ACADEMY%20OF%20ECONOMIC%20STUDIES%20DOCTORAL%20SCHOOL%20OF%20FINANCE-BANKING%20%20ORDERED%20MEAN%20DIFFERENCE%20AND%20STOCHASTIC%20DOMINANCE%20AS%20PORTFOLIO%20PERFORMANCE%20MEASURES%20with%20an%20approach%20to%20cointegration

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ACADEMY OF ECONOMIC STUDIESDOCTORAL SCHOOL OF
FINANCE-BANKINGORDERED MEAN DIFFERENCE AND
STOCHASTIC DOMINANCE AS PORTFOLIO PERFORMANCE
MEASURESwith an approach to cointegration

• Superviser Professor Moisa Altar
• MSc Student George Popescu

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Scheme

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

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THE EQUIVALENT MARGIN

• r fund return
• R benchmark return
• t penalty levied on the fund return
• x investment in fund
• investors decision problem

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The equivalent margin
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The OMD(Bowden, 2000)

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

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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)

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A and B two random variablesA second order
stochastically dominates (SSD) B up to a poverty
line z if
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Davidson and Duclos (2000) demonstrate that the
SSD condition can be written as

• The Poverty gap function

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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.

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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

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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)gt0 for all P, then the fund was
superior to the benchmark
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The equivalent margin can be written as a
weighted average of OMDs

• (for all P)

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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

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Testing for SSD

• or, in terms of the poverty gap function

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OMD for r with R as benchmark

• OMD for R with r as benchmark

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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

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The Forsythe polynomials
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The estimated equation

• The estimated values for OMD t(P)

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EMPIRICAL 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

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Initial (gross) regression equation (34
regressors)

• Only the significant regressors maintained in
  terms of t-Statistic (p-values lt0.05)

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Verifying the OLS presumptions
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Independence of explanatory variables of
residuals
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Stationarity of residuals
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COMPUTATION OF OMD

• - series sorted in ascending order after the IFM
  values

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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 OMDs, is also positive)

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• 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.

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OMD the average area between the regression of
the fund return on the benchmark return
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• 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

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Did the fund SSD the benchmark?Inverting the
benchmark

• The regression to be estimated

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Verifying the OLS presumptions
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The OMD for IFM
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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)

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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?

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VUAN and IFM series non-stationary

• VAR(3) system

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

• LR test
• Akaike and Schwartz
• lack of autocorrelation of residuals

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Apply the Johansen test to find a cointegrating
relation

• The dominance of the fund in terms of OMD
  verified by the cointegrating relation

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Remained to be developed

• Computation of OMD (the first values computed
  using few values) - Baysian approach, EWOMD
• equivalent margin - martingale measures