Transcript Statistical Properties of the Sample Semi

Statistical Properties of
the Sample
Semi-variance
Shaun A. Bond & Stephen E.
Satchell
Some History on Semi-variance
(sv)

Risk measurement
• Markowitz (1959)
• Hogan and Warren (1972) and (1974)
• Sortino and Forsey (1996)

Axiomatic based arguments
• Fishburn (1982) and (1984)

CAPM
• Bawa and Lindenberg (1977)
History con’t
Little has been said on estimating sv
from data
 Exception

• Josephy and Aczel

Popular view among practitioners
• Population sv has desirable theoretical
properties
• Sample 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 used
Guiding question in article

Are the concerns about using
downside risk estimates valid?
• Sub-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 Satchell

Problems with variance
• Systematic risk measure

Large positive and negative gains are
treated equally in optimization
• Mean-variance framework
Problems with variance
• Risk is generally viewed in terms of
downside or asymmetric risk below a
benchmark
• When returns are asymmetric, using
variance becomes a problem

Mean-variance analysis
• Utility maximization principle only holds when
quadratic utility is assumed
 Assuming a quadratic utility is a limitation
Assumptions
A1: The Xi are randomly sampled
with pdf f(x)
 A2: The Xi are randomly sampled
with symmetric pdf f(x) so that
f(x)=f(-x)

Second moment E(X2) 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 ZJ is
derived

Compared to the equivalent expression for
variance
Under symmetry
Under symmetry

Under the assumption of asymmetry
• No relationship between cf of sv and var

Under the assumption of symmetry
1
2
E ( sv )  E ( s )
2
1
1 2 2
2
var( sv )  var( s )   ( s )
2
4n
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
sv '  2sv
1 2 2
var( sv ' )  2 var( s )   ( s )
n
var( sv ' )
1 2 (s 2 )
RV 
 2
2
var( s )
n var( s 2 )
2
Under symmetry (cont.)

Expressing variance in terms of kurtosis of
the underlying, Relative Variance gets the
following form

[ x ]
4

4
1
RV  2 
,  1
 1
Under symmetry (cont.)

Under the symmetry of pdf (A2)
1
RV  2 
,  1
 1

In case of a leptokurtic distribution
1
  3, RV  2
2

In case of a platykurtic distribution
1
1   3, RV  2
2

  , RV  2
  1, RV  
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:
• Completeness
• Transitivity
• Continuity
• Independence
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 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 s2 under A1 for asymmetric
distributions:
Let I be an indicator variable such that
I(x) = 1 if x ≥ 0
0 if x < 0
Then
Sv = [ Σxj2I(xj) ] / n
For an element in sv, E(xj2I(xj)) = E(x2 and I = 0)
= E(x2|I=0)(1-p)
= E(x2|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(s2) < Var(sv)
If positive, then Var(s2) > Var(sv)
Transforming Results into
Operational Tests

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

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
Xj = xj+Ij - xjˉ(1-Ij) where Ij=1 if xj≥0, in which case
xj=xj+
Ij=0 if xj<0, in which case xj=xjˉ
And Pr(x≥0)=p
Assume xj+, xjˉ, and Ij are jointly independent
Empirical Application
Emerging market data will be used
because of asymmetry.
 Monthly returns from January 1985
to November 1997. 155 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
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
