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Financial Markets and Portfolio Management

08-10-2024

Minimum capital requirement portfolios according to the new Basel framework for market risk

Authors: Alessandro Avellone, Ilaria Foroni, Chiara Pederzoli

Published in: Financial Markets and Portfolio Management

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Abstract

In the new Basel framework for market risk finalized in January 2019, the minimum capital requirement for banks involves a liquidity-adjusted Stressed Expected Shortfall and a penalization factor that depends on the outcome of a Value at Risk-based backtesting procedure. This paper examines the optimization problem of a bank aiming to minimize its capital requirement, expressed analytically by a non-convex and non-differentiable function, being the penalization a discrete function of the number of Value at Risk violations. To address the portfolio selection problem, we implement an algorithm based on the Particle Swarm Optimization metaheuristic. In the empirical analysis, we compare portfolios obtained by minimizing the capital requirement with those selected by standard risk measures, i.e. Expected Shortfall, Value at Risk, and variance. Two data samples are considered: one from the recent Covid-19 pandemic period and the other from a quieter phase. Our findings suggest that minimizing the capital requirement during highly volatile periods, such as the global pandemic, would help contain the number of backtesting violations compared to other strategies. By contrast, during tranquil periods the inclusion of the penalization multiplier in the objective function would have no impact. However, portfolios selected to minimize Expected Shortfall, Value at Risk, and variance in both cases are far from the regulatory capital efficient frontier, resulting in varying degrees of efficiency losses.

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The innovations introduced in the Basel regulatory framework in the recent years go beyond the CR. See, for instance, Birindelli et al. (2022) for a comprehensive analysis of the impact of Basel III on banks’ stability.

The implementation of the revised market risk framework, originally planned for January \(1^\text {th}\) 2022, had been postponed to January 2023 because of the Covid-19 pandemic (BCBS 2020).

VaR is the maximum portfolio loss at a specified confidence level over a given holding period. ES is the expected loss conditional on VaR being exceeded.

The Basel Committee on Banking Supervision (BCBS 2012) mentions the reduction in the procyclicality of the minimum CR for trading books as one of the goals for revisions to the Basel framework. We refer to Alexander et al. (2014), Danielsson et al. (2001), Gordy and Howells (2006), Pederzoli et al. (2010), among the others, for comments on the potential procyclicality effects implied in Basel II.

The CR calculation rule will be shortly explained in Sect. 2.

The datasets analyzed during the current study are available in the FactSet repository.

Under standard assumptions, the Markowitz approach to portfolio optimization allows the construction of the efficient frontier in the mean-variance plane. However, the inclusion of real-world factors can alter the shape of this frontier (see, among the others, Michaud 1989; Ritter 2003; Pattitoni and Savioli 2011; Elton et al. 2014).

We refer to Righi and Borenstein (2018) for a list of risk measures adopted by researchers and practitioners in portfolio optimization and for their empirical comparison.

The interested reader may refer to Rockafellar and Uryasev (2002), which is the milestone in this field, to Rockafellar and Uryasev (2002); Lim et al. (2010).

Some papers analyze the problem of minimizing the CR of a given portfolio by choosing the most favorable risk model (the passive approach in the terminology of Santos et al. 2012). McAleer (2009) deals with the CR minimization problem where the decision consists in choosing the more suitable VaR model. In the same vein, Liu and Stentoft (2021) compare different models for the dynamic of the portfolio losses in terms of the CR in the new FRTB.

We refer to BCBS (2019) for an exhaustive description.

See Balter and McNeil (2018) for a justification of the adjustment formula.

In this presentation, we are neglecting some elements which are not relevant for our analysis. Specifically, we did not mention that: (1) Stressed ES can be obtained as a proxy from the ES estimated on the latest observations and Stressed ES over a subset of risk factors; (2) risk factors must be considered as well; (3) a default risk charge must be added to the CR; (4) a capital surcharge is added for desks falling in the amber zone of the newly introduced profit and loss attribution test; (5) the regulatory CR is determined as the maximum of the current sES estimate and the average sES over the preceding 60 business days multiplied by the penalty factor. Focusing on the last point, we note that the penalty factor applies to the average ES only: actually, since\(k\ge 1.5\) the max operator will select sES only in the case of an exceptionally high estimation of the ES in the current date. We can reasonably expect that the penalization factor enters the quantification of the CR. Moreover, under the hypothesis of our empirical work (constant portfolio and estimation via historical simulation) the 60-days average sES necessarily coincides with the current sES. We refer to BCBS (2019) for the detailed illustration of these points.

Santos et al. (2012) actually consider the evolution of the portfolio weights over a period of 8 years; however, they fix the portfolio to the currently determined one when modeling the number of VaR violations as a constraint. Our approach is essentially in line with Drenovak et al. (2017), even if they fix the asset holdings instead of the asset weights.

In other contexts, VaR estimated for \(t+1\) based on the information available in t is denoted as \(\text {VaR}_{t+1/t}\) or simply \(\text {VaR}_{t+1}\).

For the sake of precision, the ES corresponds to the so called \(\hbox {CVaR}^+\) (upper Conditional VaR), as defined in Rockafellar and Uryasev (2002). While CVaR is always convex, the same cannot be said for \(\hbox {CVaR}^+\). However, since we do not rely on techniques for convex programming, in our work the difference is actually not relevant.

Actually, by including only the stocks with long historical series available, we had at our disposal 46 stocks.

As stated in BCBS (2019) - MAR 33.4 - the 10-days return time series can be computed with overlapping observations. Therefore, we needed to go back 260 days to obtain the stress period.

In our analysis, we decided to disregard the assets held in the portfolio with a weight of less than 1%.

We recall that the sES is estimated over the financial crisis stress period and it is not influenced by the data of the actual sample under consideration (LV or HV). However the optimal solutions for problem (I) depend on the expected returns of portfolios that we have estimated over the 500 days preceding the optimization date T and, therefore, they may differ for the HV and the LV samples.

The benefits of diversification during financial crises are actually a topic of ongoing research, as highlighted in the review by Koumou (2020).

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Title: Minimum capital requirement portfolios according to the new Basel framework for market risk
Authors: Alessandro Avellone
Ilaria Foroni
Chiara Pederzoli
Publication date: 08-10-2024
Publisher: Springer Berlin Heidelberg
Published in: Financial Markets and Portfolio Management
Print ISSN: 1934-4554
Electronic ISSN: 2373-8529
DOI: https://doi.org/10.1007/s11408-024-00454-5