Tail Risk Measures Research Articles

A Theory for Measures of Tail Risk

The notion of “tail risk” has been a crucial consideration in modern risk management and financial regulation, as very well documented in the recent regulatory documents. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures that quantify the tail risk, that is, the behaviour of a risk beyond a certain quantile. Such risk measures are referred to as tail risk measures in this paper. The two popular classes of regulatory risk measures in banking and insurance, value at risk (VaR) and expected shortfall, are prominent, yet elementary, examples of tail risk measures. We establish a connection between a tail risk measure and a corresponding law-invariant risk measure, called its generator, and investigate their joint properties. A tail risk measure inherits many properties from its generator, but not subadditivity or convexity; nevertheless, a tail risk measure is coherent if and only if its generator is coherent. We explore further relevant issues on tail risk measures, such as bounds, distortion risk measures, risk aggregation, elicitability, and dual representations. In particular, there is no elicitable tail convex risk measure other than the essential supremum, and under a continuity condition, the only elicitable and positively homogeneous monetary tail risk measures are the VaRs.

Expected returns with leverage constraints and target returns

Classic mean–variance optimization is very sensitive to expected returns. An alternative and more robust approach is to calculate the implied returns given the current portfolio allocation and risk profile. Portfolio managers can then do a reality check on the implied returns and find opportunities for better allocations. The most common implied return calculation assumes normal distribution and unlimited leverage, and use volatility as risk measure and covariance matrix as model input. However, practitioners usually have leverage constraints, often use non-parametric risk models, and care about portfolio tail risk. This paper presents a new approach to calculate expected returns with leverage constraints. This approach is flexible enough to alleviate normal distribution assumption, connect with non-parametric risk models, and use tail risk measures, such as conditional VaR.

Measuring Tail Risk

In this study we comprehensively investigate the usefulness of the tail risk measures proposed in the literature, evaluating the tail risk measures on the basis of their statistical and economic validity. Our main conclusion is that the option-implied measure of Bollerslev and Todorov (2011b) outperforms all others. It performs well for all tests and can predict not only the occurrence but also the size of future crash events. In addition, the measure is priced in the market: it predicts returns both in the time-series and in the cross-section. Finally, it also has an impact on real economic activity.

What Can Volatility Smiles Tell Us About the Too Big to Fail Problem?

This paper exploits the information content of option markets to offer fresh insight into the Too-Big-to-Fail (TBTF) problem for banks. I employ option prices to construct a forward-looking measure of bank exposure to significant price drops (i.e. tail-risk) and explore cross-sectional differences between large banks with at least $50B in assets identified as systemically important and smaller banks. I document a permanent increase in the average tail-risk of the U.S. banking industry as a whole following the Global Financial Crisis (GFC), except for banks above the $50B size threshold. I argue that the stark post-crisis difference in tail-risk for banks above and below the $50B threshold is consistent with the notion that the TBTF status of above 50B banks was reinforced by the series of bailouts targeted at them during the crisis and their subsequent designation as systemically important by the Dodd-Frank Act. This in turn raised investor expectations of future bailouts for above 50B banks and reduced their perceived exposure to downside risk as captured by the tail-risk measure.

Forecasting Tail Risk Measures for Financial Time Series: An Extreme Value Approach With Covariates

The paper develops a tail risk forecasting model that incorporates the wealth of economic and financial information available to risk managers. The approach can be viewed as a regularized extension of the two-stage GARCH-EVT model of McNeil and Frey (2000) where we permit a time-varying data-driven selection of a sparse set of covariates affecting the scale of the extreme value distribution of risk. We use a rich data set from the US equity market to explore when this additional information improves Value-at-Risk and Expected Shortfall forecasts as compared to popular tail risk forecasting methods. Under an extensive set of performance criteria and tests we demonstrate that our approach produces competitive risk forecasts, particularly for short horizons and during periods of financial distress.

Stress-Testing Net Trading Income: The Case of European Banks

Net trading income is an important but volatile source of income for many euro area banks, highly sensitive to changes in financial market conditions. Using a representative sample of European banks, we study the distribution of net trading income (normalized by total assets) conditional to changes in key macro-financial risk factors. To map the linkages of net trading income with financial risk factors and capture non-linear effects, we implement a dynamic fixed effects quantile model using the method of moments approach. We use the model to empirically estimate and forecast the conditional net trading income distribution from which we quantify tail risk measures and expected losses across banks. We find a heterogeneous and asymmetric impact of the risk factors on the distribution of net trading income. Credit and interest rate spreads affect lower quantiles of the net trading income distribution while stock returns are an important determinant of the upper quantiles. We also find that the onset of the Covid-19 pandemic resulted in a significant increase in the 5th and 10th percentile expected capital shortfall. Moreover, adverse scenario forecasts show a wide dispersion of losses and a long-left tail is evident especially in the most severe scenarios. Our findings highlight strong inter-linkages between financial risk factors and trading income and suggest that this tractable methodology is ideal for use as an additional tool in stress test exercises.

Tail Risk and Expectations

We study changes in expectations by incorporating tail risk in a Bayesian learning framework with information frictions. Using a signal extraction problem, we find that economic agents behave differently in the face of tail risk. First, under tail risk, uncertainty shocks lead to a decrease in expectations, which implies more pessimistic forecasts. In comparison, in the absence of tail risks, uncertainty shocks do not influence expectations. Second, we show that individuals overreact under tail risk, that is, individuals are excessively optimistic and pessimistic as compared to a Bayesian learning framework without tail risk. Third, the magnitude of overreaction under tail risk depends on the level of uncertainty in the economy. To validate these theoretical findings, we use quantile regressions to estimate the GDP growth distribution and its measure of tail risk. By comparing the state of the economy in tail risk and non-tail risk episodes, we find that uncertainty shocks lead to a larger fall in sentiments when the economy exhibits tail risk. In addition, using an event study approach, we show that there is more overreaction under tail risk episodes. Also, consistent with the theoretical findings, overreaction is larger in periods of higher uncertainty. Our findings seek to shed light on factors driving overreaction in expectations, and to highlight the importance of uncertainty shocks in propagating macroeconomic stability.

Prediction of Extremal Expectile Based on Regression Models With Heteroscedastic Extremes

Expectile recently receives much attention for its coherence as a tail risk measure. Estimation of conditional expectile at extremal tails is of great interest in quantitative risk management. Regression analysis is a convenient and useful way to quantify the conditional effect of some predictors or risk factors on an interesting response variable. However, when it comes to the estimation of extremal conditional expectile, the traditional inference methods may suffer from considerable variation due to a lack of sufficient samples on tail regions, which makes the prediction inaccurate. In this article, we study the estimation of extremal conditional expectile based on quantile regression and expectile regression models. We propose three methods to make extrapolation based on a second-order condition for a framework of the so-called conditionally heteroscedastic and unconditionally homoscedastic extremes. In addition, we establish the asymptotic properties of the proposed methods and show their empirical behaviors through simulation studies. Finally, data analysis is conducted to illustrate the applications of the proposed methods in real problems.

Tail risk measurement in crypto-asset markets

The paper examines the relationships among market assets during stressful times, using two recently proposed econometric modeling techniques for tail risk measurement: the extreme downside hedge (EDH) and the extreme downside correlation (EDC). We extend both measures taking into account the sensitivity of asset's return to innovations not only from the overall market index, but also from its components, by means of network modeling. Applying our proposal to the cryptocurrencies market, we find that crypto-assets can be clustered in two groups: speculative assets, such as Bitcoin, which are mainly “givers” of tail contagion; and technical assets, such as Ethereum, which are mainly “receivers” of contagion.

The SKEW index: Extracting what has been left

This study disentangles a measure of implied skewness that is related to downward movements in the U.S. equity index from the corresponding implied skewness that is associated with upward movements. A positive SKEW index is constructed from S&P 500 call options, whereas a negative SKEW index is constructed from the S&P 500 put options. We show that the positive SKEW is linked to market sentiment, whereas the negative SKEW is related to existing tail risk measures. The negative SKEW is proposed as a more objective prudent tail risk measure, and it is found to be able to predict recessions, market downturns, and uncertainty indicators up to one year in advance. The predictive power of the negative SKEW is also confirmed when we control for other tail risk measures and also out-of-sample.

The SKEW Index: Extracting What Has Been Left

This study disentangles a measure of implied skewness that is related to downward movements in the U.S. equity index from the corresponding implied skewness that is associated with upward movements. A positive SKEW index is constructed from S&P500 call options, whereas a negative SKEW index is constructed from the S&P500 put options. We show that the positive SKEW is linked to market sentiment, whereas the negative SKEW is related to existing tail risk measures. The negative SKEW is proposed as a more objective prudent tail risk measure, and it is found to be able to predict recessions, market downturns, and uncertainty indicators up to one year in advance. The predictive power of the negative SKEW is also confirmed when we control for other tail risk measures and also out-of-sample.

Can the Premium for Idiosyncratic Tail Risk be Explained by Exposures to its Common Factor?

Stocks in the highest idiosyncratic tail risk decile earn 8% higher average annualized returns than in the lowest. I propose a risk-based explanation for this premium, in which shocks to intermediary funding cause idiosyncratic tail risk to follow a strong factor structure, and the factor, common idiosyncratic tail risk (CITR), comoves with intermediary funding. Consequently, firms with high idiosyncratic tail risk have high exposure to CITR shocks, and command a risk premium due to their low returns when intermediary constraints tighten. To test my explanation, I create a novel measure of idiosyncratic tail risk that is estimated using high-frequency returns, and theoretically establish its time-aggregation properties. Consistent with my explanation, CITR shocks are procyclical, are correlated to intermediary factors, are priced in assets, and explain the idiosyncratic tail risk premium. Furthermore, volume tail risk also earns a premium, follows a strong factor structure, and its common factor is priced. This duality of idiosyncratic tail risk and volume tail risk provides evidence for my risk-based explanation, and further supports the hypothesis that intermediaries' large trades cause idiosyncratic tail risk and volume tail risk from Gabaix et al. (2006).

Two Stage Decumulation Strategies for DC Plan Investors: 'A Goal Properly Set Is Halfway Reached'

Optimal stochastic control methods are used to examine decumulation strategies for a defined contribution (DC) plan retiree. An initial investment horizon of fifteen years is considered, since the retiree will attain this age with high probability. The objective function reward measure is the expected sum of the withdrawals. The objective function tail risk measure is the expected linear shortfall with respect to a desired lower bound for wealth at fifteen years. The lower bound wealth level is the amount which is required to fund a lifelong annuity fifteen years after retirement, which generates the required minimum cash flows. This ameliorates longevity risk. The controls are the withdrawal amount each year, and the asset allocation strategy. Maximum and minimum withdrawal amounts are specified. Specifying a short initial decumulation horizon, results in the optimal strategy achieving: (i) median withdrawals at the maximum rate within 2-3 years of retirement (ii) terminal wealth larger than the desired lower bound at fifteen years, with greater than $90\%$ probability and (iii) median terminal wealth at fifteen years considerably larger than the desired lower bound. The controls are computed using a parametric model of historical stock and bond returns, and then tested in bootstrap resampled simulations using historical data. At the fifteen year investment horizon, the retiree has the option of (i) continuing to self-manage the decumulation policy or (ii) purchasing an annuity.

Do Investors Care about Tail Risk? Evidence from Mutual Fund Flows

This paper examines investor attitude toward tail risk in investment decision-making. Based on a large sample of mutual funds, we show that investor flows are significantly sensitive to tail risk in the cross-section, even after controlling for fund performance and characteristics. Using terrorist attacks and COVID-19 as exogenous shocks to the investor fear level, we find that fund flows become increasingly sensitive to tail risk following the shocks, suggesting that fear is a driving force of the tail risk aversion. In particular, the flow-tail risk sensitivity during the onset of COVID-19 is about 4.5-10 times as large as that in other periods. In addition, tail risk is associated with the activeness of mutual fund investment strategies. The results are robust to alternative measures of tail risk. Overall, our findings suggest that investors care about tail risk beyond traditional risks.

Break-even Inflation Rates: The Italian Case

This paper focuses on break-even inflation rates (BEIRs), a widely used market-based measure of expected inflation, computed from government bonds. In the first part of the paper, we regress the Italian BEIR on several financial variables to assess the contribution of inflation, credit and liquidity components. In the second, in order to disentangle market participants’ inflation expectations from risk premia, we estimate a term structure model for the joint pricing of the Italian nominal and real yield curves, considering also credit and liquidity factors. The results show that BEIRs could be a misleading measure of the expected inflation due to the importance of inflation risk premium and credit risk effect. According to our estimates, the decrease in market-based measures of inflation observed in the last part of the sample period seems to reflect a lowering of both inflation expectations and risk premia. Inflation premia co-move with a measure of tail risk of the long-term inflation distribution signalling that investors become more concerned with downside risks.

Time-varying parameters realized GARCH models for tracking attenuation bias in volatility dynamics

This paper proposes novel approaches to the modeling of attenuation bias effects in volatility forecasting. Our strategy relies on suitable generalizations of the Realized GARCH model by Hansen et al. [Realized garch: A joint model for returns and realized measures of volatility. J. Appl. Econom., 2012, 27(6), 877–906] where the impact of lagged realized measures on the current conditional variance is weighted according to the accuracy of the measure itself at that specific time point. This feature allows assigning more weight to lagged volatilities when they are more accurately measured. The ability of the proposed models to generate accurate forecasts of volatility and related tail risk measures, Value-at-Risk and Expected Shortfall, is assessed by means of an application to a set of major stock market indices. The results of the empirical analysis show that the proposed specifications are able to outperform standard Realized GARCH models in terms of out-of-sample forecast performance under both statistical and economic criteria.

An alternative nonparametric tail risk measure

The proposition of tail risk as a new asset pricing factor has gained traction in recent years. Recent work by Almeida, Ardison, Garcia, and Vicente (Nonparametric tail risk, stock returns, and the macroeconomy. J. Financ. Economet., 2017, 15(3), 333–376) proxies the cross-sectional variation in returns by Fama-French portfolios, which are further summarized into a few basis assets via principal component analysis. The number of states of nature is set higher than that of the basis assets to estimate the stochastic discount factors, which in turn risk-neutralize the excess expected shortfall as a tail risk measure. As an alternative approach to this means of dimension reduction, we propose tackling the problem directly by forming portfolios that minimize the excess expected shortfall. Our direct measure exhibits greater explanatory power when applied to more liquid, nonlottery-style stock returns. More importantly, our proposed approach reveals direct exposure to downside systematic risk without the need for an additional risk-neutralization step.

Decomposing the VIX: Implications for the predictability of stock returns

AbstractThe VIX index is not only a volatility index but also a polynomial combination of all possible higher moments in market return distribution under the risk‐neutral measure. This paper formulates the VIX as a linear decomposition of four fundamentally different elements: the realized variance (RV), the variance risk premium (VRP), the realized tail (RT), and the tail risk premium (TRP), respectively. Using an innovative and nonparametric tail risk measure, we find that approximately one‐third of the VIX's formation is attributed to the TRP. In addition to VRP, RT and TRP are crucial components for predicting future returns on equity portfolios.

Tail risk of electricity futures

This paper compares the in-sample and out-of-sample performance of several models for computing the tail risk of one-month and one-year electricity futures contracts traded in the NordPool, French, German, and Spanish markets in 2008–2017. As measures of tail risk, we use the one-day-ahead Value-at-Risk (VaR) and the Expected Shortfall (ES). With VaR, the AR (1)-GARCH (1,1) model with Student-t distribution is the best-performing specification with 88% cases in which the Fisher test accepts the model, with a success rate of 94% in the left tail and of 81% in the right tail. The model passes the test of model adequacy in the 100% of the cases in the NordPool and German markets, but only in the 88% and 63% of the cases in the Spanish and French markets. With ES, this model passes the test of model adequacy in 100% of cases in all markets. Historical Simulation and Quantile Regression-based approaches misestimate tail risks. The right-hand tail of the returns is more difficult to model than the left-hand tail and therefore financial regulators and the administrators of futures markets should take these results into account when setting additional regulatory capital requirements and margin account regulations to short positions.

Tail risk and return predictability for the Japanese equity market

This paper studies the predictability of the Japanese equity market, focusing on the forecasting power of nonparametric volatility and tail risk measures obtained from options data on the S&P 500 and Nikkei 225 market indices. The Japanese market is notoriously difficult to forecast using standard predictive indicators. We confirm that country-specific regressions for Japan – contrary to existing evidence for other national equity indices – produce insignificant predictability patterns. However, we also find that the U.S. option-implied tail risk measure provides significant forecast power both for the dollar–yen exchange rate and the Japanese excess returns, especially when measured in U.S. dollars. Thus, the dollar-denominated Japanese returns are, in fact, predictable through the identical mechanism as for other equity market indices, suggesting a high degree of global integration for the Japanese financial market.