LpL_pLp regularized portfolio optimization (original) (raw)
Related papers
Lp Regularized Portfolio Optimization
SSRN Electronic Journal, 2014
Investors who optimize their portfolios under any of the coherent risk measures are naturally led to regularized portfolio optimization when they take into account the impact their trades make on the market. We show here that the impact function determines which regularizer is used. We also show that any regularizer based on the norm L p with p > 1 makes the sensitivity of coherent risk measures to estimation error disappear, while regularizers with p < 1 do not. The L 1 norm represents a border case: its "soft" implementation does not remove the instability, but rather shifts its locus, whereas its "hard" implementation (equivalent to a ban on short selling) eliminates it. We demonstrate these effects on the important special case of Expected Shortfall (ES) that is on its way to becoming the next global regulatory market risk measure.
Liquidity Risk and Instabilities in Portfolio Optimization
International Journal of Theoretical and Applied Finance, 2016
We show that including a term which accounts for finite liquidity in portfolio optimization naturally mitigates the instabilities that arise in the estimation of coherent risk measures on finite samples. This is because taking into account the impact of trading in the market is mathematically equivalent to introducing a regularization on the risk measure. We show here that the impact function determines which regularizer is to be used. We also show that any regularizer based on the norm [Formula: see text] with [Formula: see text] makes the sensitivity of coherent risk measures to estimation error disappear, while regularizers with [Formula: see text] do not. The [Formula: see text] norm represents a border case: its “soft” implementation does not remove the instability, but rather shifts its locus, whereas its “hard” implementation (including hard limits or a ban on short selling) eliminates it. We demonstrate these effects on the important special case of expected shortfall (ES) w...
Optimal liquidation strategies regularize portfolio selection
The European Journal of Finance, 2013
We consider the problem of portfolio optimization in the presence of market impact, and derive optimal liquidation strategies. We discuss in detail the problem of finding the optimal portfolio under Expected Shortfall (ES) in the case of linear market impact. We show that, once market impact is taken into account, a regularized version of the usual optimization problem naturally emerges. We characterize the typical behavior of the optimal liquidation strategies, in the limit of large portfolio sizes, and show how the market impact removes the instability of ES in this context.
Instability of Portfolio Optimization Under Coherent Risk Measures
Advances in Complex Systems, 2010
It is shown that the axioms for coherent risk measures imply that whenever there is a pair of portfolios such that one of them dominates the other in a given sample (which happens with finite probability even for large samples), then there is no optimal portfolio under any coherent measure on that sample, and the risk measure diverges to minus infinity. This instability was first discovered in the special example of Expected Shortfall which is used here both as an illustration and as a springboard for generalization.
Bias-variance trade-off in portfolio optimization under expected shortfall with ℓ 2 regularization
LSE Research Online Documents on Economics, 2019
The optimization of a large random portfolio under the Expected Shortfall risk measure with an 2 regularizer is carried out by analytical calculation. The regularizer reins in the large sample fluctuations and the concomitant divergent estimation error, and eliminates the phase transition where this error would otherwise blow up. In the data-dominated region, where the number of different assets in the portfolio is much less than the length of the available time series, the regularizer plays a negligible role, while in the (much more frequently occurring in practice) opposite limit, where the samples are comparable or even small compared to the number of different assets, the optimum is almost entirely determined by the regularizer. Our results clearly show that the transition region between these two extremes is relatively narrow, and it is only here that one can meaningfully speak of a trade-off between fluctuations and bias.
Bias-Variance Trade-Off in Portfolio Optimization under Expected Shortfall with 2 Regularization
SSRN Electronic Journal, 2017
The optimization of a large random portfolio under the Expected Shortfall risk measure with an 2 regularizer is carried out by analytical calculation. The regularizer reins in the large sample fluctuations and the concomitant divergent estimation error, and eliminates the phase transition where this error would otherwise blow up. In the data-dominated region, where the number N of different assets in the portfolio is much less than the length T of the available time series, the regularizer plays a negligible role even if its strength η is large, while in the opposite limit, where the size of samples is comparable to, or even smaller than the number of assets, the optimum is almost entirely determined by the regularizer. We construct the contour map of estimation error on the N/T vs. η plane and find that for a given value of the estimation error the gain in N/T due to the regularizer can reach a factor of about 4 for a sufficiently strong regularizer.
On the feasibility of portfolio optimization under expected shortfall
Quantitative Finance, 2007
We address the problem of portfolio optimization under the simplest coherent risk measure, i.e. the expected shortfall. As it is well known, one can map this problem into a linear programming setting. For some values of the external parameters, when the available time series is too short, the portfolio optimization is ill posed because it leads to unbounded positions, infinitely short on some assets and infinitely long on some others. As first observed by Kondor and coworkers, this phenomenon is actually a phase transition. We investigate the nature of this transition by means of a replica approach.
Sparse Portfolio Selection via Quasi-Norm Regularization
2013
In this paper, we propose ellp\ell_pellp-norm regularized models to seek near-optimal sparse portfolios. These sparse solutions reduce the complexity of portfolio implementation and management. Theoretical results are established to guarantee the sparsity of the second-order KKT points of the ellp\ell_pellp-norm regularized models. More interestingly, we present a theory that relates sparsity of the KKT points with Projected correlation and Projected Sharpe ratio. We also design an interior point algorithm to obtain an approximate second-order KKT solution of the ellp\ell_pellp-norm models in polynomial time with a fixed error tolerance, and then test our ellp\ell_pellp-norm modes on S&P 500 (2008-2012) data and international market data.\ The computational results illustrate that the ellp\ell_pellp-norm regularized models can generate portfolios of any desired sparsity with portfolio variance and portfolio return comparable to those of the unregularized Markowitz model with cardinality constraint. Our analysis of a combined model lead us to conclude that sparsity is not directly related to overfitting at all. Instead, we find that sparsity moderates overfitting only indirectly. A combined ell_1\ell_1ell1-$\ell_p$ model shows that the proper choose of leverage, which is the amount of additional buying-power generated by selling short can mitigate overfitting; A combined ell2\ell_2ell2-$\ell_p$ model is able to produce extremely high performing portfolios that exceeded the 1/N strategy and all ell1\ell_1ell1 and ell2\ell_2ell_2 regularized portfolios.
arXiv (Cornell University), 2016
The optimization of a large random portfolio under the Expected Shortfall risk measure with an 2 regularizer is carried out by analytical calculation. The regularizer reins in the large sample fluctuations and the concomitant divergent estimation error, and eliminates the phase transition where this error would otherwise blow up. In the data-dominated region, where the number N of different assets in the portfolio is much less than the length T of the available time series, the regularizer plays a negligible role even if its strength η is large, while in the opposite limit, where the size of samples is comparable to, or even smaller than the number of assets, the optimum is almost entirely determined by the regularizer. We construct the contour map of estimation error on the N/T vs. η plane and find that for a given value of the estimation error the gain in N/T due to the regularizer can reach a factor of about 4 for a sufficiently strong regularizer.
Feasibility of Portfolio Optimization under Coherent Risk Measures
2008
It is shown that the axioms for coherent risk measures imply that whenever there is an asset in a portfolio that dominates the others in a given sample (which happens with finite probability even for large samples), then this portfolio cannot be optimized under any coherent measure on that sample, and the risk measure diverges to minus infinity. This instability