Abstract
We extend the analysis of investment strategies derived from penalized quantile regression models, introducing alternative approaches to improve state– of– art asset allocation rules. We make use of the post– model– selection estimation, which builds on two important choices: the specification of the penalty function and the selection of the optimal tuning parameter. Therefore, we first investigate whether and to what extent the performance of a given portfolio changes when moving from convex to nonconvex penalty functions. Second, we compare different methods to select the optimal tuning parameter, which controls the intensity of the penalization. Empirical analyses on real– world data show that these alternative methods outperform the standard LASSO, providing improvements in terms of risk, risk– adjusted return and portfolio concentration. This evidence becomes stronger when focusing on extreme risk, which is strictly linked to quantile regression.
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Notes
To be precise, our dataset includes \(Q>T\) weekly returns for each stock. However, we divide our time series into \(Q-T\) equally sized subsamples, each of which includes T weeks, to implement a rolling window procedure in the empirical analysis (see Sect. 3 for additional details). The methods described in this section refer to a generic sample that includes the interval [1, T]; that is, the first subsample in the rolling window scheme. However, the theory described in this section equally applies to the remaining subsamples spanning the intervals \([2,T+1]\),...,\([Q-T+1,Q]\).
We recover the value– weighted returns of the 49 Industry Portfolios and 100 Portfolios Formed on Size and Book– to– Market from the library provided by Kenneth R. French, available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/. Starting from the original daily data, we compute the weekly returns for both datasets.
The data are recovered from Thomson Reuters Datastream.
By doing so, we exclude from the SP100 and SP500 datasets a subset of constituents, due to poor performance, mergers and acquisitions (M&A) or because some institutions entered the market more recently. Therefore, we could have a form of survivorship bias. Nevertheless, as highlighted by Bonaccolto et al. (2018), our analysis does not compare stock indices with allocation strategies. Therefore, our results do not suffer from any bias associated with the absence of adherence to the index basket, as the competitive asset allocation strategies are consistently applied over the same investment universe.
The in– sample results are available upon request.
The results obtained with the 100P dataset and for both sizes of the estimation window (T) are similar and available upon request.
References
Acerbi C, Tasche D (2002) Expected shortfall: a natural coherent alternative to value at risk. Econom Notes 31(2):379–388
Artzner P, Delbaen F, Eber J, Heath D (1999) Coherent measures of risk. Math Financ 9(3):203–228
Assa H (2015) Risk management under a prudential policy. Decis Econom Financ 38(2):217–230
Banz RW (1981) The relationship between return and market value of common stocks. J Financ Econom 9(1):3–18
Bassett G, Koenker R, Kordas G (2004) Pessimistic portfolio allocation and Choquet expected utility. J Financ Econom 2(4):477–492
Belloni A, Chernozhukov V (2011) L1-penalized quantile regression in high-dimensional sparse models. Ann Stat 39(1):82–130
Bonaccolto G, Caporin M, Paterlini S (2018) Asset allocation strategies based on penalized quantile regression. Comput Manag Sci 15(1):1–32
Bonaccolto G, Paterlini S (2020) Developing new portfolio strategies by aggregation. Ann Operat Res 292(2):933–971
Brodie J, Daubechies I, Mol CD, Giannone D, Loris I (2009) Sparse and stable Markowitz portfolios. PNAS 106(30):12267–12272
Chen SX (2008) Nonparametric estimation of expected shortfall. J Financi Econom 6(1):87–107
Ciliberti S, Kondor I, Mézard M (2007) On the feasibility of portfolio optimization under expected shortfall. Quant Financ 7(4):389–396
DeMiguel V, Garlappi L, Nogales FJ, Uppal R (2009) A generalized approach to portfolio optimization: improving performance by constraining portfolio norms. Manag Sci 55(5):798–812
DeMiguel V, Garlappi L, Uppal R (2009) Optimal versus naive diversification: how inefficient is the 1/N portfolio strategy? Rev Financ Stud 22:1915–1953
Duchin R, Levy H (2009) Markowitz versus the talmudic portfolio diversification strategies. J Portf Manag 35:71–74
Fan J, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 96(456):1348–1360
Fan J, Zhang J, Yu K (2012) Vast portfolio selection with gross-exposure constraints. J Am Stat Assoc 107(498):592–606
Fastrich B, Paterlini S, Winker P (2015) Constructing optimal sparse portfolios using regularization methods. Comput Manag Sci 12(3):417–434
Gasso G, Rakotomamonjy A, Canu S (2009) Recovering sparse signals with a certain family of nonconvex penalties and DC programming. IEEE Trans Signal Process 57(12):4686–4698
Gijbels I, Herrmann K (2018) Optimal expected-shortfall portfolio selection with copula-induced dependence. Appl Math Financ 25(1):66–106
Gilli M, Këllezi E (2002). A global optimization heuristic for portfolio choice with VaR and Expected Shortfall. In E. J. Kontoghiorghes, B. Rustem, and S. Siokos (Eds.), Computational Methods in Decision-Making, Economics and Finance. Applied Optimization, Volume 74, pp. 167–183. Boston, MA: Springer
Giuzio M (2017) Genetic algorithm versus classical methods in sparse index tracking. Decis Econom Financ 40(1–2):243–256
Giuzio M, Paterlini S (2018) Un–diversifying during crises: is it a good idea? Comput Manag Sci 16(3):401–432
Hastie T, Tibshirani R, Friedman J (2009) The elements of statistical learning. Springer, Berlin
Hautsch N, Schaumburg J, Schienle M (2014) Financial network systemic risk contributions. Rev Financ 19(2):685–738
Klein RW, Bawa VS (1977) The effect of limited information and estimation risk on optimal portfolio diversification. J Financ Econom 5(1):89–111
Koenker R (2005) Quantile regression. Number 38. Cambridge University press
Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46(1):33–50
Kratz M, Lok YH, McNeil AJ (2018) Multinomial VaR backtests: a simple implicit approach to backtesting expected shortfall. J Bank Financ 88:393–407
Kremer PJ, Talmaciu A, Paterlini S (2018) Risk minimization in multi–factor portfolios: what is the best strategy? Ann Operat Res 266(1–2):255–291
Ledoit O, Wolf M (2008) Robust performance hypothesis testing with the Sharpe ratio. J Empiric Financ 15(5):850–859
Lee ER, Noh H, Park BU (2014) Model selection via Bayesian information criterion for quantile regression models. J Am Stat Assoc 109(505):216–229
Leeb H, Pötscher BM (2008a) Sparse estimators and the oracle property, or the return of Hodges’ estimator. J Econom 142(1):201–211
Leeb H, Pötscher BM (2008b) Can one estimate the unconditional distribution of post-model-selection estimators? Econom Theor 24(2):338–376
Li Y, Zhu J (2008) L1-norm quantile regression. J Comput Graph Stat 17(1):163–185
Markowitz H (1952) Portfolio selection. The. J Financ 7:77–91
Patton AJ, Ziegel JF, Chen R (2019) Dynamic semiparametric models for expected shortfall (and Value–at–Risk). J Econom 211(2):388–413
Peng B, Wang L (2015) An iterative coordinate descent algorithm for high-dimensional nonconvex penalized quantile regression. J Comput Graph Stati 24(3):676–694
Rockafellar R, Uryasev S (2000) Optimization of conditional VaR. J Risk 2(3):21–41
Stattman D (1980) Book values and expected stock returns. Unpublished M.B.A. honors paper (University of Chicago, Chicago, IL)
Tibshirani R (1996) Regression analysis and selection via the LASSO. J Royal Stat Soc, Series B 58(1):267–288
Tu J, Zhou G (2011) Markowitz meets Talmud: a combination of sophisticated and naive diversification strategies. J Financ Econom 99(1):204–215
Yamai Y, Yoshiba T (2005) Value–at–risk versus expected shortfall: a practical perspective. J Bank Financ 29(4):997–1015
Yen Y, Yen T (2014) Solving norm constrained portfolio optimization via coordinate-wise descent algorithms. Comput Stat Data Analy 76:737–759
Zhang CH (2010) Nearly unbiased variable selection under minimax concave penalty. Ann stat 38(2):894–942
Acknowledgements
The author would like to thank the editor, associate editor and two anonymous reviewers of this journal, as well as Massimiliano Caporin, Sandra Paterlini and Marina Töpfer for helpful comments and suggestions.
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Bonaccolto, G. Quantile– based portfolios: post– model– selection estimation with alternative specifications. Comput Manag Sci 18, 355–383 (2021). https://doi.org/10.1007/s10287-021-00396-7
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DOI: https://doi.org/10.1007/s10287-021-00396-7