### Tail Risk Protection Trading Strategies

Frankfurt School of Finance & Management gemeinnützige GmbH

PPI AG; Think Tank Firamis

Zurich University of Applied Sciences, Center for Asset Management

Goethe University Frankfurt – Department of Finance

December 11, 2015

**Abstract: **

We develop dynamic portfolio protection trading strategies based on econometric methods. As a criterion for riskiness we consider the evolution of the value-at-risk spread from a GARCH model with normal innovations relative to a GARCH model with generalized innovations. These generalised innovations may for example follow a Student t, a Generalized hyperbolic, an alpha-stable or a Generalized Pareto (GPD) distribution. Our results indicate that the GPD distribution provides the strongest signals for avoiding tail risks. This is not surprising as the GPD distribution arises as a limit of tail behavior in extreme value theory and therefore is especially suited to deal with tail risks. Out-of-sample backtests on 11 years of DAX futures data, indicate that the dynamic tail risk protection strategy effectively reduces the tail risk while outperforming traditional portfolio protection strategies. The results are further validated by calculating the statistical significance of the results obtained using bootstrap methods.

### Tail Risk Protection Trading Strategies – Introduction

Despite the well-documented long-term equity risk premium, institutional and private investors may be underinvested in equity due to investment constraints, downside tail risk, value-at-risk limits or behavioral effects like loss aversion. The recent literature identifies a particular aversion against tail risks (Bollerslev and Todorov, 2011). The reluctance to bear tail risks leads to high prices of put options, which is reflected in the downward slope of the equity implied volatility curve as a function of the strike price (Kozhan et al., 2013), and the upward slope of VIX futures prices as a function of expiry (Zhang et al., 2010; Luo and Zhang, 2012; Eraker and Wu, 2014). Both market patterns are more pronounced than what would be reasonable by the realized return distributions and curtail a successful long-term static hedge position against tail risks.

We develop dynamic trading strategies that aim at protecting against large downturns by taking into account the time-variation and dynamics of distributional parameters of financial time series. First, accounting for the time-dependent dynamics of distributional parameters via a GARCH process allows to incorporate volatility clustering and auto-regressive behavior in volatility, both of which are well-documented stylized facts of financial time series (Cont, 2001; Engle and Patton, 2001; Thurner et al., 2012; Godin, 2015). Second, by fitting the GARCH innovations to exible distribution families incorporating both normal and extreme behavior allows to determine whether, in a given time period, extreme events are more likely to occur than suggested by e.g. normal innovations. This information can be used in several ways, for example as an early warning indicator, but also for robust portfolio building.

In our setup, the key input is the spread between value-at-risk (VaR) from a GARCH model with innovations following a generalized Pareto distribution (GPD), which is a distribution focusing on extreme risks, and a GARCH model with normally distributed innovations.We also consider Student t, generalized hyperbolic (GH) and Alpha-stable distributions as alternatives for the innovations, but found little explanatory power for extreme events. Because of the GARCH component, the magnitude of VaR, when viewed as a process over time, quickly adapts to changes in volatility. The distributional properties of the innovation process on the other hand provide information on skewness, excess kurtosis and in particular on the tail risk in the data. The resulting VaR spread can therefore be used to derive an expectation on the frequency of extreme events and, as such, may be taken to produce signals about the presence of tail risks. The principal idea of the VaR spread is therefore similar to the approach followed by Rachev et al. (2010), who focus on Alpha-stable innovations.

The choice of distributions is motivated as follows: The Student t, GH and Alpha-stable distributions incorporate the normal distribution as a special case. The GH distribution is a flexible distribution family comprising light- and medium-tailed distributions, and has been successfully applied for modelling financial time series (e.g. McNeil et al., 2005 and references therein). The Student t distribution is a special limiting case of a GH distribution, spanning the entire range of heavy-tailed behavior. On the other hand, Alpha-stable distributions are very heavy-tailed with the exception of the normal distribution, which is light-tailed. As such, all distribution families considered incorporate heavy-tailed behavior (at least in a limiting sense), which is a well-documented stylized fact of financial time series (e.g. Cont, 2001; McNeil et al., 2005). The GPD arises as the asymptotic limit of considering only (extreme) outcomes beyond a threshold (hence the name threshold exceedances for methods involving the GPD, e.g. McNeil et al., 2005). Fitting innovations to a GPD therefore provides an entirely different approach by focusing on the extreme behavior rather than the whole data.

Aside from focusing on daily returns, we include overnight returns in our analysis. This allows to capture a wider array of different signals in the data. We inspected intraday returns, but they turned out to be less useful for the analysis.

A trading strategy is generated from measuring the risk build-up in terms of the evolution of the VaR spreads of each data set. As already mentioned, the VaR spread with GPD innovations from daily returns and from overnight returns yields the most reliable results. This is not surprising, as the GPD arises as the limit distribution of the tail behavior in extreme value theory; because of its focus on tail behaviour, it is particularly well suited for a tail risk protection strategy.

In an extensive out-of-sample backtest covering more than eleven years of data we find that the tail risk protection strategy outperforms the benchmark (DAX future) in terms of mean return, standard deviation, Sharpe ratio, worst drawdown and Calmar ratio. In addition, the results are compared with the performance of two traditional protection strategies, protective put (PP) and delta-replicated put (DRP). PP is a strategy that eliminates downside risk for a specific asset and period of time at the cost of an option premium, whereas DRP replicates the delta of a hypothetical put with a dynamic trading position in the underlying without actually buying a physical option.

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