Downside risk in multiperiod tracking error models

Declaring a benchmark for a manager allows to better define the risk profile of the fund and to evaluate the relative performance against the market (Jorion (2003)). The issue of attracting potential investors in a period of financial crisis and economic recession arise the challenge of designing products which are capable of attracting the interest of investors and highlights the need for clear risk profiles for the products. The recent crisis made it evident that replicating the performance of a benchmark is not a sufficient goal to meet the expectations of usually risk-averse investors. The manager should also consider that the investor are seeking for a downside protection when the benchmark performs poorly and thus they should integrate a form of downside risk control. The objective is to attract potential investors who express an interest in high stock market returns but also are not risk-seeking enough to fully accept the volatility of this investment and require a cushion. Managing downside risk is thus a crucial part of active portfolio management. From the point of view of the portfolio management strategies the objective of providing a downside protection has the implication that part of the capital must be invested in risk-less or very low-risk fixed income financial instruments while the remaining part is invested in risky assets, eventually enabling the participation in the superior returns of the stock market. We propose a multiperiod portfolio model which combines these two goals and provide enough flexibility, in particular the control of the downside risk is carried out through the presence of a floor benchmark with respect to which we can accept different levels of shortfall. The choice of a proper measure for downside risk leads to different problem formulations and investment strategies which can reflect different attitudes towards risk (see, for example, Krokhmal et al. (2011)). The choice of fixing a benchmark and evaluating the portfolio performance with respect to this reference portfolio is quite common in the fund management practice. From the point of view of the risk control, this approach allows to measure risk in terms of deviations, shortfalls, from the benchmark implicitly assuming that the risk profile of the benchmark is known and acceptable for the investors. The benchmark can be another portfolio strategy, an index, a reference fund, a liability. It can be tradable or not allowing for different hedging strategies. In the literature there are contributions which discuss the two components separately, there are contributions which discuss the tracking error problem when a VaR, CVaR or Maximum Drawdown (MD) constraint is introduced mainly in static framework (see, for example, Jansen et al. (2000) and Michalowski and Ogryczak (2001)), but very few contributions address the dynamic portfolio management problem when both a downside risk control and a tracking error objectives are present, see for example, Dempster et al. (2004) and Ibrahim et al. (2008). In this paper we consider a multiperiod tracking error problem including a downside protection and we compare this approach with widely used non-optimized portfolio protection strategies.