Estimating portfolio risk for tail risk protection strategies

Author:David Happersberger, Harald Lohre, Ingmar Nolte
Publication Date:01 Sep 2020
© 2020 The Authors. European Financial Management Published by John Wiley & Sons Ltd.
Eur Financ Manag. 2020;26:11071146.
DOI: 10.1111/eufm.12256
Estimating portfolio risk for tail risk
protection strategies
David Happersberger
Harald Lohre
Ingmar Nolte
Centre for Financial Econometrics,
Asset Markets and Macroeconomic
Policy, Lancaster University Management
School, Lancaster, United Kingdom
Invesco Quantitative Strategies,
Frankfurt am Main, Germany
David Happersberger, Centre for
Financial Econometrics, Asset Markets
and Macroeconomic Policy, Lancaster
University Management School, Bailrigg,
Lancaster LA1 4YX, United Kingdom.
Funding information
Economic and Social Research Council,
United Kingdom
We forecast portfolio risk for managing dynamic tail risk
protection strategies, based on extreme value theory,
expectile regression, copulaGARCH and dynamic gen-
eralized autoregressive score models. Utilizing a loss
function that overcomes the lack of elicitability for
expected shortfall, we propose a novel expected shortfall
(and valueatrisk) forecast combination approach, which
dominates simple and sophisticated standalone models as
well as a simple average combination approach in
modeling the tail of the portfolio return distribution.
While the associated dynamic risk targeting or portfolio
the latter strategies suffer less from inferior risk forecasts,
given the defensive portfolio insurance mechanics.
CPPI, DPPI, expected shortfall, forecast combination, return syn-
chronization, risk modeling, tail risk protection, valueatrisk
C13; C14; C22; C53; G11
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and
reproduction in any medium, provided the original work is properly cited.
We thank the editor (John A. Doukas), two anonymous referees, Torben Andersen, Ole Christian BechMoen, Christian
Groll, Manel Kammoun, Mark Kritzman, Yifan Li, Stefan Mittnik, James Taylor, Ralf Wilke as well as the participants
in the 2017 Paris Financial Management Conference (PFMC), the 2017 CEQURA Conference on Advances in Financial
and Insurance Risk Management in Munich, the 3rd KoLa Workshop on Finance and Econometrics at Lancaster
University Management School in 2017, the 2017 Doctoral Workshop on Applied Econometrics at the University of
Strasbourg, the 2017 Global Research Meeting of Invesco Quantitative Strategies in Boston, the 2018 Frontiers of Factor
Investing Conference in Lancaster, the 2018 FMA European Conference in Kristiansand and the 2018 IFABS Porto
Conference for fruitful discussions and suggestions. This work was supported by funding from the Economic and Social
Research Council (UK). Note that this paper expresses the authors' views, which are not necessarily those of Invesco.
Tail risk hedging strategies are of vital interest to many market participants to protect
investment portfolios against extreme negative market moves. An obvious protection is the
purchase of a put option. However, such a strategy can be expensive, since the option premium
is payable each investment period, although the protection could prove unnecessary in the
majority of periods. A possible alternative are dynamic asset allocation strategies, which aim to
improve the downside risk profile of an investment strategy without jeopardizing its longterm
return potential by dynamically shifting between a risky asset (or portfolio) and a riskfree asset.
Risk targeting strategies
are one such possibility (Bollerslev, Hood, Huss, & Pedersen, 2018;
Hocquard, Ng, & Papageorgiou, 2013; Perchet, De Carvalho, Heckel, & Moulin, 2015). A risk
targeting strategy controls portfolio risk over time by taking advantage of the negative
relationship between risk and return. Specifically, the investment exposure of the portfolio is
adjusted according to updated risk forecasts in order to keep the ex ante risk at a constant target
level. A stricter way to limit downside risk is to apply portfolio insurance strategies, such as the
constant proportion portfolio insurance (CPPI) strategy (Black & Jones, 1987, 1988; Perold,
1986; Perold & Sharpe, 1988), where the investor defines a minimum capital level to be
preserved at the end of the investment period. The key element in determining the investment
exposure to the risky asset is the socalled multiplier. This represents the inverse of the
maximum sudden loss of the risky asset, so that a given risk budget will not be fully consumed
and the portfolio value will not fall below the protection level. Initially, the multiplier was
assumed to be static and unconditional (e.g. Balder, Brandl, & Mahayni, 2009; Bertrand &
Prigent, 2002; Cont & Tankov, 2009). However, given the empirical characteristics of financial
assets, such as timevarying volatility or volatility clustering (cf. Andersen, Bollerslev,
Christoffersen, & Diebold, 2006; Longin & Solnik, 1995), other studies (e.g. Hamidi, Maillet,
& Prigent, 2014) propose to model the multiplier as timevarying and conditional. The
corresponding strategy is known as dynamic proportion portfolio insurance (DPPI).
The success of both dynamic tail risk protection strategies strongly depends on the success of
forecasting tail risk (Perchet et al., 2015). Given a plethora of available risk models, we
contribute to the existing literature on tail risk protection strategies by investigating suitable
risk models for timely management of the investment exposure in dynamic tail risk protection
strategies. At the same time, we contribute to the literature on risk model evaluation by
assessing not only its statistical performance but also its economical relevance when testing the
risk forecasts in a relevant portfolio application.
Risk targeting strategies have been extensively backtested using historical data, and are
known to show superior performance compared to a simple buyandhold strategy (Cooper,
2010; Giese, 2012; Ilmanen & Kizer, 2012; Kirby & Ostdiek, 2012). Hallerbach (2012, 2015)
demonstrates that the Sharpe ratio increases, even if the portfolio mean return is constant over
time. Constant volatility portfolios not only deliver higher Sharpe ratios than their passive
counterpart but also reduce drawdowns (Hocquard et al., 2013). Similar to the risk targeting
strategy that we apply, the dynamic valueatrisk (VaR) portfolio insurance strategy of Jiang,
Ma, and An (2009) aims to control the exposure of a risky asset such that a specified VaR is not
violated. However, their strategy can only be applied to parametric locationscale models, while
Risk targeting strategies are also known as constant risk, target risk, or inverse risk weighting strategies.
For a comprehensive literature review on portfolio insurance and CPPI/DPPI multipliers, see Benninga (1990), Black and Perold (1992), Basak (2002), Dichtl
and Drobetz (2011), Hamidi et al. (2014), among others.
the one we apply is compatible with any type of risk model. In a similar vein, Bollerslev et al.
(2018) use a risk targeting strategy to compare realized volatility models to more conventional
procedures that do not incorporate the information in highfrequency intraday data.
The literature on DPPI puts forward various ways to model the conditional timevarying
multiplier. While Ben Ameur and Prigent (2007, 2014) employ generalized autoregressive
conditional heteroskedasticity (GARCH) type models, Hamidi, Jurczenko, and Maillet (2009)
and Hamidi, Maillet, and Prigent (2009) define the multiplier as a function of a dynamic
autoregressive quantile model of the VaR according to Engle and Manganelli (2004). In
contrast, Chen, Chang, Hou, and Lin (2008) propose a multiplier framework based on genetic
programming. More recently, Hamidi et al. (2014) employ a dynamic autoregressive expectile
(DARE) model to estimate the conditional multiplier.
All these papers provide evidence that
the DPPI strategy, based on a timevarying conditional risk estimate, outperforms the
traditional CPPI strategy.
We are particularly interested in comparing different ways to determine the risky investment
exposure of dynamic tail risk protection strategies, assessing various models to estimate a
portfolio's downside risk measured by VaR and expected shortfall (ES). While the literature
suggests a myriad of VaR and ES modelsAndersen et al. (2006, 2013), Kuester, Mittnik, and
Paolella (2006), and Righi and Ceretta (2015) provide thorough summaries on market risk
modelingpractitioners still only use a limited number of them. This discrepancy might be due
to complexity, (computational) efficiency or the perception that the incremental benefit of
implementing a highly sophisticated model is minor. Therefore, we examine simple methods
that are common among practitioners as well as more involved methods to predict VaR and ES.
Specifically, we consider: historical simulation (HS), CornishFisher approximation (CFA),
RiskMetrics, quantile and expectile regressions, extreme value theory, copulaGARCH and
recent generalized autoregressive score (GAS) models (including oneand twofactor GAS
models as well as the hybrid GAS/GARCH model).
The primary issue of these (standalone) risk models is that their performance and reliability
in accurately predicting risk often depend heavily on the data. While a parsimonious model can
perform well in stable markets, it might fail during a volatile period. Likewise, highly
parameterized models can be adequate during periods of high volatility, but might be easily
outperformed by simpler approaches in less turbulent times (cf. Bayer, 2018). Hence, it is often
beneficial to combine predictions originating from various approaches. Reviewing forecasting
combinations, Timmermann (2006) provides three arguments in favor of combining forecasts to
enhance the predictive performance relative to standalone models. First, there are diversifica-
tion gains arising from the combination of forecasts computed from different assumptions,
specifications or information sets. Second, combination forecasts tend to be robust against
structural breaks. Third, the influence of potential misspecification biases and measurement
errors of the individual models is reduced due to averaging over a set of forecasts derived from
several models.
While there exist various approaches to combine VaR predictions (see Bayer, 2018, for a
summary), the literature is lacking methods that combine ES predictions. This shortage relates
to the fact that ES is not elicitable,that is, there does not exist a loss function such that the
correct ES forecast is the solution to minimizing the expected loss (cf. Gneiting, 2011). This lack
Hamidi et al. (2014) model the multiplier as a function of the expected shortfall determined by a combination of quantile functions in order to reduce the
potential model error. Specifically, they combine the historical simulation approach, three methods based on distributional assumptions, and four methods
based on expectiles and conditional autoregressive specifications into the DARE approach.

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