, where U ■ (I) is the nth derivative of the
n=o '
utility function at point R. Expected utility is expressed as a function com
posed of constants, a , determined from the selected utility function where a =
n * n
U (R)/n! and the distribution moments, E(R  R) . Equation (1) describes an
2
investor whose utility is described by mean return, variance (cr ) and skewness
(IT), where the constants a Q = U(R) , a 2 = U"(R)/2!, and aj = U'"(R)/3!,
reflect the relative importance of each moment.
E[U(R)1 = a Q + a 2 0, the
ratio a<,/(a 7 ) indicates the investors skewnessrisk tradeoff. Increasing
values of a/(a„) describe investors willing to accept greater return dispersion in
exchange for positive skewness.
Previous studies have evaluated the impact of diversification on expected
utility by substituting population (average) values of R, a and h at selected
portfolio sizes into equation (1). Because the mean values ofo" and n decline
with diversification for many security populations, it has been argued that a
skewness preference will cause investors to antidiversify . However, these
studies have not considered the uncertainty about distribution moments for
portfolios smaller than the market. Inpart icular, the more uncertainty concern
ing the mean level of skew, the greater is the motivation to diversify and
increase the confidence about the actual level of skewness which will be ob
tained.
Under a naive or random investment policy of equal investment in each
security, the mean level of skewness for a portfolio of n securities (see Conine
7 8
and Tamarkin, (1981) ' can be determined by equation (2).
E(M >)  d) g 5 + [3ia=tijg. . ♦ [ ( ^ (n  2) ]M.. v (2)
n n L J n 1 J K
where: ^(M^ ) = mean skewness on a portfolio of n securities
n = number of securities in the portfolio
n ■ average skewness for a one security portfolio
M. . . = average curvilinear relationship for the
population
M. ., ■ average triplicate product for the population also
equal to the systematic skewness of an equally
weighted market portfolio, M
It is important to realize that (2) is a measure of only the mean level of
portfolio skewness at portfolio size n. For example, in a population containing
100 securities, 4950 (i.e., 100 x 99/2) unique two security portfolios can be
formed; equation (1) is the crosssectional arithmetic average of the skewness
found in these portfolios.
Whereas meanvariance studies have employed the variance (e.g., variance in
variance) as a measure of sampling risk, the comparable measure for skewness,
the variance in skewness, is insufficient to evaluate the sampling risk asso
ciated with skewness because the distribution of portfolio skews at any given
9
size is itself markedly skewed. For securities such as stocks and long op
tions, the distribution of skews is positively skewed; for portfolios of covered
calls, the distribution of skews is negatively skewed. Thus, positive (nega
tive) outliers in individual security return distributions produce positive
(negative) outliers in the structure of portfolio skews. These cases are dis
played graphically in Figures 1 and 2.
INSERT FIGURES 1 AND 2
The asymmetry of the skewness distribution has interesting implications
about the conclusions presented in prior studies concerning diversification and
skewness. Focusing on the mean as a "typical value" in a skewed distribution is
misleading (see Winkler and Hayes (1975) ) .because the mean is "pulled" in the
direction of the tail. For positively skewed distributions the mean will over
state the typical value ( in a probability sense) and mislead the investor to
hold a smaller portfolio than would be selected if the investor was aware of the
sampling risk present. For a negatively skewed distribution there is motivation
for complete diversification since mean skewness becomes less negative as port
folio size increases.
e(mJ)
\
\
\
= E(M^ = the mean level of
\
portfolio skewness at size n
— — — = the dispersion in skewness
at size n
— ~ N
Figure 1: Diversification and its effects upon the dispersion
dispersion about a declining mean level of portfolio
skewness
N
E(Mn)
Figure 2
/
/
/
■ E(M^) = the mean level
of portfolio skewness at size n
■ the dispersion in skewness
at size n
Diversification and its effects upon the dispersion
about an increasing mean level of portfolio skewness
III. Portfolio Skewness and Sampling Risk
In this section we illustrate the differences in magnitude and behavior of
portfolio skewness and sampling risk for different security populations. Three
types of assets are chosen as samples, primarily because of the diverse nature
of their return distributions; these include common stocks, atthemoney covered
option writing and atthemoney long option positions.
A. The Data and Methodology
The sample chosen includes the 136 stocks having listed options available
on December 31, 1975. Securities not having complete price data on the Compu
stat tapes over the period July 1, 1963 to December 31, 1978, were eliminated,
resulting in 102 sample securities for analysis. Although the choice of this
particular group introduces a selection bias in the study, these securities
represent almost onehalf of the population of listed option securities; thus,
these results may be inferred to the current universe of optionable stocks.
Since listed options were not available until 1973, six month atthemoney
premiums for the 102 stock sample were generated for the 15 1/2 year sample
period using the Black and Scholes option pricing model adjusted for divi
dends. Use of BlackScholes beginningofperiod option premiums is believed
necessary to generate a sample period of sufficient length and to standardize
stock price/exercise price ratios. The similarity between BlackScholes model
prices and actual premiums has been demonstrated in Bhattachrya (1980) and
Merton, et. al. (1978).
Semiannual returns (gross of commissions) on each long option position for
the thirtyone six month holding periods were calculated by dividing the begin
ning of period call value as determined by the BlackScholes option pricing
model into the intrinsic value of the option at maturity. Intrinsic value is
the maximum of zero or the difference between stock price and striking price at
option maturity.
Semiannual returns on each covered writing position were calculated by
dividing the beginning stock price less the option premium received into the sum
of stock price at the end of the period plus dividends, less the option's
intrinsic value at maturity. Semiannual holding period returns for stocks
include price appreciation plus dividends. Commissions are ignored in all
transactions .
B. Return Distribution Statistics for Alternative Portfolios
Table I presents return distribution statistics for the three security
groups examined. Line 1 reveals that average returns increase (5.01% to 17.60%)
as one goes from option writing strategies, to stocks, to long call options
2
while total risk as measured by the average security variance, a > increases
from 130. 51 to 31,869.90. The large amounts of systematic risk in long option
positions compared to stocks and covered writing portfolios is shown by the
2
2
market variance, ovr (line 3).
Average security skewness (M ) and systematic (market portfolio) skewness,
(M^) , data presented in lines 4 and 6 exhibit a wide range of values and behav
ior. Average onesecurity skewness for covered call options is negative while
onesecurity common stock skewness is positive; onesecurity long call posi
tions exhibit extreme positive skewness. The average curvilinear product, M.
and market portfolio skewness values, M^j, reveal that for stocks and long option
positions, diversification will reduce the positive skewness (M£j < M. . . < Mr and
M^j, M. . . and M > 0) whereas for option writing strategies, increasing portfolio
size will lower (a benefit) the negative skewness (M^ > M. . . > n and Mm, M. . .
_3
and M < 0). Since investors can diversify their holdings, it is instructive to
examine the behavior of the portfolio skewness and the sampling risk for these
security populations in response to changes in portfolio size
INSERT TABLE 1
C. Diversification and Changes in Portfolio Skevness
Using the summary skewness and coskewness data from Table 1, equation (2)
allows the traditional time series average skewness measures for any portfolio
size to be analytically determined. Sampling procedures such as those used by
Evans and Archer (1968) and Simkowitz and Beedles (1978) are unnecessary and not
as precise. Table 2 presents relationships between portfolio size and mean
skewness for the three security populations examined.
INSERT TABLE 2
The results reveal a wide spectrum of portfolio sizeskewness relation
ships. First, for the option writing strategy, increasing portfolio size is
beneficial because it eliminates much of the negative skewness present in these
security positions, with over 90% being potentially diversif iable ( 1— [— 156 . 96/—
2423.13] = 93.52%). On the other hand, for stocks and long call options, the
diversification process reduces the desired positive skewness. This is particu
larly evident for the long call portfolio, where over 98% of the skewness is
unsystematic and thus can be destroyed with diversification. Slight increases
in portfolio size of long call options or stock portfolios rapidly reduces the
mean level of positive skewness.
D. Diversification and the Uncertainty about Skewness
Previous studies have focused on data similar to that presented in Table 2
and have concluded that investors in stocks or long options could be expected to
hold small portfolios in an effort to capture the greatest positive skew. How
ever, this approach ignores the sampling risk in these portfolios and the upward
bias in the expected level of skewness caused by the positively skewed distribu
tions. While a tstatistic has been developed for certain skewed distributions
(see Johnson (1978)) we believe sampling risk best can be illustrated by examin
ing the sampling distributions of portfolio skews at various portfolio sizes.
One thousand portfolios were randomly selected with security replacement
for n = 3, 5, 10 20 and 40 and time series skewness values calculated for each
portfolio. The size 1 values were computed directly from the data. The portfo
lios at each n were ranked by skewness; the deciles and extreme values of
skewness in these distributions are presented in Table 3. In Table 4 the
relationship of each skewness decile is expressed as a fraction of the mean
(i.e., M 3 /E(M 3 ).
INSERT TABLES 3 AND 4
The upward bias in the "typical" skewness as measured by the mean level of
skewness is illustrated by noting that for the stock sample (panel A) at portfo
lio size 1, approximately 75% of the distribution of portfolio skews lies below
the mean value. As diversification proceeds the distribution of skews becomes
more normal and the probability that an investor will draw a portfolio of stocks
whose positive skewness is below the expected value falls to 51% at portfolio
size 40. Because the right tail of the distribution collapses with diversifica
tion, the mean (E(m )) changes more dramatically than the median (50%) value of
skew. For example, for stocks, E(>T) goes from 29,001 to 5,518 as n moves from
1 to 40, but the median (50%) value only falls from 6,681 to 5,537. In fact,
there is very little change in the median until n equals 10. These results
10
2
imply that previous studies of stocks and skewness which used E(M ) have over
stated the change in a "typical" portfolios skew with increased diversification
and thus understated the "optimal" portfolio size. Similar behavior is evidenced
by the option portfolios, which, because of their greater skewness, exhibit a
greater proportion of portfolio skews below the mean value. For example, at a
portfolio size of 1 (see panel B), over 80% of the skewness distribution lies
below the expected value. For portfolio size 40, this number is reduced to 55%.
The covered option portfolio skews shown in panel C of Table 3 reveal that the
probability of holding a portfolio which is less negatively skewed than the mean
is about 65% for onesecurity portfolios and 52% for 40security portfolios.
The importance of sampling risk to the diversification decision can be
illustrated further by examining the lowest decile and lowest extreme value of
skewness at each portfolio size as shown in Table 4. For stocks, the minimum
decile of skewness increases with diversification and represents a value 62% as
large as the mean for a 40 stock portfolio. However, it is possible to select a
stock portfolio with negatively skewed returns as shown by the lowest observed
values, even when holding five securities. Because the downside risk of holding
a portfolio with a skewness below the expected level is significantly reduced
with diversification, even investors having a preference for positive skewness
may choose to diversify. Investors in covered call portfolios also benefit
from diversification as the minimum decile level of negative skewness approaches
zero as portfolio size increases.
For the option buying strategy the diversification implications are less
clear. These assets contain such extreme levels of positive skewness that even
the lowest decile at portfolio size one is almost three times as large as the
lowest decile of all other portfolio sizes. It should be noted, however, that
the smallest observed skewness, 542,100 £ s i eS s than 3% as large as the mean
value. Furthermore, Tables 3 and 4 illustrate the wide range of skewness values
11
possible for small option portfolios and the data suggest that investors may
choose to engage in some diversification to increase the certainty of the skew
ness estimate.
IV. Conclusions and Implications
Recent papers have extended portfolio theory to include skewness along with
mean return and variance to explain security preferences. Because the positive
skewness present in many assets is rapidly reduced through diversification,
several authors have suggested that a preference for positive skewness can lead
to antidiversif ication as investors attempt to capture the greatest amount of
positive skew. However, these analyses ignore the sampling risk present when
selecting assets from skewed distributions. Because the mean value of a posi
tively skewed distribution is biased upward, an investor who ignores sampling
risk may hold a smaller portfolio than required to achieve a desired level of
expected utility.
Our data illustrates the extreme differences in distributions of portfolio
skews and sampling risk for stocks, long calls and covered option writing port
folios. Without having knowledge of an individual's preference function, it is
impossible to specify which types of securities and what portfolio sizes can be
expected to maximize investor utility. However, some general observations can
be made based on the data. First, investors who hold covered call positions
should follow a policy of complete diversification because larger portfolios
possess lower variance, less negative skewness and less sampling risk. While as
strong a statement cannot be made about common stock portfolios, diversification
will dramatically increase the probability of achieving a portfolio skewness
value at least as great as the expected value for any portfolio size. Because
of sampling risk, it appears that diversification beyond the levels suggested in
12
previous skewness literature may be appropriate. Finally, the extreme skewness
uncertainty present in call option portfolios provides motivation for diversifi
cation, thereby improving the investor's chances of holding a portfolio which
will obtain a level of skewness near its expected value.
13
FOOTNOTES
In this paper the term "skewness" will refer to a distribution's third
moment. Many authors use the term "skewness" to denote the third moment divided
by the cube of the standard deviation.
«■>
Failure to consider sampling risk in the diversification process implies
that all portfolios of a given size have the same distribution (e.g., the same
mean, variance, skewness. . . .) Because portfolio distributions do differ, the
investor is faced with sampling risk — the probability that a particular portfolio
will have return characteristics different from the averages.
3
"Sampling risk" should not be confused with "estimation risk", a phrase
popularized in work of Bawa, et. al. (1979) and others. The "estimation risk"
literature deals with the uncertainty of measuring individual security returns
and the resultant implications for optimal decisionmaking. Sampling risk, on
the other hand, measures the uncertainty that distribution moments for a partic
ular portfolio of a given size n will differ from the average values of all
portfolios of size n. Even if estimation risk is assumed to be 0, sampling risk
still is present because the moments of portfolios of a given size will differ
from the expected values.
4 .
Because sampling risk is a function of the crosssectional dispersion
among individual asset returns (see Elton and Gruber (1977), its magnitude
becomes increasing larger with higher distribution moments (e.g., skewness).
that is, the dispersion of portfolio variances is greater than the variance in
portfolio mean returns, the dispersion among portfolio skews is greater than the
variance dispersion and so on. For any distribution moment, the sampling risk
is greatest when only one security is held and is eliminated when the investor
is fully diversified, since full diversification results in only one possible
portfolio — the market portfolio.
Care must be exercised when using a truncated Taylor series to represent
expected utility. The truncation after three moments transforms the original
function into a cubic expression whose values may diverge significantly from the
original utility function (see Hasset, et. al. (1984), Levy (1969).
M 3
6 7
Because portfolios of a given size possess different levels of R, ° and
, they will also have different expected utilities. In particular, because
the third moment will have the greatest amount of sampling risk, differences in
portfolio expected utilities will be especially sensitive to differences in M .
The assumption of an equal weighting scheme is consistent with the diver
sification literature (for example, see Beedles (1979), Conine and Tamarkin
(1981), Elton and Gruber (1977), Evans and Archer (1968), Sears and Trennepohl
(1982, 1983) and Simkowitz and Beedles (1978).
8
Equation (l) is developed as follows. The skewness of any equally
weighted portfolio containing n securities is:
NNNN NNN
*  ±h <*> M + i^i jii A »> *iii * A j*i A ( n> 3 s £
jk
14
where n = E(r .  r . )
 ^3
r . <
1 1
M. . . = E[(r.  f.) 2 (r.  r.)l and
lij i i 1 j
M. ., = E[(r.  f.)(r.  r.)(r.  r, )].
ijk i ij j k k
There are n terms like n . 3n(nl) terms like M. . . and n(nl)(n2) terms like
lij
3
M. .. for a total of N term. Taking expected values:
Ljk
E(Mn) = (^)M 3 + [%=^)]M... ♦ [ (nlKn2) ]g (1)
n~ 11 J n L J lc
9
Evans and Archer (1968) found the distribution of risks (standard devia
tion) to be approximately normal, thus justifying their F test analysis. The
authors have derived the analytical expression for the variance in skewness.
While it is a reasonable approximation of sampling risk at large portfolio
sizes, it is inadequate at small n, because of the asymmetry in the
distribution.
Sterk (1983) has compared the Black and Roll adjustment procedures for
dividends and found that the Roll Technique produces slightly better results.
However, we do not believe that the Roll method would produce any significant
differences in our data due to the extreme skewness in option portfolios.
Furthermore, the Roll technique is only applicable in the strictest sense for
shortlived options having only one dividend payment.
While it would be informative to use actual premiums, we believe that
deficiencies in the historical data base could provide misleading results.
These data problems include:
a. A short time period for analysis. The CBOE began trading listed
options in 1973 on only sixteen securities.
b. Nonavailability of listed contracts for desired stock price/ exercise
price ratios. It has not been until the last few years that sufficient
varieties of stock price/ exercise price ratios have been available on
most securities.
Further research can incorporate actual premiums once the listed option market
becomes more complete and the historical data base has been generated. The
objective of our analysis was to select a sample of reasonable size and suffi
cient duration so as to provide meaningful measures of portfolio skewness.
15
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16
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17
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