Risk forecasting models and optimal portfolio selection

Abstract

This study analyses, from an investor's perspective, the performance of several risk forecasting models in obtaining optimal portfolios. The plausibility of the homoscedastic hypothesis implied in the classical Markowitz model is dicussed and more general models which take into account assymetry and time varying risk are analysed. Specifically, it studies whether ARCH-type based models obtain portfolios whose risk-adjusted returns exceed those of the classical Markowitz model. The same analysis is performed with models based on the Lower Partial Moment (LPM) which take into account the assymetry in the distribution of returns. The results suggest that none of the models achieve a clearly superior average performance. It is also found that models based on semivariance perform as well as those based on the variance, but not better than, even if the evaluation criterion is based on the Reward-to-Semivariance ratio. When attention turns to the analysis of worst case performance, the results are clearly different. Models which employ LPM with a high degree of risk aversion (n>2) as the risk measure are consistently superior to those which employ a symmetric measure, either homoscedastic or heteroscedastic.

Acknowledgements

We thank David Nawrocki, for helpful comments and suggestions. The contents of this paper are the sole responsibility of the authors.

Notes

It is also obvious that particular investors demand more sophisticated financial products as well as risk control systems since the crash of 1987. Concepts such as ‘implied volatility' and ‘Value at Risk' are now terms commonly used by even the most modest investor.

Levy and Markowitz (1979), Pulley (1981, 1985), Kroll et al. (1984), among others.

Note that the variance, employed as a risk measures does not have this property, it is symmetric and, consequently, penalizes both gains and losses.

See Michaud (1989) and Best and Grauer (1991) among others, for a review.

See, e.g. Merton (1980) and Nelson (1992).

This view was first proposed by Roy (1952) who shows that investors try to obtain returns over a minimum or catastrofic risk, this is the principle of Safety First.

For n smaller than one ‘risk lovers' is considered, n = 1 represents ‘risk neutrals' and for n higher than one it is considered that investors are ‘risk averse' (see Fishburn, 1977). Note that semivariance is equal to LPM with n equal to 2.

Note, again, that from this expression one can derive the cosemivariance, a particular case of LPM when the degree is equal to two.

Since this database is well known to the financial researchers a full description is not provided here.

The emerging markets analysed are Indonesia (INO), Korea (KOR), Malaysia (MAL), Philippines (PHI), Taiwan (TAW), Thailand (THA), Argentina (ARG), Brazil (BRA), Chile (CHE), Mexico (MEX), Greece (GRE), Jordan (JOR), and Turkey (TUR). The developed markets are Australia (AUL), Austria (AUT), Belgium (BEL), Canada (CAN), Denmark (DEN), Finland (FIN), France (FRA), Germany (GER), Hong Kong (HON), Ireland (IRE), Italy (ITA), Japan (JAP), Luxembourg (LUX), Netherlands (NET), Norway (NOR), Portugal (POR), Singapore (SIP), Spain (SPA), Sweden (SWE), Switzerland (SWI), United Kingdom (UKG) and United States (USA).

The Sharpe Ratio can be defined as follows:

where R_i is the return of portfolio i and r_f is the risk-free rate (3-month Treasury bills are employed) and σ _i is the standard deviation of portfolio i.

Another argument to employ this alternative performance measure is that the Sharpe ratio is a biased estimator (see e.g. Ang and Chua, 1979).

The Reward-to-Semivariability can be defined as: