stable distributions in the black-litterman approach to the asset allocation

8/22/2019 Stable distributions in the Black-Litterman approach to the asset allocation

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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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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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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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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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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 .

38

stable distributions in the black-litterman approach to the asset allocation

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