This paper is based on the thesis of Marcel Bräutigam, a Sorbonne Université, LPSM & ESSEC CREAR (1) doctoral student, under the supervision of Marie Kratz. Their work was recently awarded one of the two special mentions at the PRIX DES SCIENCES DU RISQUE 2020.
« The major mistake made in 2010 was imposing austerity at the worst moment ». Gilles Moëc (Chief Economist at AXA), Le Monde, January 21st, 2020
Procyclicality (i.e.the tendency of risk measurements to overestimate future risk in times of crisis, while underestimating it in normal times) is a major problem that all financial institutions must manage: insurance companies, banks, regulatory bodies… As a result, they are required to provide substantial capital in the aftermath of a financial crisis, but far less capital prior to such crises. Hence Gilles Moëc’s warning.
The issue of procyclicality arises in the context of regulation and financial risk management. Financial institutions (such as insurance companies, banks, investment funds) are required to evaluate their risk based on probabilistic and statistical models, from which risk measures are estimated. The two most common risk measures are the ValueatRisk (VaR), i.e. a quantile at a certain threshold of the distribution of portfolio returns, and Expected Shortfall (ES), the average of all returns greater than the VaR. Clearly, an accurate capital valuation is essential for risk management. Further, determining the most appropriate risk measure to evaluate the risk of financial institutions is a highly debated question, especially after the 20082009 financial crisis [see Chen (2018) and Emmer et al. (2015) for a review]. We tackle this question from another angle, namely that of the procyclicality of the risk measure estimation, and examine the two following issues: How can we quantify procyclicality? How can we explain it?
Of course, the regulatory authorities of various sectors, like the BIS (Bank for International Settlements), EIOPA (European Insurance and Occupational Pensions Authority) or ESMA (European Securities and Markets Authority), have proposed various solutions to reduce procyclicality (in the context of Basel III, Solvency II, EMIR). These approaches are based on a macroeconomic perspective on procyclicality, as is the majority of the academic literature, namely: the macroeconomy is affected by how banks react to changing macroeconomic conditions, which in turn impacts the bank’s performances, reinforcing the effects of the cyclical fluctuations of the macroeconomy.
In this study (2), taking a statistical point of view, we examine the possibility that the way of estimating capital requirements using risk measures (such as VaR and ES) is a possible source of procyclicality. This new approach complements the various economic studies carried out on the subject, since the error margin of risk measurement undoubtedly affects risk management. Starting from this very concrete question, we study it both empirically and theoretically, going back and forth between these two approaches, in order to mathematically validate (or not) the empirical facts we discovered.
We developed a methodology allowing us to quantify the procyclicality, using a new indicator called the “lookforward ratio” that compares (a posteriori) the estimated capital requirements with the actual capital required. The practical advantage of this method of measuring the accuracy of our risk prediction is that we directly obtain the degree of under or overestimation of the riskadjusted capital. If the predicted VaR is close to the actual VaR one year later, the ratio will be around 1. Otherwise, it will be either less than 1 (if the prediction was too conservative) or greater than 1 (if the VaR was insufficient). We analyze this indicator conditioned to the current market situation. To determine the state of the market, we use the volatility, as the realized volatility is high in times of crisis and low otherwise.
To quantify the procyclicality, we measure the dependence between the lookforward ratio and the realized volatility. This dependence turned out to be negative, and this empirical observation will be validated theoretically. To isolate the empirically observed effect of procyclicality, we introduce appropriate stochastic models. This permits us to identify two main factors that explain procyclicality:

the clustering and mean reversion effect of volatility, caused by fluctuations in the macroeconomic cycle (as expected), highlighted by the use of GARCH models

and, more surprisingly, the very way itself in which risk is measured, even in the case of independent variables, i.e. independent of market cycles!
The empirical results we obtained led us to study this dependence from a theoretical perspective: what can we say about the joint distribution between estimators of quantiles (as risk measures) and dispersion estimators (for volatility) for an iid (independent and identically distributed) sample or a GARCH process? We have answered this question in a more general mathematical framework (especially for the class of augmented GARCH processes, containing many wellknown GARCH processes).
This allowed us to return to our empirical study and to mathematically confirm the observed procyclicality, but also to generalize these results to other classic risk measures other than the VaR, like the ES and the expectile (a popular measure in the academic literature). The extension to the ES is especially important from a practical point of view, as the ES replaces the VaR in the Basel IV regulation.
Concrete solutions and implications of this study:

An empirical methodology for quantifying procyclicality that can be used for other risk measures (both existing and future measures)
The graph shows the average behavior of the capital requirement as a function of the state of the market (based on 11 stock market indices of the major economies). The first [yaxis] is presented in relation to a reference value of 1 (of the lookforward ratio), the second [xaxis] corresponds to the states of market volatility (in ascending order from 1 to 10). 
We can clearly see and measure the effect of procyclicality in this graph. Indeed, for low volatility, i.e. in "normal" times, we observe a value greater than 1 for the lookforward ratio. This indicates an underestimation of capital (shown in orange), which can be over 40% for very low volatility. Whereas in crisis times, represented by high volatility (states 8 to 10), the capital required from institutions is overestimated (lookforward ratio less than 1). The excess (represented by the stripped pattern) can be as high as about 30% of the capital required  and all this during a period during which companies and institutions have to face up to the crisis...

The existence of two factors explaining procyclicality, one linked to the cyclic behavior of volatility, as expected, and the other intrinsically linked to the method of estimating risk measures, regardless of which risk measure was used. The latter is more surprising and more interesting, as it goes beyond economic studies, showing the negative impact that the method of estimating risk can have on risk management.
Conclusion and perspective:
This research tackled the procyclicality of risk measures both empirically and theoretically, thereby complementing the existing macroeconomic research on the subject from a novel statistical point of view.
A key result of this thesis is the fact that procyclicality is not only caused by fluctuations in the macroeconomic cycle, like we expected, but there is also an intrinsic part due to the method used for risk measure estimation.
Further, a mathematical understanding of procyclicality opens up concrete paths for developing the next generation of regulatory risk measures (contracyclic), which we are currently working on.
Footnotes
1. With the financial support of Labex MMEDII
2. «Procyclicality of Risk Measurements Empirical Quantification and Theoretical Confirmation» (2020). Doctoral thesis of Marcel Bräutigam under the supervision of Marie Kratz, Statistics [math.ST], Sorbonne Université, <tel02954165>
References
Chen, J. M. (2018). On exactitude in financial regulation: ValueatRisk, expected shortfall, and expectiles. Risks, 6(2), 61.
Emmer, S., Kratz, M., & Tasche, D. (2015). What is the best risk measure in practice? A comparison of standard measures. Journal of Risk, 18(2).