Portfolio selection in the presence of systemic risk

Abstract

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.

Appendix

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 r_{j, 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,…, 14.⁵ 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, σ_{j,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.⁶ The procedure illustrated here permits one to generate S scenarios at time T+1 of the vector of returns.