In this article we study the portfolio selection problem in the presence of systemic risk. We propose reward-risk measures that account for systemic risk and provide a methodology to generate realistic return scenarios. The methodology involves first analyzing the empirical behavior of several MSCI country indexes, suggesting how to approximate future scenarios. Then we examine the profitability of several strategies based on the forecasted evolution of returns. In particular, we compare the optimal sample paths of future wealth obtained by performing reward-risk portfolio optimization on simulated data and we discuss the ex-post performance of the proposed portfolio strategies.
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The original paper was published in 2008 and updated in 2011.
The first window of historical observations considers daily returns from 4 January 1988 to 31 May 2007. The other windows of observations start after other subsequent 250 trading days (about one year).
The portfolio selection problem with transaction costs has been studied in several papers. See, among others, Magill and Constantinides (1976), Davis and Norman (1990), Dumas and Luciano (1991), and Sau (2012). In this article we optimize strategies with transaction costs using the same methodology proposed by Ortobelli et al (2010).
We consider these transaction costs level based on an experiment with an international trading platform (Interactive Brokers’ platform, for transaction costs see the web site: https://www.interactivebrokers.com/en/index.php?f=commission&p=stocks2). In the experiment we assume an initial budget of USD 250 000. This empirical example supported the position that we have on average proportional transaction costs slightly less than 5 basis points when we use the transaction costs applied by Interactive Brokers’ platform and the optimal weights obtained with the Co-Rachev ratio.
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We are grateful to Professor Rachev who suggested the main ideas of the article and to an anonymous referee for helpful comments. Sergio Ortobelli is grateful for the Italian funds by ex MURST 60 per cent 2014 and MIUR PRIN MISURA Project, 2013–2015 and through the Czech Science Foundation (GACR) under project 13-13142S and SP2013/3, an SGS research project of VSB-TU Ostrava, and furthermore by the European Social Fund in the framework of CZ.1.07/2.3.00/20.0296.
1holds a PhD in mathematical and empirical finance and is currently a manager at Deloitte in London. She held previous positions as a risk manager at DZ Bank International in Luxembourg, research assistant in the Department of Econometrics, Statistics, and Mathematical Finance at the University of Karlsruhe in Germany, an Assistant Professor in the Department of Mathematics and Cybernetics on the Faculty of Informatics at Technical University of Ufa in Russia, and a financial specialist in the Department of Financial Management at Moscow Credit Bank in Russia. She specializes in mathematical modeling and numerical methods.
Generation of return scenarios
The algorithm for the generation of return scenarios is as follows.
Step 1. Carry out maximum likelihood parameter estimation of ARMA(1, 1)-GARCH(1, 1) for each return rj, t (j=1, …, 14)
Since we have 2762 historical observations, we use a window of T=2762. Approximate with α j –stable distribution the empirical standardized innovations
where the residuals can be derived from the model as follows j=1,…, 14.Footnote 5 In order to value the marginal distribution of each innovation, we first simulate S stable distributed scenarios for each of the future standardized innovations series. Then we compute the sample distribution functions of these simulated series:
where (1⩽s⩽S) is the sth value simulated with the fitted α j –stable distribution for future standardized innovation (valued in T+1) of the jth return.
Step 2. Fit the 14-dimensional vector of empirical standardized innovations
with an asymmetric t-distribution V=[V1, …, V14]′ with v degree of freedom; that is,
where μ and γ are constant vectors and Y is inverse-gamma distributed IG(v/2;v/2) (see among others, Rachev and Mittnik, 2000) independent of the vector Z that is normally distributed with zero mean and covariance matrix Σ=[σ ij ]. We use the maximum likelihood method to estimate the parameters of each component. Then an estimator of matrix Σ is given by
where and cov(V) is the variance-covariance matrix of V. Since we have estimated all the parameters of Y and Z, we can generate S scenarios for Y, and independently, S scenarios for Z, and using (11) we obtain S scenarios for the vector of standardized innovations that is asymmetric t-distributed. Denote these scenarios by (V1(s), …,V14(s)) for s=1, …, S and denote the marginal distributions for 1⩽j⩽14 of the estimated 14-dimensional asymmetric t-distribution by
Then considering 1⩽j⩽14; 1⩽s⩽S, we can generate S scenarios (U1(s), …, U14(s)) s=1, …, S of the uniform random vector (U1, …, U14) (with support on the 14-dimensional unit cube) and whose distribution is given by the copula
Considering the stable distributed marginal sample distribution function of the j th standardized innovation ; j=1, …, 14 (see (10)); and the scenarios for 1⩽j⩽14;1⩽s⩽S, then we can generate S scenarios of the vector of standardized innovations (taking into account the dependence structure of the vector)
valued at time T+1 assuming
Once we have described the multivariate behavior of the standardized innovation at time T+1 using relation (9), we can generate S scenarios of the vector of the model’s residuals as follows:
where s=1, …, S, σj,T +1 are still defined by (9). Thus using relation (9) we can generate S scenarios of the vector of returns
valued at time T+1. Observe that this procedure can always be used to generate a distribution with some given marginals and a given dependence structure.Footnote 6 The procedure illustrated here permits one to generate S scenarios at time T+1 of the vector of returns.
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Biglova, A., Ortobelli, S. & Fabozzi, F. Portfolio selection in the presence of systemic risk. J Asset Manag 15, 285–299 (2014). https://doi.org/10.1057/jam.2014.30