Ацканов_резюме_ENG (Динамическая оптимизация стилизованных портфелей акций с применением копул), страница 4

Therefore, empirical distribution function (EDF) isused in this study to model residuals distribution (Patton(2012)):( )=1+11{ε ≤ }EDF are further used for building joint distribution with Vine copulas which are based onGumbel copula.The resulting standardized residuals are then used to construct a joint distribution of a copula,based on the Gumbel inverse copula, which places greater emphasis on the relationship betweennegative returns. The choice of this copula is related to several of its advantages. First, dynamiccopulas allow you to track changes in the characteristics of the joint distribution of quantities.Secondly, since the risks of the financial instruments under consideration are of the greatestinterest, and the risk can be defined as the decrease in the asset value, the choice of the reversecopula of Gumbel, which focuses on the relationship of negative returns, also appears as areasonable choice.

In that sense one may also consider other more common alternative – Claytoncopula, which also emphasizes interdependence of negative values. The decision for choosingrotated Gumbel copula over Clayton copula is based on study by Patton(2012) that proves betterperformance of the first. Patton(2012) also shows that t-copula is best in terms of performance,followed by rotated Gumbel copula. Still in terms of computation simplicity rotated Gumbelcopula is better and that is the reason for picking it over t-copula in this study.Vine - copulas allow constructing a multidimensional copula based on pair copulas of a largenumber of random variables, thus solving the problem of constructing a joint multidimensionaldistribution.

The use of vine copulas for building an optimal investment portfolio has beenaddressed in a number of studies (Deng et al (2011), Low et al (2013)). The most frequentvariations are D-Vine and C-Vine and imply a different hierarchy structure.,…,The joint distribution density of n the random variablesloss of generality using the following formula:()= (,…,)× (|)× (|can be decomposed without)× …× (,|),…,Each conditional density in the product can be further decomposed using a copula. For example,the second multiplier can be represented as:(|)=с ( (),()) ×()Thus, expanding each factor, and using the C-Vine structure, the joint density formula can berepresented in the following form (Deng et al (2011)):()=,…,( ),(| ,…|,…,, (||,…,,…,))The formula for D-Vine can be written as(),…,=( ),(| ,…|,…,, ())Where the marginal conditional distributions ( | ) for each j can be represented as:С( | )=Where is a vector,element .,(|, ( |))is a separate element of the vector, andis a vector v without theThe optimal parameters for the joint distribution are selected using the maximum likelihoodfunction.

For C-Vine:,(| ,…,|,,,…,, (,|,,,…,,)For D-Vine:,|,…,(,|,,…,,,(,|,,…,,)As noted in the work of Deng et al. (2011), when calculating the optimal parameters in theformulas above, it is first necessary to estimate the optimal parameters for each step in thehierarchy, and use them as initial data.As it is emphasized in Deng et al. (2011), D-Vine and C-Vine are suitable for different datasets.The structure of C-Vine assumes the presence of one key element that determines the behavior ofthe rest of the elements. If there is no such element, then D-Vine is more suitable. This paperstudies the relationship between returns of various stocks, issuers of which are represented invarious sectors of the economy.

Therefore it is not possible to determine some key stock thatseems to define the behavior of the others. Consequently, for this study, it is more appropriate touse D-Vine structure of a copula.It is important to mention that Vine-copulas allow using different copula models in the sameconstruction. Still, in order to keep the focus of the study on stylized portfolio optimization, thisstudy is conducted basing on only one model – rotated Gumbel copula. The problem ofcomparison between other copula models and pair constructions is left for other studies. Inaddition to Vine-copula in order to build multivariate distribution one may use Archimedianhierarchical copulas (as mentioned in Penicas(2014)), still the problem of applicability of thesestructure to stylized optimization is also left for the other studies.After fitting parameters of the copula, one can switch to building the optimal portfolio.

Since thestudy aims to develop a procedure for stylized optimization of investment portfolios, a separateprocedure was performed for each of the five styles. Nevertheless, the general optimizationprocedure can be presented as follows:1. Let the investment portfolio is being optimized at time T. The data on returns in period[T-365, T] is used to estimate the marginal distributions and parameters of the Vinecopula, which determines the joint distribution of financial instruments.2.

Based on the rating according to a specific criterion and according to the newest dataavailable at time T, the Top-30%, Top-40% or Top-50% of stocks are selected. Furthersteps are taken for these assets.3. ARMA is estimated for each of the stock in the groups defined on step 24. Joint distribution for ARMA residuals is built by optimizing Vine copula parameters5.

Residuals are generated via simulationwith copula and further used to generate assetreturn scenarios using formulas (1) and (2).……= ……Wherestock.is the return on asset i on the simulation j. There are 1000 simulations for each6. Based on the simulated scenarios, which imply the specific returns for each of the stocks,one can estimate the return on the entire portfolio at each of the simulations, assumingparticular weights:..=....×....Where PR is the portfolio return vector for given weights W = (independent simulations (∑= 1),…,) for N7.

Assuming confidence level of 99%, CVaR is defined as the average value of elements ofPR vector, which are less than 99% of the remaining values.8. The portfolio is optimized by changing weight vector W to achieve minimum CVaR ofthe portfolio:(С)Whereis CVaR of the portfolio, calculated on the basis of the PR vector, asindicated in step 4.9. The resulting weight vector W is used as the portfolio structure until the next moment Tat which the portfolio composition is revised using steps 1-8The following restriction is imposed on the weights of individual assets in the portfolio to meetthe norms of the regulation of mutual funds in Russia:15% ≥≥0In determining the rebalancing frequency of the portfolio structure, the main prerequisite was theregulation of mutual funds in Russia. In particular there is the provision that if the weight of oneissuer in the portfolio exceeds 15% due to a change in the market value, the violation must beliquidated within a month.

Accordingly, portfolios are reviewed on a monthly basis, so that thereare no serious regulatory violations.Steps 2 and 8 in the above-mentioned procedure should be altered in order to address specificstylize of the investment portfolio. The required modifications for each style are describedbelow. It is important to note that although the stylization criteria defined in this study can beconsidered rather naive, since there are fresh studies with a more rigorous set of criteria, themain purpose of this study is not to select criteria for each investment style, but to develop aprocedure for stylized portfolio optimization and introduce stylized utility functions.

Expandingthe set of criteria for defining each investment style would not only significantly dilute the focusof the research, but also complicate the calculations and reduce the number of observations dueto limited data on Russian stock market.The growth stocks in this study are defined as the stocks of companies with market value beinghigh relative to current fundamental indicators. The Price-to-Book multiple is used to classify thegrowth stocks in this study.

The higher the multiple, the more the stock fits the growth stockcriteria. This multiple was also used in papers by Graham & Dodd (1934), Fama & French(1998), Bauman et al (1998), Fama & French (2007), Bragg (2007), Bodie et al (2009), Pinto etal (2010))Stylized portfolio in this case will be optimized using the following utility function:/Whereassets.– is the average multiple of the portfolio in accordance with the weights W of theValue stocks are defined in this study as stocks of companies with market value that does notfully reflect the company's financial performance in comparison with the average for the market /sector / country, that is, the company is undervalued.

As in the previous case, the Price-to-Bookmultiple is used to find value stocks. Although in case of value, stocks with the lowest multipleare selected. The utility function for stylized value portfolio is as follows:1/×Profitability stocks are characterized by high profitability indicators. When selecting shares ofprofitability, stocks of companies with relatively high ROE (Return on Equity) or ROA (Returnon Assets) ratios are selected. Similarly, these coefficients were used in the works of Greenblatt& Tobias (2010), Asness et al. (2013), Bouchaud et al (2016) In the case of this study, the ROEcoefficient is used, since there was more data on it. The utility function for profitability portfoliois as follows:/Whereis weighted ROE value for a portfolio with weights W.Momentum stocks are characterized by a relatively high price change over a certain period oftime.