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Stable distributions in the
Black-Litterman approach to the
asset allocation
December 13, 2005
Rosella Giacometti1, Marida Bertocchi1, Svetlozar T. Rachev2 and FrankJ. Fabozzi3
Acknowledgments.The authors acknowledge the support given by researchprojects MIUR 60% 2003 Simulation models for complex portfolio alloca-tion and MIUR 60% 2004 Models for energy pricing, by research grantsfrom Division of Mathematical, Life and Physical Sciences, College of Let-
ters and Science, University of California, Santa Barbara and the DeutschenForschungsgemeinschaft.
1Department of Mathematics, Statistics,Computer Science and Applications, BergamoUniversity, Via dei Caniana, 2, Bergamo 24127, Italy
2School of Economics and Business Engineering, University of Karlsruhe, Postfach6980, 76128 Karlasruhe, Germany and Department of Statistics and Applied Probability,University of California, Santa Barbara, CA 93106-3110, USA
3Yale School of Management, 135 Prospect Street, Box 208200, New Haven, Connecti-cut 06520-8200, USA
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Stable distributions in the Black-Litterman approach tothe asset allocation
Abstract. The integration of quantitative asset allocation models and the
judgment of portfolio managers and analysts (i.e., qualitative view) dates
back to papers by Black and Litterman [4], [5], [6]. In this paper we improve
the classical Black-Litterman model by applying more realistic models for
asset returns (the normal, the t-student, and the stable distributions) and
by using alternative risk measures (dispersion-based risk measures, value at
risk, conditional value at risk). Results are reported for monthly data and
goodness of the models are tested through a rolling window of fixed size along
a fixed horizon. Finally, we find that incorporation of the views of investors
into the model provides information as to how the different distributional
hypotheses can impact the optimal composition of the portfolio.
Key Words. Black-Litterman model, risk measures, return distributions.
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Stable distributions in the Black-Litterman approach to
the asset allocation
1 Introduction
The mean-variance model for portfolio management as formulated by Markowitz
[18] is probably one of the most known and cited financial model. Despite
its introduction in 1952, there are several reasons cited by academics and
practitioners as why its use is not widespread. Some of the major reasons
are the scarcity of diversification [12] or highly concentrated portfolios and
the sensitivity of the solution to inputs (especially to estimation errors of the
mean, see Kallberg and Ziemba [16], [17], Best and Grauer [3], Michaud [21])
and the approximation errors in the solution of the maximization problem.
The integration of quantitative asset allocation models and the judgment
of portfolio managers and analysts (i.e., qualitative view) has been motivated
by various discussions on increasing the usefulness of quantitative models for
global portfolio management. The framework dates back to papers by Black
and Litterman [4], [5], [6] that led to development of extensions of the frame-
work proposed by members of both the academic and practitioner commu-
nities. Subsequent research has explained the advantages of this framework,
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what is now popularly referred to as Black-Litterman model (BL model here-
after), as well as the models main characteristics 1.
In most of these papers, there are explicit or implicit assumptions that
logaritmic returns on N asset classes are multivariate Gaussian distributed,
an assumption consistent with other mainstream theories in finance such as
the standard Black-Scholes model [7]. However, there are numerous empirical
studies 2 that show that in many cases logaritmic returns are quite far from
being normally distributed, especially for high frequency data. Many recent
papers (see Ortobelli et al. [24], [25], Bertocchi et al. [2]) show that sta-
ble Paretian distributions are suitable for the autoregressive portfolio return
process in the framework of asset allocation problem over a fixed horizon.
In this paper we investigate whether the BL model can be enhanced
by using the stable Paretian distributions as a statistical tool for asset re-
turns. We use as a portfolio of assets a subset, duly constructed, of the
S&P500 benchmark. We generalize the procedure of the BL model allowing
the introduction of dispersion matrices obtained from multivariate Gaussian
distribution, symmetric t-Student, and -stable distributions for computing1See the papers by Fusai and Meucci [10], Satchell and Scowcroft [30], He and Litterman
[13] and the books by Litterman [15] and Meucci [23].2See Embrechts et al. [8], Rachev and Mittnik [28] and the references therein, Mittnick
and Paolella [22], Panorska et al. [27], Tokat et al. [33], Tokat and Schwartz [32].
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the equilibrium returns. Moreover, three different measures of risk (variance,
value at risk and conditional value at risk) are considered. Results are re-
ported for monthly data and goodness of the models are tested through a
rolling window of fixed size along a fixed horizon. Finally, our analysis shows
that the incorporation of the views of investors into the model provides in-
formation as to how the different distributional hypotheses can impact the
optimal composition of the portfolio.
2 The -stable distribution
The -stable distributions describe a general class of distribution functions.
A random variable X is - stable distributed if it has a domain of attraction:
that is, there exists a sequence of i.i.d. random variables {Yi}i, a sequenceof positive real values {di}i and a sequence of real values {ai}i such that
as n +
1
dn
ni=1
Yi + andX (1)
where
d
points out the convergence in the distribution. Thus, the -stable random variables describe a general class of distributions including the
leptokurtic and asymmetric ones.
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The -stable distribution is identified by four parameters: the index of
stability (0, 2] which is the parameter of the kurtosis, the skewness pa-
rameter [1, 1]; and + which are, respectively, the location
and the dispersion parameter. If X is a random variable whose distribu-
tion is -stable, we use the following notation to underline the parameter
dependence
Xd=S( , , ) (2)
The stable distribution is normal, when = 2 and it is leptokurtotic
when < 2. A positive skewness ( > 0) identifies distributions with right
fat tails, while a negative skewness ( < 0) typically characterizes distrib-
utions with left fat tails. Therefore, the stable density functions synthesize
the distributional forms empirically observed in the real data. The Maximum
Likelihood Estimation (MLE) procedure used to approximate stable parame-
ters is described by Rachev and Mittnik [28]). Unfortunately the density of
stable distributions cannot be express in closed form. Thus, in order to value
the density function, it is necessary to invert the characteristic function.
In the case where the vector r = [r1, r2, . . . rn] of returns is sub-Gaussian
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-stable distributed with 1 < < 2, then the characteristic function of r
assumes the following form:
r(t) = E(exp(it
r)) = exp((tVt) 2 + itE(r)) (3)
For the dispersion matrix V = [v2ij ] we use the following estimation (see
Ortobelli and al. [26] and Lamantia et al. [14])
v2ij = ( vjj)2qA(q)
1
T
Tk=1
rik|rik|q1sgn( rjk) (4)
where rjk = rjk E(rj) is the k-th centered observation of the j-th asset,
A(q) =(1 q
2)
2q(1 q)( q+1
2)
and 1 < q <
vjj = (A(p)1
n
nk=1
|rjk |p)2p , 1 < p < 2 (5)
where rjk = rjk E(rj) is the k-th centered observation of the j-th asset.
2.1 t-Student distribution
Analogously to the normal distribution, the t-Student distribution depends
on an n - dimensional location parameter u which corresponds to the peak of
the distribution and on a nn symmetric dispersion matrix V that influences
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the shape of the distribution around the peak. Moreover, the degree of free-
dom of the distribution, , provides information on the relation between the
peak and the tails. However, as the number of degrees of freedom increases,
the t-Student distribution approaches closely the normal distribution.
The multivariate t-Student probability density function is given by:
v,u,V(x) = ()n
2( +n
2)
(2
)(|V|)12 (1 + 1
(x u)V1(x u))(+n)2
(6)
where denotes the Gamma function
(a) =+0
ua1eudu (7)
that for half integer arguments looks as follows
(n
2) =
(n 2)(n 4) . . . n2
n12
. (8)
The dispersion matrix is given by the following formula
Vt = V 2
. (9)
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3 The Black-Litterman model for asset allo-
cation and our extension
As previously mentioned, the BL model overcomes the critical step of ex-
pected return estimation, using the equilibrium returns defined as the returns
implicit in the benchmark. If the Capital Asset Pricing Model holds and if
the market is in equilibrium, the weights based on market capitalizations are
also the weights of the optimal portfolio. If the benchmark is a good proxy
for the market portfolio, its composition is the solution of an optimization
problem for a vector of unknown equilibrium returns. Moreover, the equi-
librium returns provide a neutral reference point for asset allocation. Black
and Litterman argue that the only sensible definition of neutral returns is
the set of expected returns that would clear the market if all investors had
identical views. In fact, an investor with neutral views should select a passive
strategy, tracking the benchmark portfolio. The equilibrium returns of the
stocks comprising the benchmark are obtained by solving the unconstrained
maximization problem faced by an investor with quadratic utility function
or assuming normally distributed returns
Max
x xx (10)
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where is the covariance matrix of our stocks returns.
From Kuhn-Tucker conditions on (10), and solving the reverse optimiza-
tion problem we get
= x (11)
The expected return E(r) is assumed to be normally distributed
E(r) N(, ) with the covariance matrix proportional to the historical
one, rescaled by a shrinkage factor; since uncertainty of the mean is lower
than the uncertainty of the returns themselves, the value of should be close
to zero.
An active asset manager can deviate from the benchmark tracking strat-
egy, according to his or her economic reasoning in the tactical asset alloca-
tion. The major contribution of the BL model is to combine the equilibrium
returns with uncertain views about expected returns. In particular, the op-
timal portfolio weights are moved in the direction of assets favored by the
investor. The investors views have the effect of modifying E(r) according to
the degree of uncertainty. Greater the uncertainty less the deviation from the
neutral views. To this aim the new vector of expected returns is computed
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minimizing the Mahalanobis distance between the expected returns E(r) and
the equilibrium returns which are additionally constrained by the investors
view on the expected return. This brings us to the following model:
min (E(r) )(E(r) ) (12)
s.t. constraint
PE(r) = q + e (13)
where P is a matrix with each row corresponding to one view, q is the
vector containing the specific investor views, and e a random vector of errors
in the view. If all views are independent, the covariance matrix is diagonal.
Its diagonal elements are collected in the vector e.
In our analysis we consider different problems of optimal allocation among
n risky assets with returns [r1, r2, . . . , rn] using different risk measures vari-
ance, Value at Risk (V aR), and Conditional Value at Risk (C V a R). Assume
that all portfolios r
x are uniquely determined by the neutral mean
x and
by the risk measure () that is defined alternatively as the dispersion x
Vx,
the V aR(r
x) and the CV aR(r
x). This means that instead of (10) we
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have
Max
x (rx). (14)
We recall here that V aR(X) is implicitely defined by P(X V aR(X)) =
, i.e. the percentile of the probability density function of the random vari-
able X such that the probability that the random variable assumes a value
less than x is greater than , where represents, in this framework, the
maximum probability of loss that the investor would accept. We also recall
that CV aR(X) is defined as E(X|X V aR(X)), i.e it measures the
expected value of the tail of the distribution for values less than V aR. Note
also that CV aR(X) is a coherent risk measure in the sense of Artzner et al.
[1] while V aR(X) is not.3
Notice that for elliptical distributions, following Embrechts et al [8] and
Stoyanov et al. [31], the CV aR of portfolio returns is expressed as
CVaR(r
x) =
x
VtxCV aR(Y) xu (15)
where CV aR(Y) for univariate t-distribution takes the following form
3For detailed description of C V aR see for example Rockafeller and Uryasev [29] andfor -stable see Stoyanov et al. [31]. For comparisons between CV aR and V aR seeGaivoronski and Pflug [11].
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CV aR(Y) =( +1
2)
( 2
)
( 1) (1 +V aR(Y)
2
)12 (16)
where Y is distributed according to a t-student with > 1 degree of
freedom.
Using again Stoyanov et al. [31], we can represent the C V a R(X) for
multivariate standardised -stable distribution, X S( , , ) as
CVaR(X) =
1 |V aR(X)|
2
0g()exp(|V aR(X)|
1 v())d (17)
where
g() =sin((0 + ) 2)
sin (0 + ) cos
2
sin2 (0 + ), (18)
v() = (cos 0)1
1 (cos
sin (0 + ))
1
cos(0 + ( 1))cos
, (19)
and
0 =1
arctan(tan
2), = sgn(V aR(X)). (20)
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In the case where we have a non-standardized -stable, we need to use
the following transformation
CVaR(X + ) = CV aR(X) , (21)
where
X + S( , , ). (22)
Properties similar to (15) and (21) hold for V aR too (see Lamantia et al.
[14]).
Notice that the optimization problem for C V a R risk measures using prop-
erties (15) and (21), can be written as
Max
x
(CVaR
x
Vx
E(r)
x) (23)
Similar considerations apply to V aR.
Applying first-order Kuhn-Tucker conditions to (23), the reverse opti-
mization model, and using the three different measures of risks and the
three different return distributions (Gaussian, t-student, stable) we obtain
the following equilibrium returns for the three different dispersion measures
V characterizing the three distributions:
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Risk Measure: variance
= Vx (24)
Risk Measure: CVaR
= (CVaRVxx
Vx E(r)x) (25)
Risk Measure: VaR
= (V aRVxx
Vx E(r)x) (26)
In formulas (24), (25), and (26), we will substitute the convenient estimate
for the dispersion matrix, CV aR and V aR depending on the corresponding
distribution.
Notice that the coefficient can be interpreted as a coefficient of risk
aversion: if is zero the investor is risk neutral, if > 0, the investor is risk
averse because investments with large dispersion are penalized, if < 0, the
investor is a risk seeker because investments with large dispersion are favored.
Once we found the neutral returns implied in the benchmark, we wanted to
test the goodness of these equilibrium returns over a 20 month horizon. We
thought that a reasonable way was to compute the sum of squared errors
between the neutral view return suggested by our model and the day after
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realization of return for 20 consecutive months, using a rolling window of
110 months for the parameters estimation. We compare the equilibrium
returns obtained under different distributional hypotheses and different risk
measures with a naive forecast: the unconditional mean. We recall that
for a stationary return process the best forecast of future realizations is the
unconditional mean.
But which is the optimal value of to be used? Black and Litterman
suggest, under the normal distributional hypothesis, using the market risk
premium which is 0.32 in our case. 4
If we try to determine the value of which minimizes the distance between
the optimal solution of the portfolio and the weight of the benchmark we get
= 36.29, i.e. the risk aversion parameter becomes very large. This may
be considered reasonable when we look at it from the equity premium puzzle
side (see Fama and French [9], Mehra and Prescott [19], [20]). However since
we consider three different risk measure we must consider different values of
. Indeed, in (24)-(26), the coefficient acts as a scaling factor, i.e. a larger
increases the equilibrium returns. Large values of will eventually scale
all the equilbrium return to very large values that would not be realistic.
4This is computed as the excess mean return divided by the variance.
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We set the value of equal to the solution of an optimization problem.
For each risk measure, we fix equal to the value that minimizes the sum
of the squared error under all distributional hypothesis described before,
computed for the first day of the out-of-sample analysis. We mantain the
same values for all the out-of-sample analysis. Therefore, we choose = 0.5
for the case where risk was measured by dispersion and = 0.15 for the
other cases. Finally, we tested how the forcing of a special investor view
(both under certainty and under incertainty) may influence the benchmark
composition.
4 Analysis of the data
In this section we analyze the time series data for the benchmark used in the
computational part and we estimate the parameters of the different distribu-
tions (normal, symmetric t-student, and -stable) and the related dispersion
matrices. We selected as the benchmark the S&P500 and we obtained daily,
weekly, and monthly data from July 1995 to July 2005 from DataStream.
The analysis was done for all the frequencies mentioned above. However,
only results for the monthly data are reported here.
We divided the data in two samples: the first 110 data for the parameter
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estimation and the remaining 20 data for out-of-sample analysis. The out-of-
sample analysis is repeated for 20 consecutive months using a rolling window
of length 110 to estimate the parameters for each month. In order to better
analyze the results, we choose to reduce the dimension of the benchmark
considering the most capitalized shares which account for about 50% of the
index. We collected the data for the 50 most capitalized shares and rescaled
the weights to sum up to 1. We use the new weights to construct a synthetic
index that we will refer to as the S&P50 in the following. We also tested
that S&P50 returns are almost perfectly correlated ( = 0.98) to the S&P500
along the considered horizon.
In Table 1 we report for each of the 50 shares included in the synthetic
index, the ticker, the name of the company, the weight in the S&P500, the
new weight in the S&P50, the mean, the volatility, the skewness, and the
kurtosis of each share in the sample. Recall that we selected monthly data
because in an asset allocation problem the reasonable time horizon should not
be too short. Because of the frequency selected, we tested for the absence of
autocorrelation in the returns and squared returns, but we found no evidence
of it. Based on the Bera-Jarque test, we rejected for 19 of the 50 shares the
null hypothesis of normal distribution at the 5% significance level, for 21
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of the 50 at the 10% significance level, and for 23 of the 50 at the 15%
significance level. From the results reported in Table 1 we can observe that
about half of the 50 stocks could be well described by a normal distribution.
In Table 2 we report the average of the estimated parameters , , for
the -stable distribution and the average of the degree of freedom for the
t-student computed as the mean of 20 estimations over the rolling window.
A similar analysis is done for a rolling window of increasing size, with no
significant changes in the results. Therefore, we do not report those results.
From Table 2 we note that only 9 stocks show a value of the parameter
equals to 2. The -stable distribution looks more appropriate in describing
the behavior of the remaining stocks returns.
In order to assess the hypothesis of non-normal behavior of the stocks
and the statistical significance of the -stable parameters, we estimated the
following AR(1, 1) model in order to construct a 90% confidence level for
and
yt = a1 + a2yt1 + t
a32
(27)
where yt is the (or ) series, a1 and a2, a3 the parameters to be esti-
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mated.
The estimated coefficients, together with their statistical significance, are
reported in Table 3 for and in Table 4 for . We do not consider stocks with
=2 and =1 or = 1 for all the out-of-sample period. So we exclude
9 stocks from Table 3 and 17 stocks from Table 4. Each table contains the
ticker, the estimated coefficients of the AR(1, 1) process, the ratio between
the value of the coefficients and the standard error, the value of the likelihood
function, the 5-th, the 50-th, and the 95-th percentiles for the 40 stocks. We
observe that the autoregressive coefficient is significant for 80% of the stocks
for and for 70% of the stocks for (see columns Ta2 ). We used the model
given by (27) to create 5000 scenarios for each of the parameters and
and each stock in the benchmark, thus obtaining the related distributions:
we report in Tables 3 and 4 the median, the 5-th and 95-th percentiles of
those ones to costruct the 90% confidence level.
The analysis of the 90% confidence interval confirms that for 82% of the
stocks considered the true value of is less than 2, suggesting the presence
of leptokurtic behavior. Only for 9 stocks is the normal distribution suitable.
Moreover, the upper value of the confidence level of is less then 0 for 19
stocks and is equal to -1 for 6 stocks, suggesting a left fat tail distribution
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for 50% of the stocks. The lower value of the confidenze level of exceeds
0 for 7 stocks and is equal to 1 for 2 stocks over the 50 stocks considered
suggesting for a rigth fat tail for 18% of the stocks. Only for 7 stocks does
the confidence interval include the null value suggesting that at most 16
stocks can show a symmetric behavior. That explains the poor behavior of
a symmetric t-student distributional model which indeed seems to give the
same result as the normal distribution. This is a further confirmation that
the -stable distribution is suitable to describe the returns of our data.
5 Computational results
In Table 5 we report the equilibrium returns for the different distributions and
the different risk measures for January 28, 2004, computed with = 0.5 when
risk is measured by dispersion and = 0.15 for the other cases. In Table 6 we
report the optimal composition under equilibrium returns (or neutral views)
for the different distributions and the different risk measures. We observe
that equilibrium returns under the -stable hypothesis are in general less
than the equilibrium returns under the normal and t-student distributions.
Moreover, the corresponding optimal portfolio is less diversified involving
17 stocks against the 29 stocks under normal and t-student distributions.
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In general under the normal and t-student distributions we get very similar
equilibrium returns and portfolio composition under all the different risk
measures while the -stable hyphotesis implies different equilibrium returns
and portfolio composition.
We consider the equilibrium returns as a forecast of the future returns.
Of course, we assume that when we compare the forecast with the future
realizations, the data that we observe in the future are the products of a
market in equilibrium. In Table 7 we report the sum of squared errors for
20 months between the neutral view and realization of the day after using a
rolling window of 110 months. Note that we reestimate the parameters of the
distribution as we move the rolling window. We observe that the hypothesis
of a stable distribution and the use of dispersion as a risk measure gives the
best combination 5. For 13 months of the 20 months, it is the combination
that gives the best forecast (65% success rate). The second best combination
is the -stable distribution and the use ofC V a R as a risk measure. For this
combination 3 of the 20 months gives the best forecast (15% success rate).
5The use of a symmetric risk measure is not particularly surprising since we are dealing
with a model of strategic allocation to find the optimal composition of our portfolio ona relative long time horizon. Generally this phase is followed by a tactical allocationstrategy where it is more likely that a relative V aR can be considered by the market,i.e. the additional tail risk that we accept when we move from the benchmark replicationstrategy assuming specific risk.
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The third best combination is the unconditional mean which resulted in 2 of
the 20 months (10% success rate). Finally, following Satchell and Scowcroft
[30], we report again in Table 8 the optimal composition for the January
28, 2004 under a specific view (asset 2 overperforming asset 5 by 0.1%)
for the different distributions and the different risk measures with absolute
certainty. In Table 9 we report again the optimal composition for the same
date (January 28, 2004) under the same view (asset 2 overperforming asset
5 by 0.1%) but this time we are not absolutely certaint about our view
(with uncertainty of 0.61). According to neutral view, asset 2 is expected to
overperform asset 5 for an average of 3.45% with a maximum of 4.85% for
dispersion under the normal distribution and a minimum of 1.62% under the
same risk measure but -stable hypothesis (see Table 5). If we compute the
equilibrium return with the additional constraint that the difference between
the two returns is 0.1% with certainty, we get the equilibrium return in
Table 10. We can observe that the difference between the new returns and
neutral view equilibrium are larger for the dispersion measure and normal
returns. Indeed, this is the case with the highest variation in the portfolio
composition. Once again the -stable hypothesis with the same risk measure
lead to a more stable portfolio. If we consider the same view with uncertainty,
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we have effects similar but mitigated by the uncertainty that we put in our
view.
6 Conclusions
The purpose of our work is twofold. The first is to improve the classical BL
model by applying more realistic models for asset returns. We compare the
BL model under the normal, t-student, and the stable distributions for asset
returns. The second is to enhance the BL model by using alternative risk
measures which are currently used in risk management and portfolio analysis.
They include dispersion-based risk measures, value at risk, and conditional
value at risk.
For the stocks in our sample, only a minority can be characterized as
having a normal return distribution based on statistical tests we performed.
As a result of incorporating heavy-tailed distribution models for asset re-
turns and alternative risk measures, we obtained the following results: (1)
the appropriateness of the -stable distributional hypothesis is more evident
when we compute the equilibrium returns and (2) the combination of -
stable distribution and the choice of dispersion risk measure provides the
best forecast.
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ticker company % of SP500 % of SP50 mea n v ol atil ity s kewn es s kurtosis BJ (p-vaXOM Exxon Mobil Corp. 3.33 6.86 11.73% 0.90 0.20 3.90 0.361
GE General Electric 3.31 6.82 10.96% 0.60 0.12 2.93 0.870MSFT Microsoft Corp. 2.4 4.94 13.41% 1.24 0.19 5.58 0.000
C CityGroup Inc. 2.16 4.45 17.07% 1.08 0.02 4.23 0.246WMT Wal-Mart Stores 1.81 3.73 12.15% 0.84 0.39 3.28 0.128PFE Pfizer Inc. 1.81 3.73 11.46% 0.89 0 2.68 0.811JNJ Johnson & Johnson 1.73 3.56 11.97% 0.67 0.08 3.16 0.930
BAC Bank of America Corp. 1.62 3.34 10.79% 1.06 0.41 4.46 0.057INTC Intel Corp. 1.45 2.99 10.85% 1.50 0.88 5.74 0.000AIG American Intl Group 1.37 2.82 11.44% 0.87 0.01 3.40 0.600MO Altria Group Inc. 1.21 2.49 8.92% 1.11 0.33 3.85 0.216PG Procter & Gamble 1.18 2.43 10.61% 0.85 1.69 11.18 0.000
JPM JPMorgan Chase & Co. 1.11 2.29 6.62% 1.25 0.18 5.41 0.001CSCO Cisco Systems 1.09 2.24 16.87% 1.62 0.63 4.60 0.000
IBM Int. Business Machines 1.08 2.22 10.11% 1.17 0.37 4.86 0.003CVX Chevron Corp. 1.07 2.20 7.87% 0.63 0.38 3.31 0.140WFC Wells Fargo 0.93 1.92 13.53% 0.80 0.05 3.82 0.737
KO Coca Cola Co. 0.91 1.87 2.36% 0.84 0.29 3.02 0.376VZ Verizon Communications 0.86 1.77 1.56% 0.83 0.00 3.32 0.990
DELL Dell Inc. 0.85 1.75 33.87% 1.72 0.66 4.89 0.000PEP PepsiCO Inc. 0.8 1.65 8.67% 0.73 0.02 4.14 0.183HD Home Depot 0.76 1.57 13.12% 1.08 0.67 4.38 0.001
COP Conoco Philips 0.74 1.52 11.47% 0.76 0.18 3.32 0.839
SBC SBC Communications Inc. 0.71 1.46 0.09% 0.93 0.08 4.91 0.017TWX Time Warner Inc. 0.7 1.44 20.97% 1.96 0.43 3.72 0.340UPS United Parcel Service 0.69 1.42 6.86% 0.95 0.60 4.39 0.008ABT Abbot Labs 0.68 1.40 8.89% 0.76 0.32 3.70 0.162
AMGN Amgen 0.68 1.40 17.35% 1.05 0.58 4.64 0.002MRK Merck & Co. 0.61 1.26 2.68% 1.00 0.69 5.34 0.691
ORCL Oracle Corp. 0.61 1.26 13.80% 1.68 0.13 3.41 0.752HPQ Hewlett-Packard 0.61 1.26 4.03% 1.46 0.25 4.76 0.021
CMCSA Compcast Corp. 0.6 1.24 10.99% 1.22 0.06 5.52 0.000UNH United Health Group Inc. 0.6 1.24 20.27% 1.27 2.31 14.11 0.000AXP Amercian Express 0.6 1.24 13.44% 0.94 0.34 4.26 0.059LLY Lilly (Eli) & Co. 0.56 1.15 9.80% 0.97 0.03 3.48 0.654
MDT Medtronic Inc. 0.56 1.15 15.04% 0.97 0.07 3.89 0.698WYE Wyeth 0.54 1.11 7.69% 1.13 1.69 12.61 0.000TYC Tyco International 0.53 1.09 14.57% 1.43 1.00 6.17 0.000
MWD Morgan Stanley 0.52 1.07 13.5% 1.35 0.33 3.85 0.174FNM Fannie Mae 0.51 1.05 8.61% 0.79 0.36 2.98 0.280
MMM 3M Company 0.5 1.03 9.47% 0.71 0.20 3.17 0.559QCOM Qualcomm Inc. 0.49 1.01 23.8% 1.97 0.16 4.11 0.353
BA Boeing Company 0.48 0.99 6.28% 1.04 0.35 3.87 0.198UTX United Technologies 0.47 0.97 14.82% 0.89 0.13 5.50 0.000MER Merryl Linch 0.47 0.97 13.20% 1.30 0.36 4.80 0.007
VIA.B Viacom Inc. 0.47 0.97 2.23% 1.11 0.08 3.10 0.870DIS Walt Disney Co. 0.46 0.95 2.81% 1.05 0.01 3.40 0.940
G Gillette Co. 0.45 0.93 3.94% 0.94 0.50 4.05 0.004BMY Bristol-Myers Squibb 0.44 0.91 7.94% 0.41 0.36 3.67 0.264BLS Bell South 0.44 0.91 4.32% 0.80 0.13 3.54 0.769
Table 1: Statistics on the single time seriesNote: From the table we observe the skewness and kurtosis of the single stocks on the complete sample. We performe
Bera-Jarque test and we cannot reject the null hypothesis of normal distribution for 19 over 50 at 5% significance level,for 21 over 50 at 10% significance level, and for 23 over 50 at 15% significance level.
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numbering ticker company % of SP500 % of SP50 degree offreedom
1 XOM Exxon Mobil Corp. 3.33 6.86 1.95 -0.91 0.64 182 GE General Electric 3.31 6.82 2.00 0.96 0.42 1003 MSFT Microsoft Corp. 2.4 4.94 1.71 0.49 0.79 74 C CityGroup Inc. 2.16 4.45 1.88 -0.18 0.74 105 WMT Wal-Mart Stores 1.81 3.73 1.85 -1.00 0.58 226 PFE Pfizer Inc. 1.81 3.73 2.00 -0.85 0.64 1007 JNJ Johnson & Johnson 1.73 3.56 2.00 -0.14 0.48 548 BAC Bank of America Corp. 1.62 3.34 1.48 -0.44 0.58 49 INTC Intel Corp. 1.45 2.99 1.81 -0.86 0.97 6
10 AIG American Intl Group 1.37 2.82 2.00 -0.20 0.61 1711 MO Altria Group Inc. 1.21 2.49 1.90 -0.84 0.78 1012 PG Procter & Gamble 1.18 2.43 1.82 -1.00 0.51 413 JPM JPMorgan Chase & Co. 1.11 2.29 1.69 -0.14 0.76 514 CSCO Cisco Systems 1.09 2.24 1.77 -0.69 1.05 515 IBM Int. Business Machines 1.08 2.22 1.77 0.39 0.73 616 CVX Chevron Corp. 1.07 2.20 1.86 1.00 0.42 1117 WFC Wells Fargo 0.93 1.92 1.87 -0.01 0.56 818 KO Coca Cola Co. 0.91 1.87 1.95 -1.00 0.59 9219 VZ Verizon Communications 0.86 1.77 2.00 -0.19 0.61 3720 DELL Dell Inc. 0.85 1.75 1.76 -0.69 1.08 521 PEP PepsiCO Inc. 0.8 1.65 1.80 -0.02 0.47 622 HD Home Depot 0.76 1.57 1.83 -1.00 0.72 9
23 COP Conoco Philips 0.74 1.52 2.00 0.57 0.53 1024 SBC SBC Communications Inc. 0.71 1.46 1.81 -0.36 0.61 725 TWX Time Warner Inc. 0.7 1.44 1.82 0.91 1.30 1426 UPS United Parcel Service 0.69 1.42 1.46 -0.45 0.51 427 ABT Abbot Labs 0.68 1.40 1.94 -0.93 0.53 1828 AMGN Amgen 0.68 1.40 1.84 0.88 0.69 429 MRK Merck & Co. 0.61 1.26 1.93 -1.00 0.67 1630 ORCL Oracle Corp. 0.61 1.26 2.00 -0.40 1.26 3631 HPQ Hewlett-Packard 0.61 1.26 1.74 0.05 0.94 532 CMCSA Compcast Corp. 0.6 1.24 1.73 -0.16 0.75 433 UNH United Health Group Inc. 0.6 1.24 1.54 0.15 0.54 334 AXP Amercian Express 0.6 1.24 1.53 -0.84 0.54 835 LLY Lilly (Eli) & Co. 0.56 1.15 1.94 -0.22 0.68 1436 MDT Medtronic Inc. 0.56 1.15 1.81 0.27 0.62 637 WYE Wyeth 0.54 1.11 1.65 -0.56 0.62 438 TYC Tyco International 0.53 1.09 1.71 -0.59 0.87 439 MWD Morgan Stanley 0.52 1.07 1.82 -0.75 0.92 1040 FNM Fannie Mae 0.51 1.05 1.91 -1.00 0.54 10041 MMM 3M Company 0.5 1.03 1.96 1.00 0.50 5742 QCOM Qualcomm Inc. 0.49 1.01 1.76 0.19 1.28 643 BA Boeing Company 0.48 0.99 1.72 -0.42 0.67 644 UTX United Technologies 0.47 0.97 1.67 -0.43 0.53 545 MER Merryl Linch 0.47 0.97 1.81 -0.35 0.86 646 VIA.B Viacom Inc. 0.47 0.97 2.00 -0.55 0.83 10047 DIS Walt Disney Co. 0.46 0.95 2.00 0.89 0.78 6148 G Gillette Co. 0.45 0.93 1.84 -0.87 0.65 749 BMY Bristol-Myers Squibb 0.44 0.91 1.87 -0.97 0.60 750 BLS Bell South 0.44 0.91 1.97 0.69 0.57 19
Table 2: Estimated parameters with -stable and t-student distributionsNote: In this table we report the average estimate of the -stable computed on a rolling window.
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numbering ticker a1 a2 a3 Ta1 Ta2 Ta3 LLF perc.5% perc.50% perc.95%
1 XOM 0.724 0 .629 - 1.997 3 .376 2.136 79.249 1.9357 1.9433 1.9507
2 GE - - - - - - - - - -3 MSFT 0.349 0.793 0.005 0.831 3.497 3.406 25.658 1.4907 1.5982 1.70954 C 0.072 0.961 - 0.345 8.717 3.338 82.661 1.8551 1.8614 1.86805 WMT 0.361 0.805 - 1.130 4.677 1 .543 70.703 1.8483 1.8598 1.87116 PFE - - - - - - - - - -7 JNJ - - - - - - - - - -8 BAC 0.232 0.842 - 0.962 5.265 2.959 47.765 1.4178 1.4538 1.49059 INTC 0.600 0.668 - 1.192 2.403 2.805 62.988 1.7805 1.7978 1.8144
10 AIG - - - - - - - - - -11 MO 0.122 0.934 - 0.297 4.306 3.432 49.386 1.7882 1.8219 1.856712 PG 0.015 0.993 - 0.062 7.664 1.876 78.473 1.8378 1.8457 1.853413 JPM 0.187 0.889 - 0.612 4.949 3.775 59.791 1.6460 1.6667 1.687314 CSCO 0.827 0.532 - 2.981 3.400 2 .789 48.111 1.7286 1.7640 1.799115 IBM 0.145 0 .917 - 0.366 4 .103 3.847 6 0.213 1.7101 1.7295 1.749116 CVX 0.991 0.468 - 3.667 3.203 3 .582 64.867 1.8459 1.8610 1.877117 WFC 0.096 0.946 0.001 0.262 4.979 3.766 37.453 1.7114 1.7738 1.834818 KO 0.674 0.655 - 1.560 2.967 2.501 82.666 1.9445 1.9510 1.957519 VZ - - - - - - - - - -20 DELL 0.025 0.980 0.003 0.031 2.158 2.347 29.273 1.4427 1.5345 1.626721 PEP 0.296 0.834 - 1.214 6.148 1.750 58.839 1.7261 1.7466 1.768022 HD 0.505 0.724 - 1.671 4.380 1.710 73.083 1.8217 1.8320 1.842123 COP - - - - - - - - - -
24 SBC 0.275 0 .848 - 1.880 4 .920 3.815 77.941 1.8063 1.8145 1.822825 TWX 0.191 0.894 0.001 0.434 3.700 3.648 46.213 1.7440 1.7825 1.820626 UPS 0.181 0.877 0.001 0.900 6.435 2.987 45.839 1.4429 1.4828 1.522727 ABT 1.853 0.045 - 0.620 0.029 1.511 57.538 1.9183 1.9411 1.963028 AMGN 0.695 0.622 0.002 1.501 2.358 2.351 33.662 1.7800 1.8543 1.929829 MRK 0.258 0.866 - 0.176 1.149 1.944 51.033 1.8728 1.9033 1.934130 ORCL - - - - - - - - - -31 HPQ 0.133 0.922 - 0.519 6.250 3.758 57.519 1.6797 1.7025 1.724832 CMCSA 1.107 0.358 - 2.150 1.203 3.041 57.611 1.7001 1.7224 1.745233 UNH 0.412 0.733 0.001 0.791 2.153 2.913 43.604 1.4799 1.5252 1.571334 AXP 0.258 0.832 - 1.178 5.811 2.236 53.223 1.5128 1.5392 1.567535 LLY 0.168 0.912 0.001 0.411 4.361 3.423 44.314 1.8213 1.8654 1.908536 MDT 0.210 0.880 0.003 0.477 3.724 3.118 29.431 1.5760 1.6674 1.759837 WYE 0.484 0.706 - 0.806 1.943 3.730 51.040 1.6047 1.6354 1.665938 TYC 0.166 0.903 - 0.622 5.778 3 .323 56.193 1.6705 1.6939 1.717839 MWD 0.675 0.630 0.001 1.041 1.808 1.784 40.896 1.7651 1.8172 1.870240 FNM 0.545 0.716 - 1.616 4.065 2.147 65.505 1.9010 1.9161 1.932141 MMM 0.456 0.767 - 1.552 5 .105 2 .480 73.090 1.9474 1.9578 1.968342 QCOM 0.268 0.847 0.001 0.619 3.463 3.689 44.868 1.6753 1.7170 1.759343 BA 0.647 0.624 0.001 2.402 3.948 2.584 45.355 1.6726 1.7132 1.753844 UTX 0.464 0.723 - 1.491 3.890 2.847 58.561 1.6539 1.6762 1.697345 MER 0.243 0.865 - 0.546 3.524 4.650 69.746 1.7862 1.7982 1.810346 VIA.B - - - - - - - - - -47 DIS - - - - - - - - - -48 G 1.100 0.401 - 1.505 1.011 2.053 55.033 1.8080 1.8329 1.857849 BMY 1.618 0.135 0.001 0.256 0.040 1.787 37.088 1.7882 1.8497 1.911250 BLS 1.299 0 .339 - 2.196 1 .143 1.333 54.315 1.9363 1.9633 1.9895
Table 3: Estimated parameters of the AR(1,1) process on valuesNote: The analysis of the 90% confidence interval confirm that for the 82% of the stocks considered (i.e. those with
different from 2) the true value of is smaller than 2, suggesting the presence of leptokurtic behavior. Only for 9 stocksthe normal distribution is suitable.
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numbering ticker a1 a2 a3 Ta1 Ta2 Ta3 LLF perc.5% perc.50% perc.95%
1 XOM 0.016 0.990 0.014 0.060 2.859 2.032 14.274 -0.5791 -0.3036 -0.1924
2 GE - - - - - - - - - -3 MSFT 0.261 0.472 0.192 1.036 1.658 1.895 -11.859 -0.1468 0.5880 1.31064 C -0.066 0.623 0.002 - 1.289 2.198 3.043 35.618 -0.2072 -0.1405 -0.07465 WMT - - - - - - - - - -6 PFE - - - - - - - - - -7 JNJ - - - - - - - - - -8 BAC -0.014 0.959 - -0.151 4.331 3.426 48.559 -0.4008 -0.3660 -0.33139 INTC -0.015 0.975 0.001 -0.091 5.404 2.399 37.926 -0.7761 -0.7170 -0.6570
10 AIG - - - - - - - - - -11 MO -0.192 0.757 0.017 -0.720 2.353 1.905 12.488 -0.7972 -0.5816 -0.370912 PG - - - - - - - - - -13 JPM -0.037 0.737 0.001 -1.552 4.105 3.456 47.167 -0.1706 -0.1342 -0.096414 CSCO -0.195 0.712 0.005 -1.527 4.227 2.171 24.582 -0.7508 -0.6355 -0.519215 IBM 0.004 0.966 0.003 0.041 3.872 4.751 30.636 0.1086 0.1927 0.279416 CVX - - - - - - - - - -17 WFC 0.096 0.946 0.001 0.262 4.979 3.766 37.453 1.7114 1.7738 1.834818 KO - - - - - - - - - -19 VZ - - - - - - - - - -20 DELL -0.011 0.936 0.051 -0.011 0.703 1.182 1.441 -0.3535 0.0161 0.379121 PEP -0.008 0.599 0.001 -0.884 1.510 3.668 38.309 -0.0925 -0.0346 0.026222 HD - - - - - - - - - -23 COP - - - - - - - - - -
24 SBC -0.120 0.668 - -1.930 3.780 3.230 47.633 -0.3885 -0.3516 -0.313825 TWX 0.199 0.785 0.005 0.782 2.638 2.198 23.887 0.8665 0.9836 1.104626 UPS -0.190 0.577 - -2.036 2.759 3.232 54.503 -0.4741 -0.4477 -0.421927 ABT -0.977 -0.055 0.104 -0.056 -0.003 0.390 -5.739 -1.4412 -0.9204 -0.392628 AMGN 0.333 0.624 0.055 1.732 1.612 1.853 0.586 0.5667 0.9582 1.343229 MRK - - - - - - - - - -30 ORCL - - - - - - - - - -31 HPQ 0.012 0.728 0.001 0.848 3.593 1.909 43.827 -0.0299 0.0151 0.060432 CMCSA 0.005 0.980 0.002 0.146 5.190 5.146 34.298 -0.0683 0.0044 0.079033 UNH 0.026 0.860 0.003 0.405 2.431 3.941 29.404 0.1545 0.2446 0.335134 AXP -0.179 0.788 - -0.600 2.217 2.260 63.458 -0.8579 -0.8406 -0.824235 LLY 0.007 0.897 0.110 0.095 5.689 4.415 -6.319 -0.1457 0.4035 0.952036 MDT 0.196 0.259 0.006 3.042 1.386 2.744 23.535 0.1365 0.2571 0.378937 WYE -0.176 0.687 0.001 -1.140 2.481 3.373 47.454 -0.6063 -0.5695 -0.531238 TYC -0.036 0.930 0.002 -0.241 3.828 4.387 35.015 -0.5459 -0.4764 -0.408139 MWD -0.255 0.653 0.013 -0.392 0.910 0.878 15.149 -0.8484 -0.6553 -0.464740 FNM - - - - - - - - - -41 MMM - - - - - - - - - -42 QCOM 0.034 0.808 0.001 0.815 3.773 3.859 37.019 0.0862 0.1486 0.211043 BA -0.321 0.244 - -2.092 0.681 2.442 57.426 -0.4422 -0.4200 -0.397544 UTX -0.150 0.648 0.001 -1.570 2.889 2.430 45.302 -0.4441 -0.4042 -0.362845 MER -0.049 0.852 0.001 -0.520 3.123 3.812 45.335 -0.3261 -0.2847 -0.245246 VIA.B - - - - - - - - - -47 DIS - - - - - - - - - -48 G -0.249 0.708 0.004 -0.955 2.390 2.783 25.784 -0.8737 -0.7651 -0.651649 BMY -1.972 -1.000 0.014 -0.001 -0.001 0.109 14.410 -1.7079 -1.5127 -1.321250 BLS 0.496 0.287 0.359 0.918 0.378 0.503 -18.137 -0.2001 0.7659 1.7537
Table 4: Estimated parameters of the AR(1,1) process on valuesNote: We can observe that, at 90% percent confidence level, the 50% of the stocks show a left fat tail, 18% a rigth
fat tail. Since for 9 stocks, that account for 18% of the total, = 0 by definition since = 2, and only for 7 stocks theconfidence interval include the null value, we can suggest that at most 16 stocks can show a symmetric behavior.
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dispersion VaR CVaR uncond. meanticker normal t-stud stable normal t-stud stable normal t-stud stable
XOM 22.64 21.80 9.21 16.35 15.98 16.26 20.98 20.73 25.13 12.48GE 8.81 8.48 3.98 5.76 5.62 6.51 7.56 7.46 10.34 8.85
MSFT 25.30 24.36 11.56 17.88 17.47 20.27 23.05 22.78 31.41 16.56C 30.17 29.05 12.08 21.16 20.67 20.66 27.33 27.00 32.30 20.82
WMT 13.66 13.15 5.60 8.75 8.53 8.78 11.54 11.39 14.17 14.98PFE 16.83 16.21 7.60 11.09 10.82 12.50 14.53 14.35 19.81 16.38JNJ 10.77 10.37 4.49 6.91 6.73 7.08 9.11 8.99 11.40 11.73
BAC 22.17 21.36 8.17 16.15 15.79 14.39 20.68 20.44 22.26 11.34INTC 28.46 27.41 11.11 20.74 20.28 19.70 26.56 26.25 30.41 14.45AIG 20.37 19.62 8.04 14.22 13.89 13.65 18.39 18.17 21.39 14.50MO 7.11 6.85 2.39 4.45 4.34 3.43 5.91 5.83 5.73 8.47PG 6.01 5.79 3.10 3.11 3.01 4.37 4.34 4.27 7.36 11.56
JPM 32.61 31.41 12.16 25.02 24.49 22.70 31.69 31.34 34.41 8.19CSCO 35.21 33.91 13.25 25.04 24.46 22.78 32.24 31.85 35.55 22.05
IBM 24.67 23.76 9.14 17.86 17.46 15.98 22.90 22.63 24.79 13.35CVX 7.36 7.09 2.76 5.00 4.88 4.50 6.50 6.42 7.16 6.18WFC 14.86 14.32 6.31 9.64 9.40 10.09 12.68 12.52 16.17 15.48
KO 10.46 10.07 4.57 7.73 7.56 8.30 9.87 9.75 12.70 4.61VZ 13.07 12.59 4.80 10.22 10.01 9.16 12.89 12.75 13.78 2.01
DELL 26.80 25.81 10.65 15.85 15.42 15.25 21.34 21.04 25.51 38.14PEP 10.70 10.30 4.37 7.34 7.17 7.33 9.53 9.41 11.54 8.47HD 21.27 20.48 8.68 15.03 14.68 15.00 19.38 19.14 23.36 13.94
COP 9.48 9.13 3.51 6.63 6.47 5.90 8.56 8.46 9.28 6.69SBC 17.00 16.37 6.28 13.56 13.28 12.23 17.03 16.85 18.28 0.85
TWX 44.13 42.50 18.82 31.64 30.93 33.17 40.67 40.19 51.30 25.87UPS 17.54 16.89 6.36 13.05 12.77 11.46 16.64 16.45 17.59 7.12ABT 10.51 10.12 4.56 7.03 6.86 7.54 9.18 9.06 11.93 9.54
AMGN 11.57 11.14 4.40 6.42 6.23 5.77 8.78 8.66 10.00 19.30MRK 16.74 16.13 7.27 12.38 12.11 13.22 15.81 15.62 20.22 7.30
ORCL 18.72 18.03 8.85 12.71 12.40 15.05 16.54 16.33 23.57 15.72HPQ 25.94 24.98 8.71 20.27 19.84 16.54 25.57 25.29 24.93 4.10
CMCSA 19.82 19.09 7.09 13.90 13.58 11.90 17.96 17.74 18.73 13.69UNH 10.49 10.11 3.50 5.78 5.61 4.22 7.93 7.82 7.59 17.74AXP 24.65 23.74 9.52 17.64 17.24 16.54 22.68 22.41 25.72 14.66LLY 11.68 11.25 5.48 7.30 7.11 8.69 9.69 9.56 13.98 13.99
MDT 15.09 14.54 6.90 9.59 9.35 11.03 12.68 12.52 17.68 17.04WYE 21.58 20.78 7.98 16.10 15.75 14.44 20.51 20.27 22.13 8.49TYC 28.17 27.13 10.63 20.29 19.83 18.54 26.05 25.74 28.78 15.90
MWD 35.17 33.87 14.18 25.78 25.21 25.39 32.98 32.59 39.05 16.82FNM 12.14 11.69 5.43 7.86 7.66 8.77 10.34 10.21 14.00 12.74
MMM 8.22 7.92 3.68 4.74 4.60 5.37 6.42 6.33 8.92 12.54QCOM 24.31 23.41 10.80 15.77 15.38 17.46 20.75 20.48 27.87 25.31
BA 16.61 16.00 6.75 12.95 12.69 12.86 16.35 16.17 19.36 2.77UTX 19.05 18.34 6.83 12.84 12.53 10.95 16.73 16.52 17.53 16.62MER 32.03 30.84 12.68 23.41 22.89 22.60 29.96 29.61 34.82 15.82
VIA.B 19.41 18.70 7.73 14.75 14.43 14.34 18.72 18.51 21.79 5.86DIS 18.02 17.35 6.70 14.16 13.87 12.86 17.85 17.65 19.32 2.26
G 17.52 16.87 6.86 13.17 12.89 12.58 16.76 16.56 19.19 6.17BMY 9.68 9.33 4.12 6.97 6.81 7.28 8.95 8.84 11.25 5.53BLS 11.67 11.24 4.76 8.54 8.35 8.52 10.93 10.80 13.11 5.68
Table 5: Equilibrium returnsNote: The Table shows equilibrium returns for the different distributions and the different risk measures for January
28, 2004 computed with = 0.5 when risk is measured by dispersion and = 0.15 for the other cases. We observe thatequilibrium returns under the -stable hypothesis are in general smaller that the equilibrium returns under normal andt-student distributions.
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dispersion VaR CVaRnumbering ticker normal t-stud stable normal t-stud stable normal t-stud stable
1 XOM - - - 11.96% 11.59 3.01 13.99 13.89 7.252 GE 5.89 5.52 5.74 - - 4.38 - - -3 MSFT 5.52 5.37 3.19 3.57 3.61 13.08 3.30 3.31 14.084 C - - - - - - 3.39 2.94 11.515 WMT 4.35 4.30 3.30 0.20 0.59 - - - -6 PFE 0.07 - - - - 5.56 - - 3.007 JNJ 0.77 0.58 - - - - - - -8 BAC 2.81 2.63 - 0.78 0.90 3.426 - - -9 INTC 1.25 1.14 - 3.32 3.19 2.31 3.44 3.44 1.91
10 AIG - - - - - - - - -11 MO 2.02 2.02 1.66 - - - - - -12 PG 7.32 7.65 11.00 - - - - - -13 JPM - - - 12.04 11.57 2.50 16.76 16.69 -14 CSCO - - - - - - - - -15 IBM - - - 3.88 3.92 - 3.65 3.68 -16 CVX 12.43 13.03 20.89 - 0.21 - - - -17 WFC 3.26 3.08 - - - 3.8 - - -18 KO - - - 8.25 8.56 9.67 2.39 2.76 1.0819 VZ - - - 5.19 5.43 - 0.16 0.43 -20 DELL 1.67 1.77 2.04 - - - - - -21 PEP 1.65 1.60 - - - - - - -22 HD - - - - - 3.55 - - 0.17
23 COP 7.42 7.60 5.95 - - - - - -24 SBC - - - 7.76 7.74 2.60 8.09 8.10 -25 TWX - - - 2.43 2.12 5.41 5.75 5.58 12.0526 UPS - - - 2.90 3.01 0.02 - - -27 ABT 0.61 0.57 0.79 - 0.02 2.99 - - -28 AMGN 4.22 4.32 4.53 - - - - - -29 MRK - - - 5.43 5.35 15.02 5.60 5.63 17.7330 ORCL 4.24 4.28 5.29 - - 3.19 - - 1.5931 HPQ 0.25 0.16 - 4.30 4.33 - 5.02 4.98 -32 CMCSA 0.21 0.09 - - - - - - -33 UNH - - - - - - - - -34 AXP - - - - - - - - -35 LLY 5.22 5.25 6.03 - - - - - -36 MDT 3.11 3.15 2.74 - - 0.18 - - -37 WYE - - - 6.45 6.35 - 7.47 7.47 -38 TYC - - - - - - - - 0 .2239 MWD - - - 4.88 4.69 11.29 6.93 6.884 19.2140 FNM 4.80 4.73 4.37 - - - - - -41 MMM 5.43 5.69 9.67 - - - - - -42 QCOM 0.72 0.64 - - - - - - -43 BA - - - 6.22 6.15 9.22 7.29 7.26 10.1844 UTX - - - - - - - - -45 MER 0.17 - - - - - - - -46 VIA.B 1.07 1.00 - 1.99 2.15 0.81 - - -47 DIS 3.28 3.32 0.38 4.28 4.38 - 1.94 2.13 -48 G - - - 4.17 4.14 - 4.83 4.84 -49 BMY 4.11 4.13 5.64 - - 1.41 - - -50 BLS 6.13 6.36 6.78 - - - - - -
Table 6: Optimal composition with neutral viewNote: We observe that the optimal composition under the -stable hypothesis are in general different from optimal
composition under normal and t-student distributions. The last two are very similar. Moreover, optimal portfolio is lessdiversified involving 17 stocks against the 29 stocks under normal and t-student distributions. In general under normaland t-student distributions we get similar equilibrium returns and portfolio composition under the three different riskmeasures.
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dispersion VaR CVaR unc. meanDate normal t-stud stable normal t-stud stable normal t-stud stable
01/28/04 9.08 9.04 9.18 8.90 8.91 9.03 8.99 8.98 9.37 9.1802/25/04 22.89 22.77 21.74 22.24 22.21 22.07 22.71 22.68 23.01 21.8103/24/04 47.70 47.09 38.47 43.12 42.87 42.89 46.35 46.18 49.26 41.7204/21/04 16.34 16.36 17.26 16.81 16.84 16.64 16.57 16.58 16.19 15.8705/19/04 22.58 22.21 17.26 19.81 19.67 19.60 21.70 21.60 23.29 19.6306/16/04 10.60 10.78 14.26 12.19 12.29 12.30 11.05 11.10 10.23 12.7507/14/04 27.42 26.98 20.90 24.27 24.10 23.97 26.55 26.42 28.54 22.6908/11/04 29.53 29.15 24.04 26.63 26.48 26.62 28.60 28.49 30.69 26.5909/08/04 17.30 17.44 20.40 18.74 18.83 18.79 17.79 17.84 17.04 18.2310/06/04 44.86 44.79 44.51 44.49 44.48 44.61 44.74 44.72 45.32 44.4711/03/04 15.68 15.55 14.38 14.77 14.73 14.88 15.33 15.30 16.29 15.3212/01/04 14.33 14.42 16.59 15.31 15.37 15.54 14.69 14.72 14.49 14.7312/29/04 9.31 9.37 11.13 10.01 10.05 10.11 9.51 9.53 9.24 10.1501/05/05 25.65 25.23 19.38 22.54 22.37 22.39 24.72 24.60 27.27 21.7602/23/05 22.58 22.41 20.54 21.36 21.31 21.35 22.15 22.10 23.32 21.7103/23/05 19.71 19.42 15.55 17.58 17.47 17.39 19.05 18.97 20.85 17.2304/20/05 35.37 34.96 29.25 32.27 32.11 31.85 34.38 34.26 36.63 32.2605/18/05 15.82 16.03 20.07 17.71 17.81 18.01 16.42 16.49 15.31 17.5806/15/05 12.24 12.13 11.22 11.53 11.50 11.61 12.00 11.97 12.92 11.5907/13/05 11.21 11.11 10.19 10.59 10.57 10.54 11.04 11.01 11.74 10.23
Table 7: Squared errors among the optimal composition and the unconditional meanover the rolling window horizon
Note:The hypothesis of stable distribution and the use of dispersion as risk measure gives the best combination.For 13 of the 20 considered months, it is the combination that gives the b est forecast(65% success rate). The second bestcombination is the -stable distribution and the use of C V a R as risk measure. For this combination 3 of the 20 monthsgives the best forecast (15% success rate). The third best combination is the unconditional mean resulted in the bestforecast in 2 of the 20 months(10% success rate).
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dispersion VaR CVaRnumbering ticker normal t-stud stable normal t-stud stable normal t-stud stable
1 XOM - - - 11.85% 11.60 3.44 13.39 13.35 7.432 GE 10.41 10.34 7.70 0.25 0.29 10.11 - - -3 MSFT 5.38 4.70 3.10 3.54 3.65 11.57 2.66 2.75 13.914 C - - - - - - 2.02 1.56 9.345 WMT - 0.23 1.63 - - - - - -6 PFE 0.12 0.21 - - - 6.40 - - 4.467 JNJ 1.12 1.86 - - - - - - -8 BAC 2.64 2.07 - - - - - - -9 INTC 1.28 1.32 - 2.26 2.23 2.91 2.22 2.23 1.93
10 AIG - - - - - - - - -11 MO 2.08 1.98 1.76 - - - - - -12 PG 7.34 6.92 10.89 - - - - - -13 JPM - - - 11.21 10.93 1.92 15.71 15.49 -14 CSCO - - - - - - - - -15 IBM - - - 6.00 5.74 - 7.09 7.07 -16 CVX 12.61 12.55 2 1.05 1.37 1.95 - - - -17 WFC 2.88 1.96 - - - 3.8 - - -18 KO - 0.42 - 9.15 9.31 11.51 4.23 4.58 3.5019 VZ - 0.72 - 3.31 3.61 - - - -20 DELL 1.72 1.96 2.06 - - - - - -21 PEP 1.82 2.20 - - - - - - -22 HD - - - - - - - - -
23 COP 7.26 7.20 5.76 - 0.11 - - - -24 SBC - - - 10.24 10.12 2.40 10.58 10.66 -25 TWX - - - 1.95 1.67 5.05 4.95 4.79 10.8426 UPS - 0.78 - 3.92 3.96 1.91 - 0.17 -27 ABT 0.22 0.47 0.81 - - 1.28 - - -28 AMGN 4.07 4.46 4.46 - - - - - -29 MRK - - - 7.89 7.83 14.81 8.31 8.34 17.8230 ORCL 4.22 4.13 5.39 - - 3.09 - - 0.4231 HPQ 0.32 0.24 - 4.52 4.55 - 4.37 4.37 -32 CMCSA 0.15 0.40 - - - - - - -33 UNH - - - - - - - - -34 AXP - - - - - - - - -35 LLY 5.06 4.61 6.24 - - - - - -36 MDT 3.10 2.88 2.65 - - 0.12 - - -37 WYE - - - 4.18 4.11 - 5.02 4.99 -38 TYC - - - - - - - - -39 MWD - - - 4.03 3.81 12.12 6.49 6.47 20.3040 FNM 5.14 4.77 4.35 - - 0.42 - - -41 MMM 5.59 5.86 9.41 - - - - - -42 QCOM 0.65 0.78 - - - - - - -43 BA - - - 4.71 4.64 9.75 5.30 5.31 10.0744 UTX - - - - - - - - -45 MER 0.31 0.20 - - - - - - -46 VIA.B 1.02 0.95 - 0.51 0.67 0.16 - - -47 DIS 3.35 3.34 0.39 6.73 6.80 1.01 4.91 5.07 -48 G - - - 2.38 2.42 - 2.74 2.80 -49 BMY 4.03 4.10 5.79 - - - - - -50 BLS 6.10 5.41 6.57 - - - - - -
Table 8: Optimal composition with a view of asset 2 overperforming asset 5of 0.1% with certainty
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dispersion VaR CVaRnumbering ticker normal t-stud stable normal t-stud stable normal t-stud stable
1 XOM - - - 11.48% 11.25 3.20 13.41 13.40 7.412 GE 9.18 8.99 6.63 - - 7.09 - - -3 MSFT 5.39 5.22 3.15 3.54 3.60 12.49 2.92 2.98 14.044 C - - - - - - 2.60 2.14 10.585 WMT 1.25 1.32 2.54 - - - - - -6 PFE 0.07 - - - - 5.97 - - 3.757 JNJ 1.10 0.97 - - - - - - -8 BAC 2.68 2.61 - - - - - - -9 INTC 1.31 1.24 - 2.63 2.60 2.58 2.63 2.66 1.97
10 AIG - - - - - - - - -11 MO 2.05 2.60 1.70 - - - - - -12 PG 7.28 7.46 10.95 - - - - - -13 JPM - - - 11.20 10.93 2.25 15.85 15.74 -14 CSCO - - - - - - - - -15 IBM - - - 5.47 5.28 - 5.99 5.96 -16 CVX 12.49 12.88 2 0.97 0.99 1.62 - - - -17 WFC 2.98 2.82 - - - 1.61 - - -18 KO - - - 9.07 9.31 10.76 3.64 3.99 2.2919 VZ - - - 4.24 4.49 - - - -20 DELL 1.76 1.83 2.05 - - - - - -21 PEP 1.85 1.82 - - - - - - -22 HD - - - - - 1.59 - - -
23 COP 7.22 7.32 5.86 - 0.07 - - - -24 SBC - - - 8.98 8.86 2.53 9.77 9.82 -25 TWX - - - 2.06 1.83 5.22 5.20 5.05 11.5126 UPS - - - 3.79 3.87 1.05 - - -27 ABT 0.26 0.23 0.80 - - 2.41 - - -28 AMGN 4.07 4.17 4.50 - - - - - -29 MRK - - - 6.89 6.81 14.89 7.40 7.44 17.6530 ORCL 4.24 4.31 5.33 - - 3.18 - - 1.0831 HPQ 0.24 0.16 - 4.45 4.49 - 4.62 4.61 -32 CMCSA 0.19 0.13 - - - - - - -33 UNH - - - - - - - - -34 AXP - - - - - - - - -35 LLY 5.14 5.22 6.13 - - - - - -36 MDT 3.06 3.05 2.70 - - 0.09 - - -37 WYE - - - 5.13 5.05 - 5.89 5.90 -38 TYC - - - - - - - - -39 MWD - - - 4.69 4.43 11.76 6.60 6.59 19.5840 FNM 5.10 5.03 4.37 - - - - - -41 MMM 5.56 5.79 9.55 - - - - - -42 QCOM 0.68 0.61 - - - - - - -43 BA - - - 5.36 5.27 9.75 6.03 6.07 10.1444 UTX - - - - - - - - -45 MER 0.29 - - - - - - - -46 VIA.B 1.06 1.04 - 0.92 1.10 0.60 - - -47 DIS 3.30 3.30 0.38 5.78 5.82 0.44 3.90 4.05 -48 G - - - 3.32 3.30 - 3.52 3.59 -49 BMY 4.13 4.23 5.71 - - 0.64 - - -50 BLS 6.05 6.17 6.68 - - - - - -
Table 9: Optimal composition with a view of asset 2 overperforming asset 5of 0.1% with uncertainty of 0.61
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dispersion VaR CVaR uncond. meanticker normal t-stud stable normal t-stud stable normal t-stud stable
XOM 22.32 21.49 9.13 16.15 15.79 16.15 20.72 20.47 24.94 12.48GE 10.25 9.87 4.50 6.66 6.49 7.22 8.75 8.64 11.52 8.85
MSFT 24.45 23.55 11.22 17.35 16.96 19.81 22.36 22.09 30.64 16.56C 29.40 28.31 11.85 20.68 20.20 20.35 26.70 26.37 31.78 20.82
WMT 10.15 9.77 4.40 6.56 6.39 7.12 8.65 8.54 11.42 14.98PFE 16.80 16.18 7.66 11.07 10.80 12.59 14.51 14.33 19.96 16.38JNJ 10.60 10.21 4.49 6.80 6.63 7.08 8.97 8.85 11.40 11.73
BAC 21.29 20.51 7.99 15.60 15.25 14.13 19.96 19.72 21.84 11.34INTC 27.04 26.04 10.76 19.86 19.42 19.23 25.39 25.10 26.92 14.45AIG 19.81 19.08 7.84 13.87 13.55 13.38 17.93 17.71 20.94 14.50MO 8.44 8.13 2.90 5.28 5.15 4.13 7.00 6.91 6.90 8.47PG 6.22 5.99 3.13 3.23 3.13 4.41 4.50 4.43 7.42 11.56
JPM 31.40 30.24 11.77 24.27 23.76 22.17 30.69 30.35 33.53 8.19CSCO 33.75 32.50 12.96 24.12 23.57 22.38 31.03 30.66 34.88 22.05
IBM 25.13 24.21 9.34 18.14 17.74 16.26 23.28 23.01 25.25 13.35CVX 7.98 7.68 3.00 5.38 5.25 4.84 7.01 6.92 7.72 6.18WFC 14.12 13.60 6.02 9.18 8.95 9.70 12.07 11.92 15.51 15.48
KO 10.96 10.56 4.84 8.04 7.86 8.67 10.28 10.16 13.32 4.61VZ 13.02 12.54 4.73 10.19 9.98 9.05 12.85 12.71 13.60 2.01
DELL 23.94 23.06 9.75 14.07 13.68 14.00 18.98 18.72 23.44 38.14PEP 10.69 10.29 4.37 7.33 7.16 7.34 9.52 9.40 11.55 8.47HD 18.61 17.92 7.67 13.37 13.07 13.61 17.19 16.98 21.06 13.94
COP 10.14 9.77 3.74 7.04 6.88 6.22 9.11 9.00 9.80 6.69SBC 17.26 16.63 6.25 13.72 13.44 12.19 17.25 17.06 18.21 0.85
TWX 42.35 40.78 18.08 30.53 29.84 32.15 39.20 38.74 49.61 25.87UPS 17.43 16.79 6.38 12.98 12.70 11.48 16.55 16.36 17.63 7.12ABT 10.40 10.01 4.52 6.96 6.79 7.48 9.08 8.97 11.84 9.54
AMGN 10.57 10.18 4.02 5.79 5.62 5.24 7.96 7.84 9.13 19.30MRK 16.91 16.28 7.32 12.48 12.21 13.28 15.94 15.76 20.33 7.30
ORCL 16.54 15.93 8.23 11.35 11.08 14.21 14.74 14.56 22.17 15.72HPQ 25.63 24.68 8.63 20.07 19.65 16.42 25.31 25.03 24.74 4.10
CMCSA 19.26 18.55 6.80 13.55 13.24 11.51 17.49 17.28 18.08 13.69UNH 10.99 10.58 3.75 6.09 5.92 4.57 8.34 8.22 8.17 17.74AXP 24.18 23.29 9.36 17.35 16.96 16.32 22.30 22.03 25.34 14.66LLY 11.02 10.61 5.42 6.89 6.71 8.60 9.14 9.02 13.82 13.99
MDT 15.11 14.55 6.95 9.60 9.36 11.09 12.69 12.53 17.79 17.04WYE 20.72 19.96 7.81 15.56 15.23 14.20 19.80 19.58 21.74 8.49TYC 27.07 26.07 10.30 19.60 19.16 18.08 25.14 24.85 28.02 15.90
MWD 33.98 32.73 13.87 25.05 24.49 24.96 32.00 31.63 38.34 16.82FNM 12.26 11.81 5.47 7.94 7.74 8.83 10.44 10.31 14.10 12.74
MMM 8.65 8.33 3.77 5.00 4.86 5.49 6.77 6.67 9.12 12.54QCOM 22.19 21.37 9.99 14.45 14.09 16.35 19.00 18.75 26.03 25.31
BA 16.31 15.71 6.68 12.77 12.51 12.77 16.11 15.93 19.21 2.77UTX 18.99 18.29 6.78 12.80 12.50 10.89 16.69 16.48 17.42 16.62MER 31.33 30.17 12.44 22.97 22.46 22.26 29.38 29.04 34.26 15.82
VIA.B 18.39 17.71 7.42 14.11 13.81 13.92 17.87 17.67 21.09 5.86DIS 18.44 17.76 6.87 14.43 14.13 13.08 18.20 18.00 19.68 2.26
G 17.32 16.68 6.80 13.05 12.77 12.49 16.59 16.40 19.04 6.17BMY 9.07 8.73 3.95 6.58 6.44 7.04 8.44 8.34 10.86 5.53BLS 11.91 11.47 4.80 8.69 8.50 8.57 11.12 10.99 13.20 5.68
Table 10: Equilibrium returns with views under certaintyNote: The Table shows equilibrium returns for the different distributions and the different risk measures for January
28, 2004, computed with = 0.5 when risk is measured by dispersion and = 0.15 for the other cases with a view ofasset 2 overperforming asset 5 of 0.1% with certainty.
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dispersion VaR CVaR uncond. meanticker normal t-stud stable normal t-stud stable normal t-stud stable
XOM 22.59 21.76 9.21 16.32 15.95 16.25 20.94 20.69 25.12 22.59GE 9.01 8.68 4.01 5.89 5.74 6.55 7.73 7.63 10.42 9.01
MSFT 25.17 24.25 11.54 17.80 17.40 20.24 22.95 22.68 31.36 25.17C 30.06 28.95 12.07 21.09 20.60 20.64 27.24 26.91 32.27 30.06
WMT 13.15 12.68 5.52 8.43 8.23 8.67 11.12 10.99 13.99 13.15PFE 16.83 16.21 7.60 11.09 10.82 12.50 14.53 14.35 19.82 16.83JNJ 10.74 10.34 4.49 6.89 6.72 7.08 9.09 8.97 11.40 10.74
BAC 22.05 21.24 8.16 16.07 15.71 14.37 20.58 20.34 22.23 22.05INTC 28.25 27.21 11.09 20.61 20.16 19.67 26.39 26.08 30.36 28.25AIG 20.29 19.54 8.02 14.17 13.84 13.63 18.32 18.10 21.36 20.29MO 7.30 7.03 2.42 4.57 4.45 3.47 6.07 5.98 5.80 7.30PG 6.04 5.82 3.10 3.12 3.03 4.38 4.36 4.29 7.37 8.04
JPM 32.44 31.24 12.13 24.91 24.39 22.66 31.55 31.20 34.35 32.44CSCO 35.00 33.71 13.24 24.90 24.34 22.76 32.06 31.68 35.51 35.00
IBM 24.74 23.82 9.15 17.90 17.50 16.00 22.96 22.69 24.82 24.74CVX 7.45 7.17 2.77 5.05 4.93 4.52 6.58 6.49 7.20 7.45WFC 14.76 14.21 6.29 9.58 9.34 10.07 12.59 12.44 16.13 14.76
KO 10.53 10.14 4.59 7.77 7.60 8.32 9.93 9.81 12.74 10.53VZ 13.06 12.58 4.80 10.21 10.00 9.15 12.88 12.74 13.77 13.06
DELL 26.39 25.43 10.59 15.60 15.18 15.17 20.99 20.72 25.37 26.39PEP 10.70 10.30 4.37 7.34 7.17 7.33 9.53 9.41 11.54 10.70HD 20.88 20.12 8.62 14.79 14.46 14.91 19.06 18.84 23.21 20.88
COP 9.58 9.22 3.52 6.69 6.53 5.92 8.64 8.54 9.31 9.58SBC 17.04 16.41 6.28 13.58 13.30 12.23 17.06 16.88 18.28 17.04
TWX 43.87 42.26 18.77 31.48 30.78 33.10 40.45 39.98 51.19 43.87UPS 17.53 16.88 6.37 13.04 12.76 11.46 16.63 16.43 17.59 17.53ABT 10.49 10.11 4.56 7.02 6.85 7.54 9.17 9.05 11.93 10.49
AMGN 11.43 11.01 4.38 6.33 6.15 5.73 8.66 8.54 9.95 11.43MRK 16.77 16.15 7.27 12.40 12.12 13.22 15.83 15.64 20.23 16.77
ORCL 18.40 17.73 8.81 12.51 12.22 15.00 16.27 16.08 23.48 18.40HPQ 25.89 24.94 8.71 20.24 19.82 16.53 25.53 25.25 24.92 25.89
CMCSA 19.74 19.02 7.07 13.85 13.53 11.88 17.89 17.68 18.69 19.74UNH 10.57 10.17 3.51 5.83 5.66 4.24 7.99 7.87 7.63 10.57AXP 24.58 23.67 9.51 17.60 17.20 16.53 22.62 22.36 25.69 24.58LLY 11.58 11.16 5.48 7.24 7.06 8.69 9.61 9.49 13.97 11.58
MDT 15.10 14.54 6.90 9.59 9.35 11.03 12.66 12.52 17.68 15.10WYE 21.45 20.67 7.97 16.02 15.67 14.43 20.41 20.18 22.11 21.45TYC 28.01 26.98 10.61 20.19 19.74 18.51 25.92 25.62 28.73 28.01
MWD 35.00 33.71 14.16 25.68 25.11 25.36 32.83 32.46 39.01 35.00FNM 12.16 11.71 5.43 7.87 7.67 8.77 10.36 10.22 14.00 12.16
MMM 8.28 7.97 3.69 4.77 4.64 5.38 6.47 6.38 8.93 8.28QCOM 24.00 23.13 10.75 15.58 15.20 17.39 20.49 20.24 27.75 24.00
BA 16.57 15.96 6.74 12.93 12.66 12.86 16.32 16.14 19.35 16.57UTX 19.04 18.34 6.83 12.83 12.52 10.95 16.73 16.52 17.52 19.04MER 31.93 30.75 12.67 23.34 22.83 22.57 29.87 29.53 34.78 31.93
VIA.B 19.26 18.56 7.71 14.65 14.35 14.31 18.59 18.39 21.74 19.26DIS 18.08 17.41 6.71 14.20 13.91 12.87 17.90 17.70 19.34 18.08
G 17.49 16.84 6.86 13.16 12.87 12.58 16.73 16.54 19.18 17.49BMY 9.60 9.24 4.11 6.91 6.76 7.27 8.87 8.77 11.23 9.60BLS 11.70 11.27 4.77 8.56 8.37 8.53 10.95 10.83 13.12 11.70
Table 11: Equilibrium returns with views under uncertaintyNote: The Table shows equilibrium returns for the different distributions and the different risk measures for January
28, 2004, computed with = 0.5 when risk is measured by dispersion and = 0.15 for the other cases with a view ofasset 2 overperforming asset 5 of 0.1% with uncertainty of 0 .61 .
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