Investor Preferences and Portfolio Selection: Is Diversification an Appropriate Strategy?
C. James Hueng 1 and Ruey Yau 2
1.
Department of Economics, Western Michigan University, Kalamazoo, MI 49008, U.S.A.
2.
Department of Economics, National Central University, Taoyuan, Taiwan 32001, R.O.C.
Email: James.Hueng@wmich.edu and ryau@mgt.ncu.edu.tw
Abstract:
This paper analyzes the relationship between diversification and several distributional
characteristics that have risk implications for stock returns. We develop a flexible three-parameter
distribution to model the stock returns. Using data on the current 30 DJIA stocks, we show that an
investor’s strategy on diversification depends on the measures of risk for particular concerns. For
example, investors who desire to increase positive skewness would hold a less diversified portfolio,
while those who care more about extreme losses would hold a more diversified portfolio.
Experimenting with a more general pool of stocks yields the same conclusions.
JEL Classification: C51, G11, G12.
Keywords: diversification; asymmetric generalized t distribution; skewness.
1. Introduction
Research conducted in the mean-variance framework has long shown that diversification
offers the benefit of reducing unsystematic risk (e.g., Sharpe, 1964, Lintner, 1965, Evans and
Archer, 1968, and Fielitz, 1974). However, empirical evidence has revealed that investors do not
tend to hold fully diversified portfolios. For example, Goetzmann and Kumar (2002) examine more
than 40,000 equity investment accounts over a six-year period from 1991 to 1996 and find that the
average investor has four stocks in her/his portfolio, while less than five percent of investors hold
more than ten stocks in their portfolios. Furthermore, more than 25 percent of portfolios contain
only one stock, and more than 50 percent contain fewer than three. Given the benefits typically
associated with holding a diversified portfolio, the apparent lack of diversification among most
investors is somewhat puzzling.
The perception that investors do not weigh downside risk equally with upside potential
provides an explanation to this phenomenon. In this case, the variance is no longer an appropriate
measure for risk.
Kraus and Litzenberger (1976) build a three-factor CAPM framework by
explicitly incorporating skewness in investors’ preference. Malevergne and Sornette (2005) derive
a modified efficient frontier where higher moments (up to the eighth order) replace variance as the
measure of risk. They demonstrate how the traditional portfolio optimization framework can be
improved. Cooley (1977) conducts an experiment to test the perception of risk on the part of
institutional investors and finds that, among 56 institutional investors who were asked to rate
distributions according to perceived risk, at least 29 associated the asymmetry of return distributions
with risk.
In particular, the investors associated increases in risk with increases in negative
skewness, indicating a preference for positive skewness. 1 Harvey and Siddique (2000) and
1
Arrow (1974) theoretically shows that risk averse investors with non-increasing risk aversion
prefer positively-skewed investment positions.
1
Premaratne and Tay (2002) explore the role of skewness in asset returns. If an asset contributes
positive skewness to a diversified portfolio, then that asset will be valuable and has a higher price
(or lower expected return).
Investors who prefer positive skewness would seek to construct portfolios that have this
characteristic. However, this complicates the portfolio selection decision, because a desire on the
part of investors to obtain positive skewness may not be compatible with the familiar method of
constructing a diversified portfolio in order to reduce risk. As shown in Simkowitz and Beedles
(1978), even though portfolio variance decreases as diversification occurs, skewness may either
increase or decrease with diversification. Based on computations of the raw skewness, they find
that skewness is rapidly reduced by diversification for a pool of 549 common stocks. Aggarwal and
Aggarwal (1993) and Cromwell, Taylor, and Yoder (2000) also record similar results.
Aside from variance and skewness, there exist other risk measurements that are of particular
concern to specific investors. For example, recent research on Value at Risk (VaR) indicates that
investors may be concerned strongly about the likely maximum loss, i.e. the left tail of the
distribution of a portfolio’s expected returns. Of particular interest to us is the relationship between
diversification and the other information contained in the distribution beyond variance and
skewness. For example, investors may have a stronger preference for a lower expected extremeloss than for a higher skewness. As such, the relationship between expected extreme losses and
diversification adds another dimension to the portfolio selection process. Indeed, Gaivoronski and
Pflug (2000) conclude that investors who are concerned with VaR will not achieve comparable
results using a portfolio selection methodology that relies on another risk measure such as variance;
rather, VaR is a very different measure that deserves its own place in the portfolio selection process.
To achieve our stated goals, we propose a parametric model to estimate the population
distribution of stock returns. It is well known that stock returns are not normally distributed.
2
Therefore, to model the higher moments in stock returns, we develop the “Asymmetric Generalized
t (AGT) distribution,” an asymmetric version of McDonald and Newey’s (1988) generalized t (GT)
distribution. Although this is a simple three-parameter distribution, it is very flexible in that it
permits very diverse levels of higher moments.2 This distribution nests several popular distributions
often seen in the literature, including the normal and the Student’s t distributions. This general
distribution allows us to estimate the population distribution of the returns and provides us with
information on a wide range of distributional characteristics that have risk implications.
We focus on the current 30 stocks in the Dow Jones Industrial Average (DJIA) for both
computational tractability and its desired nature of a well-diversified portfolio. The results show
that first, consistent with the previous studies using sample moments, our results show a trade-off
between low variance and high skewness: diversification reduces the portfolio variance, but at the
same time also reduces the skewness. Second, expected extreme losses become smaller when the
portfolio size increases. As such, an investor’s strategy of diversification depends on the measures
of risk for particular concerns. For example, investors who want to increase positive skewness
would hold a less diversified portfolio, while those who care more about extreme losses would hold
a more diversified portfolio.
2
Other flexible alternatives include the exponential generalized beta of the second kind (EGB2),
transformations of normally-distributed variables discussed by Johnson (1949), and a family of
modified Weibull distributions proposed by Malevergne and Sornette (2004). Wang et al. (2001)
apply the EGB2 to GARCH models. The AGT distribution proposed in this paper is at least as
equally flexible as the EGB2 distribution. Whereas the EGB2 distribution imposes limited ranges
on higher moments, the AGT distribution has no such limits. The modified Weibull distribution is
characterized by only two parameters, but it does not nest the Student’s t distribution, the most
often used statistical distribution to capture the fat-tail behaviors in asset returns.
3
The next section outlines the model and the properties of the AGT distribution. Section 3
presents the empirical results by using the DJIA component stocks. Section 4 experiments with a
more general pool of stocks to test the robustness of our conclusions.
Section 5 offers the
conclusion.
2. The Model and Methodology
The conditional mean of the stock returns is modeled as a simple AR(m) process in an
attempt to estimate the zero-mean, serially-uncorrelated residuals:
yt = μ +
m
∑φ y
i
t −i
+ εt ,
(1)
i =1
where yt represents the daily stock returns and lag length m is chosen by the Ljung-Box Q tests as
the minimum lag that renders serially-uncorrelated residuals (at the 5% significance level) up to 30
lags from an OLS regression. Note that the OLS regression is used only to determine the number of
lagged returns to be included in the conditional mean. The coefficients in the conditional mean
equation will be jointly estimated with the conditional variance equation using the full information
maximum-likelihood (FIML) estimation.
The conditional variance, denoted as ht ≡ E (ε t2 | t − 1) , follows a GARCH process. It is
well documented in the finance literature that stock returns have asymmetric effects on predictable
volatilities; see among others, Glosten, Jaganathan, and Runkle (1993) and Bekaert and Wu (2000).
Therefore, we use an asymmetric GARCH model proposed by Glosten, Jaganathan, and Runkle (the
GJR model), which is claimed to be the best parametric model among a wide range of predictable
volatility models experimented by Engle and Ng (1993):
ht = κ + α ⋅ ht −1 + β ⋅ ε t2−1 + γ ⋅ I t+−1 ⋅ ε t2−1 ,
where I t+−1 = 1 if ε t −1 >0 and I t+−1 = 0 otherwise.
4
(2)
The relationship between the return residual and its conditional variance can be specified as
ε t = ht vt , where vt is a zero-mean and unit-variance random variable. The specification of the
distribution of the stochastic process {vt} determines the distribution of yt. The most commonlyused distribution for vt is the standard normal distribution, which is symmetric and has a kurtosis
coefficient of three. While the GARCH specification makes some allowance for unconditional
excess kurtosis, according to Bollerslev, Chou, and Kroner (1992), it is still unable to adequately
model the fat-tailed properties of the stock returns. In response to the levels of kurtosis found in
stock return data, Bollerslev (1987) combines a t-distribution and a GARCH (1,1) model. The tdistribution has fatter tails than are found in the normal distribution.
McDonald and Newey (1988) propose the Generalized-t (GT) distribution, which is a more
general distribution that can accommodate both leptokurtosis (thicker tailed than the normal
distribution) and platykurtosis (thinner tailed than the normal distribution). The pdf of the GT
distribution is:
p
f GT ( xt ; ω , p, q ) =
p
xt ⎞⎟
⎛ 1 ⎞⎛⎜
2ωq B ⎜⎜ , q ⎟⎟ 1 +
qω p ⎟
⎝ p ⎠⎜⎝
⎠
1
p
q+
1
p
,
where B(.) is the beta function. The parameters p and q are positive and p×q>n in order for the nth
moment to exist. The mean is zero and the variance is ω 2 q 2 / p B (3 / p, q − 2 / p ) / B(1 / p, q ) . The
kurtosis is [ B (1 / p, q ) / B (3 / p, q − 2 / p )]2 [ B (5 / p, q − 4 / p ) / B(1 / p, q)] with p×q>4. Special cases
of the GT include the power exponential or Box-Tiao (BT) (q → ∞ ), Student’s t (p=2), normal (p=2
5
and q → ∞ ), and Laplace (p=1 and q → ∞ ). The apparent advantage of the GT distribution over the
t-distribution is its flexibility in modeling higher moments.3
While the GT distribution is more flexible than the popular normal and student’s t
distributions in modeling the fourth moment, it is still symmetric and unable to model the skewness
in stock returns. A distribution with the ability to capture all of the first four moments could
provide more flexibility for modeling returns.
To obtain such a distribution, we develop an
asymmetric version of the generalized t distribution, where we define:
⎧
⎛ zt ⎞
⎟
⎪ f GT ⎜⎜
⎟
⎪⎪
⎝1+ r ⎠
h( zt ) = ⎨
⎪
⎛ zt ⎞
⎟
⎪ f GT ⎜⎜
⎟
⎪⎩
⎝1− r ⎠
for
zt ≥ 0 ,
for
zt < 0 ,
where -1<r<1. 4 This specification is a proper pdf and allows different rates of descent for zt>0 and
zt<0, and therefore, it allows for skewness. Next, we scale zt by defining xt = zt / A, where A is a
positive scaling constant for simplifying the notation.
The density function of the asymmetric generalized t (AGT) distribution can be written as:
3
The GT distribution does not restrict the level of kurtosis. In addition, it can be shown that the
square of the skewness is less than one plus the kurtosis [(SK)2 < KU + 1]. Therefore, a wider range
for kurtosis also allows a wider range for skewness.
4
Hansen (1994) uses the same technique to develop an asymmetric t-distribution. In a different
framework, Theodossiou (1998) also develops a skewed version of the GT distribution, but his
distribution has four parameters and a more complicated pdf.
6
⎧
⎛ A ⋅ xt
⎪ f GT ⎜⎜
⎪⎪
⎝ 1+ r
∂z t
= h( A ⋅ xt ) ⋅ A = ⎨
f AGT ( xt ) = h( z t ) ⋅
∂xt
⎪
⎛ A ⋅ xt
⎪ f GT ⎜⎜
⎪⎩
⎝ 1− r
⎞
⎟⋅ A
⎟
⎠
for
xt ≥ 0 ,
⎞
⎟⋅ A
⎟
⎠
for
xt < 0 .
Specifically, let A = [ p / 2ωq1/ p B(1 / p, q)]−1 , and the pdf becomes:
⎧
1
⎪⎧ ⎛
⎪ ⎪⎪ ⎜ 2 B( p , q ) ⋅ x t
⎪ ⎨1 + ⎜
⎪ ⎪ ⎜ p (1 + r )
⎪ ⎪ ⎜⎝
⎪⎩
f AGT ( x t ; p, q, r ) = ⎨
⎪
1
⎪⎧ ⎛
⎪ ⎪⎪ ⎜ 2 B( p , q) ⋅ xt
⎪ ⎨1 + ⎜
⎪ ⎪ ⎜ p(1 − r )
⎪ ⎪ ⎜⎝
⎩⎩
⎞
⎟
⎟
⎟
⎟
⎠
p
⎞
⎟
⎟
⎟
⎟
⎠
p
⎫
⎪
⎪
⎬
⎪
⎪⎭
−q−
⎫
⎪
⎪
⎬
⎪
⎪⎭
−q−
1
p
for
xt ≥ 0 ,
for
xt < 0.
1
p
As is apparent, the AGT distribution nests the GT, BT, Laplace, t, and normal distributions. (See
the Technical Appendix for more details on this distribution.) Like in the GT distribution, p and q
control the height and tails of the density. The additional parameter r controls the rate of descent of
the density around x=0. Specifically, when r>0, the mode of the density is to the left of zero and
the distribution skews to the right, and vice versa when r<0.
When r=0, the distribution is
symmetric. Figure 1 plots this density for several different parameterizations.
Let vt = (xt-μ)/σ. The standard AGT distribution with zero mean and unit variance has the
following probability density function:
7
1
⎧
p −q− p
⎫
⎧
1
⎪
⎞
⎛
⎪ ⎪⎪ ⎜ 2 B ( p , q ) ⋅ σ ⋅ vt + μ ⎟ ⎪⎪
⎟ ⎬
⎪σ ⎨1 + ⎜
⎟ ⎪
⎜
p
(
1
r
)
+
⎪ ⎪
⎟ ⎪
⎪ ⎪ ⎜⎝
⎠ ⎭
⎪ ⎩
f SAGT ( vt ; p, q, r ) = ⎨
1
⎪
p −q− p
⎫
⎧
⎪
1
⎞
⎛
⎪ ⎪ ⎜ 2 B( , q) ⋅ σ ⋅ vt + μ ⎟ ⎪
p
⎟ ⎪⎬
⎪σ ⎪⎨1 + ⎜
⎟ ⎪
⎜
p(1 − r )
⎪ ⎪
⎟ ⎪
⎪ ⎪ ⎜⎝
⎠ ⎭
⎩ ⎩
for
σ ⋅ vt + μ ≥ 0 ,
(3)
for
σ ⋅ vt + μ < 0 .
The conditional mean equation (1) and the GJR model (2) with the conditional distribution implied
by (3) are jointly estimated by the full information maximum-likelihood (FIML) method. When
estimating the model, we restrict p×q>2 for the variance to exist. 5 The likelihood function is
obtained as:
g AGT (ε t | ht ) = f SAGT ( v t ) ×
∂v t
ε
1
.
= f SAGT ( t ) ×
∂ε t
ht
ht
3. Empirical Results
Data and Model Estimation
5
We use a logistic transformation θ = L +
(U − L)
to set constraints on the parameters. With
1 + exp(−ω )
this transformation, even if the parameter ω is allowed to vary over the entire real line, θ will be
constrained to lie in the region [L, U]. Specifically, p and q are restricted to be between 0 and 50
and r is between -1 and 1.
8
We focus on the current 30 components of the DJIA for both computational tractability and
the prominence of the index in the investment arena. 6 The DJIA is very widely quoted in
investment news and is among the most scrutinized indicators of U.S. stock market performance. It
includes a wide variety of industries and is composed of blue chip stocks that are typically industry
leaders.7 Many investors choose to invest in these stocks in order to achieve diversification and
long-term growth. We collect daily return data from the CRSP database from January 1990 to
December 2002, for a total of 3280 observations for each stock.
Panel A of Table 1 reports the sample mean and standard deviation of individual daily stock
returns (in percentage). The average daily return ranges from 0.038% to 0.143%. The daily
standard deviations are generally high, ranging from 1.457% to 2.915%. Panel B reports the
estimated values of several key parameters in the model. In the conditional variance equation, as
expected, the estimated α’s and β’s are statistically significant at the 5% level and show that the
volatilities of the returns are highly persistent for all 30 stocks. In addition, the estimated GJR
parameters ( γ ’s) are all negative and most of them are statistically significant, showing that the
6
Note that we are using the 30 stocks that are currently the components of the Dow Jones Index.
Some of them were not in the Index in the earlier part of the sample. As of the date this paper is
written, the latest change of the components in the DJIA was on April 8, 2004. Table 1 includes a
list of the stock symbols.
7
The industries represented by the DJIA include materials, electronics, food/beverages/tobacco,
financial services, aviation/aerospace, heavy equipment, chemicals, petroleum, automobiles, retail,
computer hardware/software/services, pharmaceuticals, household supplies, telecommunications,
and entertainment.
9
return shocks have asymmetric effects on predictable variance. Negative shocks tend to cause more
volatilities than positive shocks do.
Most of the estimated distributional parameters are significantly different from zero at the
5% level. Of particular interest is the estimate of r. Recall that r governs the asymmetry of the
distribution. When r>0, the distribution is positively skewed. When r<0, the distribution is
negatively skewed. All the 30 estimated r’s are positive, and 22 of them are significantly different
from zero at the 5% level.
To test the fit of the model, we conduct Newey’s (1985) GMM specification test by using
orthogonality conditions implied by correct specifications. Correct specifications require that the
standard AGT-distributed vt be i.i.d. and have zero mean and unit variance. Therefore, we test the
moment conditions E(vt)=0 and E( vt2 )=1, as well as the serial correlation in vt at lags one through
four. Panel C of Table 2 reports the t-statistics for the six selected orthogonality conditions, along
with the chi-square statistic for the joint test. The chi-square test shows that our model fits very
well, with all but one (stock code JPM) of the statistics being statistically insignificant at the
conventional significance level.
Considered individually, almost all of the six orthogonality
conditions are insignificantly different from zero at the 5% level, with only four exceptions.
Diversification and Risk Measures
To investigate how the behaviors of portfolios’ distributional characteristics vary with
increasing diversification, we construct portfolios containing different numbers of stocks in the
following manner. First, a stock is randomly selected from the pool of the 30 DJIA stocks and the
returns from this one-stock portfolio are used to estimate the model. Another stock randomly
chosen from the remaining 29 stocks is then added to form a two-stock portfolio. The arithmetic
average of the returns of these two stocks is used to estimate the model. We keep adding one stock
10
at a time to form 30 portfolios with sizes ranging from one to 30, where the portfolio with size n is a
subset of the portfolio with size n+1.
Several distributional characteristics with risk implications are next calculated from the
estimations: the unconditional variance, the skewness parameter r, and the 5% and 95% quantiles.8
Since we do not assume the existence of higher moments beyond the variance, the usual measures
of skewness and kurtosis may not exist.
However, our parametric model provides all the
information needed for the distribution.
The parameter r measures the asymmetry of the
distribution directly, and the 5% and 95% quantiles show the thickness of the tails. These risk
measures are plotted against the portfolio sizes. We call these curves the “diversification structures
of risk.” By doing this we are able to observe how a risk measure varies with the portfolio size (the
degree of diversification). For example, if the plot is downward sloping, then the risk measure is
decreasing with diversification.
The whole process is repeated 100 times to yield 100 simulated samples. Therefore, there
are 100 simulated curves for each risk measure. Note that the results for the 30-stock portfolio are
identical across these 100 simulations. These curves are plotted in Figures 2(a)–2(d), which show a
similar pattern: the absolute value of the slope is very big when the portfolio size is small, quickly
declines as the portfolio size is approaching ten, and stays almost unchanged afterward.
8
There are two approaches to calculate the distributional characteristics of risks for a portfolio’s
returns. The first method is to apply estimation schemes to the portfolio returns directly. This is the
method employed in this paper and in Campbell et al. (2001). The second approach is to compute
the distributional characteristics for the portfolio returns from a multivariate model, in which the
dependence structure among individual stock returns needs to be identified beforehand. Examples
of this are Guidolin and Timmermann (2006) and Malevergne and Sornette (2004).
11
The risk measures apparently are all non-linearly associated with portfolio sizes. Therefore,
to model the diversification structure of risk, we propose the following specification:
Ri ,n = ai + bi ⋅
1
n
+ ci ⋅
+ ei ,n ,
exp(n )
exp( n )
(4)
where R is one of the four risk measures, i = 1,…,100 is the sample index, and n = 1, … , 30 is the
portfolio size.9 According to this specification, the risk measure has an asymptote a. The second
term
1
n
decays rapidly as n increases. The third term
also decays rapidly, but not as
exp( n )
exp(n )
fast as the second term. Hence, the speed of decay depends on the relative magnitudes of the
estimated b and c. This specification allows monotonic, humped, or S shapes, depending on the
values of b and c.
According to this structure, the “diversifiable” risk is
b+c
, and
exp(1)
b+c ⎡
1
n ⎤
− ⎢b ⋅
+ c⋅
shows the diversified risk by holding n stocks.
exp(1) ⎣ exp( n )
exp( n ) ⎥⎦
In the first four columns of Panel (A) in Table 2, we report (for each risk measure) the
average R-squared and estimated coefficients over 100 samples. The model fits quite well as the
average R-squared’s are high and almost all the average coefficients, with only one exception, are
statistically significant at the conventional level. Figures 3(a)-3(d) plot the estimated diversification
structure of risk, along with its upper and lower one-standard deviation bounds, for each risk
measure using the averages of the estimated coefficients. It can be seen that, first of all, there is a
reduction tendency in return variance and return skewness as the portfolio size increases. This is
consistent with the findings in previous studies using sample raw moments. Based on the fact that
investors prefer lower return variance and more positive skewness, there exists a trade-off between
9
This specification has been used in the literature on yield curves. See, for example, Nelson and
Siegel (1987).
12
lower variance and higher skewness when the portfolio is diversified. Second, both tails in the
distribution are thicker when the portfolio is less diversified, as shown in the results for the 5% and
95% quantiles.
The first four columns of Panel (B) in Table 2 report the diversifiable risk and the
diversified risk by diversifying the portfolio with n stocks. The first, third, and fourth columns
show that over 90% of the diversifiable risk can be diversified (the benefit of diversification) by
holding five stocks in the portfolio. The second column shows that 90% of the diversifiable
skewness is reduced (the cost of diversification) by holding five stocks. The gains or losses from
diversification all vanish very quickly and are ignorable when the portfolio size is greater than ten.
Value-at-Risk Analysis
The goal of a quality risk system is to generate valid forecast distributions to enhance
executive decision-making. Hence, most risk systems require models to generate valid ex-ante
estimates of the forecast distribution, especially the left tail of the distribution. Recent research on
Value at Risk (VaR) indicates that investors may be concerned strongly about the likely maximum
loss, i.e. the left tail of the distribution of a portfolio’s expected returns. The evaluation of such
large losses is a topic of increasing importance to investors operating in today’s tumultuous market
environment. One major advantage of the proposed AGT distribution in this paper is its flexibility
in estimating this risk measure.
Of particular interest to us is the relationship between this
maximum loss concern and the diversification strategy.
The estimation results from the previous subsection are used to calculate the 1% and 5%
conditional VaR’s for a one-day horizon. 10 Figures 4(a)–4(b) plot the estimated diversification
10
We also use weekly data to estimate the 1% and 5% conditional VaR’s at the weekly horizon.
All qualitative patterns observed based on the daily data are well preserved in the weekly data.
Specifically, the average expected maximum 1% and 5% weekly losses are 10.455% and 6.692%,
13
structure of risk along with its upper and lower one-standard deviation bounds. The last two
columns of Table 2 report the regression results based on equation (4). It is shown that the expected
extreme losses tend to be smaller as the portfolio size increases. The average expected maximum
1% and 5% daily losses are 4.805% and 3.030%, respectively, for a single-stock portfolio. The
diversifiable losses are 1.493% and 0.945%, respectively. The average gains of diversifying the
portfolio to a two-stock fund are 0.536% and 0.281%, respectively.
More than 90% of the
diversifiable losses are gone with a five-stock portfolio. The reduction in the extreme losses fades
away as the portfolio size goes beyond five.
4. Alternative Data for Robustness Check
Market Pool
Even though the DJIA stocks are the most popular stocks held by investors and provide a
well-diversified portfolio, one may argue that they still do not provide a representative picture of
the market. To test the robustness of our conclusions, we replicate the empirical analysis for a more
general set of stocks. This data set consists of all NYSE and AMEX firms that have complete
return data (no missing value) during the sample period (January 1990 to December 2002). We
follow the conventional method and exclude stocks that do not have a CRSP share code of 10 or 11,
i.e., we only include ordinary common shares and exclude REITs, closed-end funds, primes, and
scores. There are a total of 457 stocks in this pool.
The random selection methods used before are applied to this new pool of data. We extend
the maximum portfolio size to 50 to see whether our conclusions change when the portfolios
include more than 30 stocks. The simulation results analogous to those in Figures 2-4 are plotted in
respectively, for a single-stock portfolio.
The diversifiable losses are 3.393% and 2.019%,
respectively. More than 92% of the diversifiable losses are gone with a five-stock portfolio.
14
Figures 5-7. Table 3 shows that the results are analogous to those in Table 2. As expected, the
simulations from the market pool yield a wider range of each risk measure.
However, the
qualitative results are consistent with those from the DJIA dataset. First, the absolute value of the
slope is very big when the portfolio size is small, quickly declines as the portfolio size is
approaching ten, and stays almost unchanged afterward, even beyond the portfolio size of 30. Our
model of diversification structure of risk still fits quite well.
Secondly, variance and skewness decline as the portfolio size increases, i.e., there exists a
trade-off between lower variance and higher skewness when the portfolio is diversified.
In
addition, both tails in the distribution are thicker when the portfolio is less diversified, and the
expected extreme losses (VaR) tend to be smaller as the portfolio size increases.
Finally, 90% of the diversifiable risk can be diversified (the benefit of diversification) by
holding five to six stocks, and 90% of the diversifiable skewness is reduced (the cost of
diversification) by holding six stocks in the portfolio. The gains or losses from diversification all
vanish very quickly and are ignorable when the portfolio size is greater than ten. Therefore,
extending the sample to the market pool does not change our conclusions.
Characteristics-Based Diversification
Investors can achieve diversification by investing in the DJIA stocks, because the DJIA
includes a wide variety of industries. However, these stocks are typically industry leaders and have
high market capitalizations and trading volumes. Studies such as Campbell et al. (2001) suggest
that returns on small firms be more volatile and therefore, to reduce the higher risk (measured by
volatility) of small firms, it may be optimal to hold 20-30 stocks in a portfolio. This suggestion
clearly is not supported by our simulations of industry-wise diversification. However, it would be
interesting to see whether diversification strategies based on stock characteristics such as firm sizes
15
and liquidities would yield results different from those with industry-wise diversification. This
section experiments these two alternative strategies to test the robustness of our conclusions.
We first rank those 457 stocks used in the previous section according to their average sizes
(market values) over the sample period. The seven stocks with the smallest sizes are dropped and
the rest are put into 30 size-based groups, ranging from the smallest-size group to the biggest-size
group, with each group including 15 stocks of similar sizes.
We then construct portfolios
containing different numbers of stocks in the following manner. First, a stock is randomly selected
from the smallest-size group and the returns from this one-stock portfolio are used to estimate the
model.11 Next, a stock is randomly selected from the second-to-the-smallest-size group and added
to form a two-stock portfolio. The arithmetic average of the returns of these two stocks is used to
estimate the model. We keep adding one stock at a time from a bigger size group to form 30
portfolios with portfolio sizes ranging from one to 30, where the portfolio with size n is a subset of
the portfolio with size n+1. The thirtieth stock is randomly selected from the biggest-size group.
Therefore, diversification is achieved by adding stocks from bigger size groups. The whole process
is again repeated 100 times to yield 100 simulated samples, and the diversification structure of risk
11
We select stocks from size-ordered groups (from the smallest to the biggest), because we want to
see the diversification effect when bigger-size stocks are added to the portfolio, in an attempt to
compare our results with those from studies such as Campbell et al. (2001). A more general
approach is to randomly select a size-group, randomly select another size-group from the other 29
groups, and so on. We also experiment with this more general approach and obtain similar results.
The same sampling approach is also applied to the liquidity-based strategy experimented upon later.
The results from both experiments do not change our conclusions and are available from the authors
upon requests.
16
model is estimated using these 100 samples. Figures 8 (a)-(f) plots the estimated diversification
structure of risk for the six risk measures.
The second diversification strategy we experiment with in this section is based on the
liquidity of the stocks, which is measured by the turnover ratios (trading volume divided by shares
outstanding). We rank those 457 stocks according to their average turnover ratios over the sample
period and conduct simulations similar to what we have done in the size-based diversification
strategy. Figures 9 (a)-(f) plots the estimated diversification structure of risk for those six risk
measures.
Figures 8 and 9 show that our conclusions are not changed by using different
diversification strategies based on stock characteristics: Variance, skewness, kurtosis, and extreme
losses all decrease with increasing diversification, and the gains or losses from diversification all
vanish very quickly and are ignorable when the portfolio size is greater than ten.
5. Discussion and Conclusion
This paper addresses additional dimensions in the analysis of portfolio diversification and
risk. In addition to the conventional measures of risk, namely variance and skewness, we propose a
parametric model to estimate the whole distribution of asset returns and investigate the relationships
between diversification and other distributional characteristics that have risk implications.
Our results indicate that variance, kurtosis, and extreme losses decrease with increasing
diversification. As such, the goal of an investor who wants to decrease variance and extreme losses
would be to hold a more-diversified portfolio. However, portfolio skewness also decreases with
increasing diversification. Therefore, an investor who desires higher skewness would choose to
hold a less-diversified portfolio. These results indicate that an investor’s strategy of diversification
depends on which risk measure is the main concern.
17
Our results also shed some light on the empirical puzzle that investors do not tend to hold
fully diversified portfolios. A possible explanation is that most investors do not weigh downside
risk equally with upside potential and prefer more positively skewed returns over low return
variance and extreme losses. Furthermore, taking into account the transaction cost and information
cost embedded in managing a more diversified portfolio, investors are likely not to hold more than
ten stocks in their portfolios since the benefit of diversification beyond ten stocks is limited. The
cost outweighs the benefit of a highly diversified portfolio.
Possible extensions of this paper are considered. First, the literature on the risk-return
tradeoffs using a three-moment CAPM indicates that stocks which decrease the skewness of a
portfolio should have higher expected returns, that is, stocks with lower systematic skewness (i.e.,
coskewness with the market portfolio) outperform stocks with higher systematic skewness [see, for
example, Kraus and Litzenberger (1976) and Harvey and Siddique (2000)].
In other words,
investors prefer higher systematic skewness. Idiosyncratic skewness, on the other hand, does not
affect expected returns. Therefore, it would be interesting to decompose the skewness measure into
systematic skewness and idiosyncratic skewness and see whether the former decreases as the
portfolio size increases, i.e., diversification decreases coskewness.12 Unfortunately, we are unable
to do this decomposition using the current parametric model, because it would require a bivariate
AGT distribution to jointly model the individual stock return and the market return, which is out of
the scope of this paper. We leave this task to future studies.
12
In a more recent strand of the literature, Barberis and Huang (2005) and Kumar (2005) argue that
models with cumulative prospect-theoretic preferences imply that idiosyncratic skewness should be
priced as well. In this case, the relationship between diversification and idiosyncratic skewness is
also an interesting topic.
18
Another possible extension of the paper is to analyze the actual holdings of stocks by
investors in the market, rather than using the simulation method proposed in this paper. This would
provide us with evidence from the real world, rather than from counterfactual simulations. For
example, a unique dataset used by Barber and Odean (2000) and Mitton and Vorkink (2004), which
consists of the investments of 78,000 households from January 1991 to December 1996, would
serve this purpose.
Since this dataset is not publicly available, we leave this extension to
researchers who have access to it.
Acknowledgement
We are grateful to the editor, two anonymous referees, and the participants in the 2003 Conference
of High-Frequency Financial Data in Taipei for helpful comments. Naturally, all remaining errors
are ours.
Yau acknowledges the research support from the National Science Council of the
Republic of China (NSC93-2415-H-008-007).
19
Technical Appendix: Specifics of the Asymmetric Generalized t Distribution.
The generalized t (GT) distribution in McDonald and Newey (1988) has the following pdf:
p
f GT ( xt ; ω , p, q ) =
x ⎞
⎛ 1 ⎞⎛
2ωq B ⎜⎜ , q ⎟⎟⎜1 + t p ⎟
qω ⎟
⎝ p ⎠⎜⎝
⎠
1
p
p
q+
1
p
,
where B(.) is the beta function. To transform this symmetric distribution to an asymmetric one, we
define:
⎧ ⎛ zt ⎞
⎪ f GT ⎜ 1 + r ⎟
⎠
⎪ ⎝
h( zt ) = ⎨
⎪ ⎛ | zt | ⎞
⎟
⎪ f GT ⎜
⎩ ⎝1− r ⎠
for
zt ≥ 0 ,
for
zt < 0 ,
where -1<r<1. This specification allows different rates of descent for zt≥0 and zt<0. Next, we scale
zt by defining xt = zt / A, where A is a scaling constant in order to simplify the notation. The density
function of the asymmetric generalized t (AGT) distribution can be written as:
⎧ ⎛ A ⋅ xt
⎪ f GT ⎜⎜
⎪⎪ ⎝ 1 + r
∂z t
= h( A ⋅ xt ) ⋅ A = ⎨
f AGT ( xt ) = h( z t ) ⋅
∂xt
⎪
⎛ A ⋅ xt
⎪ f GT ⎜⎜
⎪⎩
⎝ 1− r
⎞
⎟⎟ ⋅ A
⎠
for
xt ≥ 0 ,
⎞
⎟⎟ ⋅ A
⎠
for
xt < 0 .
p⋅ A
⎧
⎪
1
⎪
⎛ ⎛ A⋅ x
t
⎪ 2ωq p B ⎛⎜ 1 , q ⎞⎟⎜1 + ⎜
⎟
⎜
⎜
⎜
⎪
⎝ p ⎠⎝ ⎝ 1 + r
⎪
Specifically, f AGT ( xt ; A,ω , p, q, r ) = ⎨
⎪
p⋅ A
⎪
1
⎪
⎛
⎪ 2ωq p B ⎛⎜ 1 , q ⎞⎟⎜1 + ⎛⎜ A ⋅ xt
⎜ p ⎟⎜ ⎜ 1 − r
⎪
⎠⎝ ⎝
⎝
⎩
20
p
⎞ 1
⎟
⎟ qω p
⎠
p
⎞ 1
⎟
⎟ qω p
⎠
⎞
⎟
⎟
⎠
q+
1
p
⎞
⎟
⎟
⎠
q+
1
p
for
xt ≥ 0 ,
for
xt < 0.
1
p
1
2ωq B( , q)
p
; as such, the pdf becomes:
Let A =
p
⎧
⎧ ⎛
1
⎪
⎪ ⎜ 2 B( , q) ⋅ xt
⎪
p
⎪ ⎜
⎪ f1 = ⎨1 +
⎪
⎪ ⎜⎜ p(1 + r )
⎪
⎪⎩ ⎝
⎪
f AGT ( x t ; p, q, r ) = ⎨
⎪
⎧ ⎛
⎪
1
⎪ ⎜ 2 B( , q) ⋅ xt
⎪
p
⎪ f 2 = ⎪⎨1 + ⎜
⎜
⎪
⎪ ⎜ p(1 − r )
⎪
⎪⎩ ⎝
⎩
⎞
⎟
⎟
⎟
⎟
⎠
p
⎞
⎟
⎟
⎟
⎟
⎠
p
⎫
⎪
⎪
⎬
⎪
⎪⎭
−q−
⎫
⎪
⎪
⎬
⎪
⎪⎭
1
p
−q−
for
xt ≥ 0 ,
for
xt < 0 .
1
p
The nth raw moment of xt is:
∞
Mn =
∫
∞
∫
−∞
∞
0
x n f AGT ( x )dx = x n f1 ( x )dx +
∫
∫
−∞
0
∞
∫
x n f 2 ( x )dx = x n f1 ( x )dx + ( −1) n x n f 2 ( x )dx.
0
0
n +1
⎞
⎛ n +1
n + 1 ⎞⎛⎜
⎟⎟ p ⋅ k p ⎟
x (1 + k ⋅ x ) dx = B ⎜⎜
,m −
⎟
p ⎠⎜
⎝ p
0
⎝
⎠
∞
Using
the
formula
∫
p −m
n
p
−1
and
letting
p
1
1
⎛
⎞
⎛
⎞
⎜ 2 B( , q) ⎟
⎜ 2 B( , q) ⎟
p
p
⎟ , k =⎜
⎟ , and m = q + 1 , the nth raw moment becomes:
k1 = ⎜
2
⎜ p(1 + r ) ⎟
⎜ p(1 − r ) ⎟
p
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
n +1
⎛ n +1
n + 1 ⎞⎛
⎞
,m −
M n = B ⎜⎜
⎟⎟⎜ p ⋅ k1 p ⎟
p ⎠⎝
⎠
⎝ p
= (1 + r )
n +1
n +1
⎛ n +1
n + 1 ⎞⎛
⎞
,m −
+ ( −1) n B ⎜⎜
⎟⎟⎜ p ⋅ k 2 p ⎟
p ⎠⎝
⎠
⎝ p
n +1
n ⎡
1 ⎤
p B(
, q − ) ⎢2 B ( , q )⎥
p
p ⎣
p ⎦
n
= [( −1) (1 − r )
n
−1
n +1
+ (1 + r )
n +1
− n −1
+ ( −1) (1 − r )
n
n +1
n +1
n ⎡
1 ⎤
] p B(
, q − ) ⎢2 B ( , q )⎥
p
p ⎣
p ⎦
n
21
−1
n +1
n ⎡
1 ⎤
p B(
, q − ) ⎢2 B ( , q )⎥
p
p ⎣
p ⎦
n
− n −1
.
− n −1
−2
⎛2
1 ⎞ ⎡ ⎛ 1 ⎞⎤
Therefore, the mean is μ ≡ M 1 = 4 rp ⋅ B ⎜⎜ , q − ⎟⎟ ⎢2 B ⎜⎜ , q ⎟⎟⎥ ; the variance is σ 2 ≡ M 2 − μ 2 ;
p ⎠ ⎣ ⎝ p ⎠⎦
⎝p
the skewness (SK) is:
E ( x − μ )3
σ3
=
E ( x 3 − 3μx 2 + 3μ 2 x − μ 3 )
σ3
=
M 3 − 3μ (σ 2 + μ 2 ) + 3μ 2 μ − μ 3
σ3
=
M 3 − 3μσ 2 − μ 3
σ3
;
and the kurtosis (KU) is:
E ( x − μ )4
σ4
=
E ( x 4 − 4 μx 3 + 6μ 2 x 2 − 4 μ 3 x + μ 4 )
=
M 4 − 4 μM 3 + 6μ 2 M 2 − 4 μ 3 μ + μ 4
σ4
σ4
M − 4 μ ( M 3 − 3μσ 2 − μ 3 ) − 6μ 2σ 2 − μ 4 M 4 − 4 μ ⋅ SK ⋅ σ 3 − 6μ 2σ 2 − μ 4
.
= 4
=
σ4
σ4
∞
Note
that
Pr( x ≥ 0) =
∫
0
0
Similarly, Pr( x < 0) =
∫f
−∞
2 ( x )dx
=
∞
⎡
⎤
1
1
f1 ( x )dx = x f1 ( x )dx = (1 + r ) B ( , q ) ⎢2 B ( , q )⎥
p
p
⎣
⎦
0
∫
0
−1
=
1+ r
2
.
1− r
, which shows the asymmetry of the distribution when r ≠
2
0. Furthermore, Pr(x≥0)+Pr(x<0)=1 confirms that fAGT is a proper pdf.
22
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25
Table 1: Estimated Coefficients and GMM Specification Test Statistics.
(A) Sample
Statistics
Stock Mean Std Dev
AA
(B) Estimated Coefficients
a
(C) GMM Specification Tests
b
α
β
γ
p
q
r
(1)
(2)
t-Statistics
(3)
(4)
χ
(5)
(6)
2
Statistic
0.058 2.100 0.943* 0.071* -0.043 2.054* 3.443* 0.132* 0.011 -0.175 0.810
0.645
0.697 -2.549*
AIG
0.074 1.747 0.933* 0.078* -0.039* 2.028* 3.784* 0.090* -0.149 -0.210 0.752
0.293
0.438
0.434
1.246
AXP
0.072 2.256 0.924* 0.098* -0.062* 2.408* 2.378* 0.076* -0.234 -0.281 1.454
0.686 -1.948 -1.239
6.916
BA
0.043 2.039 0.928* 0.075* -0.025 2.312* 1.885* 0.052* -0.126 -0.066 0.728
0.781
1.142 -0.534
2.643
C
0.117 2.334 0.923* 0.091* -0.077* 2.257* 2.658* 0.065* -0.003 0.283
0.557 -0.013 -1.266
2.662
0.919
8.510
CAT
0.065 2.058 0.969* 0.043* -0.029* 2.124* 2.359* 0.126* -0.622 -0.632 1.276 -0.410 -1.415 -0.479
4.716
DD
0.052 1.854 0.966* 0.040* -0.021* 2.075* 3.334* 0.099* -0.403 -0.125 0.528 -0.001 -1.887 0.311
3.700
DIS
0.041 2.072 0.971* 0.040* -0.031* 2.409* 1.955* 0.089* -0.494 0.024
0.566 -1.786 -1.933 0.345
7.822
GE
0.070 1.749 0.942* 0.086* -0.068* 2.349* 3.092* 0.054
-0.535 -0.158 0.487 -0.123 0.137 -0.002
0.600
GM
0.038 2.068 0.917* 0.086* -0.053* 1.898* 5.224* 0.118* -0.245 -0.236 0.825 -2.106* -1.463 0.347
7.581
HD
0.107 2.314 0.927* 0.094* -0.073* 2.315* 2.609* 0.055* -0.649 0.268
0.827
0.160 -0.230 0.432
1.204
HON
0.064 2.226 0.866* 0.162* -0.112* 2.249* 1.968* 0.047
0.690 -1.259 -1.258 -1.308
5.616
HPQ
0.082 2.752 0.972* 0.040* -0.028* 2.446* 1.729* 0.060* -0.389 -0.009 0.610
IBM
0.067 2.126 0.927* 0.101* -0.083* 2.551* 1.474* 0.071* -0.061 -0.868 0.558 -1.156 0.285
INTC 0.125 2.915 0.950* 0.058* -0.038 3.007* 1.372* 0.047
0.237
0.410
-1.191 -0.637 0.046
JNJ
0.081 1.647 0.893* 0.110* -0.088* 2.345* 2.816* 0.052* -0.443 0.059
JPM
0.072 2.446 0.932* 0.114* -0.100* 2.001* 3.912* 0.068* 0.123
KO
0.499
0.526 -0.319 -1.028
1.220
1.783
3.576
0.128 -0.751 0.017
2.839
0.170 -0.589 -0.620
1.099
0.174 3.625* -0.297 -1.405 -0.088
13.980*
0.066 1.692 0.934* 0.083* -0.066* 2.381* 2.363* 0.078* -0.448 -0.108 1.299
0.013 -0.536 0.188
MCD 0.038 1.758 0.968* 0.032* -0.014 2.332* 2.336* 0.098* -0.580 -0.131 0.635
0.261 -1.458 -1.104
3.881
MMM 0.059 1.543 0.980* 0.021
0.278
2.998
MO
-0.004 1.885* 2.864* 0.063* -0.302 -0.048 1.486
0.063 2.064 0.812* 0.163* -0.040 2.564* 1.319* 0.028
0.199 -0.633
2.157
-0.983 0.362
0.780
1.049 -0.130 1.428
4.419
MRK 0.070 1.781 0.938* 0.074* -0.062* 2.211* 2.894* 0.064* -0.759 0.005
0.514
0.363 -0.411 0.398
1.095
MSFT 0.143 2.393 0.870* 0.132* -0.087* 2.303* 2.688* 0.093* -0.231 -0.061 0.858
0.720 -1.588 -0.452
4.082
PFE
0.098 1.968 0.922* 0.084* -0.046* 2.284* 3.224* 0.051
-0.424 -0.107 0.955 -0.425 -2.092* -1.948
8.630
PG
0.072 1.731 0.916* 0.079* -0.047 2.351* 2.246* 0.048
-0.467 0.601
0.071
0.118
0.850
SBC
0.047 1.842 0.917* 0.099* -0.052* 1.779* 7.564* 0.056* -0.119 0.027
0.576
0.747 -0.572 0.443
1.531
UTX
0.073 1.874 0.907* 0.121* -0.085* 1.963* 3.523* 0.048
-0.178 0.513
0.931
0.552 -0.136 -0.397
1.619
VZ
0.043 1.799 0.944* 0.074* -0.045* 2.049* 3.886* 0.069* -0.067 0.055
0.306
0.114 -0.052 -0.447
0.292
WMT 0.090 2.053 0.944* 0.071* -0.042* 2.150* 3.681* 0.073* -0.052 -0.138 0.447 -0.442 0.037 0.213
XOM 0.056 1.457 0.934* 0.073* -0.034* 2.073* 4.350* 0.047 -0.031 0.077 1.154 0.008 -0.157 0.204
0.411
1.570
0.402 -0.639
The asterisk * indicates significance at the 5% level.
a. The parameters are for equations (2) and (3). All computations were performed using the GAUSS
MAXLIK module. The estimated standard errors were calculated with the robust standard errors
corresponding to results summarized in Greene (2003, p.520).
b. The null hypotheses are (1) E(vt) = 0, (2) E( v t2 -1) = 0, (3) E(vt ⋅ vt-1) = 0, (4) E(vt ⋅ vt-2) = 0, (5) E(vt ⋅
vt-3) = 0, and (6) E(vt ⋅ vt-4) = 0, where vt is the standardized residual from the AR-GARCH-AGT
model.
c. This is a joint test for the null hypothesis that the model is correctly specified based on the moment
conditions. It has a χ2(6) distribution. The covariance matrix of the orthogonality conditions is
calculated using the consistent estimator proposed by Newey and West (1987, 1994) with a Bartlett
kernel.
26
c
Table 2: Estimation Results for Equation (4): Ri ,n = ai + bi ⋅
1
n
+ ci ⋅
+ ei ,n ; DJIA
exp( n )
exp( n )
Stocks
Risk
Measure
Variance
Skewness
5% quantile 95% quantile
1% VaR
5% VaR
0.683
0.651
(A) Estimation Results
Avg R2
0.880
0.758
0.805
0.857
Average of estimated coefficients and their t-statistics (in parentheses)
â
1.587
(116.4)
-0.019
(-19.27)
-1.975
(-225.5)
2.085
(243.5)
3.311
(246.1)
2.085
(224.2)
b̂
6.308
(9.917)
-0.033
(-1.900)
-1.860
(-8.706)
2.147
(9.422)
1.042
(2.110)
0.230
(0.726)
ĉ
1.460
(3.891)
0.273
(16.78)
-1.179
(-6.541)
1.525
(7.688)
3.017
(6.379)
2.340
(7.456)
(B) Diversifiable and Diversified Risk
⎛
Average risk measure for n=1 (in percentages): ⎜ aˆ +
⎜
⎝
4.445
Diversifiable risk:
2.858
n
2
3
4
5
6
7
8
9
10
11
12
13
14-30
Diversified risk:
1.609
2.326
2.635
2.766
2.820
2.843
2.852
2.855
2.857
2.857
2.857
2.858
2.858
0.069
bˆ + cˆ ⎞
⎟
exp(1) ⎟
⎠
-3.093
3.436
4.805
3.030
-1.118
1.351
1.493
0.945
0.536
0.991
1.253
1.385
1.446
1.473
1.485
1.490
1.492
1.493
1.493
1.493
1.493
0.281
0.584
0.770
0.865
0.910
0.930
0.939
0.943
0.944
0.945
0.945
0.945
0.945
bˆ + cˆ
exp(1)
0.088
⎧⎪ bˆ + cˆ ⎡
1
n ⎤ ⎫⎪
− ⎢ bˆ ⋅
+ cˆ ⋅
⎨
⎬
exp(1) ⎣ exp( n )
exp( n ) ⎥⎦ ⎪
⎩⎪
⎭
0.019
0.049
0.069
0.079
0.084
0.086
0.087
0.088
0.088
0.088
0.088
0.088
0.088
-0.547
-0.849
-0.997
-1.066
-1.096
-1.109
-1.114
-1.116
-1.117
-1.118
-1.118
-1.118
-1.118
0.648
1.016
1.200
1.285
1.323
1.339
1.346
1.349
1.350
1.351
1.351
1.351
1.351
Note: All risk measures, except for skewness, are in percentage.
27
Table 3: Estimation Results for Equation (4): Ri ,n = ai + bi ⋅
1
n
+ ci ⋅
+ ei ,n ; Market
exp( n )
exp( n )
Pool
Risk
Measure
Variance
Skewness
5% quantile 95% quantile
1% VaR
5% VaR
0.676
0.647
(A) Estimation Results
Avg R2
0.879
0.601
0.818
0.843
Average of estimated coefficients and their t-statistics (in parentheses)
â
1.142
(70.427)
-0.050
(-32.268)
-1.687
(-139.910)
1.731
(151.558)
2.863
(106.910)
1.801
(105.085)
b̂
9.963
(5.973)
-0.374
(-13.888)
-1.828
(-4.556)
1.811
(3.962)
-1.962
(-1.824)
-1.719
(-2.613)
ĉ
3.797
(4.607)
0.686
(25.250)
-3.168
(-9.981)
4.104
(11.626)
8.838
(9.418)
5.797
(10.036)
(B) Diversifiable and Diversified Risk
⎛
Average risk measure for n=1 (in percentages): ⎜ aˆ +
⎜
⎝
6.204
Diversifiable risk:
5.062
n
Diversified risk:
0.065
bˆ + cˆ ⎞
⎟
exp(1) ⎟
⎠
-3.525
3.907
5.393
3.302
-1.838
2.176
2.530
1.500
0.403
1.307
1.918
2.245
2.403
2.475
2.507
2.520
2.526
2.528
2.529
2.530
0.164
0.720
1.107
1.317
1.418
1.465
1.485
1.494
1.498
1.499
1.500
1.500
bˆ + cˆ
exp(1)
0.115
⎧⎪ bˆ + cˆ ⎡
1
n ⎤ ⎫⎪
− ⎢ bˆ ⋅
+ cˆ ⋅
⎨
⎬
exp(1) ⎣ exp( n )
exp( n ) ⎥⎦ ⎪
⎩⎪
⎭
2.686
-0.020
-0.733
0.820
3.999
0.031
-1.274
1.473
4
4.601
0.071
-1.572
1.842
5
4.867
-1.719
2.026
0.094
6
4.981
0.105
-1.786
2.111
7
5.029
0.111
-1.816
2.148
8
5.049
0.113
-1.829
2.165
9
5.057
0.114
-1.834
2.171
10
5.060
0.114
-1.837
2.174
11
5.061
0.115
-1.837
2.175
12
5.062
0.115
-1.838
2.176
13-50
5.062
0.115
-1.838
2.176
Note: All risk measures, except for skewness, are in percentage.
2
3
28
Figure 1 (a)
Generalized Student t Density for p=2, q=100
___ r = 0 ---- r = 0.5 …. r = -0.5
Figure 1 (b)
Generalized Student t Density for r=0
___ p=2 q = 3 (t distribution)
---- p=2 q = 100 (Normal)
…. p=1 q = 100 (Laplace)
29
Figure 2: Risk Measures against Portfolio Sizes
(100 Samples) (DJIA)
Figure 3: Estimated Diversification
Structure of Risk (DJIA)
(a)
(a)
(b)
(b)
(c)
(c)
(d)
(d)
30
Figure 4 (a): Estimated Diversification Structure of Risk (1-day horizon 1% VaR) (DJIA)
Figure 4 (b): Estimated Diversification Structure of Risk (1-day horizon 5% VaR) (DJIA)
31
Figure 5: Risk Measures against Portfolio Sizes
(100 Samples) (Market Pool)
Figure 6: Estimated Diversification
Structure of Risk (Market Pool)
(a)
(a)
(b)
(b)
(c)
(c)
(d)
(d)
32
Figure 7 (a): Estimated Diversification Structure of Risk (1-day horizon 1% VaR) (Market Pool)
Figure 7 (b): Estimated Diversification Structure of Risk (1-day horizon 5% VaR) (Market Pool)
33
Figure 8: Estimated Diversification
Structure of Risk
(Size-Based Diversification)
(a)
Figure 9: Estimated Diversification
Structure of Risk
(Liquidity-Based Diversification)
(a)
(b)
(b)
(c)
(c)
(d)
(d)
(e)
(e)
(f)
(f)
34