Noise sensitivity of portfolio selection under various risk measures

Abstract

We study the sensitivity to estimation error of portfolios optimized under various risk measures, including variance, absolute deviation, expected shortfall and maximal loss. We introduce a measure of portfolio sensitivity and test the various risk measures by considering simulated portfolios of varying sizes N and for different lengths T of the time series. We find that the effect of noise is very strong in all the investigated cases, asymptotically it only depends on the ratio N/T , and diverges at a critical value of N/T , that depends on the risk measure in question. This divergence is the manifestation of a phase transition, analogous to the algorithmic phase transitions recently discovered in a number of hard computational problems. The transition is accompanied by a number of critical phenomena, including the divergent sample to sample fluctuations of portfolio weights. While the optimization under variance and mean absolute deviation is always feasible below the critical value of N/T , expected shortfall and maximal loss display a probabilistic feasibility problem, in that they can become unbounded from below already for small values of the ratio N/T , and then no solution exists to the optimization problem under these risk measures. Although powerful filtering techniques exist for the mitigation of the above instability in the case of variance, our findings point to the necessity of developing similar filtering procedures adapted to the other risk measures where they are much less developed or nonexistent. Another important message of this study is that the requirement of robustness (noise-tolerance) should be given special attention when considering the theoretical and practical criteria to be imposed on a risk measure.

Replica approach to mean-variance portfolio optimization

“…The value of this exponent seems to be the same in a number of similar problems: it was obtained by numerical simulations in the case of portfolio optimization for the variance as risk measure in refs. [22,25,27,36], for Expected Shortfall in [47,49,50,52], for the minimax risk measure in [52], even for non-stationary underlying processes in a GARCH framework [37]. The invariance of this exponent for these different "Hamiltonians" is analogous to the independence of the usual critical exponents of microscopic details.…”

“…In addition to being a well defined portfolio optimization problem [11], the minimax has been shown [36] to be equivalent to the following problem in high dimensional random geometry: What is the probability that T random hyperplanes thrown into and N -dimensional space make a convex polytope? The answer was found by Schmidt and Mattheiss [68] for any N and T .…”

“…The independence of the estimation error from the covariances and expected returns is the manifestation of a surprising degree of universality, analogous to the independence of the critical point of the minimax risk measure from the underlying probability distribution (see [36]), but also to the universality discovered in a wide class of high dimensional random geometric phase transitions by Donoho and Tanner [57].…”

Maximum Expected Utility of Markovian Predicted Wealth

“…Moreover we also observe that we get better results using three states. We believe that this fact can be justified by a more robust approximation of the forecasted final wealth (see Papp et al (2005), Kondor et al (2007)). These results are further confirmed by Fig.…”

“…In the empirical comparisons, we consider the optimal allocation among 20 assets components of the Dow Jones Industrials 1 on the period from 1/3/1985 till 5/1/2008 for a total of 5884 daily observations. The work of Kondor et al (2007) on the sensitivity to estimation error of portfolios optimized under various risk measures suggests that we need a large number of observations when we want to propose portfolio models considering rare events. The output of this analysis is represented in Tables 2, 3, Fig.…”