Estimating portfolio risk for tail risk protection strategies

• Published date: 01 September 2020
• Date: 01 September 2020
• DOI: http://doi.org/10.1111/eufm.12256

© 2020 The Authors. European Financial Management Published by John Wiley & Sons Ltd.

Eur Financ Manag. 2020;26:1107–1146. wileyonlinelibrary.com/journal/eufm

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DOI: 10.1111/eufm.12256

ORIGINAL ARTICLE

Estimating portfolio risk for tail risk

protection strategies

David Happersberger

1,2

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Harald Lohre

1,2

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Ingmar Nolte

1

1

Centre for Financial Econometrics,

Asset Markets and Macroeconomic

Policy, Lancaster University Management

School, Lancaster, United Kingdom

2

Invesco Quantitative Strategies,

Frankfurt am Main, Germany

Correspondence

David Happersberger, Centre for

Financial Econometrics, Asset Markets

and Macroeconomic Policy, Lancaster

University Management School, Bailrigg,

Lancaster LA1 4YX, United Kingdom.

Email: d.happersberger@lancaster.ac.uk

Funding information

Economic and Social Research Council,

United Kingdom

Abstract

We forecast portfolio risk for managing dynamic tail risk

protection strategies, based on extreme value theory,

expectile regression, copula‐GARCH 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 value‐at‐risk) 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

insurancestrategiesprovideeffectivedownsideprotection,

the latter strategies suffer less from inferior risk forecasts,

given the defensive portfolio insurance mechanics.

KEYWORDS

CPPI, DPPI, expected shortfall, forecast combination, return syn-

chronization, risk modeling, tail risk protection, value‐at‐risk

JEL CLASSIFICATION

C13; C14; C22; C53; G11

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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 Bech‐Moen, 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.

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INTRODUCTION

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 long‐term

return potential by dynamically shifting between a risky asset (or portfolio) and a risk‐free asset.

Risk targeting strategies

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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 so‐called 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 time‐varying 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 time‐varying and conditional. The

corresponding strategy is known as dynamic proportion portfolio insurance (DPPI).

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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 buy‐and‐hold 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 value‐at‐risk (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 location‐scale models, while

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Risk targeting strategies are also known as constant risk, target risk, or inverse risk weighting strategies.

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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.

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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 high‐frequency intraday data.

The literature on DPPI puts forward various ways to model the conditional time‐varying

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.

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All these papers provide evidence that

the DPPI strategy, based on a time‐varying 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 models—Andersen et al. (2006, 2013), Kuester, Mittnik, and

Paolella (2006), and Righi and Ceretta (2015) provide thorough summaries on market risk

modeling—practitioners 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), Cornish‐Fisher approximation (CFA),

RiskMetrics, quantile and expectile regressions, extreme value theory, copula‐GARCH and

recent generalized autoregressive score (GAS) models (including one‐and two‐factor 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

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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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