weighting to the first values

Did the fund SSD the benchmark? Inverting the benchmark: The regression to be estimated:

31 Verifying the OLS presumptions:

32 The OMD for IFM

33 Interpretation: OMD is not negative for all the fund return values the fund did not SSD the market (represented by the benchmark) not always the poverty gap was less for the fund than for the benchmark the fund SSD the benchmark only for the greater values of the fund returns the fund was preferred especially by the more risk averse investors (who fix lower levels they wish the fund to attain)

34 AN APPROACH TO COINTEGRATION both the OMD measure and the cointegration theory describe long run behaviour the fund is allowed to have temporary fall below the benchmark, but these falls do not affect the overall conclusion if the long run behaviour indicates the superiority of the fund Does exist a cointegration relation between VUAN and IFM that verifies the superiority of the fund?

VUAN and IFM series: non-stationary VAR(3) system

36 VAR(3) and not VAR(2) because of: LR test Akaike and Schwartz lack of autocorrelation of residuals

37

38 Apply the Johansen test to find a cointegrating relation: The dominance of the fund in terms of OMD verified by the cointegrating relation

39 Remained to be developed: Computation of OMD (the first values: computed using few values) - Baysian approach, EWOMD equivalent margin - martingale measures