COVARIANCE
STABILITY AND THE 2008 FINANCIAL CRISIS: THE IMPACT IN THE PORTFOLIO OF THE 10
BIGGEST COMPANIES IN BM&FBOVESPA BETWEEN 2004 E 2012
Leticia Naomi Ono Maeda
UNICAMP  Universidade Estadual de Campinas, Brazil
Email: leticiamaeda@gmail.com
Johan Hendrik Poker
UNICAMP  Universidade Estadual de Campinas, Brazil
Email: johan.poker@fca.unicamp.br
Submission: 15/10/2015
Revision: 08/11/2015
Accept: 20/11/2015
ABSTRACT
This study's purpose is to
analyze the influence of the covariance fluctuation between assets over the
structure of a portfolio of investments. To accomplish that, the covariances between the daily returns of the 10 biggest
participants of the BM&FBovespa stock market are
analyzed, before, during and after the 2008 financial crisis. The procedure of
this research includes: (1) collection of returns of the selected stocks
between 2004 and 2012; (2) composition of the classical portfolio proposed by
Markowitz’s theory (1952); and (3) the measurement of the covariances
instability effect between the 10 selected assets over the maintenance of a
portfolio’s risk and return, according to the preferences of a hypothetical
investor. We discover that the asset’s covariance vary over time and affect the
correlations among the assets, especially in financial crisis periods.
Consequently, both risk and return of the portfolio may change greatly if the
asset’s weights are not recalculated periodically. This supports the idea that
portfolio theory might benefit from the development of stability weighted
techniques.
Keywords: portfolio theory; asset risk
management; financial crisis.
1. INTRODUCTION
The ability to predict the future value of
assets in the financial market was always desirable, and there are currently
many ways to choose assets which compose a given investment portfolio,
evaluating the assets characteristics such as their expected return, risk,
investment period, liquidity, among others. One of the possible financial
instruments that analyze the relationship between two of these characteristics
– precisely, risk and return –to elect the best investment option is the
Markowitz model (1952).
However, it is crucial to clarify that the
covariance stability between the companies is assumed over time, so that the
chosen investment portfolio according to the Markowitz model (1952) is
maintained during the investment period. Thus, if the covariances
are unstable, possible commitments related to the expected portfolios results
may occur.
Since covariances
are dynamic and dependent on economy variations in general and, specifically,
on the financial market, this study is justified by the need to assess to what
extent and in what kind of scenario it would be unwise to use the Markowitz
model (1952) – especially in economic instability situations, as in the recent
2008 crisis – with no use of improvements, in order not to put at risk results
expected from a portfolio.
2. THEORETICAL FOUNDATION
2.1.
Markowitz model (1952)
According
to the Markowitz model (1952), an investor tries to predict the future outcome
of assets basically through the analysis of expected return and risk of the asset.
The latter, in turn, is considered, according to Luenberg
(1997), as random variables, since the asset can take different future values,
each with a given probability of occurrence, considering that the future asset
value is not known upon purchase.
Thus, mathematically speaking, the expected
return is basically the sum of the possible asset returns weighted by their probabilities of occurence, whereas the
risk is in the variance – or on the square of the variance (standard deviation)
which is most routinely used – of the aforesaid return, that is, the
calculation of how far a value is from its expected value. Both described in
the following formulas,


(1) 
Where

_{} = expected value
of asset X;

_{} = value of asset X in time i;

_{} = probability of the value of asset X in time i.


(2) 
Where

_{} = standard deviation of the asset X;

_{} = value of asset X in the time i;

_{} = expected value
of the asset X;

_{} = probability of the value of asset X in the
time i.
Usually, however, one does not invest in a
single asset but in a set of assets, named assets portfolio or investment
portfolio. The preference for an investment portfolio to only one asset occurs
due to the need to diminish the risk of an investment. According to Bodie et al.,
(2010), the risk may be classified as nondiversifiable risk and diversifiable
risk. The first is the risk inherent in the market as a whole, whereas the
second is closely related to one or more specific parts of the market and,
therefore, may be minimized by diversifying assets, that is, an investment of a
specific amount in different assets of the financial market. Markowitz (1952,
p. 89) describes this phenomenon as follows:
"In
an attempt to reduce variance, investing in various assets is not enough. One
must avoid that the investment is made in assets with high covariance between
them. We must diversify investment among industries, particularly industries
with different economic characteristics, since companies from different
industries have lower covariance than companies in the same industry."
In this respect, composing an assets portfolio
decreases diversifiable risk significantly, increasing the probability of an
asset to obtain a certain expected value, or in other words, reducing risk.
However, we still need to understand how we should select some on them among
the various assets available in the market, which can form what Markowitz
(1952) named as efficient portfolio investment. A portfolio is effective for a
given return, there is no other portfolio with less risk, or, similarly, for a
given risk, there is no other portfolio with a higher return. This concept can also be interpreted by
the Sharpe Dominance Principle (1965):
"An investor should choose their optimal
portfolio from the set of portfolios that:
1. Offers maximum expected return for different
levels of risk, and
2. Offers minimal risk for different levels of
expected return."
Thus, in order to calculate the expected return
and the risk of a portfolio, it is assumed that an investor distributes an
amount between n assets, each with a weight in the portfolio, whereas and where is the amount invested in the i ^{th}
asset, it follows that the total return of the portfolio is given by:


(3) 
Where:

_{} = portfolio total expected return;

_{} = weight of the asset i;

_{} = total expected return of the
asset i.
In order to calculate a portfolio
variance, the covariance and correlation concept is necessary. Both the
covariance and the correlation can be clarified as the interdependence of two
random variables. With respect to correlation, it follows that:

If
= 0, then X and Y are uncorrelated;

If
= 1, then X and Y are perfectly correlated;

If
= 1, then X and Y are negatively correlated.
Furthermore, the covariance between
two X and Y assets can be mathematically defined by:


(4) 
By knowing the covariance value
between two variables it is possible to calculate the standard deviation (risk)
of a portfolio of two assets, which is given by:


(5) 
Where:

_{} = portfolio standard deviation;

_{} = weight of the asset X;

_{} = weight of the asset Y;

_{} = standard deviation of asset X;

_{}= standard deviation of asset Y

_{} = covariance between assets X and Y.
However, if we wish to know the variance of a
portfolio with more than two assets, we just need to use, according to Luenberger (1997), the formula:


(6) 
Where:

_{} = portfolio total variance;

_{} = weight of the asset i;

_{} = weight of the asset j;

_{} = covariance between asset i
and j.
Thus, we can reject that the variance of the
portfolio is calculated from the covariance between pairs of assets. Recalling
that .
Using these return and portfolio risk concepts
we can relate them to a chart whose abscissa corresponds to the risk and the
orderly, to the expected return. The points of the chart correspond to an
investment portfolio involving certain assets. The points corresponding to
these investment portfolios form returnrisk curves. In this curve are
presented investment portfolios composed of the same assets, however with
different weights for each portfolio (point) of the
curve.
Figure
1: ReturnRisk Curves.
Source: Adapted from Hieda and Oda (1998)
After
choosing assets that will compose the portfolio, the corresponding RiskReturn
curve to depreciated assets is found. Thus,
we would obtain the following RiskReturn curve whose inner area is called
feasible region. Both at the curved line as well as at the feasible region are
all possible portfolios of the same assets, however with different weights.
Figure
2: Feasible region
Source: Adapted from Luenberger
(1997)
However
the only part of the RiskReturn curve that follows the Dominance principle, cited
above, corresponds to the curved line that goes from point "E" of
minimal risk to point "S" of maximum return.
Figure 3: Efficient frontier
Source: Adapted from Hieda and Oda
(1998)
The
curve is called efficient frontier. Such boundary
defines all the possible efficient portfolio investments, that is, those that
for a given level of return have the minimum possible risk.
Finally,
it is necessary to point out the relationship that the assets weights have with
their correlation index. Assuming a portfolio composed of two X and Y assets,
we form several X and Y combinations, each with a different correlation index
among the same and different weights. Thus, short selling is not possible (weight of X + weight of Y = 1).
Chart 1:
Correlation index
Source: Adapted from Bodie, Marcus and Kane (2010)
In it there is the following information on the assets
correlation influence on the diversification effects:
 when the correlation between
assets is positively perfect (), there is no diversification
effect of assets;
 when
the correlation between assets is imperfect ,
there are imperfect effects of asset diversification;
 when the correlation between
assets is negatively perfect (), there is a perfect effect
of assets diversification shown by the scope of a risk equal to zero.
As
the correlation between the two assets changes, the assets weights of the
portfolio must also be changed in order to maintain a certain level of risk.
For example, if it were necessary to maintain a minimum standard deviation, the
asset X should correspond to approximately 25%, 37.5% and 43.75% of the total
portfolio, if the correlations were, respectively, 0, 0.30 and 1 – assets of
correlation perfect in this case would not reach the aforementioned level of
risk.
2.2.
Investor preferences
Although
the efficient frontier points the best investment combination alternatives,
there is nothing on which combination or which portfolio should be selected,
since this decision is up to each investor according to their personal
characteristics.
According to Danthine (2005),
such preferences may take into account several variables: the wealth degree of
the investor, uncertainty in the investment time, among others. However, a good
instrument for assessing the preference of an investor regarding the choice
between risk assets is the indifference curve.
Figure 4: Risk aversion
Source: Adapted from Bodie,
Marcus and Kane (2010)
The indifference curve measures the risk aversion degree
of an investor, that is, the amount of additional return they need to accept
one more risk unit. In the above, we observe three indifference curves. The
steeper the curve, the greater the risk aversion degree is. Thus, the curves
"A", "B" and "C" correspond to investors of, respectively,
greater risk aversion, moderate risk aversion and lower risk aversion.
Similarly, risk aversion
can also be calculated by the Sharpe index:


(7) 
Where

_{} = return of asset X;

_{} = standard deviation of asset X.
It measures how much more return is given for each
additional unit of risk.
3. METHODOLOGY
3.1.
General data
The methodology involves the quantitative model analysis,
exploring the financial and statistical data from the top 10 companies of BM
&, that is, companies with relatively large amounts of shares traded. This
information was drawn from the company's Thomson Reuters Eikon
database, a leader in collection and distribution of information on the
business market.
To carry out this work we considered the following
periods as precrisis, crisis and postcrisis scenarios.
Table 1: Analyzed periods
Period 
Scenario 
January/2004 to June/2007 
Precrisis 
July/2007 to June/2009 
Crisis 
July/2009 to December/2012 
Postcrisis 
Source:
The author
The
ten companies chosen for this study with their codes of their actions are
listed in Table 2.
Table 2 – Analyzed companies
Action
Code 
Company 
BBAS3.SA 
Banco
do Brasil 
BBDC4.SA 
Banco
Bradesco 
CCRO3.SA 
Companhia
de Concessões Rodoviárias 
CMIG4.SA 
Companhia Energética
de Minas Gerais, 
CSNA3.SA 
Companhia
Siderúrgica Nacional 
EMBR3.SA 
Embraer 
GGB 
Gerdau 
ITUB.K 
Itaú Unibanco
Holding 
PETR4.SA 
Petrobrás 
VALE5.SA 
Vale 
Source:
The author
3.2.
Analysis of covariance
In
order to identify the covariance behavior over time we structured semiannual
covariance matrices between the companies’ share returns. Each matrix has the covariances of returns of the companies within a semester
over eight years (20042012). Furthermore, for matrices calculation we used the
"COVARIAÇÃO.S" tool from the Microsoft Excel program. The result of
this formula is the deviation average of products of each pair of points in two
datasets, in this case, two sets return of two different companies. Therefore,
the matrix is composed of covariances of all possible
pairwise combinations of the ten aforementioned companies.
3.2.1. Constructing
hypothetical portfolio
This
section of the study primarily aims to quantify the influence of the covariances instability in a theoretical portfolio, by
changing the assets weights over time.
First
of all, we identified, for each of the three periods studied (precrisis,
crisis and postcrisis) their returns, standard deviations, and covariance
matrices. From these variables, we built six hypothetical portfolios. Three
portfolios have restrictions such as the preference of a Sharpe index of a
hypothetical investor equal to 15% in the three periods. The three other
portfolios must maintain constant their weights, to quantify the Sharpe index
variation.
In
order to find the returns, standard deviations and covariance matrices, we used
formulas showed in the theoretical foundation of this article. Whereas the
construction of portfolios that meet a Sharpe index of 15% were made by the
SOLVER tool from Microsoft Excel, under the following restrictions:


4. ANALYSIS OF COVARIANCE
The covariance matrices of the ten companies in the study
are as follows.
Table 3: Covariance matrix of
1^{st} semester of 2004
Source:
The author
Table 4: Covariance matrix of
2^{nd} semester 2004
Source:
The author
Table 5: Covariance matrix of
1^{st} semester of 2005
Source:
The author
Table 6: Covariance matrix of 2^{nd} semester of 2005
Source:
The author
Table 7: Covariance matrix of 1^{st} semester of 2006
Source:
The author
Table 8: Covariance matrix of 2^{nd} semester of 2006
Source:
The author
Table 9: Covariance matrix of 1^{st} semester of 2007
Source:
The author
Table 10: Covariance matrix of 2^{nd} semester of
2007
Source:
The author
Table 11: Covariance matrix of 1^{st} semester of
2008
Source: The author
Table 12: Covariance matrix of 2^{nd} semester of
2008
Source:
The author
Table 13: Covariance matrix of 1^{st} semester of
2009
Source:
The author
Table 14: Covariance matrix of 2^{nd} semester of
2009
Source:
The author
Table 15: Covariance matrix of 1^{st} semester of
2010
Source:
The author
Table 16: Covariance matrix of 2^{nd} semester of
2010
Source:
The author
Table 17: Covariance matrix of 1^{st} semester of
2011
Source:
The author
Table 18: Covariance matrix of 2^{nd} semester of
2011
Source:
The author
Table 19: Covariance matrix of 1^{st} semester of
2012
Source:
The author
Table 20: Covariance matrix of 2^{nd} semester of
2012
Source:
The author
Based on
these matrices (Table 3 to 20), we can infer that the covariances are not stable over time, which
would put at risk the maintenance over time of portfolios of investment
according to the Markowitz model (1952). It is also important to note that
these variabilities further increase in periods of crisis, when a significant
increase in covariance is observed among the majority of shares in 2007 and,
especially, in 2008.
There is a reasonable peak
increase in the 2^{nd} semester of 2007, followed by a slight drop in
the 1^{st} semester of 2008. And later, a substantially higher peak –
approximately, 700% higher – in the 1^{st} semester of 2008, reaching
the maximum covariance of the nine years studied in this work. Thus, in general,
the tables present a growing instability in the 1^{st} semester of 2004
until the 2^{nd} semester of 2007, when the summit is reached in 2008. Consecutively, from the 1^{st}
semester of 2009 until the 2^{nd} semester of 2012, instabilities are
perceived and they still exist, although decreasing.
Another secondary result is the stable relation of CMIG4
and CCRO3 have when compared to the others. Using a simple measurement of
dispersion, namely the interval of variation, calculated by the difference
between the maximum and minimum in the observation dataset, we found that CMIG4
and CCRO4 had 6,29% and 6,38% of variation respectively. As opposed to the GGB
share with a very unstable variation of 45,38%.
In the following chart such a conclusion can be
more visually observed.
Chart 2: Behavior of
covariance between shares returns over time
Source: The author
Chart 3: CCRO2 to other assets covariance in
time
Source: The author
Chart 4: CMIG4
to other assets covariance in time
Source: The author
Chart 5: GGB to
other assets covariance in time
Source:
The author
5. CONSTRUCTING HYPOTHETICAL PORTFOLIO
After concluding that the
instability of the covariances between the shares not
only exist, but it is also significant, a more particular evaluation of these
oscillations is necessary, from the construction of six hypothetical portfolios
in relation to the periods of precrisis, crisis and postcrisis. These
portfolios seek to identify the influence of covariance instability in the
maintenance cost of the portfolios.
We formed two species of hypothetical
portfolios for each period aforementioned. Both types of portfolios do not
allow short selling, that is, _{}. However, one of them has
as a constraint, obtaining a Sharpe index equivalent to 15% for its formation. The other
portfolio genre should present constant asset weight over time, and it starts
with a distribution that generates a Sharpe index also equivalent to 15%. In
the following we present the data to obtain each portfolio – returns, standard
deviations and covariance matrices – as well as the construction of portfolios
with the respective weights of each asset.
5.1.
Hypothetical portfolio precrisis:
January/2004 to June/2007
Table
21: Data for individual assets in the precrisis period
Assets 
μ 
σ 
BBAS3 
253,20% 
365,21% 
BBDC4 
270,12% 
499,17% 
CCRO3 
416,51% 
143,23% 
CMIG4 
129,92% 
152,94% 
CSNA3 
150,99% 
179,99% 
EMBR3 
20,23% 
282,37% 
GGBR4 
410,43% 
248,11% 
ITUB.K 
352,20% 
339,76% 
PETR4 
160,81% 
422,94% 
VALE5 
187,09% 
538,25% 
Source:
The author
Table 22: Covariance between assets in the precrisis period

BBAS3 
BBDC4 
CCRO3 
CMIG4 
CSNA3 
EMBR3 
GGBR4 
ITUB.K 
PETR4 
VALE5 
BBAS3 
13.34 
17.44 
5.03 
5.26 
5.82 
8.82 
8.77 
12.02 
14.54 
18.95 
BBDC4 
17.44 
24.92 
6.73 
7.41 
7.29 
11.78 
11.63 
16.65 
20.62 
24.60 
CCRO3 
5.03 
6.73 
2.05 
2.03 
2.20 
3.36 
3.32 
4.67 
5.53 
7.39 
CMIG4 
5.26 
7.41 
2.03 
2.34 
2.18 
3.32 
3.47 
4.99 
6.21 
7.50 
CSNA3 
5.82 
7.29 
2.20 
2.18 
3.24 
4.14 
4.06 
5.18 
6.21 
8.73 
EMBR3 
8.82 
11.78 
3.36 
3.32 
4.14 
7.97 
6.03 
8.02 
9.94 
12.44 
GGBR4 
8.77 
11.63 
3.32 
3.47 
4.06 
6.03 
6.16 
8.12 
9.91 
12.35 
ITUB.K 
12.02 
16.65 
4.67 
4.99 
5.18 
8.02 
8.12 
11.54 
13.86 
17.15 
PETR4 
14.54 
20.62 
5.53 
6.21 
6.21 
9.94 
9.91 
13.86 
17.89 
20.21 
VALE5 
18.95 
24.60 
7.39 
7.50 
8.73 
12.44 
12.35 
17.15 
20.21 
28.97 
Source: The author
Table 23: Assets portfolios in the precrisis period

IS = 15% 
Constant weights 
Assets 
Wi 

BBAS3 
0,00% 
0,00% 
BBDC4 
0,00% 
0,00% 
CCRO3 
0,00% 
0,00% 
CMIG4 
0,00% 
0,00% 
CSNA3 
0,00% 
0,00% 
EMBR3 
83,83% 
83,83% 
GGBR4 
0,00% 
0,00% 
ITUB.K 
0,00% 
0,00% 
PETR4 
4,14% 
4,14% 
VALE5 
12,03% 
12,03% 
Σwi 
100,00% 
100,00% 
μp 
46,12% 
46,12% 
σp 
307,46% 
307,46% 
IS 
15,00% 
15,00% 
Source: The author
5.2.
Crisis hypothetical portfolio:
July/2007 to June/2009
Table 24: Data for individual
assets in the precrisis period
Assets 
μ 
σ 
BBAS3 
20,72% 
387,24% 
BBDC4 
25,78% 
311,41% 
CCRO3 
22,57% 
101,15% 
CMIG4 
5,70% 
74,18% 
CSNA3 
86,07% 
498,03% 
EMBR3 
56,67% 
432,06% 
GGBR4 
35,35% 
511,16% 
ITUB.K 
33,09% 
344,08% 
PETR4 
51,00% 
570,36% 
VALE5 
22,14% 
701,37% 
Source: The author
Table 25: Covariance between
assets in the precrisis period
Source: The author
Table 26: Assets portfolios in
the crisis period

IS = 15% 
Constant weights 
Assets 
Wi 

BBAS3 
0,00% 
0,00% 
BBDC4 
3,46% 
0,00% 
CCRO3 
20,09% 
0,00% 
CMIG4 
14,89% 
0,00% 
CSNA3 
28,16% 
0,00% 
EMBR3 
0,00% 
83,83% 
GGBR4 
9,22% 
0,00% 
ITUB.K 
13,46% 
0,00% 
PETR4 
10,72% 
4,14% 
VALE5 
0,00% 
12,03% 
Σwi 
100,00% 
100,00% 
μp 
43,69% 
42,73% 
σp 
291,27% 
429,97% 
IS 
15,00% 
9,94% 
Source: The author
5.3.
Hypothetical portfolio
postcrisis: July/2009 to December/2012
Table 27: Data for individual
assets in the postcrisis period
Assets 
μ 
σ 
BBAS3 
4,74% 
214,79% 
BBDC4 
24,48% 
225,07% 
CCRO3 
118,61% 
296,50% 
CMIG4 
47,94% 
358,54% 
CSNA3 
50,82% 
527,22% 
EMBR3 
56,65% 
174,90% 
GGBR4 
47,29% 
289,20% 
ITUB.K 
31,06% 
285,78% 
PETR4 
42,40% 
360,04% 
VALE5 
8,45% 
300,85% 
Source: The author
Table 28: Covariance between
assets in the postcrisis period
Source: The author
Table 29: Assets portfolios in
the postcrisis period

IS = 15% 
Constant weights 
Assets 
Wi 

BBAS3 
0.00% 
0.00% 
BBDC4 
0.00% 
0.00% 
CCRO3 
0.00% 
0.00% 
CMIG4 
0.00% 
0.00% 
CSNA3 
0.34% 
0.00% 
EMBR3 
0.00% 
83.83% 
GGBR4 
39.48% 
0.00% 
ITUB.K 
29.28% 
0.00% 
PETR4 
30.90% 
4.14% 
VALE5 
0.00% 
12.03% 
Σwi 
100.00% 
100.00% 
μp 
41.04% 
46.75% 
σp 
273.60% 
147.23% 
IS 
15.00% 
31.75% 
Source: The author
Where:

= return of a given asset in the corresponding
period;

= standard deviation of a given asset in the
corresponding period;

SI
= Sharpe index;

Σwi = total sum of assets weights;

= total expected return of the portfolio;

= total standard deviation of a portfolio.
The
preferences of return and risk of an investor are one of the most important
factors to be considered in assembling portfolios, as seen in the theory. From
this analysis it is evident that in order to maintain such preferences, in the
case of a rate of beyond 15% of return for each additional unit of risk, it is
necessary to change periodically the assets weights in the hypothetical portfolio.
If the investor does not recalculate the assets weights of their portfolio as
shown by the type of hypothetical portfolio of constant weights, their
preference regarding return and risk expected is not met over time.
Nevertheless, if the investor wishes to rescue the application in times of
crisis or postcrisis, they will have a loss of 42.73% or 46.75%,
respectively, of the initial investment made in January 2004 (precrisis). The
behavior of returns and risks during the studied period can be verified
according to the following charts.
Charts 3: Behavior of
portfolio returns of constant weights over time
Source: The author
Chart 4: Behavior of portfolios risk of constant weights over time
Source: The author
It
is interesting to note that the Petrobras share remains in the three
hypothetical portfolios whose premise is a constant Sharpe
index equal to 15%.
6. CONCLUSION
As we could observe in the theoretical foundation, the
correlation index is a measure resulting from the ratio of the covariance and
standard deviations of the analyzed elements. Thus, keeping everything else
constant, as their covariance changes, the correlation between them also
changes. Consequently, knowing that the riskreturn curve has its curvature defined
by the correlation itself, this curvature will depend on the changes undergone
by the covariance statistics.
According
to the charts we can clearly observe that the covariances,
originated from the returns of the companies, are not stable over time, and in
times of crisis they vary even more. These changes reflect in the riskreturn
curves, so to modify the possible sets of portfolios to be assembled and,
therefore, the asset allocation within these portfolios. This observation is
properly shown on chart 1.
This
means that the portfolios assembly according to the Markowitz model works only
for a period – which is lower or higher in accordance to the economic turmoil
which oscillates the covariance between the returns of the companies – the portfolio
must be constantly recomposed. In other words, given a specific set of shares,
the holding of each share must be periodically recalculated, as demonstrated,
in order to always adequate the preferences of a particular investor. As
demonstrated in this research, share weights will change more in the portfolio
in times of crisis, in which the covariances between
companies change substantially.
Periodically
recalculate the holding of each asset of the investment portfolio is a possible
solution to the instability problem of covariance over time. However, it might
increase the maintenance costs of portfolio which will be as costly as larger
are the covariance volatilities. Another interesting solution to be explored in
future studies is by identifying portfolios according to the degree of
stability of their covariances which could be
measured most precisely with the aid of a statistical hypothesis test.
REFERENCES
ALMEIDA, N.; SILVA, R.
F.; RIBEIRO, K.(2010) Aplicação do modelo de Markowitz na seleção de
carteiras eficientes: uma análise de cenários no mercado de capitais brasileiro. XIII Seminários em administração, Uberlândia.
Setembro.
HIEDA, A.; ODA, A. L.
(1998) Um estudo sobre a utilização de dados históricos no modelo de
Markowitz aplicado a Bolsa de Valores de São Paulo. In: Seminários de Administração, 3, Out. 1998, São Paulo. Anais do III
SEMEAD. São Paulo: Faculdade de Economia, Administração e Contabilidade da USP.
BODIE, Z.; MARCUS, A. J.; KANE, A. (2010) Investimentos. New York: Artmed.
DANTHINE, J.; DONALDSON, J. B. (2005) Intermediate financial theory. California: Elsevier.
LUENBERGER, D. G. (1997) Investment Science. USA: Oxford University
Press. p. 137172.
MARKOWITZ, Harry M. (1952) Portfólio selection.
Journal of Finance,
v. 7, n. 1, p. 7791. Mar.
PENTEADO, M. A.; FAMÁ,
R. (2002) Será que o beta que temos é o beta que queremos?. Cadernos de pesquisas em administração, São Paulo,
v. 09, n. 3, julho/setembro.
SHARPE, William F.; ALEXANDER, GORDON J.; BAILEY, JEFFERY V. (1995) Investments. NewJersey: Prentice Hall.
Article Metrics
_{Metrics powered by PLOS ALM}
Refbacks
 There are currently no refbacks.
Copyright (c) 2016 Leticia Naomi Ono Maeda
This work is licensed under a Creative Commons Attribution 4.0 International License.
LIBRARIES BY  