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Statistical Properties of the Sample Semi-variance Shaun A. Bond & Stephen E. Satchell.

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Presentation on theme: "Statistical Properties of the Sample Semi-variance Shaun A. Bond & Stephen E. Satchell."— Presentation transcript:

1 Statistical Properties of the Sample Semi-variance Shaun A. Bond & Stephen E. Satchell

2 Some History on Semi-variance (sv)  Risk measurement Markowitz (1959)Markowitz (1959) Hogan and Warren (1972) and (1974)Hogan and Warren (1972) and (1974) Sortino and Forsey (1996)Sortino and Forsey (1996)  Axiomatic based arguments Fishburn (1982) and (1984)Fishburn (1982) and (1984)  CAPM Bawa and Lindenberg (1977)Bawa and Lindenberg (1977)

3 History con’t  Little has been said on estimating sv from data  Exception Josephy and AczelJosephy and Aczel  Popular view among practitioners Population sv has desirable theoretical propertiesPopulation sv has desirable theoretical properties Sample sv has high volatilitySample sv has high volatility  Impractical as empirical measure of risk

4 History con’t  Grootveld and Hallerbach (1999) Estimation error is more likely in portfolios where downside risk measures are usedEstimation error is more likely in portfolios where downside risk measures are used

5 Guiding question in article  Are the concerns about using downside risk estimates valid? Sub-level concernSub-level concern  Statistical properties of sv are not well understood

6 Properties of sample sv  Special case of a general class of risk measures These were done by Stone, Fishburn, Holthausen, and Pedersen and SatchellThese were done by Stone, Fishburn, Holthausen, and Pedersen and Satchell  Problems with variance Systematic risk measureSystematic risk measure  Large positive and negative gains are treated equally in optimization Mean-variance frameworkMean-variance framework

7 Problems with variance Risk is generally viewed in terms of downside or asymmetric risk below a benchmarkRisk is generally viewed in terms of downside or asymmetric risk below a benchmark When returns are asymmetric, using variance becomes a problemWhen returns are asymmetric, using variance becomes a problem  Mean-variance analysis Utility maximization principle only holds when quadratic utility is assumedUtility maximization principle only holds when quadratic utility is assumed  Assuming a quadratic utility is a limitation

8 Assumptions  A1: The X i are randomly sampled with pdf f(x)  A2: The X i are randomly sampled with symmetric pdf f(x) so that f(x)=f(-x)  Second moment E(X 2 ) exists and is finite  The pdf is consistent with the axiomatic presentation of Fishburn

9 Understanding sv  Must know how the measure is distributed Characteristic function (cf) of Z J is derivedCharacteristic function (cf) of Z J is derived  Compared to the equivalent expression for variance

10 Under symmetry

11  Under the assumption of asymmetry No relationship between cf of sv and varNo relationship between cf of sv and var  Under the assumption of symmetry

12 Under symmetry (cont.)  Correct comparison of risk measures is between two estimators with common expectation when underlying is symmetrically distributed.  If we rescale semi-variance

13 Under symmetry (cont.)  Expressing variance in terms of kurtosis of the underlying, Relative Variance gets the following form

14 Under symmetry (cont.)  Under the symmetry of pdf (A2)  In case of a leptokurtic distribution  In case of a platykurtic distribution  Under A2, the appropriate estimator is variance. Semi-variance has inefficiency of at least 2.

15 Stochastic Dominance Condition  This section tries to prove whether or not expected utility theory can be a basis for comparing different measures of risk  The authors focus on probability distributions of the sample versions of two risk measures

16 Sample Variance vs. Sample Semi-Variance  The sample variance will dominate the sample semi-variance if taken from a symmetric, iid probability distribution, and the semi-variance is adjusted to have the same expectation as the variance  Therefore, anyone that uses a concave von Neumann utility function will prefer variance to semi-variance

17 Sample Variance vs. Sample Semi-Variance  This preference will occur in populations where the second moment exists, but higher moments may not.  In these cases the variance of the risk measures cannot be used to decide which risk measure is least desirable  A von Neumann utility function is assumed for the decision makers

18 von Neumann Utility Function  A von Neumann utility function assumes that the following axioms of preferences are satisfied: CompletenessCompleteness TransitivityTransitivity ContinuityContinuity IndependenceIndependence

19 Proposition 2 Interpretations  Variance is a preferred measure of risk to semi-variance Assuming that returns are iid and therefore both risk measures have the same meanAssuming that returns are iid and therefore both risk measures have the same mean  Sample variance is preferred to the sample semi-variance for any concave utility functions

20 Under Asymmetry  The distribution is asymmetric about the origin if A1 holds but A2 does not  Properties of sv and s 2 under A1 for asymmetric distributions: Let I be an indicator variable such that I(x) = 1 if x ≥ 0 0 if x < 0 0 if x < 0Then Sv = [ Σx j 2 I(x j ) ] / n For an element in sv, E(x j 2 I(x j )) = E(x 2 and I = 0) = E(x 2 |I=0)(1-p) = E(x 2 |I=0)(1-p) = E(x 2 |x<0)(1-p) = E(x 2 |x<0)(1-p)

21 Comparison of the Variances  Under A1 there is no simple proportionality adjustment, so the suggested approach is to express the relative variances as a ratio  Comparing the variances allows for the examination of whether the volatility of semi-variance is too high to be a practical measure  Must determine the sign of the numerator in the second half of the expression  If negative, then Var(s 2 ) < Var(sv)  If positive, then Var(s 2 ) > Var(sv)

22 Transforming Results into Operational Tests  Define a target  Outcomes below the target are risky and undesirable  Outcomes above the target are non-risky  Unfavorable subset: Xˉ = {x[X:x<0}  Favorable subset: X + = {x[X:x≥0}  The set of historical returns over time are viewed as containing elements of either the favorable or unfavorable subsets X j = x j + I j - x j ˉ(1-I j ) where I j =1 if x j ≥0, in which case x j =x j + I j =0 if x j <0, in which case x j =x j ˉ I j =0 if x j <0, in which case x j =x j ˉ And Pr(x≥0)=p Assume x j +, x j ˉ, and I j are jointly independent

23 Empirical Application  Emerging market data will be used because of asymmetry.  Monthly returns from January 1985 to November observations for 20 series.  Double gamma pdf is used  Variance is found to be more volatile  Semi-variance looks inefficient with symmetric distribution

24 Conclusions  Variance is more efficient when symmetric distribution of returns is assumed Second order stochastic dominance of svSecond order stochastic dominance of sv  With asymmetric returns if the means are not adjusted the variance is a more volatile risk measure.  Semi-variance vs. variance in portfolio optimisation is not discussed


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