## 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.
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.

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## 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:

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:

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):

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, \(\sigma ^2_p(w) \mathop {=}\limits ^{{\text {def}}} w_t^{\top } \Sigma _t w_t\),

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)

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