Asymmetric jump beta estimation with implications for portfolio risk management

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Abstract

We evaluate the impact of extreme market shifts on equity portfolios and study the difference in negative and positive reactions to market jumps with implications for portfolio risk management. Employing high-frequency data for the constituents of the S&P500 index over the period 2 January 2003 to 30 December 2017, we investigate to what extent the portfolio exposure to the downside and upside jumps can be mitigated. We contrast the risk exposure of individual stocks with those of the portfolios as the number of holdings increases. Varying the jump identification threshold, we show that the number of holdings required to stabilise portfolios’ sensitivities to negative jumps is higher than when positive jumps are considered and that the asymmetry is more prominent for more extreme events. Ignoring this asymmetry results in under-diversification of portfolios and increases exposure to sudden extreme negative market shifts.

Introduction

The recent 2007–2008 and 2011 financial crises have revived a considerable degree of scepticism of portfolio theory. The availability of high-frequency data has led to great improvements in our ability to measure risk, allowing us to separate risk into its contributing factors. As a result, asset allocators are making changes in how they manage risk. Portfolio managers are becoming increasingly aware of the pitfalls of approaches that fail to address downside risk, or more specifically, extreme negative events. Risk factor diversification is becoming the focus. In this paper, we separate positive and negative jumps, using each family of jumps as risk factors, and analyse portfolio sensitivities to these risk factors as the number of portfolio holdings varies.

An important feature explored in our study is the asymmetry in portfolios’ behaviour during extreme market downturns versus extreme market upsurges. Sudden large market shifts are rare events, but have substantially higher impacts than the diffusive price movements. Hedging against these extreme shifts is difficult, unless the portfolios are large enough to diversify away such risks. Recent studies by Bollerslev et al. (2008), Jacod and Todorov (2009) and Mancini and Gobbi (2012) have all argued for the presence of common jump arrivals across different assets, thus possibly inducing stronger dependencies in the “extreme”. However, Bajgrowicz, Scaillet, and Treccani (2016) argue that no co-jump affects all stocks simultaneously, suggesting jump risk is diversifiable. Our analysis shows stronger concordance between market and portfolio returns during extreme market downturns than during market upsurges, and reveals a large disparity in recommended number of portfolio holdings.

In evaluating the impact of extreme negative and positive market shifts on portfolios we investigate the extent the downside and upside jump risk can be diversified away. In particular, we address the following questions: How many stocks should investors hold on average, in order to reduce the sensitivity to market jumps to a certain level? How does the recommended portfolio size changes with the magnitude of extreme events? Are there any differences in recommended portfolio sizes for investors seeking to diversify against negative extreme events only? We find that more stocks are required to stabilise portfolio sensitivities to extreme negative market jumps than to extreme positive ones.1 In addition, the more we focus on the tails of distributions, the larger the difference we anticipate in their behaviours. When defining sudden extreme shifts, we explore several thresholds and consider the asymmetry effects for different levels of extreme market movements. We expect the difference in recommended portfolio sizes to be larger in the presence of more extreme jumps. Ignoring asymmetry results in under-diversification of portfolios and increases portfolio exposure to extreme negative market jumps.

Our analysis combines the developments from two strands of literature: modelling of extreme events and jump identification. Our contribution is, firstly, the evaluation of extreme negative vs extreme positive market shifts separately and, secondly, its impact in a portfolio setting. Some research has been done on the asymmetric tail risk (Ang, Chen, & Xing, 2006; van Oordt and Zhou, 2016),2 including a companion paper by Alexeev et al. (2016) which only contrasts continuous and discontinuous systematic risks. In this paper, we use the inferential procedure of Li, Todorov, and Tauchen (2017) to extend the single jump beta to the positive and negative jump betas. To the best of our knowledge no papers have investigated the behaviour of signed systematic jump risk in a portfolio setting and the implications it would have in portfolio risk management and diversification to extreme market events.

In the past two decades, modelling extreme events has become mainstream in risk management practice.3 Regulators are attentive to market conditions during a crisis because they are concerned with the protection of the financial system against catastrophic events. Bates (2008) formalises the intuition that investors treat extreme events differently than they treat more common and frequent ones. The increased availability of high-frequency data amplified the interest in the analysis of these tail events.4 Modelling rare and extreme events often explains the high observed equity risk premia by taking into account the premia for rare events, provided that these events are sufficiently severe (Barro, 2006; Bates, 2008; Rietz, 1988).5

The jumps in the high-frequency literature may be rare events when considered spatially, but often appear too frequently in calendar time to be considered extreme or disastrous. The need for a refined classification of jumps according to their magnitude and its association with extreme events is apparent. Mounting empirical evidence in the high-frequency literature suggests that jumps occur on 4%–13% of days per year on average (e.g., Andersen et al., 2007; Patton & Verardo, 2012; Alexeev et al., 2017; among others). It can be argued, however, that events that occur this frequently can hardly be classified as “extreme”.6 The jump identification literature offers a number of methods that can sieve out less extreme events by varying the threshold used in its detection (e.g., Mancini, 2001; Mancini & Renò, 2011; Davies and Tauchen, 2015). Thus, in our empirical application, in addition to threshold levels commonly used in the jump identification literature, we allow for the thresholds high enough to investigate the most severe occasions only. In line with Christensen et al. (2014), we detect far fewer jumps than what is usually found in the literature. More importantly, it is only at the higher jump detection threshold that we begin to observe the asymmetry in portfolio sensitivities to market jumps.

This asymmetry has crucial implications for portfolio allocation decisions. Investors typically perceive downside and upside extreme events differently.7 It is believed that the fear of large negative shocks is a component that drives asset prices, because investors expect compensation for the risk that such a rare event occurs. It is not only the occurrence of rare events but also the very fear of them that influences investors’ behaviour and market prices. Bollerslev and Todorov (2011) explore pricing implications for jump risk in periods of extreme downside losses as opposed to extreme upside gains, and check if investors demand additional compensation for holding stocks with high sensitivities to these movements. Results show that, although the behaviour of the two tails is clearly related, the contributions to the overall risk premium are far from symmetric. Further evidence on asymmetric effects of jump risk measures can be found in Guo, Wang, and Zhou (2015) and Audrino and Hu (2016). These findings highlight the importance of considering the asymmetry effects of extreme events in portfolio risk management.

In this paper, we provide equity portfolio size recommendations to stabilise portfolio jump betas. We find that the number of stocks required to stabilise portfolio exposure to sudden large negative price changes in the market is substantially greater than under a scenario where only positive jumps are considered, or when the asymmetry is not taken into account. We show that correlations between extreme market returns and corresponding stock returns increases during crises years, and are more pronounced for negative jumps. Thus, for a naive portfolio, using conservative conjecture on how many stocks to hold (i.e., the maximum number of recommended holdings across jump types, and threshold levels), we recommend at least 54 holdings. For example, based on portfolio holdings data from the Centre for Research in Security Prices (CRSP), the interquantile range of the number of holdings for domestic equity mutual funds is 44–132 with median of 75 holdings. Consequently, funds with less than 54 holdings, representing 35 percent of all equity mutual funds, are unnecessarily exposed to extreme market drops.

The remainder of the paper is organised as follows. Section 2 sets up the model framework. The data and our empirical investigation are detailed in Sections 3 and 4. We investigate the behaviours of systematic negative and positive jump risk factors in portfolios of assets in Section V. Section VI concludes.

Section snippets

Model setup

We start with a panel of N assets over a fixed time interval [0,T]. Following the convention in the high-frequency financial econometrics literature, we assume the log-price pi,t of the ith asset follows a semi-martingale plus jumps process in continuous time. In turn, the log-return of any asset, ri,t, has the following representation:ri,tdpi,t=bi,tdt+σi,tdWi,t+κi,tdμi,t,t[0,T],i=1,2,,N,where bi,t is a locally bounded drift term, σi,t denotes the non-zero spot volatility, Wi,t is a standard

Data

We investigate the behaviour of the βid+ and βid estimates over the period from January 2, 2003 to December 30, 2017. This period includes the financial crisis associated with the bankruptcy of Lehman Brothers in September 2008 and the subsequent period of turmoil in US and international financial markets. The underlying data are tick-by-tick price observations on 501 stocks drawn from the constituent list of the S&P500 index during our sample period, obtained from the Thomson Reuters

Empirical analysis

In this section we analyse the statistical properties of betas estimated based on the overall market returns, and based on the upside and downside market returns separately.

Portfolio simulation

The concept of portfolio diversification is straightforward: the level of portfolio risk falls as the number of holdings in a portfolio increases.20 In the previous section we explored the behaviour of continuous and jump risks for individual stocks. In this section we investigate how fast these systematic risks dissipate in portfolios.

In the last two decades, the availability of high-frequency

Conclusion

In this paper, we studied jump dependence between processes using high-frequency data focusing on observations that are informative for jump inference. In particular, we investigated the relationship between jumps in a process for an asset (or portfolio of assets) and an aggregate market factor, and analysed co-movements in jumps of the two processes. We examined a linear relationship between jumps and assessed the sensitivity of jumps in (portfolios of) assets to jumps in the market. Using

Funding

We thank Simona Boffelli, Mardi Dungey, Bart Frijns, Artem Prokhorov, George Tauchen, Viktor Todorov, Marcel Scharth, Ayesha Scott, Lars Winkelmann, Marcin Zamojski, and participants at the 10th Annual SoFiE Conference (New York, June 2017), the Midwest Finance Association Annual Meeting (Chicago, March 2017), Discipline of Business Analytics seminar at the University of Sydney (Sydney, October , 2016), the Financial Econometrics and Empirical Asset Pricing Conference (Lancaster, July , 2016),

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