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Un-diversifying during crises: Is it a good idea?

  • Margherita GiuzioEmail author
  • Sandra Paterlini
Original Paper
  • 17 Downloads

Abstract

High levels of correlation among financial assets and extreme losses are typical during crises. In such situations, investing in few assets might be a better choice than holding diversified portfolios. We show that constraining the sparse \(\ell _q\)-norm of portfolio weights automatically controls diversification and selects portfolios with a small number of active weights and low risk, in presence of high correlation and volatility. We highlight the diversification relationships between the minimum variance portfolio, risk budgeting strategies and diversification-constrained portfolios. Finally, we show empirically that the \(\ell _q\)-strategy can successfully cope with bear markets by shrinking portfolio weights and total amount of shorting.

Keywords

Diversification Regularization methods Minimum variance Sparsity 

Mathematics Subject Classification

91G10 91G70 91-08 

Notes

Acknowledgements

We would like to thank the two anonymous referees and the Associate Editor for providing us with constructive and detailed comments that have improved the quality of our paper. Sandra Paterlini gratefully acknowledges financial support from ICT COST Action IC1408 “Computationally-intensive methods for the robust analysis of non-standard data”.

References

  1. Bauer D, Zanjani G (2016) The marginal cost of risk, risk measures, and capital allocation. Manag Sci 62:1431–1457CrossRefGoogle Scholar
  2. Behr P, Guettler A, Miebs F (2013) On portfolio optimization: imposing the right constraint. J Bank Finance 37:1232–1242CrossRefGoogle Scholar
  3. Benoit S, Colletaz G, Hurlin C, Perignon C (2013) A theoretical and empirical comparison of systemic risk measures. HEC Paris Research Paper (FIN-2014-1030)Google Scholar
  4. Bodie Z, Kane A, Marcus A (1999) Investments, 4th edn. Irwin/McGraw-Hill, BostonGoogle Scholar
  5. Boyle P, Garlappi L, Uppal R, Wang T (2012) Keynes meets Markowitz: the trade-off between familiarity and diversification. Manag Sci 58:253–272CrossRefGoogle Scholar
  6. Brands S, Brown S, Gallagher D (2005) Portfolio concentration and investment manager performance. Int Rev Finance 5:149–174CrossRefGoogle Scholar
  7. Brodie J, Daubechies I, De Mol C, Giannone D, Loris I (2009) Sparse and stable Markowitz portfolios. Proc Natl Acad Sci 106:12267–12272CrossRefGoogle Scholar
  8. Bruder B, Roncalli T (2012) Managing risk exposures using the risk budgeting approach. Working paperGoogle Scholar
  9. Buffett W (1979) Chairman’s Letter. http://www.berkshirehathaway.com/letters/1979.html
  10. Carrasco M, Noumon N (2012) Optimal portfolio selection using regularization. Working paper, University of MontrealGoogle Scholar
  11. Cazalet Z, Grison P, Roncalli T (2014) The smart beta indexing puzzle. J Index Invest 5:97–119CrossRefGoogle Scholar
  12. Chen C, Li X, Tolman C, Wang S, Ye Y (2013) Sparse portfolio selection via quasi-norm regularization, preprint. arXiv:1312.6350
  13. Chopra VK, Ziemba WT (1993) The effect of errors in means, variances, and covariances on optimal portfolio choice. J Portf Manag 19:6–11CrossRefGoogle Scholar
  14. Choueifaty Y, Coignard Y (2008) Toward maximum diversification. J Portf Manag 34:40–51CrossRefGoogle Scholar
  15. Daniel K, Grinblatt M, Titman S, Werme R (1997) Measuring mutual fund performance with characteristic-based benchmarks. J Finance 52:1035–1058CrossRefGoogle Scholar
  16. De Miguel V, Nogales FJ (2009) Portfolio selection with robust estimation. Oper Res 57:560–577CrossRefGoogle Scholar
  17. De Miguel V, Garlappi L, Nogales F, Uppal R (2009a) A generalized approach to portfolio optimization: improving performance by constraining portfolio norm. Manag Sci 55:798–812CrossRefGoogle Scholar
  18. De Miguel V, Garlappi L, Uppal R (2009b) Optimal versus naive diversification: how inefficient is the 1/n portfolio strategy? Rev Financ Stud 22(5):1915–1953CrossRefGoogle Scholar
  19. Doganoglu T, Hartz C, Mittnik S (2007) Portfolio optimization when risk factors are conditionally varying and heavy tailed. Comput Econ 29:333–354CrossRefGoogle Scholar
  20. Fan J, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 96:1348–1360CrossRefGoogle Scholar
  21. Fan J, Zhang J, Yu K (2012) Vast portfolio selection with gross-exposure constraints. J Am Stat Assoc 107:592–606CrossRefGoogle Scholar
  22. Fastrich B, Paterlini S, Winker P (2014) Cardinality versus q-norm constraints for index tracking. Quant Finance 14:2019–2032CrossRefGoogle Scholar
  23. Fastrich B, Paterlini S, Winker P (2015) Constructing optimal sparse portfolios using regularization methods. Comput Manag Sci 12:417–434CrossRefGoogle Scholar
  24. Figueiredo M, Nowak R, Wright S (2007) Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J Sel Top Signal Process 1:586–597CrossRefGoogle Scholar
  25. Frank I, Friedman J (1993) A statistical view of some chemometrics regression tools. Technometrics 35:109–135CrossRefGoogle Scholar
  26. Gasso G, Rakotomamonjy A, Canu S (2009) Recovering sparse signals with a certain family of nonconvex penalties and DC programming. IEEE Trans Signal Process 57:4686–4698CrossRefGoogle Scholar
  27. Giuzio M (2017) Genetic algorithm versus classical methods in sparse index tracking. Decis Econ Finance 40:243–256CrossRefGoogle Scholar
  28. Goto S, Xu Y (2015) Improving mean variance optimization through sparse hedging restrictions. J Financ Quant Anal 50:1415–1441CrossRefGoogle Scholar
  29. Grinold RC, Kahn R (1999) Active portfolio management, 2nd edn. McGraw-Hill, New YorkGoogle Scholar
  30. Guidolin M, Rinaldi F (2013) Ambiguity in asset pricing and portfolio choice: a review of the literature. Theory Decis 74:183–217CrossRefGoogle Scholar
  31. Huang J, Horowitz J, Ma S (2008) Asymptotic properties of bridge estimators in sparse high-dimensional regression models. Ann Stat 30:587–613CrossRefGoogle Scholar
  32. Jagannathan R, Ma T (2003) Risk reduction in large portfolios: why imposing the wrong constraints helps. J Finance 58:1651–1684CrossRefGoogle Scholar
  33. Kacperczyk M, Sialm C, Zheng L (2005) On the industry concentration of actively managed equity mutual funds. J Finance 60:1983–2011CrossRefGoogle Scholar
  34. Knight K, Fu W (2000) Asymptotics for lasso-type estimators. Ann Stat 28:1356–1378CrossRefGoogle Scholar
  35. Kolm PN, Tütüncü R, Fabozzi F (2014) 60 years following Harry Markowitz’s contribution to portfolio theory and operations research. Eur J Oper Res 234:343–582CrossRefGoogle Scholar
  36. Kotkatvuori-Örnberg J, Nikkinen J, Äijö J (2013) Stock market correlations during the financial crisis in 2008–2009: evidence from 50 equity markets. Int Rev Financ Anal 28:70–78CrossRefGoogle Scholar
  37. Ledoit O, Wolf M (2004) A well-conditioned estimator for large-dimensional covariance matrices. J Multivar Anal 88:365–411CrossRefGoogle Scholar
  38. Ledoit O, Wolf M (2012) Nonlinear shrinkage estimation of large-dimensional covariance matrices. Ann Stat 40:1024–1060CrossRefGoogle Scholar
  39. Maillard S, Roncalli T, Teïletche J (2010) The properties of equally weighted risk contribution portfolios. J Portf Manag 36:60–70CrossRefGoogle Scholar
  40. Mainik G, Mitov G, Rüschendorf L (2015) Portfolio optimization for heavy-tailed assets: extreme risk index vs. Markowitz. J Empir Finance 32:115–134CrossRefGoogle Scholar
  41. Markowitz H (1952) Portfolio selection. J Finance 7:77–91Google Scholar
  42. Merton R (1980) On estimating the expected return on the market: an exploratory investigation. J Financ Econ 8:323–361CrossRefGoogle Scholar
  43. Michaud R (1989) The Markowitz optimization enigma: is optimized optimal? Financ Anal J 45:31–45CrossRefGoogle Scholar
  44. Murphy KP (2012) Machine learning: a probabilistic perspective. MIT Press, CambridgeGoogle Scholar
  45. Statman M (1987) How many stocks make a diversified portfolio? J Financ Quant Anal 22:353–363CrossRefGoogle Scholar
  46. Tibshirani R, Saunders M, Rosset S, Zhu J, Knight K (2005) Sparsity and smoothness via the fused lasso. J R Stat Soc Ser B 67:91–108CrossRefGoogle Scholar
  47. Tütüncü R, Koenig M (2004) Robust asset allocation. Ann Oper Res 132:157–187CrossRefGoogle Scholar
  48. Weston J, Elisseeff A, Schölkopf B (2003) Use of the zero-norm with linear models and kernel methods. J Mach Learn Res 3:1439–1461Google Scholar
  49. Xing X, Hub J, Yang Y (2014) Robust minimum variance portfolio with L-infinity constraints. J Bank Finance 46:107–117CrossRefGoogle Scholar
  50. Yen Y, Yen T (2014) Solving norm constrained portfolio optimization via coordinate-wise descent algorithms. Comput Stat Data Anal 76:737–759CrossRefGoogle Scholar
  51. You L, Daigler R (2010) Is international diversification really beneficial? J Bank Finance 34:163–173CrossRefGoogle Scholar
  52. Zou H (2006) The adaptive lasso and its oracle properties. J Am Stat Assoc 101:1418–1429CrossRefGoogle Scholar
  53. Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J R Stat Soc 67:301–320CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.European Central BankFrankfurtGermany
  2. 2.Department of Finance and AccountingEBS Universität für Wirtschaft und RechtWiesbadenGermany
  3. 3.Department of Economics and ManagementUniversity of TrentoTrentoItaly

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