# Asset allocation strategies based on penalized quantile regression

## Abstract

It is well known that the quantile regression model used as an asset allocation tool minimizes the portfolio extreme risk whenever the attention is placed on the lower quantiles of the response variable. By considering the entire conditional distribution of the dependent variable, we show that it is possible to obtain further benefits by optimizing different risk and performance indicators. In particular, we introduce a risk-adjusted profitability measure, useful in evaluating financial portfolios from a ‘cautiously optimistic’ perspective, as the reward contribution is net of the most favorable outcomes. Moreover, as we consider large portfolios, we also cope with the dimensionality issue by introducing an \(\ell _1\)-norm penalty on the assets’ weights.

## Keywords

Quantile regression \(\ell _1\)-Norm penalty Asset allocation## Notes

### Acknowledgements

The authors thank the participants of the “9th Financial Risks International Forum” in Paris, organised by Institut Louis Bachelier, the “9th International Conference on Computational and Financial Econometrics” in London, the “SOFINE-CEQURA Spring Junior Research Workshop” in Nesselwang, the “Financial Econometrics and Empirical Asset Pricing Conference” in Lancaster, the seminar organized by the University of Palermo for the helpful comments and stimulating discussions. M. Caporin acknowledges financial support from the European Union, the Seventh Framework Program FP7/2007–2013 under Grant Agreement SYRTO-SSH-2012-320270, the MIUR PRIN project MISURA-Multivariate Statistical Models for Risk Assessment, the Global Risk Institute in Financial Services and the Louis Bachelier Institute. S. Paterlini acknowledges financial support from ICT COST ACTION 1408-CRONOS.

## Supplementary material

## References

- Acerbi C, Tasche D (2002) Expected shortfall: a natural coherent alternative to value at risk. Econ Notes 31(2):379–388CrossRefGoogle Scholar
- Aït-Sahalia Y, Xiu D (2015) Principal component estimation of a large covariance matrix with high-frequency data. Technical report, Princeton University and The University of ChicagoGoogle Scholar
- Alexander G, Baptista AM (2002) Economic implications of using a mean–var model for portfolio selection: a comparison with mean–variance analysis. J Econ Dyn Control 26(7–8):1159–1193CrossRefGoogle Scholar
- Ando T, Bai J (2015) Asset pricing with a general multifactor structure. J Financ Econom 13(3):556–604CrossRefGoogle Scholar
- Artzner P, Delbaen F, Eber J, Heath D (1999) Coherent measures of risk. Math Finance 9(3):203–228CrossRefGoogle Scholar
- Azzalini A (2014) The skew-normal and related families. IMS monograph series. Cambridge University Press, CambridgeGoogle Scholar
- Basak S, Shapiro A (2001) Value-at-risk based risk management: optimal policies and asset prices. Rev Financ Stud 14(2):371–405CrossRefGoogle Scholar
- Bassett G, Koenker R, Kordas G (2004) Pessimistic portfolio allocation and choquet expected utility. J Financ Econom 2(4):477–492CrossRefGoogle Scholar
- Belloni A, Chernozhukov V (2011) L1-penalized quantile regression in high-dimensional sparse models. Ann Stat 39(1):82–130CrossRefGoogle Scholar
- Britten-Jones M (1999) The sampling error in estimates of mean–variance efficient portfolio weights. J Finance 54(2):655–671CrossRefGoogle Scholar
- Brodie M (1993) Computing efficient frontiers using estimated parameters. Ann Oper Res 45(1):21–58CrossRefGoogle Scholar
- Brodie J, Daubechies I, Mol CD, Giannone D, Loris I (2009) Sparse and stable markowitz portfolios. PNAS 106(30):12267–12272CrossRefGoogle Scholar
- Caporin M, Jannin G, Lisi F, Maillet B (2014) A survey on the four families of performance measures. J Econ Surv 28(5):917–942CrossRefGoogle Scholar
- Chen SX (2008) Nonparametric estimation of expected shortfall. J Financ Econom 6(1):87–107CrossRefGoogle Scholar
- Chopra VK, Ziemba T (1993) The effect of errors in means, variances and covariances on optimal portfolio choice. J Portfolio Manag 19(2):6–11CrossRefGoogle Scholar
- Ciliberti S, Kondor I, Mezard M (2007) On the feasibility of portfolio optimization under expected shortfall. Quant Finance 7(4):389–396CrossRefGoogle Scholar
- Cont R (2001) Empirical properties of asset returns: stylized facts and statistical issues. Quant Finance 1(2):223–236CrossRefGoogle Scholar
- Davino C, Furno M, Vistocco D (2014) Quantile regression: theory and applications. Wiley, LondonGoogle Scholar
- 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):798–812CrossRefGoogle Scholar
- Fan J, Zhang J, Yu K (2012) Vast portfolio selection with gross-exposure constraints. J Am Stat Assoc 107(498):592–606CrossRefGoogle Scholar
- Fan J, Furger A, Xiu D (2016) Incorporating global industrial classification standard into portfolio allocation: a simple factor-based large covariance matrix estimator with high frequency data. J Bus Econ Stat. doi: 10.1080/07350015.2015.1052458 Google Scholar
- Farinelli S, Ferreira M, Rossello D, Thoeny M, Tibiletti L (2008) Beyond sharpe ratio: optimal asset allocation using different performance ratios. J Bank Finance 32(10):2057–2063CrossRefGoogle Scholar
- Fastrich B, Paterlini S, Winker P (2015) Constructing optimal sparse portfolios using regularization methods. Comput Manag Sci 12(3):417–434CrossRefGoogle Scholar
- Fitzenberger B, Winker P (2007) Improving the computation of censored quantile regressions. Comput Stat Data Anal 1(52):88–108CrossRefGoogle Scholar
- Gotoh J, Takeda A (2011) On the role of norm constraints in portfolio selection. Comput Manag Sci 8(4):323–353CrossRefGoogle Scholar
- Härdle WK, Nasekin S, Chuen DLK, Fai PK (2014) Tedas—tail event driven asset allocation. SFB 649 discussion papers SFB649DP2014-032, Sonderforschungsbereich 649, Humboldt University, Berlin, GermanyGoogle Scholar
- Hastie T, Tibshirani R, Friedman J (2009) The elements of statistical learning. Springer, BerlinCrossRefGoogle Scholar
- Jagannathan R, Ma T (2003) Risk reduction in large portfolios: why imposing the wrong constraints helps. J Finance 54(4):1651–1683CrossRefGoogle Scholar
- Keating C, Shadwick WF (2002) A universal performance measure. The Finance Development Centre, LondonGoogle Scholar
- Kocherginsky M (2003) Extensions of the Markov chain marginal bootstrap. Ph.D. thesis, University of Illinois Urbana-ChampaignGoogle Scholar
- Koenker R (2005) Quantile regression, vol 38. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46(1):33–50CrossRefGoogle Scholar
- Konno H, Yamazaki H (1991) Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Manag Sci 37(5):519–531CrossRefGoogle Scholar
- Kourtis A, Dotsis G, Markellos RN (2012) Parameter uncertainty in portfolio selection: shrinking the inverse covariance matrix. J Bank Finance 36(9):2522–2531CrossRefGoogle Scholar
- Krokhmal P, Palmquist J, Uryasev S (2002) Portfolio optimization with conditional value-at-risk objective and constraints. J Risk 4(2):43–68CrossRefGoogle Scholar
- Ledoit O, Wolf M (2008) Robust performance hypothesis testing with the sharpe ratio. J Empir Finance 15:850–859CrossRefGoogle Scholar
- Li Y, Zhu J (2008) L1-norm quantile regression. J Comput Graph Stat 17(1):163–185CrossRefGoogle Scholar
- Lintner J (1965a) Security prices, risk and maximal gains from diversification. J Finance 20(4):587–615Google Scholar
- Lintner J (1965b) The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Rev Econ Stat 47(1):13–37CrossRefGoogle Scholar
- Mansini R, Ogryczak W, Speranza M (2007) Conditional value at risk and related linear programming models for portfolio optimization. Ann Oper Res 152(1):227–256CrossRefGoogle Scholar
- Markowitz H (1952) Portfolio selection. J Finance 7(1):77–91Google Scholar
- Mossin J (1966) Equilibrium in a capital asset market. Econometrica 35(4):768–783CrossRefGoogle Scholar
- Ortobelli S, Stoyanov S, Fabozzi F, Biglova F (2005) The proper use of risk measures in portfolio theory. Int J Theor Appl Finance 8(8):1107–1133CrossRefGoogle Scholar
- Rockafellar R, Uryasev S (2000) Optimization of conditional var. J Risk 2(3):21–41CrossRefGoogle Scholar
- Sharpe W (1964) Capital asset prices: a theory of market equilibrium under conditions of risk. J Finance 19(3):425–442Google Scholar
- Statman M (1987) How many stocks make a diversified portfolio. J Financ Quant Anal 22(3):353–363CrossRefGoogle Scholar
- Tian S, Yu Y, Guo H (2015) Variable selection and corporate bankruptcy forecasts. J Bank Finance 52:89–100CrossRefGoogle Scholar
- Tibshirani R (1996) Regression analysis and selection via the lasso. J R Stat Soc Ser B 58(1):267–288Google Scholar
- Xing X, Hu J, Yang Y (2014) Robust minimum variance portfolio with l-infinity constraints. J Bank Finance 46:107–117CrossRefGoogle Scholar
- Yen Y, Yen T (2014) Solving norm constrained portfolio optimization via coordinate-wise descent algorithms. Comput Stat Data Anal 76:737–759CrossRefGoogle Scholar