# Calculating the Efficient Frontier for the Portuguese Stock Market.

IntroductionThis paper calculates the efficient frontier (EF) (1) and the capital market line (CML), using data from companies listed in PSI 20, considering two different periods: before and after the Global Financial Crisis. Under the assumption of the economic rationality, investors choose to keep efficient portfolios, i.e., portfolios that maximize the return for a certain level of risk or that minimize the risk for a certain level of expected return. According to Markowitz (1952, 1959), these efficient portfolios are in the opportunity set of investments, that we can refer to as the set of all portfolios that can be constructed from a given set of risky assets.

Modern portfolio theory posits that investment strategy must be conducted in a risk diversification perspective, what leads to an analysis of the combination of assets considering the return and the risk, and also the covariance of the returns between the assets. Mean-variance theory is a solution to the portfolio selection problem, assuming that investors make rational decisions. Conceptually, the mean-variance analysis links diversification with the notion of efficiency because optimal diversification is achieved along the EF (Markowitz 1987). The optimal portfolio should be the tangency portfolio between the EF and the highest indifference curve or, in other words, the efficient portfolio with maximum expected utility. Under the economic theory of choice, an investor chooses among the opportunities by specifying the indifference curves or utility function. These curves are constructed so that along the same curve, the investor is equally happy, what leads to an analysis of the assumed investor's profile. Nevertheless, limitations of diversification are increasingly prominent (Amenc et al. 2011).

By combining the risk-free asset with a portfolio on the EF, one can construct portfolios with better outcomes than the ones simply on EF (Tobin 1958). This is represented by the CML, which is a tangent line from the risk-free asset to the risky assets region, i.e. the set of investment possibilities created by all combinations of the risky and riskless assets. The tangency point originated by the combination of the CML and the EF represents the searched market portfolio.

The asset weights that provide the optimal risk return tradeoff on the CML require the means, the variances and covariances of the underlying asset returns that, in practice, are often estimated from historical data. As a consequence, this has led to portfolios that may perform poorly and have counter-intuitive asset allocation weights, referred to as the "Markowitz optimization enigma" (Michaud 1989; Black 1993; Chopra and Ziemba 1993; Lai et al. 2009). In addition, few implementations of portfolio theory have been performed in the literature and the majority of portfolio managers do not use it. According to Kolm et al. (2014), the reasons for this is the existence of an irrational relationship between inputs and outputs as well as the great sensitivity of the portfolio allocation to changes in inputs. Indeed, one of the main problems when trying to apply mean-variance portfolio optimization in practice is its high input sensitivity. Problems arise with the choices of risky assets, length of time and time period. To improve expected return estimates, numerous methods can be found in the literature (Michaud and Michaud 2008; Michaud et al. 2013), but this important question is beyond the scope of our paper.

This paper focuses attention on the estimation and sensitivity analysis of mean-variance-efficient portfolios to changes in the input data, estimated from historical data before and after the Global Financial Crisis, for Portuguese-listed companies included in PSI 20.

Basic Inputs

To calculate the EF, an estimate of the expected returns and the covariance matrix for the set of risky securities is needed. This estimate is difficult to obtain, and the optimal portfolio calculation is extremely sensitive to it. For this reason, an EF-based portfolio is difficult to successfully implement in practice. However, deriving the EF is nevertheless very useful and provides intuition about diversification and the relationship between risk and return. For example, Costello et al. (2008), examine whether Australia offers any unique diversification benefits to an international investor constructing the efficient frontier of optimal portfolios (minimum risk for a given level of return) with and without an Australian index. In this section, concepts of modern portfolio theory are presented, that have been discussed at length in the literature and in finance textbooks (Bodie et al. 2009; Elton et al. 2010).

The return of a financial security is the rate computed based on what an investment generates during a certain period of time, where we include the capital gains/losses and the cash-flows it may generate (dividends, in the case of stocks). We can calculate the returns as the difference between an asset price at the end and in the beginning of a selected period, divided by the price of the asset at the beginning of the selected period, [mathematical expression not reproducible], where [R.sub.it] is the return of asset i on moment t; [P.sub.it] is the asset i price on moment t; and [P.sub.it-1] is the asset i price on moment t-1. If t represents a week time interval, weekly returns data are obtained. In addition, the arithmetic mean of the return of the period is calculated as [mathematical expression not reproducible], where T is the number of observations.

Markowitz (1952, 1959) introduced the concept of risk and assumed that risk is measured by the variance (or by the standard deviation from the average return), given by [mathematical expression not reproducible]. Covariance enhances the influence of an asset on other assets, with different characteristics, in the determination of the variance of a portfolio. It measures how returns on assets move together. In this context, the correlation is a simple measure to standardize covariance, scaling it with a range of -1 to +1. The covariance concept (Markowitz 1952) puts in evidence the importance of the diversification in the choice of the optimal portfolio.

Another theory supporting this concept is the mean-variance theory, which shows up as a solution to the portfolio selection problem. Markowitz (1952) demonstrates that the expected return of a portfolio is based on the mean of the asset's expected returns, while the standard deviation is not simply a mean of the individual asset's standard deviation, but considers the covariance between tthe asset's returns. In this context, investors always want to maximize the expected return given a determined level of risk or to minimize the risk given a determined level of expected return. Implicitly, investors are risk averse and assume as selection criterion the mean-variance theory, i.e., the mean and the standard deviation of the returns. Therefore, assuming economic rationality, all investors choose to have efficient portfolios. The selected portfolio must be above the global minimum variance (GMV) portfolio or in the concave portion of the portfolio possibility curve. The optimum portfolio will always depend on the investor preferences, mainly on his risk aversion profile, that can be characterized by the utility function. This is based on the economic theory of choice, where an investor chooses among the opportunities by specifying a series of curves that are called indifference curves. These so-called curves are constructed so that everywhere, along the same curve, the investor is assumed to be equally satisfied. This way, for each investor, the optimal portfolio should be the tangency portfolio between the EF and the indifference curve, i.e., the efficient portfolio with greater expected utility from the investor perspective (Markowitz 1987; Merton 1972).

Formally, the portfolio expected return, with N risky assets, is the weighted average of the expected returns of the single assets that comprise the portfolio and can be represented as [mathematical expression not reproducible], where N is the number of assets in the portfolio, w is the weight of each asset in the portfolio, and [R.sub.i] is the average return of asset i Assuming equal weights, this yields [mathematical expression not reproducible].

The portfolio variance is simply the expected value of the squared deviations of the return on the portfolio from the mean return on the portfolio, that is [mathematical expression not reproducible] (2). Substituting in this general formula the expression for return on a portfolio and expected return on a portfolio, yields [mathematical expression not reproducible], where [mathematical expression not reproducible] is the variance of the return on asset i and cry is the covariance between the returns on asset i and asset j, given by [mathematical expression not reproducible]. Hence, the portfolio variance is the sum of individual asset variances plus the covariances between them, considering the weight of each asset in the portfolio.

The effect from diversification is obtainable by increasing the number of assets, which reduces significantly the value of the first term, approaches zero, and the value of the second term, approaching the average covariance. However, diversification is not able to eliminate the total portfolio risk, because returns on securities are not perfectly (negatively or positively) correlated, that is [mathematical expression not reproducible], where [mathematical expression not reproducible].

Concluding, to reduce total portfolio risk, an investor should diversify. The payoff from diversification is higher, the lower is the correlation coefficient between assets. The main goal of any investor will be to decrease total risk without affecting a certain desired level of return. Portfolio diversification is the key to solve this problem. This portfolio strategy allows the investor to reduce exposure to risk by combining different assets (stocks, bonds, futures, etc.).

The EF consists of the set of efficient portfolios, which are portfolios that have the highest return for a certain level of risk. The EF coincides with the top portion of the minimum-variance portfolio set. The portfolio with the lowest risk is the GMV portfolio, considering that, for each level of expected return, we can vary the portfolios weights of the individual assets to determine the minimum variance portfolio, i.e., the one with the lowest risk. The best combination of assets must now be determined. Assuming investors are risk averse, they prefer the portfolio that has the greatest expected return when choosing among portfolios that have the same risk. The optimal portfolio along the EF is selected considering the investor's utility function and attitude towards risk (Elton et al. 2010; Girard and Ferreira 2005).

The introduction of the risk-free asset in this analysis means that finding an optimal portfolio for a given level of risk tolerance can be separated into two easier problems: first finding an optimal mix of market securities that does not vary with risk tolerance, and then combining it with an appropriate amount of risk-free assets (Buiter 2003). The CML takes into account the inclusion of a risk-free asset in the portfolio. The CML is the tangent line drawn from the point of the risk-free asset to the feasible region for risky assets. The tangency point represents the market portfolio, so named since all rational investors (minimum variance criterion) should hold their risky assets in the same proportions as their weights in the market portfolio. Therefore, the CML is considered to be superior to the EF since all points along the CML have superior risk-return profiles to any portfolio on the EF, with the exception of the market portfolio, the point on the EF to which the CML is the tangent. This portfolio is composed entirely of the risky asset, the market, and has no holding of the risk-free asset, i.e., money is neither invested in, nor borrowed from the money market account.

The CML represents the possible combinations of the market portfolio and the risk-free asset. Similarly, the CML is denned as a risk-return trade-off derived by combining the market portfolio with risk-free borrowing and lending, being all portfolios between the risk-free and the tangency point considered efficient. Formally, [mathematical expression not reproducible], which represents a linear function, where the slope is the compensation in terms of expected return for each additional unit of risk, risk premium, and the intercept point is the risk-free rate. The optimal portfolio for each investor will be obtained through the highest indifference curve that is tangent to the CML.

Alternatively, it is possible to choose a portfolio using performance measures that permit ranking the portfolios. Although it is beyond the scope of this paper to describe in detail those measures, it is important to mention the Sharpe ratio (Dowd 2000; Jagric et al. 2007; Sharpe 1994) is defined as [mathematical expression not reproducible]. A negative value indicates that the risk-free asset would have a better performance than the actual portfolio. Important to note is that all of the portfolios on the CML have the same Sharpe ratio as that of the market portfolio, i.e., [mathematical expression not reproducible]. In fact, the slope of the CML is the Sharpe ratio of the market portfolio. Therefore, a frequently used stock-picking rule of thumb is to buy assets whose Sharpe ratio will be above the CML and sell those whose Sharpe ratio will be below.

There are many different techniques to deduce the EF (Elton et al. 1977; Bawa et al. 1979; Feldman and Reisman 2003).). Historical data is usually used to obtain estimates of the inputs to the portfolio selection process, although analysts might modify these historical estimates so that they better reflect beliefs about the future. Therefore, the EF technology is highly sensitive to input data estimates with consequences for the asset allocation decisions. On the other hand, different risky assets will result in different draws of the EF and the CML, meaning that there is no unique EF and CML, but as many as the number of possible risky assets sets one might consider to represent the market. This paper is a first attempt, as far as we know, to build and delineate the EF and the CML using the risky assets included in the PSI 20 Index for two periods, before and after the Global Financial Crisis.

Brief Analysis of Portuguese Stock Index

Portugal may be included in the group of bank-oriented countries with a universal bank system, strongly concentrated in a few financial groups, which means that the money flows essentially through financial institutions (Allen and Gale 2000; Garcia and Guerreiro 2016). Banks and government dominate as a source of financing but have not been immune to the financial crisis that started with the bankruptcy of the Lehman Brothers, in September of 2008 (Nanto 2009). In addition, Portugal, like other European countries, had a financial assistance program in order to solve their debt and economic structural problems. Hence, bank funding conditions were affected by sovereign credit risk (BIS 2011).

Considering the capital market, the Portuguese Stock Index (PSI 20) is the national benchmark stock index, constituted by the 20 biggest companies listed on Lisboa Stock Exchange (Euronext Lisbon). The liquidity of each listed company is measured by the transaction volume in the stock exchange. The supervising institution, the Comissao do Mercado de Valores Mobiliarios (CMVM), requires financial reporting as the available information needed for investors. According to the regulations of the CMVM, short sales are allowed, but there are some restrictions on these operations. (2) Hence, considering that, we assumed that short selling is not possible.

The base value of the PSI 20 was 3000 points and started in December 31, 1992. In the first week of May 2015, the PSI 20 index included only 18 listed companies (Altri, Banco BPI, BANIF, BCP, CTT, EDP, EDP Renovaveis, GALP Energia, Impresa, Jeronimo Martins, Mota-Engil, NOS, Portucel, Portugal Telecom, REN, Semapa, Sonae SGPS and Teixeira Duarte). Therefore, the empirical work considers 20 risky assets, for the first period (from May 2000 to the end of 2008) and 18 risky assets, for the second period (from January 2009 to May 2015) to derive the EF and the CML.

Figure 1 shows weekly prices of the PSI 20 index for the entire period under analysis, from May 2000 to May 2015. A similar behavior, a notorious decrease, is detectable around three events: the internet bubble in 2000, September 11, 2001, and the Lehman Brothers collapse in 2008.

Data Set and Methodology

Historical data of weekly prices of all the stocks included in the PSI 20 were taken from the Datastream platform (Thomson Reuters 2018), from the first week of May 2000 to the first week of May 2015. Then, we split it into two different time periods to derive the EF in those periods. The first period is from the first week of May 2000 to the last week of 2008 (pre 2009), and the second period is from the first week of 2009 to the first week of May 2015 (post 2009). As previously mentioned, there is no ideal time period horizon to obtain reliable input data (average returns, standard deviations and covariances), so we considered periods between 5 to 10 years. In addition, an alternative first period was considered to disentangle the impact of the year 2008. Therefore, the sensitivity of the results was analysed with the exclusion of the year 2008 in the first period.

The Euribor rate for 1 year was used as the risk-free rate of return proxy for the calculations (Euribor 2015). The registered average values were 3.025%, for the first period, and 0.169%, for the second one. The first period considers weekly returns of the 20 companies included in the PSI 20 and the PSI 20 index. This represents a sample of 451 observations, for each time series. The second period considers weekly returns of the 18 companies included in PSI 20, and of the PSI 20 index, representing a sample of 333 weekly observations, for each time series. The alternative first period, comprises 19 companies and 399 observations for each time series. Hence, the weekly return rates were calculated, as well as the means, the variances and covariances, for each stock i 0=1,..., N) and the PSI 20.

Historical means and standard deviations of each weekly time series were then annualized through the transformations [mathematical expression not reproducible], where [R.sub.y] represents the average annual return and [R.sub.w] is the weekly average return, and [[sigma].sub.i,y]. = [[sigma].sub.i,w] x [square root of (52)]. where [[sigma].sub.i,y] represents the annual standard deviation and [[sigma].sub.i,w] is the standard deviation of weekly returns.

Table 1 presents the descriptive statistics for the periods under analysis. Column 2 shows annual means and standard deviations for the stocks included in PSI 20 for the first period. Fourteen out of 20 stocks registered negative annual means, with standard deviations higher than 20%. Column 3 shows annual means and standard deviations for the stocks included in PSI 20 for the second period. In this period, nine out of 18 stocks registered negative annual means, with standard deviations also higher than 20%, displaying a relative recovery in comparison with the previous period. The variance-covariance matrix for each period was obtained, displaying mostly positive imperfect correlations.

Once the input data were estimated, the construction of the EF and CML followed. To obtain efficient portfolios, we minimized the total risk of the portfolio, given by [mathematical expression not reproducible], for a given level of expected return (Markowitz 1952).

In order to align with the Portuguese stock market features, some restrictions were assumed. No borrowing or lending is allowed, so the objective was to maximize the objective function [mathematical expression not reproducible], subject to the constraint [mathematical expression not reproducible]. In addition, investors are not allowed to short-sell, which means that all assets have positive or zero investment, [w.sub.i] [greater than or equal to] 0, i = 1, ..., N. Finally, in order to avoid the complete domination of only one asset in our portfolio and to achieve Markowitz diversification, no asset was allowed to have a weight higher than 10% in the final portfolio, that is [w.sub.t] [less than or equal to] 10. This assumption was further specified in the results, with evidence that, without this assumption, efficient portfolios could be composed of a single stock.

The Solver function was used to create several portfolios with different average return rates and variances, from the moment we achieved a minor portfolio return up to the moment of a maximum possible return. This led to the construction of the efficient frontier, considering the relationship between the returns and the standard deviations of the "solved" portfolios.

The CML was derived through the consideration of the equation [mathematical expression not reproducible]. This represents a linear function from the risk-free asset rate of return up to the point of the market portfolio rate. Therefore, the CML was derived by drawing a tangent line from the intercept point on the efficient frontier to the point where the expected return equals the risk-free rate of return.

Results

The estimation results of the EF are depicted in Fig. 2, for the first and the second periods considered. Clearly, it is possible to observe a noticeable shift in the configuration of the EF between the two periods, with a movement to the northeast. In the period post-2009, the GMV portfolio registered a rate of return of 4.24%, which compares with the negative rate of return of -3.28% of the GMV portfolio obtained in the pre-2009 period.

The maximization of the Sharpe ratio led to the determination of CML. In fact, the slope of the CML was the Sharpe ratio of the market portfolio. The CML resulted from the combination of the risk-free asset and the market portfolio. All points along the CML had superior risk-return profiles to any portfolio on the efficient frontier, with the exception of the market portfolio, the point on the efficient frontier to which the CML was the tangent. The tangent portfolios presented a rate of return of 9.80%, in the post 2009 period, and a rate of return of 1.59%, in the pre-2009 period. On the other hand, the tangent portfolio of the first period registered a rate of return lower than the risk-free rate. Consequently, the estimated CML presents a negative slope, contrasting with the concept of an efficient portfolio. Indeed, independent of investor preferences, the optimal portfolio would totally consist of the risk-free asset.

In addition, concerning the second period, if the weights restriction was relaxed, the portfolio with 100% invested on asset CTT registered the higher rate of return as well as the higher risk. Assuming that restriction, it is possible to reduce the risk from 41.29% (risk of CTT stocks) to 3.72% through the process of diversification. Hence, assuming the maximum of 10% weight of each stock, it is possible to reduce the risk.

We also constructed the EF and the CML under an alternative first period that excluded the year 2008, resulting in a northwest movement (meaning higher expected return and lower risk) and in a positive CML slope, aligned with theory (Fig. 3). It follows therefore, that the year of 2008 seems to be the real booster of the bad results previously obtained in the first period. The tangent portfolio displayed a return of 10.09% and a standard deviation of 1.66%. Moreover, the GMV portfolio had a positive rate of return of 4.71%.

Conclusion

The concepts of EF and CML are most relevant in portfolio construction. To build an EF and the corresponding CML is a natural step after studying modern portfolio theory. However, few studies exist on the application of the theory to concrete capital markets. This topic has been neglected in existing literature. This is a first attempt concerning the Portuguese case. The study tries, in addition, to derive some conclusions on asset allocation decisions after the Global Financial Crisis. Indeed, it is possible to register huge and striking differences between the two sets of EF and CML, according to the two periods under analysis.

In the period before 2009, the results lead to the conclusion that the best asset allocation is 100% investment on the risk-free asset with a rate of return of 3.025%. This situation is odd in the financial market framework, since the risk-free rate is above the PSI20 average rate of return, -9.04% for that period. A further construction of the EF and CML, considering only the period between 2000 and 2007, that is, removing 2008 from the data, shows that the 2008 data year seems to be the real booster for the bad results based on data from the first period.

In the second period, very different results were obtained in the deduction of the EF. Indeed, optimal combinations are composed of at least 50% of the risky assets, yielding a rate of return higher than 5% and a standard deviation between 2 and 3.5%. These possible asset allocations contrast with the entire investment in the risk-free asset, previously recommended. A rational investor would hold a combination of risk-free assets and the optimal risky portfolio. The slope of the CML, or the SR of the market portfolio, represents an increment of 2.64% on the rate of return considering the increment of 1% on risk. It seems that after the 2008 crisis the stock market was able to get back to better results, showing some recovery.

This paper estimates the EF and CML as instruments of portfolio investment decision in the Portuguese case. However, the estimation is sensitive to the time period, as shown for the first period when the 2008 data year was excluded. This enhances the sensitivity of portfolio theory applications to the input data estimations as a guide to investors' asset allocation decisions. Further research should consider other periods and the impact that different periods might have on optimal portfolios, periodically. Moreover, statistical tests should be applied to determine whether the EF and CML shifts are significant, for example, an F-test for equality of variance at the point of maximum difference between the two EFs and the Gibbons, Ross and Shanken test to determine whether the slope of the CML is significantly different in the various cases (Gibbons et al. 1989; Costello et al. 2008).

Acknowledgements UECE is financially supported by FCT (Fundacao para a Ciencia e a Tecnologia), Portugal. Financial support from national funds by FCT (Fundacao para a Ciencia e a Tecnologia). This article is part of the Strategic Project: UID/ECO/00436/2013.

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Maria Teresa Medeiros Garcia (1,2) [iD] * Daniel Alexandre Bourdain dos Santos Borrego (1)

[??] Maria Teresa Medeiros Garcia mtgarcia@iseg.utl.pt

(1) ISEG, Lisbon School of Economics and Management, Universidade de Lisboa, Rua Miguel Lupi, 20, 1249-078 Lisboa, Portugal

(2) UECE (Research Unit on Complexity and Economics), Lisboa. Portugal

(1) Also known as the risky assets efficient frontier or Markowitz efficient frontier.

(2) CMVM (2008) establishes that for the short sales to be allowed, it is necessary that the investor already has some assets, at least of the same value as those selected for short sale. In addition, there is the information obligation.

https://doi.org/10.1007/s11294-018-9705-9

Table 1 PSI 20 descriptive statistics. 2009-2015, 2000-2008 and 2000-2007 Asset January 2009-May2015 Asset [R.sub.y] (%) [[sigma].sub.i,y] (%) Altri 15.80 34.22 Altri Banco BPI -10.37 46.77 Banco BPI BAN1F -48.89 132.26 BCP BCP -26.38 51.31 BES CTT 26.61 41.29 Brisa EDP 1.73 25.87 Cimpor EDP Renovaveis -0.60 28.94 EDP GALP Energia 3.04 31.20 EDP Renovaveis Impresa -9.29 45.44 GALP Energia Jeronimo Martins 14.86 30.78 Jeronimo Martins Mota-Engil -5.01 42.88 Mota-Engil NOS 4.31 31.97 NOS Portucel 13.76 23.90 Portucel Portugal Telecom -37.73 42.10 Portugal Telecom REN -1.52 17.63 REN Scmapa 8.13 26.46 Semapa Sonae SGPS 12.71 30.98 Sonae SGPS Tcixcira Duarte -1.24 58.51 Sonaecom Sonae Industria Teixeira Duarte PSI20 -2.94 22.67 PSI20 Risk free rate 0.169 Risk free rate Asset May 2000- May 2000-December December 2008 2007 [R.sub.y](%) [[sigma].sub.i,y](%) [R.sub.y] (%) Altri 24.80 45.54 46.46 Banco BPI -10.94 27.89 3.76 BAN1F -20.60 30.08 -8.06 BCP -7.61 21.36 3.59 CTT 1.62 20.56 11.32 EDP -2.61 27.06 6.03 EDP Renovaveis -7.07 26.48 0.03 GALP Energia 5.34 37.95 - Impresa 11.25 42.00 30.38 Jeronimo Martins -3.61 35.98 2.01 Mota-Engil -1.67 29.21 10.76 NOS -26.93 37.46 -19.45 Portucel -2.91 25.28 3.21 Portugal Telecom -9.64 31.26 -5.02 REN 10.80 36.88 17.19 Scmapa 3.21 25.73 8.58 Sonae SGPS -18.20 37.03 -2.23 Tcixcira Duarte -22.10 53.46 -10.28 -20.58 33.77 -4,78 -16.74 38.17 -1,07 PSI20 -9.04 18.90 -0.41 Risk free rate 3.03 4.03 Asset May 2000-December 2007 [[sigma].sub.i,y](%) Altri 42.45 Banco BPI 21.98 BAN1F 24.63 BCP 14.70 CTT 16.66 EDP 23.04 EDP Renovaveis 22.04 GALP Energia - Impresa 38.59 Jeronimo Martins 31.27 Mota-Engil 25.82 NOS 35.68 Portucel 22.93 Portugal Telecom 28.69 REN 37.70 Scmapa 25.33 Sonae SGPS 33.52 Tcixcira Duarte 53.56 29.64 30.66 PSI20 15.40 Risk free rate Source: Datastrcam (Thomson Reuters 2018) and Euribor (2015)

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Author: | Garcia, Maria Teresa Medeiros; dos Santos Borrego, Daniel Alexandre Bourdain |
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Publication: | International Advances in Economic Research |

Article Type: | Report |

Geographic Code: | 4EUPR |

Date: | Nov 1, 2018 |

Words: | 5631 |

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