*Peter Meier, Andreas Ruckstuhl and Marc Weibel show that optimising for expected shortfall and the Sharpe-Omega ratio can improve risk-adjusted returns from traditional assets and core-satellite portfolios that integrate alternative investments*

The classical Markowitz portfolio optimisation was a powerful intellectual concept with epochal effects on portfolio management. Nevertheless, its practical use is restricted by its many limitations. The weights of constituents of a Markowitz-optimised portfolio are extremely sensitive to return estimations. Furthermore, such optimised portfolios lose a lot of their diversification advantages during stress periods, because some assets have fat-tailed return distributions, and correlations increase.

Although the problems of the mean-variance approach of Markowitz are well addressed, few broadly accepted or commercially available approaches exist to mitigate them. A research team at Zurich University (ZHAW), supported by the Swiss Commission for Technology & Innovation, Complementa, the Cantonal Bank of Zurich, the City of Zurich Pension Plan and Alternative Soft, has developed an optimiser with many features to overcome the weaknesses of mean-variance optimisation.

**Expected shortfall and tail dependence**

The risks of the portfolio and of its constituents are measured and optimised with expected shortfall based on non-normal return distributions. The expected shortfall aims to estimate the average loss below a certain loss limit, and therefore, the modelling of the return distributions, and in particular fat-tail losses, becomes crucial.

Expected shortfall is very sensitive to the type of the return distribution, and it is well-recognised that it might be heavily underestimated within the framework of a normal distribution, where the skewness and the kurtosis of the distribution is not considered. The customised optimiser uses generalised hyperbolic distributions and estimates the sub-type of the distribution for each asset separately. An alternative non-parametric risk measure implements the put option coming from the denominator of the Sharpe-Omega ratio.

Another innovative feature is the treatment of the well-known correlation risk, the problem that diversification is not effective when it is most needed – when markets are crashing.

The copula technique is used to model the dependence of assets in the tails, and this tail-dependence can even be varied for scenario analysis. If an investor is worried about the markets, and potentially correlated price disruptions, he can assume heavy tail-dependence, and consequently simulate a sufficiently conservative portfolio.

**Risk parity and the core-satellite concept**

Tail-risk and tail-dependence measures are integrated into a core-satellite approach and can be optimised with risk parity or other optimisation targets. Assets with pure market risks are usually taken together in the core and alternative investments in the satellite. In the absence of return forecasts for the core assets, risk parity is used to search for the optimal weights of the core portfolio. For each satellite investment, alpha and beta exposures are estimated with respect to the core portfolio, and finally satellite investments are optimised and integrated into the total portfolio as far as they improve its risk-alpha profile.

The experience and the results with this advanced optimisation technology are convincing and lead to more realistic portfolios, compared with Markowitz optimisation. The results, given in the table, also demonstrate better risk-adjusted performance.

The core portfolio of the test case consists of five core and five satellite assets. The core assets are global and emerging market equity as well as government, corporate and high-yield bonds. The satellite assets are commodities and hedge fund indices: fixed-income arbitrage, global macro, short selling and CTA/managed futures. Single satellite investments are restricted to a maximum of 5% of the portfolio and the total satellite portfolio is restricted to 20% of the portfolio. Transaction costs are set to five basis points for the core assets and to 100 for satellites, and the portfolio turnover has to be less than 50% per annum.

The performance of monthly-optimised portfolios is represented in the table for the period from January 2003 to April 2012, based on the past 72-month periods. Or, more precisely, the first optimisation is done with data from January 1997 to December 2002 and the portfolio weights for December 2002 are used to calculate the portfolio return for January 2003, based on the asset returns of January 2003. Then the same procedure is applied for each month until April 2012.

All three procedures execute risk parity optimization: ‘Markowitz’ within a mean-variance and covariance context and ‘Expected shortfall’ and ‘Omega’ within a non-normal return distributions and tail-dependency framework. Looking at the cumulated returns of the upper panel ‘core portfolio only’ the Markowitz optimisation delivers higher returns than expected shortfall, but lower returns than Omega. However, the risk measured by expected shortfall is highest for the Markowitz portfolios, and hence the risk-adjusted performance is lowest for the Markowitz portfolio.

These results suggest that risk-parity-optimal portfolios with risk measures based on non-normal returns and modelling fat tails deliver better risk-adjusted performance than mean-variance optimal portfolios.

The lower panel of the table exhibits the results for the combined core-satellite portfolio. Markowitz portfolios have higher returns, but the other two portfolios have considerably lower risks and better risk-adjusted performances.

While these results cannot be generalised, they suggest that the use of risk measures based on non-normal distributions and a measurement of the tail dependence with copulas is promising. Furthermore, the core-satellite separation presented is advantageous. It allows the use of risk parity optimisation for core assets, and at the same time it permits to integrate satellite investments with proven alpha capacity and beta exposures into the optimisation. Therefore, this approach is particularly useful when integrating alternative investments into an existing portfolio, helping to optimise its share with respect to a core portfolio and to other alternative assets.

Peter Meier is head of the Centre for Alternative Investment and Risk Management; Andreas Ruckstuhl is professor of applied statistics in the Institute of Data Analysis and Process Design; and Marc Weibel works on the quantitative risk management project at the School of Engineering, all at Zurich University of Applied Sciences

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