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Tail Event Driven ASset allocation: evidence from equity and mutual funds’ markets

Abstract

The correlation structure across assets and opposite tail movements are essential to the asset allocation problem, since they determine the level of risk in a position. Correlation alone is not informative on the distributional details of the assets. Recently introduced TEDAS—Tail Event Driven ASset allocation approach determines the dependence between assets at different tail measures. TEDAS uses adaptive Lasso-based quantile regression in order to determine an active set of negative coefficients. Based on these active risk factors, an adjustment for intertemporal correlation is made. In this research, authors aim to develop TEDAS, by introducing three TEDAS modifications differing in allocation weights’ determination: a Cornish–Fisher Value-at-Risk minimization, Markowitz diversification rule or naïve equal weighting. TEDAS strategies significantly outperform other widely used allocation approaches on two asset markets: German equity and Global mutual funds.

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Notes

  1. 1.

    To define the rate, we turn to Edelen et al. (2013), who report that, on average, the equity mutual funds in their sample incur 0.80% of fund value annually in trading costs per unit. Commission charges schedule, which was obtained from Barclays Stockbrokers, proves the feasibility of such rate as well.

References

  1. Alexander, C. 2001. A Primer on the Orthogonal GARCH Model. Reading: ISMA Centre, University of Reading. Unpublished manuscript.

  2. Bai, Z., H. Liu, and W.K. Wong. 2009. Enhancement of the Applicability of Markowitz’s Portfolio Optimization by Utilizing Random Matrix Theory. Mathematical Finance 19(4): 639–667.

    Article  Google Scholar 

  3. Banz, R.W. 1981. The Relationship Between Return and Market Value of Common Stocks. Journal of Financial Economics 9(1): 3–18.

    Article  Google Scholar 

  4. Bassett, G., and R. Koenker. 1978. Regression Quantiles. Econometrica 46: 33–50.

    Article  Google Scholar 

  5. Belloni, A., and V. Chernozhukov. 2011. \(L_1\)-Penalized Quantile Regression in High-Dimensional Sparse Models. Annals of Statistics 39(1): 82–130.

    Article  Google Scholar 

  6. Bender, J., Briand, R., Fachinotti, G., and S. Ramachandran. 2005. Small Caps? No Small Oversight: Institutional Investors and Global Small Cap Equities. MSCI Research Insight, March 2012. http://www.msci.com/www/research-paper/small-caps-no-small-oversight/014391548. Accessed Feb 2015.

  7. Borke, L., and W.K. Härdle. 2016. Q3-D3-LSA, Berlin: Humboldt Universität zu Berlin. SFB 649 Discussion Paper 2016-049.

  8. Brandimarte, P. 2006. Numerical Methods in Finance and Economics: A MATLAB-Based Introduction, 2nd ed. New York: Wiley.

    Book  Google Scholar 

  9. Cornish, E.A., and R.A. Fisher. 1960. The Percentile Points of Distributions Having Known Cumulants. Technometrics 2(2): 209–225.

    Article  Google Scholar 

  10. Crain, M. 2011. A Literature Review of the Size Effect. Working Paper, 29 October 2011. http://www.ssrn.com/abstract_id=1710076. Accessed Feb 2015.

  11. DeMiguel, V., L. Garlappi, F.J. Nogales, and R. Uppal. 2009a. A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms. Management Science 55(5): 798–812.

    Article  Google Scholar 

  12. DeMiguel, V., L. Garlappi, F.J. Nogales, and R. Uppal. 2009b. Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy? Revew of Financial Studies 22: 1915–1953.

    Article  Google Scholar 

  13. Driessen, J., and L. Laeven. 2007. International Portfolio Diversification Benefits: Cross-Country Evidence From a Local Perspective. Journal of Banking and Finance 31(6): 1693–1712.

    Article  Google Scholar 

  14. Edelen, R., R. Evans, and G. Kadlec. 2013. Shedding Light on “Invisible” Costs: Trading Costs and Mutual Fund Performance. Financial Analysts Journal 69(1): 33–44.

    Article  Google Scholar 

  15. European Fund and Asset Management Association. 2016. Worldwide Regulated Open-Ended Fund Assets and Flows. Trends in the Fourth Quarter of 2015. European Fund and Asset Management association (EFAMA), March 2016. https://www.efama.org. Accessed June 2016.

  16. Engle, R. 2002. Dynamic Conditional Correlation: A Simple Class of Multivariate GARCH Models. Journal of Business and Economic Statistics 20(3): 339–350.

    Article  Google Scholar 

  17. Fan, J., J. Zhang, and K. Yu. 2012. Vast Portfolio Selection with Gross-Exposure Constraints. Journal of the American Statistical Association 107(498): 592–606.

    Article  Google Scholar 

  18. Fastrich, B., S. Paterlini, and P. Winker. 2015. Constructing Optimal Sparse Portfolios Using Regularization Methods. Computational Management Science 12(3): 417–434.

    Article  Google Scholar 

  19. Favre, L., and J.-A. Galeano. 2002. Mean-Modified Value-at-Risk Optimization with Hedge Funds. The Journal of Alternative Investments 5(2): 21–25.

    Article  Google Scholar 

  20. Franke, J., W.K. Härdle, and C.M. Hafner. 2015. Statistics of Financial Markets: An Introduction, 4th ed. Berlin: Springer.

    Google Scholar 

  21. Frankfurter, G.M., H.E. Phillips, and J.P. Seagle. 1971. Portfolio Selection: The Effects of Uncertain Means, Variances and Covariances. Journal of Financial and Quantitative Analysis 6: 1251–1262.

    Article  Google Scholar 

  22. Ghalanos, A., E. Rossi, and G. Urga. 2015. Independent Factor Autoregressive Conditional Density Model. Econometric Reviews 34(5): 594–616.

    Article  Google Scholar 

  23. Geczy, C. 2014. The New Diversification: Open Your Eyes to Alternatives. Journal of Portfolio Management. doi:10.3905/jpm.2014.40.5.146.

    Google Scholar 

  24. Härdle, W.K., Nasekin, S., Lee, D.K.C. and K.F. Phoon. 2014. TEDAS—Tail Event Driven Asset Allocation. Berlin: Humboldt Universität zu Berlin. SFB 649 Discussion Paper 2014-032.

  25. Investment Company Institute. 2016. Investment Company Fact Book: 2016. Investment company institute (ICI), May 2016. https://www.icifactbook.org. Accessed June 2016.

  26. Jobson, J.D., B. Korkie, and V. Ratti. 1979. Improved Estimation for Markowitz Portfolios Using James–Stein Type Estimators. Proceedings of the American Statistical Association, Business and Economics Statistics 41: 279–284.

    Google Scholar 

  27. Jobson, J.D., and B. Korkie. 1980. Estimation for Markowitz Efficient Portfolios. Journal of the American Statistical Association 75: 544–554.

    Article  Google Scholar 

  28. Jorion, P. 1985. International Portfolio Diversification with Estimation Risk. Journal of Business 58(3): 259–278.

    Article  Google Scholar 

  29. Kan, R., and G. Zhou. 2007. Optimal Portfolio Choice With Parameter Uncertainty. Journal of Financial and Quantitative Analysis 42(3): 621–656.

    Article  Google Scholar 

  30. Kazemi, H. 2012. An Introduction to Risk Parity. Alternative Investment Analyst Review. Chartered Alternative Investment Analyst Association, April 2012. https://www.caia.org. Accessed Feb 2015.

  31. Koenker, R., and K.F. Hallock. 2001. Quantile Regression. The Journal of Economic Perspectives 15(4): 143–156.

    Article  Google Scholar 

  32. Lee, D.K.C., F.P. Kok, and Y.W. Choon. 2006. Moments Analysis in Risk and Performance Measurement. The Journal of Wealth Management 9(1): 54–65.

    Article  Google Scholar 

  33. Lee, E.R., N. Hohsuk, and U.P. Byeong. 2014. Model Selection via Bayesian Information Criterion for Quantile Regression Models. Journal of the American Statistical Association 109(505): 216–229.

    Article  Google Scholar 

  34. Lehmann, B.N., and D.M. Modest. 1987. Mutual Fund Performance Evaluation: A Comparison of Benchmarks and Benchmark Comparisons. The Journal of Finance 42(2): 233–265.

    Article  Google Scholar 

  35. Maillard, S., T. Roncalli, and J. Teiletche. 2010. The Properties of Equally Weighted Risk Contribution Portfolios. Journal of Portfolio Management 36(4): 60–70.

    Article  Google Scholar 

  36. Matallin-Saez, J. 2007. Portfolio Performance: Factors or Benchmarks? Applied Financial Economics 17: 1167–78.

    Article  Google Scholar 

  37. Markowitz, H. 1952. Portfolio Selection. The Journal of Finance 7(1): 77–91.

    Google Scholar 

  38. McNamara, J.R. 1998. Portfolio Selection Using Stochastic Dominance Criteria. Decision Sciences 29(4): 785–801.

    Article  Google Scholar 

  39. Nathan, A. (ed.) 2013. Bond Bubble Breakdown. Commodities and Strategy Research, 22 April. http://www.nber.org. Accessed Feb 2015.

  40. Reinganum, M.R. 1981. A New Empirical Perspective on the CAPM. The Journal of Financial and Quantitative Analysis 16(4): 439–462.

    Article  Google Scholar 

  41. Schwarz, G.E. 1978. Estimating the Dimension of a Model. Annals of Statistics 6(2): 461–464.

    Article  Google Scholar 

  42. Simaan, Y. 1997. Estimation Risk in Portfolio Selection: The Mean Variance Model Versus the Mean Absolute Deviation Model. Management Science 43(10): 1437–1446.

    Article  Google Scholar 

  43. Swensen, D.F. 2009. Pioneering Portfolio Management: An Unconventional Approach to Institutional Investment, Fully Revised and Updated. New York: Free Press.

    Google Scholar 

  44. Tibshirani, R. 1996. Regression Shrinkage and Selection via the Lasso. Journal of Royal Statistical Society 58(1): 267–288.

    Google Scholar 

  45. Yen, Y., and T. Yen. 2014. Solving Norm Constrained Portfolio Optimization via Coordinate-Wise Descent Algorithms. Computational Statistics and Data Analysis 76: 737–759.

    Article  Google Scholar 

  46. Zheng, Qi, C. Gallagher, and K.B. Kulasekera. 2013. Adaptive Penalized Quantile Regression for High-Dimensional Data. Journal of Statistical Planning and Inference 143(6): 1029–1038.

    Article  Google Scholar 

  47. Zou, H. 2006. The Adaptive Lasso and Its Oracle Properties. Journal of Statistical Planning and Inference 101(476): 1418–1429.

    Google Scholar 

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Correspondence to Alla Petukhina.

Technical Appendix

Technical Appendix

Adaptive LASSO Quantile regression (ALQR)

Introduced in Bassett and Koenker (1978) quantile regression (QR) estimates conditional quantile functions–models in which quantiles of the conditional distribution of the response variable are expressed as functions of observed covariates (see Koenker and Hallock 2001).

\(L_1\)-penalty is considered to nullify “excessive” coefficients (Belloni and Chernozhukov 2011). Simple lasso-penalized QR optimization problem is:

$$\begin{aligned} \hat{ \beta }_{\tau ,\lambda } = {\text {arg}}\,\underset{ \beta \in \mathbb {R}^p}{{\text {min}}} \sum _{i=1}^n \rho _{\tau } (Y_i - X_i^{\top } \beta ) + \lambda \Vert \beta \Vert _1 \end{aligned}$$
(3)

The adaptive Lasso in Zou (2006) yields a sparser solution and is less biased. Using this result, Zheng et al. (2013) replaced \(L_1\)-penalty by a re-weighted version:

$$\begin{aligned} \hat{ \beta }_{\tau , \lambda _n}^{\text {adapt}} = {\text {arg}}\,\underset{ \beta \in \mathbb {R}^p}{{\text {min}}} \sum _{i=1}^n \rho _{\tau } (Y_i - X_i^{\top } \beta ) + \lambda _n \Vert \hat{\omega }^{\top } \beta \Vert _{1} \end{aligned}$$
(4)

here \(\tau \in \mathbb (0,1)\) is a quantile level, \(\rho _{\tau }(u) = u\{\tau - {\mathbf {I}}(u < 0)\}\) piecewise loss function, \(\lambda _n\) regularization parameter. Weights \(\hat{\omega } = 1/|\hat{ \beta }^{{\text {init}}}|\), \(\hat{\beta }^{{\text {init}}}\) is obtained from (3). In TEDAS set-up, \(Y \in \mathbb {R}^n\) represents core log-returns (DAX or S&P500 indices) and \(X \in \mathbb {R}^{n \times p}\)—satellites’ log-returns (German stocks or mutual funds), \(p > n\). The choice of the regularization parameter is critical. In the quantile regression literature, two criteria are used the Bayes information criterion (BIC) (Schwarz 1978; Lee et al. 2014) and the cross-validation criterion (e.g. Belloni and Chernozhukov 2011; Zou 2006). In TEDAS application, BIC is applied to choose \(\lambda _n\) parameter.

Cornish–Fisher VaR optimization

A modification of VaR via the Cornish–Fisher (CF) expansion (Cornish and Fisher 1960) improves its precision adjusting estimated quantiles for non-normality. To obtain asset allocation weights, the following VaR-minimization problem is solved (for details, see Favre and Galeano 2002; Härdle et al. 2014):

$$\begin{aligned} &\underset{w \in \mathbb {R}^d}{\text {minimize}}\,\,W_t\{ - q_{\alpha }(w_t) \cdot \sigma _p(w_t)\} \\ & {\text {subject to}}\,\,w_t^{\top } \mu = \mu _{p}, w_t^{\top }1 = 1, w_{t,i} \ge 0\\ \end{aligned}$$
(5)

here \(W_t \mathop {=}\limits ^{{\text {def}}} W_0 \cdot \prod _{j=1}^{t-1} w_{t-j}^{\top }(1+r_{t-j})\), \(\tilde{w}\), \(W_0\) initial wealth, \(\sigma ^2_p(w) \mathop {=}\limits ^{{\text {def}}} w_t^{\top } \Sigma _t w_t\),

$$\begin{aligned} q_{\alpha }(w_t) \mathop {=}\limits ^{{\text {def}}} z_{\alpha } + (z_{\alpha }^2-1)\frac{S_p(w_t)}{6} + (z_{\alpha }^3 - 3 z_{\alpha })\frac{K_p(w_t)}{24} - (2 z_{\alpha }^3 - 5 z_{\alpha }) \frac{S_p(w_t)^2}{36}, \end{aligned}$$
(6)

here \(S_p(w_t)\) is skewness of the portfolio, \(K_p(w_t)\) excess kurtosis of the portfolio, \(z_{\alpha }\) \({\text {N}}(0,1)\) \(\alpha\)-quantile. If \(S_p(w_t)\), \(K_p(w_t)\) are zero, then the problem reduces to the Markowitz case.

Mean-variance optimization procedure (Markowitz diversification rule)

Mean-variance optimization procedure is based on four inputs: the weights of total funds invested in each security \(w_i, i = 1,\ldots ,d\), the expected returns \(\mu\) approximated as averages \(\overline{r}\), volatilities (standard deviations) \(\sigma _i\) associated with each security and covariances \(\sigma _{ij}, j = 1,\ldots ,d; i \ne j\) between returns. Portfolio weights \(w_i\) are obtained from the quadratic optimization problem, see Brandimarte (2006, p. 74) 

$$\begin{aligned}&\underset{w \in \mathbb {R}^d}{\text{minimize}}\,\,\sigma _p^2(w_t) \mathop {=}\limits ^{{\text {def}}} w_t^{\top } \Sigma w_t\\ &{\text {subject to}}\,\,w_t^{\top } \mu = r_{T},\\ &\sum _{i=1}^d w_{i,t}= 1,\\&w_{i,t} \ge 0 \end{aligned}$$
(7)

where \(\Sigma \in \mathbb {R}^{d \times d}\) is the covariance matrix for d portfolio asset returns and \(r_{T}\) is the “target” return for the portfolio assigned by the investor.

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Härdle, W.K., Lee, D.K.C., Nasekin, S. et al. Tail Event Driven ASset allocation: evidence from equity and mutual funds’ markets. J Asset Manag 19, 49–63 (2018). https://doi.org/10.1057/s41260-017-0060-9

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Keywords

  • Adaptive lasso
  • Portfolio optimization
  • Quantile regression
  • Value-at-Risk
  • Tail events

JEL Classification

  • C00
  • C14
  • C50
  • C58