Translation-invariant and positive-homogeneous risk measures and optimal portfolio management in the presence of a riskless component

Risk portfolio optimization, with translation-invariant and positive-homogeneous risk measures, leads to the problem of minimizing a combination of a linear functional and a square root of a quadratic functional for the case of elliptical multivariate underlying distributions. This problem was recently treated by the authors for the case when the portfolio does not contain a riskless component. When it does, however, the initial covariance matrix Σ becomes singular and the problem becomes more complicated. In the paper we focus on this case and provide an explicit closed-form solution of the minimization problem, and the condition under which this solution exists. The results are illustrated using data of 10 stocks from the NASDAQ Computer Index. ► Optimal portfolio is sought in the case of translation-invariant and positive homogeneous risk measure. ► The problem leads to the minimization of a combination of a linear and a square root of a quadratic functionals. ► Elliptical multivariate distribution is assumed. ► When the portfolio contains a riskless component the optimization becomes more complicated due to singularity of the covariance matrix. ► Explicit closed-form solution for this problem is provided.