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

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)

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

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

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

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

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

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

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

Under symmetry

 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

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

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

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.

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

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

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

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

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

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)

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)

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

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

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