Download presentation

Presentation is loading. Please wait.

Published byNataly Ventre Modified about 1 year ago

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

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google