A statistically optimal estimator of semivariance

European Journal of Operational Research 67 (1993) 267-271 North-Holland

267

Theory and Methodology

A statistically optimal estimator of semivariance Norman

H. Josephy

and Amir D. Aczel

Department of Mathematical Sciences, Bentley College, Waltham, MA 02145, USA Received December 1990; revised June 1991 Abstract: This p a p e r develops a statistical estimator for the semivariance. The estimator is shown to be asymptotically unbiased, consistent, and computationally efficient. The estimator is superior to mere approximations to semivariance that have appeared in the literature recently. The suggested use of semivariance by Harry Markowitz in 1959 in cases where distributions are not symmetric should now be easily implementable by workers in portfolio analysis and other areas where risk is measured. Keywords: Risk measurement; Finance; Statistical estimation

I. Introduction A substantial portion of the theoretical analysis of returns on risky assets is based on the two-parameter (mean-variance) normal distribution model (Fama, 1976; Markowitz, 1959; Sharpe, 1981). From time to time, research work has been reported in the literature in which this model is questioned. For example, Markowitz (1959, Ch. 9) devotes a chapter to the concept of semivariance and its usefulness as an alternative measure of risk. Without pursuing its statistical properties, he utilizes a sample m o m e n t estimator in his discussion of how to use semivariance to determine efficient portfolios. The need for an alternative measure of risk is quite evident if we consider what the standard deviation really measures. R a n d o m House Dictionary (1978) defines risk as " . . . exposure to the chance of injury or loss." Sharpe (1978) points

Correspondence to: Amir D. Aczel, Department of Mathematical Sciences, Bentley College, Waltham, MA 02154, USA.

out that a good procedure for measuring risk would focus on the probability of a negative return. Typical descriptions of risk categories identify risk with loss. For example Stevenson et al. (1988) discusses the unsystematic risk associated with a particular firm or industry performing poorly despite a good ecomony, the purchasing power risk that future earnings will not command the goods and services possible with current dollars, the interest rate risk of principal loss due to changes in the general level of interest rates. All are risk categories that emphasize the ' b a d ' consequences of an uncertain investment. In order to quantify this notion, a measure of risk should gauge the magnitude and likelihood of returns below the expected, or mean value. Yet the universal measure of risk, the standard deviation, considers equally the returns above the mean, as well as returns below the mean. As Sharpe (1981) points out, But why count happy surprises (those above the expected value) at all in a measure of risk? Why not just consider the deviations below the expected re-

0377-2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

268

N.H. Joseph),, A.D. Aczel / Statistically optimal estimator

turn? Measures that do so have much to recommend them. But if a distribution is symmetric, the results will be the same, since the left side is a mirror image of the right! And in general, a list of portfolios ordered on the basis of 'downside risk' will differ little if at all from one ordered on the basis of standard deviation. The use of standard deviation implicitly assumes a symmetric distribution for returns, an assumption which may not be justified. As F a m a (1976) points out, For the investor, the form of the distribution is a major factor in determining the risk of investment. For example, although two different possible distributions for returns may have the same mean and standard deviation, the probability of returns much different from the mean may be much greater for one than for the other. If we allow for the possibility that returns are not symmetrically distributed, then the standard deviation may not adequately capture a risk concept based on the 'downside' possibilities. In order to explore the notion of risk based on 'downside possibilities', we choose the semivariance as the population p a r a m e t e r for risk, and study the properties of an estimator for this parameter. Markowitz (1959) points out one advantage of the semivariance in the context of portfolio analysis: Analyses based on semivariances tend to produce better portfolios than those based on variances. Variance considers extremely high and extremely low returns equally undesirable. An analysis based on variance seeks to eliminate both extremes. An analysis based on semivariance, on the other hand, concentrates on reducing losses. Our estimator consists of the sum of squared sample deviations from the sample mean for those observations below the sample mean, divided by a weighting coefficient chosen to guarantee asymptotic unbiasedness. An alternative computational approximation to the semivariance, given that the mean, variance and probability of being below the mean are already known, is derived by Choobineh and Branting (1986) from a quadratic optimization model with linear and quadratic equality constraints. Their optimization model approximates the underlying density as two point masses and solves for their placement under constraints on the mean are variance. Their formula is useful when the population mean, variance and

probability mass below the mean is known, although no statistical properties of their estimator are given. In addition to the semivariance, risk measures other than the standard deviation have been proposed in the literature. The Telser (1956) safetyfirst model has been proposed when protection against poor performance is a major concern, such as the m a n a g e m e n t of defined benefits pension portfolios. Risk of disaster is 'avoided' by rejecting from consideration entries whose ex ante probability of disastrous returns is not sufficiently small. Hagigi and Kluger (1987) discuss both the safety-first model and the semivariance model. Another measure of risk is proposed by Garcia and Gould (1987) in their investigation of drawdown in accrued wealth. They compute the maxim u m drop in percentage return over a selection of time horizons, and compare these draw-downs to similarly computed drops for the S & P 500. They perform this analysis, in part, because We believe that variability created by explosive upside movement is not something the risk-averse investor will wish to avoid. The Value Line Safety Rank is a measure of risk that combines a price stability index and a financial strength rating into a stock ranking from one to five. Fuller and Wong (1988) compare this measure to standard deviation and the C A P M beta measure of risk. In the remainder of this paper, we present the semivariance as a p a r a m e t e r and derive the statistical properties of an estimator consisting of the scaled sum of squared deviations from the mean using sample values below the mean. The scaling factor n / ( n - 1) 2 is determined by imposing the asymptotic unbiased condition on the estimator. The asymptotic mean squared error is shown to be zero, establishing strong consistency for the lower variance estimator.

2. Semivariance

The semivariance of the distribution of a random variable x having finite mean ~ and variance ~2 is a measure of spread to the left of the mean: dF(X)

(1)

N.H. Josephy, A.D. Aczel / Statistically optimal estimator where F is the population distribution function and Iz is the population mean. To determine an estimator for the semivariance, let x l , . . . , x n be a random sample of n independent, identically distributed r a n d o m variables with finite m e a n E x i = /z and variance Var(xi) = 002. We will also assume that the sample has finite fourth moment. Let t~ denote an estimator of lower variance given a random sample x 1. . . . . xn. We consider estimators of the following form:

t , ( x 1, x z . . . . . x , )

(Xi--'~) 2

=C n " E

(2)

x i~

where £ denotes the sample m e a n and c n is a constant to be determined. Our first task is to determine a constant value that will result in t being asymptotically unbiased. Let 1A denote the indicator function of the set A. The lower variance of a zero-mean random variable y can be written as the expected value of y2 over the non-positive real axis R . Denoting expectation by E, we have 00Z_=El~ ( y ) .y2

269

To simplify notation, define m i and I i to be m i = 1~_( Yi)' (9)

I i = 1 (0,~ 1(Yi) - 1 (~,0 1(Yi). Then

(10)

1R ( y i - - Y ) = l i + m i

and the population lower variance o-z_ and its estimator t can be written as

00z_= E m i . y2,

(11)

tn_~Cn " ~ ( m i + l i ) i=1

(12)

.(yi_~)2.

Thus,

Etn/cn=E

mi(Yi--y)2+li(Yi--y)

2

i

= ~_,Emi(Yi)2-2 ~EmiYi; i=l i-1

(3)

n

+ Y'~ Emi(YJ) z + E ~_, l i "(Yi _ y ) 2 .

where

i=1

1, O,

1~ ( y ) =

(13)

yO.

(4)

Replacing the original r a n d o m variables with random variables centered about their m e a n IX,

Yi = xi - ~,

(5)

y=x-lz,

(6)

we can express the estimator t in terms of the new sample y:

t , ( x , , x 2. . . . . x , ) =Cn" ~ a . _ ( X i _ £ ) . ( X i-1

i=1

By independence,

[EmiYiZ=002_, ErniYJYk = [ 0,

E m i Y i Y = -E m i Y i Y i = --0 °2 , hi= 1 n

~Emiy,

1 yk = -n-20 .2-.

(15)

(16)

j=l k=l

i_.~)2

_;)2

(14)

Hence

ErniY 2= - 1~ ~

= c , . ~ 1R_(y~- y ) . (yi

i=j=k, otherwise.

(7)

Thus, the expected value of the estimator can be written as

i=1

n

In order to relate the definition of the population lower variance 002 to the estimator t, we note that the two sets (y~ < 0) and (y~ < ~ ) differ when 0 < y~ < y or y < yi < 0. It follows that 1~_( Yi - Y) = 1R_(Yi) + 1 (o,y 1(Yi) - 1 (~,o 1(Yi).

(8)

Et n//C n = ( n - 1)200Z_/n + E ~, I i " ( y i _ f i ) 2 . i=1

(17) I_~tting c n = n / ( n - 1) 2 results in

Etn=002+n/(n-l)2"E~"~ti'(yi-y) 2. i~l

(18)

N.H. Josephy,A.D. Aczel / Statistically optimal estimator

270

(We are indebted to a referee for the observation that c n is not unique. For example, cn can be modified by any multiplicative factor of order

O(n).)

EmiYi9 3 = u4/n 3,

(21d)

Emi9 4 = u4/n 4,

(21e)

Emi(Yi - 9 )

(21f)

2 = o 2 ( ( n - 1 ) / n ) 2.

Now,

(Yi--Y) I"(Y'-9)2=

i I

,

_ 2 -(Yi-Y), 0,

These moments will be used to compute the estimator mean squared error:

0
Y
(19)

m,s.e. =

E(t n -

=E

Hence, I l l . ( Y i - 9) 21 < 9 2 and thus

0"2_) 2

G" ~.mi'(Yi-9)2-o"2_ i=1

IEli" ( y i - y ) 2]
+c n" ~ _ , l i ' ( Y i - 9 )

e,=

(n/(n

as

1)),

which is the standard multiplier 1 / ( n - 1) modified by the term n / ( n - 1). Strong consistency of t depends upon the lower moments of the underlying distribution. Let y denote any one of the zero mean, i.i.d, random samples Yl, Y 2 , ' " , Y n " Let u i denote the ith lower moment of y, that is, the expectation of yi over the non-positive reals. Then

2 .

(22)

i=1

Since the Yi constitute a zero mean, i.i.d, sequence of random variables, Ey 2 goes to zero as n goes to infinity. Thus, the estimator t is asymptotically unbiased. Weak consistency follows from the fact that convergence in mean implies convergence in probability. Note that the normalizing coefficient can be written as - 1))*(1/(n-

)2

Letting ~li'(Yi-9)

z,

Wi = mi" ( Yi - 9) 2,

i=I

(23) the mean squared error expression can be written

m.s.e. = E G G + 2Ecne n • c~

w i - cr2i=1

+E

(enid1) . w i - 0"2-

(24)

The terms containing e n can be bounded above by higher moments of Y:

E m i m ~ y i y j = v 2,

(20a)

[e,l
E m i m f f i Y j ( Y i + Yj)9 = ( 2 / n ) ( v l V 3 + u2),

(20b)

IGGI < (n/(n-

1))21912 ,

(25b)

E m i m j y i Y i 9 2 = (2/n2)(VlV3 + v2),

(20c)

E I c n G [ <_ ( n / ( n

- 1 ) ) 2 E l y[ 2,

(25c)

1))aEI 9[ 4

(25d)

E IGe .[2 < (n/(n-

Emim~( y i + y j ) 2 y 2 = (2/nZ)(v0v4 + 4VlV3 + 3VEZ),

(20d)

E m i m i ( Y i + Y j ) Y 3 = (2/n3)(UoU4 + 4UlV 3 + 3v22), (20e)

EmimjY 4 = (2/n4)(UoU4 + 4v,v 3 + 3v22)

(25a)

It follows from the i.i.d, and finiteness of the fourth moment assumptions that the fourth moment of 9 goes to zero (Chow and Teicher, 1988, p. 299). Thus, E I G G [ and E I c~G [ 2 go to zero as n goes to infinity. It remains to show that

(20f)

E

and

EmiY 4 =

2,

wi _ ~2

(21a)

also goes to zero with n. Noting that

E m i y 3 9 = un/n ,

(21b)

V~ = E w = E m i ( Yi - 9) 2

E m i y 2 9 z = u 4 / n 2,

(21c)

v4,

= 0"2((n - 1 ) / n ) 2 = 0"2-/(nc~)

(26)

N.H. Josephy, A.D. Aczel / Statistically optimal estimator

it follows that E

271

which goes to zero as n goes to infinity. Thus, the m.s.e, goes to zero as n goes to infinity. Our estimator is thus strongly consistent.

wi

3. C o n c l u s i o n

k = l 1=1 = C 2 ( k =~l V a r ( k ) +

k~= l l ~ k E C ° v ( k ' / ) )

(27)

and, using the moments computed above, the Coy(i, j) are given by Cov(i,

j)

t 4,nn~ ,

6 1ti ;t

=v~ 1

+ (~-~

n-l(2n-1 + 4u1o3 - n- 2

)

n

References

1

n

1 [n-l]

2

= coy (independent of i and j, i ~ j )

(28)

and the Var(i) are given by Var(i) = E ( w i - ~)2 =

- -

n

v4-v

)

= var (independent of i).

(29)

Noting that the expression for coy goes to zero as n goes to infinity, we have Wi -- O"

=ca

1Var(k) + =

=c~(n.var+n.(n

In this paper we proposed the lower variance as a meaningful measure of risk. We derived a sample estimator for this parameter, and showed that it was asymptotically unbiased and strongly consistent, thus possessing qualities of a 'good' estimator. We note that when the underlying distribution is symmetric, either the variance or lower variance may perform well as a risk measure. We recommend that the lower variance be used when the underlying distribution is skewed.

k = l l~k -

1).cov)

=(U4--U~)/n+ ( n / ( n -

1)) 3 coy

(30)

Choobineh, F., and Branting, D. (1986), "A simple approximation for semivariance", European Journal of Operational Research 27, 364-370. Chow, Y.S., and Teicher, H. (1988), Probability Theory (2nd. ed.), Springer-Verlag, New York, 299. Fama, E.F. (1976), Foundations of Finance, Basic Books, New York. Fuller, J.R., and Wong, G.W. (1988), "Traditional versus theoretical risk measures", Financial Analysts Journal 44, March-April, 52-67. Garcia, C.B., and Gould, F.J. (1987), "A note on the measurement of risk in a portfolio", Financial Analysts Journal 43, March-April, 61-69. Hagigi, M., and Kluger, B. (1987), "Alternative performance measure", Journal of Portfolio Management Summer 1987, 13, 34-40. Markowitz, H. (1959), Portfolio Selection, John Wiley, New York. Random House Dictionary of the English Language (1978), Random House, New York. Sharpe, W.F. (1978), Investments, Prentice-Hall, Englewood Cliffs, NJ, 73-74. Sharpe, W.F. (1981), Investments (2nd. ed.), Prentice-Hall, Englewood Cliffs, NJ. Stevenson, R.A., Jennings, E.H., and Loy, D. (1988), Fundamentals of lnvestments. West Publishing, St. Paul, 14-17. Telser, L.G. (1956), "Safety-First Hedging", Review of Economic Securities XXIII, 1-16.