Sensitivities-based method and expected shortfall for market risk under FRTB: its impact on options risk capital.
| Autor | Grajales, Carlos Alexander |
-
Introduction
Basel Committee on Banking Supervision (BCBS) consolidated in 2019 the Fundamental Review of the Trading Book in BIS (2019) (FRTB) to improve the design and consistency of capital standards for market risk. FRTB regulation introduces a new approach to the current market risk management models, whose implementation is planned as of 1 January 2023. The regulation incorporates three fundamental aspects.
The first aspect differentiates between the trading book and the banking book. It accurately identifies which instruments are assigned to each and avoids arbitrage opportunities among the estimated capital requirements.
The second change considers the internal models approach (IMA) to measure market risk, a scheme that the regulator must previously approve. At this point, the regulation replaces the value at risk (VaR) and stressed value at risk (sVaR) measures established by Basel 2.5. Instead, it estimates the total capital requirement based on the aggregation of the following three requirements: expected shortfall (ES), non-modellable risk factors (NMRF) and default risk capital (DRC).
The third aspect is the mandatory adoption of the standardised approach (SA) to determine capital for market risk. The total capital requirement is, in turn, defined in terms of the aggregation of the following three expositions: sensitivities-based method (SbM) through the measure of delta, vega and curvature risks; default risk capital (DRC); and residual risk add-on (RRAO).
In BCBS (2019), based on a sample of banks, the future impact of FRTB implementation was estimated in January 2019. From there, risk-weighted assets increased from 4.4% in Basel 2.5 to 5.3% in FRTB, concerning total risk-weighted assets in Basel III. In addition, the average increase in risk capital is 22% compared to Basel 2.5. In FRTB, the risk capital under SA is considerably higher (over 60% on average) than the respective capital under IMA. Moreover, further increases in risk capital are estimated for small banks using a simplified version of the standardised approach.
Some progress has been made regarding the study of market risk. For example, concerning the FRTB internal models approach, ES metric is a consistent risk measure based on Artzner et al's (1999) and Rockafellar and Uryasev's (2000, 2002, 2013) works. Furthermore, a comparative analysis between the best known VaR metric and ES can be seen in Embrechts et al (2018, 2020), and different estimation methods and ES models are documented, for example, in Patton et al. (2019), Nadarajah et al (2014), Chen (2008) and Scaillet (2004). Further, FRTB requires the implementation of back-testing procedures to test the reliability of VaR measures. In this direction, a comprehensive and practical study of different kinds of tests appears in Nieto and Ruiz (2016). Finally, Menendez and Hassani (2021) present several methodologies to achieve data augmentation in the tails of portfolio loss distributions to ensure robustness in ES estimations. They state that ES risk measure, as an FRTB benchmark, implies the renunciation of elliptic distributions for modelling the losses.
In turn, Laurent et al (2016) analyse the theoretical foundations and implications of the default risk capital under the framework of the FRTB internal models, particularly in portfolios sensitive to credit risk. In Orgeldinger (2018), some recent advances are also presented to implement the FRTB regulation industrially under SA and IMA. The author defines implementation stages; analyses technological issues, unprecedented computational demands and associated costs; and highlights the need to reconcile SA with IMA models. In line with the above, Pederzoli and Torricelli (2021) estimate and compare FRTB impacts on the capital requirement under both approaches, SA and IMA, based on a stylised portfolio with different risk factors. They report a more significant impact on banks adopting SA. In turn, Porretta and Agnese (2021) examine FRTB impacts on capital the requirement for different banking groups, classified according to their tier-1 capital size. They analyse risk capital changes from the current to the revised regulation and for both approaches, SA and IMA.
However, under FRTB, logical reasonings should be put forward to support possible SbM implementations for a general portfolio. Logical reasonings respond to the requirement of building quantitative relations between SA-SbM and IMA-ES, as well as measuring risk capital impacts via SbM (Orgeldinger (2018) and Porretta and Agnese (2021)). Additionally, ES metric's statistical performance and risk capital impacts should be investigated, particularly for financial options portfolios. Concerning the above literature, although data augmentation in Menendez and Hassani (2021) is a viable way to obtain reliable ES measures, using periods of financial crises is essential for constructing stress scenarios under internal models.
Consequently, this article develops two proposals. First, it proposes an algorithm from the FRTB standardised approach to estimate the market risk capital relative to delta, vega and curvature risks through the sensitivities-based method. This systematic reasoning is related to and extends the work of Orgeldinger (2018) since the algorithm enables to foresee fundamental conditions for its implementation, such as the involved processes, its complexity and its high computational demand. In detail, for any portfolio, the algorithm allows us, according to the new regulation, to identify its risk factors and risk classes involved, estimate sensitivities and correlations between all the assets, perform risk aggregation and financial stress analyses, and calculate the overall risk capital. Also, it helps to analyse the interactions between the parts of the SbM scheme and experiment with hypothetical portfolios to measure impacts on risk capital. Furthermore, the algorithm is a design that can connect to desirable outcomes regarding risk capital levels or impacts, implying that it supports the creation of knowledge rules in the financial risk management field.
Second, to build communication bridges between the standardised and internal approaches under FRTB, a methodology is proposed to estimate the ES metric in a portfolio of financial options and evaluate its performance and the risk capital impacts. The robustness of our methodology is accomplished in three ways. First, it generates stress scenarios with periods of crisis, which feed loss distributions with extreme data; second, it adapts VaR and ES metrics to IMA; and third, it evaluates the metrics' performances via back-testing proofs. Risk capital impacts are studied by changing option tenors and liquidity horizons. At this point, the methodology constitutes a formal adaptation of ES to IMA for option portfolios. Furthermore, the proposed relationships between ES and FRTB are a step forward in the literature that can become a technical document annexe in BCBS (2019).
Subsequently, a numerical illustration is provided to promote an understanding of the sensitivities-based method and estimate a portfolio's specific impacts by following the proposed algorithm. Then, for supporting evidence of the validity of our methodology, an application is developed where the impact of ES and VaR under FRTB versus conventional VaRis measured in a simple portfolio of currency options. This development considers stress scenarios from the 2007-9 and 2020-1 crises and back-testing procedures.
The paper is organised as follows. Section 2 reviews the theoretical framework, and Section 3 defines the methodology. Subsequently, Section 4 addresses the results, and the most relevant findings are discussed in Section 5. The last section presents the conclusions and limitations of the work.
-
Literature review
2.1 Market risk measures
2.1.1 Value at risk, VaR. The monetary measure (in dollars) $VaR for a portfolio over a time horizon and confidence level [alpha] satisfies the equation,
p [loss > $VaR] = 1 - [alpha] (1)
If the loss is assumed to be -[product]R, where [product] is the initial value of the portfolio and R its return over the time horizon, then equation (1) is equivalent to
p[R
where VaR = [VaR.sub.[alpha]] = $VaR/[product] and such risk measure is relative to the initial value of the portfolio.
2.1.2 Expected shortfall, ES. The internal models approach in FRTB has chosen to follow the ES metric to quantify market risk. Conceptually and for simple computations, if -[DELTA][product] is the loss portfolio from [product] in a given time horizon, then [ES.sub.[alpha]] is the conditional expectation of those losses that exceed [VaR.sub.[alpha]] in the loss distribution of -[DELTA][product], where [alpha] is the confidence level of the risk measure.
More precisely, following Rockafellar and Uryasev (2002), let us consider that x [member of] X [subset] [??] represents the vector of assignments in portfolio [product] for a set of constraints X, and y [member of] Y [subset] [??] denotes the vector of future values of m market variables that affect the portfolio profit and loss. Then, the portfolio loss function can be represented as z = f(x;y). Moreover, if p(y) is the probability density of y over a time horizon T, then [ES.sub.[alpha]] for portfolio [product](x) at T is defined by
[Please download the PDF to view the mathematical expression] (3)
where [Please download the PDF to view the mathematical expression].
Alternatively, other expressions that define [ES.sub.[alpha] in terms of the VaR measure appear in McNeil et al (2005). Keeping their notation, these are given by
[Please download the PDF to view the mathematical expression] (4)
[Please download the PDF to view the mathematical expression] (5)
[Please download the PDF to view the mathematical expression] (6)
where L is an integrable function with continuous distribution that represents the portfolio losses, so that L = -[DELTA][product], and the...
Para continuar leyendo
Solicita tu pruebaCOPYRIGHT GALE, Cengage Learning. All rights reserved.
