# 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.

### Keywords

Adaptive lasso Portfolio optimization Quantile regression Value-at-Risk Tail events### JEL Classification

C00 C14 C50 C58## Introduction

Asset allocation and security selection go hand in hand with risk management. They are not only important concepts in a quantitative finance and applied statistics, but also significant determinants for long-term portfolio construction for private and institutional investors. Absence of significant correlation among various asset classes is the essential motivation for traditional portfolio allocation. In reality, some strategies contradict this principle, such as the traditional 60 equity/40 bond portfolio approach: in recent years, the correlation between the bond market and the stock market was 0.98 (Geczy 2014). During the Global Financial Crisis, the Endowment Model underperformed due to increased correlation across assets, Swensen (2009). The risk parity strategy recommended a significant allocation to bonds amidst the implementation of quantitative easing and performed poorly because of interest rate volatility (Kazemi 2012; Nathan 2013).

A pillar in portfolio theory, mean-variance (MV) portfolio optimization by Markowitz (1952) proposed to study semi-variance even though the optimization was not straightforward given the low computation power at that time. As computing capacities increased, later models incorporated optimization involving higher and time-varying moments. The mean-variance and subsequent refined models did not perform well during volatile periods. In the case the number of assets (*p*) exceeds the number of observations (*n*) results of Markowitz portfolios are not stable: Bai et al. (2009) proved that the asset return estimate given by the Markowitz MV model was always larger than the theoretical return and the rate of the difference was related to *p*/*n*, the ratio of the dimension to the sample size. Jobson et al. (1979) and Jobson and Korkie (1980) showed that the Markowitz mean-variance efficient portfolios were highly sensitive to *p*/*n*. They suggested to shrink the number of estimators or assets. From this point of view, the Least Absolute Shrinkage and Selection Operator (LASSO) (Tibshirani 1996) provides a feasible solution and good out-of-sample results of mean-variance portfolios, see e.g. DeMiguel et al. (2009a), Fan et al. (2012), Yen and Yen (2014) and Fastrich et al. (2015).

Taking into consideration observed return co-movement spikes for strong negative-return times, modelling tail dependence can be viewed as more flexible approach to portfolio allocation. Härdle et al. (2014) introduced a new asset allocation strategy, Tail Event Driven Asset Allocation (TEDAS). TEDAS exploits negative co-movement of alternative assets in the tail with a core asset to hedge downside risk. The subset of alternative or satellite assets performs the role of downside protection. Successful protection of the portfolio by limiting the downside risk during volatile periods allows the portfolio returns to recover sooner. TEDAS, with smaller drawdowns, outperforms more traditional methods that suffer larger drawdowns during extreme events.

TEDAS can also be viewed as an application of the core-satellite model, given that a subset of satellite assets is chosen from a larger universe of assets. For big data, where the number of possible alternatives is larger than the number of observations, the adaptive LASSO quantile regression (ALQR) is used to pursue variable selection and measure relations between variables at tail quantiles simultaneously. In order to deal with changing volatility and correlation structure problem, Dynamic Conditional Correlation (DCC) is used. TEDAS applies Cornish–Fisher VaR (Value-at-Risk) measure for portfolio optimization, utilizing accordingly higher moments of distributions of asset returns, such as skewness and kurtosis. Here, we extend TEDAS by introducing three modifications, which we call “*TEDAS gestalts*”: *TEDAS basic*, *TEDAS naïve*, which places equal weights on every satellite asset, and *TEDAS hybrid*, which uses the most common Markowitz mean-variance rule to select the weights.

In this paper, we provide an extensive empirical analysis, and we compare the performance of the TEDAS strategies with a wide range of benchmark portfolio strategies, applied to global mutual fund and German stock market data. TEDAS portfolios yield robust and consistent results with various assets, time periods, parameter frequencies, and in big and small data. Differently from Härdle et al. (2014), in the analysis we introduce, portfolio rebalancing framework assumes incorporation of transaction costs.

The rest of the paper is organized as follows. In “TEDAS—Tail Event Driven ASset allocation strategy” section, we introduce the framework of TEDAS. In “The choice of satellite assets and data description” and “Empirical results” sections, we apply the methods to different markets and compare the performance between different models. In fifth presents the conclusion and discussion. All codes and datasets are available as Open image in new window Quantlets on Quantnet (Borke and Härdle 2016).

## TEDAS: Tail Event Driven ASset allocation strategy

The basic elements of TEDAS are presented in Härdle et al. (2014). The proposed tool set has important implications for portfolio risk management and asset allocation decisions. Along with the basic set-up, we propose two modifications: *TEDAS naïve* and *TEDAS hybrid*.

The TEDAS strategy is based on a simple idea widely used in practice—core-satellite approach. The core of portfolio (or core asset) consists of passive investments that track major market indices, e.g. the DAX30 or S&P500 (in practice the role of core can play exchange traded fund—ETF). Initially, the portfolio consists of the core asset only. The core portfolio is chosen by the fund manager, and the satellite assets are chosen by TEDAS to limit the downside risk of the core portfolio during extreme events.

At a high level, TEDAS consists of two related stages. The first stage is an asset selection. Portfolio constituents are selected using the tail event approach. The tail dependence of core and satellite assets is identified via an adaptive LASSO quantile regression (ALQR), Zheng et al. (2013). At the second stage, portfolio selection, portfolio weights for active assets are determined.

- 1.
Consider a data vector \(Y \in \mathbb {R}^T\) of core asset log-returns and a matrix \(X \in \mathbb {R}^{T \times p}\) of satellites’ log-returns,

*T*is a number of all observations and number of assets in the initial set. - 2.
For each period \(t = n,\ldots ,T\) obtain empirical \(\tau\)-quantiles \(\hat{q}_{\tau } \mathop{=}\limits ^{{\text {def}}}F_n^{-1}(\tau )\) from the core log-returns’ empirical distribution \(F_n\), where \(\tau _{j=1,\ldots ,5} = (0.05,0.15,0.25,0.35,0.50)\), where

*n*is equal to the width of the moving window and \(p > n\). - 3.
Determine core asset log-return \(r_{t}\), set \(\tau _{t} = \widehat{F}_n(r_t)\).

- 4.
Select \(\tau _{t}\) according to the right-hand side \(\hat{q}_{\tau _{j,t}}\) in: \(r_{t} \le \hat{q}_{{\tau _{1,t}}}\) or \(\hat{q}_{{\tau _{1,t}}} < r_{t} \le \hat{q}_{{\tau _{j,t}}}\).

- 5.Obtain the ALQR estimator by Zheng et al. (2013) for \(\hat{\beta }_{\tau _{t}, \lambda _n}\) using the observations \(X_t \in \mathbb {R}^{t-n+1,\ldots ,t\times p}\), \(Y_t \in \mathbb {R}^{t-n+1,\ldots ,t\times 1}\)ALQR is applied to select relevant hedging assets for a new portfolio. This technique allows simultaneously to solve two challenges for portfolio managers. It shrinks the high-dimensional universe of satellite assets to potential candidates for portfolio’s constituents: the \(L_1\) penalty \(\lambda _n \Vert \hat{\omega }^{\top } \beta \Vert _{1}\) in (1) nullifies “excessive” coefficients. ALQR also provides information concerning the dependence between the core portfolio and satellites at different quantiles (for various \(\tau\)). Additional technical details are in “Adaptive LASSO Quantile regression (ALQR)” section.$$\begin{aligned} \hat{ \beta }_{\tau _{t}, \lambda _n}^{\text {adapt}} = {\text {arg}}\,\underset{ \beta \in \mathbb {R}^p}{{\text {min}}} \sum _{i=1}^n \rho _{\tau } (Y_t - X_t^{\top } \beta ) + \lambda _n \Vert \hat{\omega }^{\top } \beta \Vert _{1} \end{aligned}$$(1)
- 6.
Choose \(X_t^{\mathcal {J}}\), where \(\mathcal {J}\) is the index set corresponding to negative coefficients \(\hat{\beta }_{\tau _{t}, \lambda _n}^{\text {adapt}}\). These assets with negative ALQR coefficients (formula 1), i.e. assets adversely moving with the core for chosen level of a tail event, are constituents of a new rebalanced portfolio.

*TEDAS basic*gestalt employs the dynamic conditional correlation model (DCC) to account for time-varying covariance structure and correlation shifts in returns’ covariance, see Engle (2002), Franke et al. (2015). The weights of satellites are defined based on the Cornish–Fischer Value-at-Risk (VaR) minimization rule, Favre and Galeano (2002) (“Cornish–Fisher VaR optimization section”).

TEDAS gestalts

TEDAS gestalt | Dynamics modelling | Weights optimization |
---|---|---|

TEDAS naïve | NO | Equal weights |

TEDAS hybrid | NO | Mean-variance optimization of weights |

TEDAS basic | DCC volatility | CF-VaR optimization |

The *TEDAS naïve* gestalt assigns to every satellite asset an equal portfolio weight.

The *TEDAS hybrid* gestalt employs the simplest approach to estimate the covariance structure of assets’ returns, the historical covariance matrix. Portfolio weights are calculated according to classical mean-variance optimization procedure with long-only constraint (Markowitz diversification rule), Markowitz (1952). Calculation details are in “Mean-variance optimization procedure (Markowitz diversification rule)” section.

- 7.
Apply one of TEDAS Gestalts to \(X_t^{\mathcal {J}}\), selected on step 6, obtain vector of weights \(\hat{w}_t\) to invest in each security.

- 8.Determine the realized portfolio wealth for \(t+1\)$$\begin{aligned} W_{t+1} = W_t\times exp({\hat{w}_t^{\top }X_t^{\mathcal {J}}}) \end{aligned}$$(2)

^{1}of portfolio wealth is used to pay transaction fees.

## The choice of satellite assets and data description

The empirical analysis of TEDAS application to equity market focuses on the German stock market. The universe of potential hedging assets consists of 123 constituents of indices SDAX, MDAX and TecDAX—small- and mid-cap stocks. The collected data cover 157 trading weeks from 1 January 2013 to 5 January 2016 (Source: *Datastream*). The performance of TEDAS strategy for German equity market was analysed on 77 eighty-week moving windows.

The role of mutual funds in world economy has increased in the 20 years. As of 2015, US$ 37.2 trillion was invested in regulated open-end fund assets worldwide and US$ 17.8 trillion in US mutual funds (Investment company institute 2016). In addition, mutual funds account for 88% of investment companies in total. The popularity of mutual funds is due to their perceived safety compared to alternatives, notably stocks. This perception has resulted in a situation where almost half specifically, 44.1% of US households have participated in such funds in the end of 2015. The number of mutual funds also grows up rapidly. The number of regulated worldwide open-ended funds amounted 52 746 in 1999 and reached 100 494 in the end of 2015. (European Fund and Asset Management Association 2016). All this underlines the sheer size and the importance of the US mutual find market which, therefore, should provide us with an important test case for the evaluation of the performance of TEDAS strategy and show whether TEDAS can handle cases of big data.

The potential of diversification, an important determinant for asset allocation, makes mutual funds attractive for institutional and retail investors. In the 2015 US market, 41% of all industry assets were held by domestic equity funds and an additional 15% by world equity funds, those that invest significantly in shares of non-US corporations. Moreover, Bond funds held 21% of US mutual fund, whereas money market funds, hybrid funds and other funds such as those that invest primarily in commodities held the remainder (23%). Finally, it has been observed that in the US there has been a tendency towards equity mutual funds regarding portfolio diversification, which means increased investment rates in foreign (non-US) markets (Investment company institute 2016).

*Lipper of Thomson Reuters*. Our empirical analysis uses monthly returns on a sample of global equity mutual funds over the 25-year period from December 1990 to December 2015 (300 monthly log-returns). We require funds to have contiguous returns time series over the chosen testing time period. Thus, our final data sample for TEDAS strategies analysis include monthly returns of 739 actively managed equity mutual funds, covering 180 moving windows with length 120 months. Figure 2 illustrates a geographical focus assignment of the mutual funds in our testing sample, defined by the predominant country or region of the fund’s portfolio (mutual fund holds at least 50% of its assets in a specific country/region). The majority of mutual funds is concentrated on American and European markets, significantly fewer—on Asian markets.

To compare the performance of TEDAS strategies with the protective put hedging approach, we used information about prices of European-style put options of DAX 30 for period from 15 July 15 2014 to 18 December 2015 and S&P 500 for period from October 2002 to December 2015 (Source: *Datastream*).

## Empirical results

### Results for German equity market

We provide a comparative analysis of TEDAS performance with three alternative widely used strategies: mean-variance strategy, 60/40 portfolio and risk parity portfolio. The rebalancing setting is equivalent for all benchmark and TEDAS portfolios. The mean-variance portfolio selection is a common benchmark for every newly introduced asset allocation strategy. The traditional Markowitz portfolio optimization approach as has been shown in the previous literature has some drawbacks especially for the case when *p* > *n*. The portfolio formed by using the classical mean-variance approach always results in extreme portfolio weights (Jorion 1985) that fluctuate substantially over time and perform poorly in the sample estimation (for example, Frankfurter et al. 1971; Simaan 1997; Kan and Zhou 2007) as well as in the out-of-sample forecasting.

Great number of studies provide various observations and suggestions concerning reasons driving MV optimization estimate so far away from its theoretic counterpart. Vast majority of authors believe that the reason behind this outcome is that the “optimal” return is formed by a combination of returns from an extremely large number of assets (see e.g. McNamara 1998). Efficiency of Markowitz optimization procedure depends on whether the expected return and the covariance matrix can be estimated accurately. In our comparative study, we used portfolio covariance matrix modelled by the basic orthogonal GARCH method to solve the dynamic Markowitz risk-return optimization problem. The orthogonal GARCH model (MV OGARCH) was first proposed in Alexander (2001), and is based on principal components analysis (PCA).

60/40 portfolio allocation strategy implies the investing of 60% of the portfolio value in stocks (often via a broad index such as S&P500) and 40% in government or other high-quality bonds, with regular rebalancing to keep proportions steady. German market’s 60/40 portfolio is constructed with DAX30 and eb.rexx Government Germany index, which contains the 25 most liquid German government bonds with residual maturities of between 1.5 and 10.5 years.

The risk parity portfolio strategy is based on allocation by risk, not by capital. In this case, the portfolio manager defines a set of risk budgets and then computes the weights of the portfolio such that the risk contributions match the risk budgets. We employ so-called equal risk contribution (ERC) portfolios. The idea of the ERC strategy is to find a risk-balanced portfolio with equal risk contributions for all assets (for details, see e.g. Maillard et al. 2010).

*TEDAS naïve*and alternative strategies, demonstrated in Fig. 4, shows that TEDAS performs significantly better than other considered approaches (Table 2).

Strategies risk and performance measures overview: German stocks sample

TEDAS basic | TEDAS naïve | TEDAS hybrid | ERC | 60/40 portfolio | MV OGARCH | DAX30 | |
---|---|---|---|---|---|---|---|

Volatility | 0.03 | 0.03 | 0.03 | 0.02 | 0.02 | 0.02 | 0.03 |

Skewness | −0.34 | 0.48 | −0.34 | 0.09 | −0.29 | 0.30 | −0.22 |

Kurtosis | 8.44 | 5.29 | 8.27 | 3.27 | 2.80 | 2.66 | 2.84 |

VaR 5% | −0.03 | −0.03 | −0.03 | −0.03 | −0.03 | −0.03 | −0.05 |

Expected shortfall 5% | −0.06 | −0.04 | −0.06 | −0.04 | −0.04 | −0.04 | −0.06 |

Sharpe ratio | 0.40 | 0.46 | 0.40 | 0.26 | 0.06 | 0.17 | 0.05 |

Maximum drawdown | 0.12 | 0.07 | 0.12 | 0.08 | 0.14 | 0.10 | 0.21 |

Cumulative return | 2.57 | 2.59 | 2.53 | 1.58 | 1.08 | 1.38 | 1.11 |

*TEDAS naïve*portfolio strategy as the most attractive for investors: it reaches 257% of cumulative return providing at the same time the highest excess return, 0.46%, for the extra volatility. In general TEDAS strategies show better risk-adjusted returns than all other analysed benchmarks and have comparatively the same level of risk. Based on results of the Wilcoxon signed-rank test (Table 3), we find three TEDAS strategies do not significantly differ from each other, but the difference from other benchmarks is statistically significant. Hence, TEDAS strategies performance is mainly driven by the asset selection decision rather than the portfolio selection decision.

Difference between TEDAS and benchmark strategies: Wilcoxon test *p* values

TEDAS basic | TEDAS naïve | TEDAS hybrid | ERC | 60/40 portfolio | MV OGARCH | DAX30 | |
---|---|---|---|---|---|---|---|

TEDAS basic | 1.00 | 0.88 | 0.52 | 0.00 | 0.00 | 0.00 | 0.00 |

TEDAS naïve | 0.88 | 1.00 | 0.68 | 0.00 | 0.00 | 0.00 | 0.00 |

TEDAS hybrid | 0.52 | 0.68 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 |

The most frequently selected satellites—German Stocks

Company name | Short ticker | Index | Industry | Frequency |
---|---|---|---|---|

Sartorius AG Vz. | SRTX | TecDAX | Industrial Machinery | 20 |

Celesio AG | CLS1X | MDAX | Drug Retailers | 13 |

Hawesko Holding AG | HAWX | SDAX | Specialty Retailers | 9 |

Patrizia Immobilien AG | P1ZX | SDAX | Real Estate Hold, Dev | 5 |

GFK | GFKX | SDAX | Media Agencies | 3 |

Grenke AG | GLJ | SDAX | Business Support Services (IT) | 3 |

Kabel Deutschland Holding AG. | KD8X | MDAX | Broadcast and Entertain | 3 |

Descriptive statistics on continuously compounded weekly returns for 10 most frequent satellites—German stocks

SRTX | CLS1X | HAWX | P1ZX | GFKX | GLJX | KD8X | |
---|---|---|---|---|---|---|---|

Maximum | 0.16 | 0.18 | 0.11 | 0.12 | 0.08 | 0.08 | 0.13 |

Upper percentile | 0.05 | 0.02 | 0.02 | 0.07 | 0.04 | 0.05 | 0.03 |

Median | 0.01 | 0.00 | 0.00 | 0.01 | 0.00 | 0.01 | 0.00 |

Mean | 0.01 | 0.00 | 0.00 | 0.01 | 0.00 | 0.01 | 0.00 |

Lower percentile | −0.03 | −0.02 | −0.02 | −0.04 | −0.05 | −0.04 | −0.02 |

Minimum | −0.09 | −0.15 | −0.09 | −0.13 | −0.14 | −0.09 | −0.09 |

Volatility | 0.04 | 0.03 | 0.02 | 0.04 | 0.03 | 0.03 | 0.02 |

Pairwise satellite stocks’ correlations of returns separately for positive (*upper triangular matrix*) and negative (lower triangular matrix) market-movement days, as defined by returns on DAX30

SRTX | CLS1X | HAWX | P1ZX | GFKX | GLJX | KD8X | |
---|---|---|---|---|---|---|---|

SRTX | 0.10 | −0.08 | 0.12 | −0.08 | −0.24 | 0.04 | |

CLS1X | 0.00 | 0.10 | −0.22 | −0.07 | −0.11 | −0.06 | |

HAWX | 0.16 | −0.07 | −0.06 | 0.08 | 0.11 | −0.07 | |

P1ZX | 0.11 | −0.09 | 0.09 | 0.05 | 0.20 | −0.02 | |

GFKX | 0.08 | 0.03 | 0.19 | 0.20 | 0.33 | 0.00 | |

GLJX | 0.12 | −0.16 | 0.31 | 0.32 | 0.24 | 0.15 | |

KD8X | −0.16 | 0.03 | 0.01 | 0.03 | 0.01 | 0.11 |

All TEDAS gestalts, applied to the universe of German stocks, outperform both traditional benchmark strategies, such as Markowitz rule or 60/40 and more sophisticated ones, such as the ERC model. Our analysis leads us to believe that using the ALQR technique delivers good results in reducing the dimensionality of the asset universe for more effective portfolio allocation.

### Results for global mutual funds

Since the number of satellites after filtering (\(p=739\)) is very large, the length moving window for mutual funds sample is adjusted to 120 months. We allocated 1 unit of money in December 2000 using each strategy and calculated the 180 monthly cumulative returns until December 2015.

*TEDAS naïve*yields the highest return, 1193%.

*TEDAS basic*and

*TEDAS hybrid*set-ups show similar returns of 1184 and 1175%, respectively (Fig. 7).

*p*values (Table 7) are all smaller than 5%, that indicates significant difference of TEDAS strategies from benchmark portfolios, which lies in line with German stocks sample results. Table 8 documents an investigation of riskiness of strategies, proving that higher absolute performance of TEDAS approach is supplemented with better risk-adjusted performance metrics.

Difference between TEDAS and benchmark strategies: Wilcoxon test *p* values

TEDAS basic | TEDAS naïve | TEDAS hybrid | ERC | 60/40 portfolio | MV OGARCH | S&P500 | |
---|---|---|---|---|---|---|---|

TEDAS basic | 1.00 | 0.77 | 0.91 | 0.00 | 0.00 | 0.00 | 0.00 |

TEDAS naïve | 0.77 | 1.00 | 0.80 | 0.00 | 0.00 | 0.00 | 0.00 |

TEDAS hybrid | 0.91 | 0.80 | 1.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Strategies risk and performance measures overview: mutual funds sample

TEDAS basic | TEDAS naïve | TEDAS hybrid | ERC | 60/40 portfolio | MV OGARCH | S&P500 | |
---|---|---|---|---|---|---|---|

Volatility | 0.03 | 0.03 | 0.03 | 0.05 | 0.02 | 0.04 | 0.06 |

Skewness | 0.34 | 0.28 | 0.35 | −1.60 | −1.06 | −0.03 | −0.91 |

Kurtosis | 4.52 | 4.20 | 4.52 | 8.60 | 6.80 | 3.02 | 7.03 |

VaR 5% | −0.02 | −0.03 | −0.02 | −0.09 | −0.03 | −0.05 | −0.10 |

Expected shortfall 5% | −0.04 | −0.05 | −0.04 | −0.13 | −0.04 | −0.07 | −0.15 |

Sharpe ratio | 0.51 | 0.50 | 0.51 | 0.04 | 0.06 | 0.14 | 0.04 |

Maximum drawdown | 0.14 | 0.16 | 0.14 | 0.58 | 0.24 | 0.39 | 0.64 |

Cumulative return | 11.84 | 11.94 | 11.75 | 1.38 | 1.19 | 2.52 | 1.44 |

The most frequently selected satellites—mutual funds

Company name | Short ticker | Geographical focus | Frequency |
---|---|---|---|

Swedbank Robur Ethica Global | SWESVGB | Global | 9 |

ESPA Stock Vienna A EUR | ESPASEV | Austria | 8 |

Lloyds Investment European GBP | TSBEURI | Europe exc UK | 8 |

Oddo Immobilier CR-EUR | ODDIMMC | Europe | 8 |

UTI Mastershare Unit Scheme-Income | UPNGFAD | India | 5 |

Sydinvest Europa Ligeveagt & Value KL | SYIEULVKL | Europe | 3 |

Lloyds Investment International GBP | TSBITEI | Global | 2 |

Petercam L Equities Opportunity B | PAEQOPP LX | Global | 2 |

Pioneer Funds Austria—Asia Stock A | PACFSTK | Asia Pacific | 2 |

Descriptive statistics on continuously compounded monthly returns of satellites—mutual funds

Maximum | Upper percentile | Median | Mean | Lower percentile | Minimum | Volatility | |
---|---|---|---|---|---|---|---|

ORBOPEF | 0.10 | 0.03 | 0.00 | 0.01 | −0.02 | −0.11 | 0.02 |

SWESVGB | 0.19 | 0.08 | 0.01 | 0.01 | −0.06 | −0.25 | 0.06 |

ESPASEV | 0.17 | 0.07 | 0.01 | 0.00 | −0.08 | −0.45 | 0.07 |

TSBEURI | 0.14 | 0.07 | 0.01 | 0.01 | −0.06 | −0.31 | 0.06 |

ODDIMMC | 0.16 | 0.06 | 0.01 | 0.00 | −0.06 | −1.60 | 0.11 |

UPNGFAD | 0.33 | 0.10 | 0.00 | 0.00 | −0.12 | −0.57 | 0.10 |

SYIEULVKL | 0.18 | 0.07 | 0.00 | 0.00 | −0.06 | −1.37 | 0.10 |

TSBITEI | 0.11 | 0.06 | 0.01 | 0.00 | −0.05 | −0.18 | 0.04 |

PAEQOPP LX | 0.15 | 0.06 | 0.01 | 0.01 | −0.05 | −0.30 | 0.05 |

PACFSTK | 1.56 | 0.07 | 0.00 | 0.00 | −0.08 | −0.25 | 0.11 |

Pairwise satellite stocks’ correlations of returns separately for positive (*upper triangular matrix*) and negative (*lower triangular matrix*) market-movement days, as defined by returns on S&P500

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | ORBOPEF | 0.09 | 0.14 | −0.05 | 0.10 | 0.07 | 0.13 | −0.06 | 0.26 | 0.12 | |

2 | SWESVGB | 0.01 | 0.04 | 0.00 | 0.02 | 0.11 | 0.05 | −0.00 | 0.08 | 0.06 | |

3 | ESPASEV | 0.27 | 0.66 | 0.07 | 0.05 | 0.05 | 0.10 | 0.06 | 0.26 | 0.09 | |

4 | TSBEURI | 0.13 | 0.83 | 0.78 | 0.06 | −0.02 | 0.03 | 1.00 | 0.14 | 0.09 | |

5 | ODDIMMC | 0.28 | 0.62 | 0.76 | 0.70 | 0.18 | 0.05 | 0.04 | 0.18 | 0.09 | |

6 | UPNGFAD | 0.13 | 0.48 | 0.52 | 0.57 | 0.49 | 0.05 | −0.04 | 0.31 | 0.32 | |

7 | SYIEULVKL | 0.12 | 0.67 | 0.63 | 0.76 | 0.66 | 0.42 | 0.02 | 0.18 | 0.08 | |

8 | TSBITEI | 0.13 | 0.78 | 0.70 | 0.82 | 0.56 | 0.52 | 0.66 | 0.11 | 0.08 | |

9 | PAEQOPP LX | 0.08 | 0.64 | 0.74 | 0.75 | 0.75 | 0.46 | 0.69 | 0.58 | 0.41 | |

10 | PACFSTK | 0.10 | 0.26 | 0.20 | 0.24 | 0.16 | 0.22 | 0.18 | 0.14 | 0.20 |

## Conclusion

This study represents an extensive empirical analysis of the performance of asset allocation strategies based on the TEDAS approach applied to global mutual funds and German stocks. TEDAS focuses on the co-movement of the core and the universe of satellite assets during extreme events. The degree of extremeness is defined by empirical quantile of returns distribution. Reduction in the universe of satellite assets to a manageable subset and at the same time having the properties of negative correlation with the core during extreme event is an innovative approach in asset allocation.

The main contribution of this paper is to demonstrate the practical significance of the TEDAS tool set for both institutional and private investors in various settings. We conducted an empirical study on the performance of TEDAS strategy applied to multiple configurations of core and satellites. TEDAS incorporates “least absolute shrinkage and selection” which does variable selection and shrinks noise coefficients to zero and simultaneously discloses causal relationships between tail events of the benchmark asset and the covariate hedge funds returns in the high-dimensional framework. TEDAS selects portfolio weights through the minimization of the portfolio objective risk function modelled as the variance–covariance Value-at-Risk adjusted for higher portfolio return distribution moments such as skewness and kurtosis via the Cornish–Fisher expansion; the variance–covariance return structure is allowed to be time-varying and is modelled with the dynamic conditional correlation (DCC) approach to capture the effect of possible asymmetric volatility clustering.

The results of three modifications of TEDAS adopted in this study are robust. Nevertheless, there is clearly room for further additional research. Theory says that TEDAS basic and TEDAS hybrid should perform better than TEDAS naïve. However, we do not observe it in our empirical study. This finding lies in line with previous researches: e.g. DeMiguel et al. (2009b) demonstrated underperformance of fourteen MV models in comparison with 1/*N* rule. Possible solution to such issue could be, for instance, incorporation of time-varying modelling of higher portfolio moments as in Ghalanos et al. (2015).

The testing of TEDAS strategy for global mutual funds and German equity data leads to conclusion that TEDAS is meaningful for geographically different markets (global and Germany), using weekly and monthly returns as well as for different levels of dimensionality of the assets universe. This paper demonstrates the power of the TEDAS strategy for different asset markets, such as equity, mutual funds and hedge funds (Härdle et al. (2014)). Furthermore, compared with four conventional benchmark allocation approaches, TEDAS cumulative returns are significantly higher. Investigation of risk-adjusted TEDAS returns also demonstrates better results than other benchmark strategies.

## Footnotes

- 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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