superviser: professor moisă altăr msc student: george popescu

ACADEMY OF ECONOMIC STUDIESDOCTORAL SCHOOL OF FINANCE-BANKING

ORDERED MEAN DIFFERENCE AND STOCHASTIC DOMINANCE AS PORTFOLIO PERFORMANCE

MEASURESwith 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 MARGINTHE EQUIVALENT MARGIN

• r: fund return

R: benchmark return

t: penalty levied on the fund return

x: investment in fund

• investor’s decision problem:

RxtrxUE Rr )1()(max ,

The equivalent margin:

])1()([maxarg , RxtrxUEt RrU

)]('[

)](')[(

RUE

RURrEtU

)(Rer , where e(R) = E[r|R] and E[ε] = 0.

.

)]('[

)(')(

RUE

RURReEtU

5

The OMDThe OMD(Bowden, 2000)

= the special case when, in the equivalent margin formula, the utility function has the form of a put pay-off

PR

PRPRRPRU P ,0

,),0max()(

P

6

Motivation for this kind of utility functionMotivation 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 variablesA second order stochastically dominates (SSD) B up to a poverty line z if:

dyyDdyyDx

B

x

A )()(0

1

0

1

for all xz, where

x

A

x

AAA dyyfydFxFxD00

1 )()()()(

x

B

x

BBB dyyfydFxFxD00

1 )()()()(

:22BA DD

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

The Poverty gap function:

BASSD

)()(00

ydFyxydFyx B

x

A

x

zyzyzyzg ,min0,max),(

x

ydFyxxD0

2 )()()( is the average poverty gap

up to z

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

),0max()( RPRU P

0,max),( yzyzg

The OMDThe OMD

Introducing the utility function in the equivalent margin formula gives:

PdRRfRRe

PFPt )(])([

)(

1)(

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 axisIf 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

dPPtPwtU )()( , where

)()('

)("

2

1)( PF

REU

PUPw

(for all P)

0)( Pw ;

1)( dPPw .

00)( UtPt

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)

tU : independent of the degree of aversion to risk

Testing for SSD

or, in terms of the poverty gap function:

PPSSD

dRRFdrrGRr )()(

P PSSD

RdFRPrdGrPRr )()()()(

OMD for r with R as benchmark:

OMD for R with r as benchmark:

)('

)(')()(

REU

RURReEPt

P

PrR

)('

)(')()(

rEU

rUrrEPt

P

Prr

)(]|[ ReRrE

)(]|[ rrRE

RrPPtSSD

r ,0)(

16

THE ECONOMETRIC MODELTHE 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:

2,1,, kikkikiki ggRg , where

N

iki

N

ikii

k

g

gR

1

21,

1

21,

and

N

iki

N

ikikii

k

g

ggR

1

22,

12,1,

,

for k = 1, 2, …K.

The estimated equation:

The estimated values for OMD [t(P)]

iii Ger ),( , where ki

K

kki GGe ,

0

),(

.

j

iiij Re

jt

1

ˆ1ˆ , where )̂,(ˆ ii Gee .

19

EMPIRICAL APPLICATIONEMPIRICAL APPLICATION

• Data:– r: Capital Plus return (VUAN series)– R: mutual fund index return (IFM series)

• Period:– 3 January 2000 - 1 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):

iiiiii GGGGVUAN 33,202,21,10,0 ...

14,148,87,7

6,65,53,31,10,0

iii

iiiiii

GGG

GGGGGVUAN

iiii GGG 26,2625,2515,15

21

Verifying the OLS presumptions:Test Distribution Value 5% critical

valueNon-autocorrelation between residualsDurbin-Watson – 2.036234 (2.00)

χ2 (4) 0.875133 9.49χ2 (8) 9.214078 16.92

χ2 (16) 16.78303 26.30

Breusch-Godfrey(LM Test)

χ2 (32) 30.48619 45.91Ljung-Box χ2 (4) 0.7470 9.49

χ2 (8) 7.5695 16.92χ2 (16) 13.060 26.30

(Q-Statistic)

χ2 (32) 35.822 45.91Homoscedasticity

χ2 (41) 19.91572 56.66WhiteR2 0.168777 –

Koenkar-Basset χ2 (8) 0.032296 14.07χ2 (4) 0.052959 9.49χ2 (8) 0.533095 15.51

χ2 (16) 1.212054 26.30

Engle (ARCH LM)

χ2 (32) 19.88728 45.91Standard normality of residualsSkewness – 0.488062 (0.00)Kurtosis – 4.332561 (3.00)Jarque-Berra χ2 (2) 13.30163 5.99

Independence of explanatory variables of residuals:

G G0 G1 G3 G5 G6 G7RESID -1.04E-32 -3.24E-18 -8.61E-20 4.83E-20 2.15E-20 -1.49E-21

G G8 G14 G15 G25 G26RESID -8.46E-22 -3.78E-26 -1.93E-26 6.22E-35 -4.15E-36

23

Stationarity of residuals

ADF Test Statistic 1% Critical Value* 5% Critical Value

-3.4890-2.8870

-3.744255

10% Critical Value -2.5802*MacKinnon critical values for rejection of hypothesis of a unit root.

PP Test Statistic 1% Critical Value* 5% Critical Value

-3.4870-2.8861

-10.93526

10% Critical Value -2.5797*MacKinnon critical values for rejection of hypothesis of a unit root.Lag truncation for Bartlett kernel: 4 ( Newey-West suggests: 4 )

COMPUTATION OF OMDCOMPUTATION OF OMD

j

iii Re

jt

1

ˆ1ˆ )ˆ,(ˆ ii Gee

- series sorted in ascending order after the IFM values

25

0.0

0.2

0.4

0.6

0.8

1.0

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

0

0,5

1

1,5

2

2,5

0 0,2 0,4 0,6 0,8 1 1,2 1,4

ifm

vuan

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:

drrfrrPF

PtP

rr )()(

)(

1)(

8,86,65,5

4,43,32,21,10,0

iii

iiiiii

GGG

GGGGGIFM

ii

iii

G

GGG

20,20

16,1614,1410,10

31

Verifying the OLS presumptions:Test Distribution Value 5% critical

valueNon-autocorrelation between residualsDurbin-Watson – 1.462238 –

χ2 (16) 29.73198(0.019443)

26.30

χ2 (18) 30.01623(0.037289)

28.75

χ2 (19) 30.01635(0.051591)

30.14

LM (Breusch-Godfrey)

χ2 (20) 30.03075(0.069357)

31.41

Homoscedasticityχ2 (32) 11.13074 45.91White

R2 0.094328 –χ2 (9) 19.15946

(0.023871)16.92

χ2 (10) 19.15025(0.038395)

18.31

χ2 (11) 19.50315(0.052638)

19.68

ARCH LM (Engle)

χ2 (12) 19.26912(0.082237)

21.03

Standard normality of residualsSkewness – 0.863407 (0.00)Kurtosis – 9.057571 (3.00)Jarque-Berra χ2 (2) 195.0739

(0.0000)5.99

32

The OMD for IFM

-0.2

0.0

0.2

0.4

0.6

0.8

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 COINTEGRATIONAN 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-stationaryHypothesis VUAN IFM McKinnon critical valuesI(1) vs. I(0) -1.721831 -2.075862I(2) vs. I(1) -8.475460 -7.039432

1%5%10%

-3.4895-2.8872-2.5807

VAR(3) system

tttt

tttt

tttt

tttt

IFMIFMIFM

VUANVUANVUANIFM

IFMIFMIFM

VUANVUANVUANVUAN

,2326225124

3232221212

,1316215114

3132121111

36

VAR(3) and not VAR(2) because of:

• LR test

• Akaike and Schwartz

• lack of autocorrelation of residuals

37

VAR(3) VAR(2)VUAN IFM VUAN IFM

DistributionCriticalvalue

DW 1.981266 1.968314 2.198261 2.138932 – (2.00)

Q(1) 0.0059 0.0100 2.4452 0.6026 χ2 (1) 3.84Q(2) 0.3807 0.0479 2.4772 0.8041 χ2 (2) 5.99Q(3) 1.5806 3.2863 6.6508 0.8046 χ2 (3) 7.82Q(4) 1.8126 5.0729 9.0332 1.3943 χ2 (4) 9.49

BG(2) 0.969355 0.216748 14.07406 4.232258 χ2 (2) 5.99BG(3) 5.207536 7.043206 14.16348 4.927487 χ2 (3) 7.82BG(4) 6.387876 15.02648 14.27377 6.733587 χ2 (4) 9.49BG(6) 7.358067 18.56603 15.08536 7.051775 χ2 (6) 12.59

White 31.6205[χ 2(27)]

81.92332[χ 2(27)]

35.87982[χ 2(14)]

16.12749[χ 2(14)]

χ2 (27)χ2 (14)

40.1123.69

ARCH-LM(2) 4.698600 6.781785 7.482439 3.278474 χ2 (2) 5.99ARCH-LM(3) 4.744544 7.043206 10.03964 3.637078 χ2 (3) 7.82ARCH-LM(4) 6.781491 8.752868 10.63870 4.196441 χ2 (4) 9.49

38

Apply the Johansen test to find a cointegrating relation:

0218465.1 IFMVUAN

• 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