Abstract
Purpose
This paper measures different market risk impacts on options portfolios under the new Fundamental Review of the Trading Book (FRTB) regulation, issued in Basel and coming into effect in 2023.
Design/methodology/approach
This paper first suggests an algorithm for implementing the FRTB standardised approach via the sensitivities-based method to estimate a portfolio's risk capital and presents an illustration applied to an option position. Second, it proposes a methodology to estimate the expected shortfall in options portfolios from the FRTB internal models approach. In this regard, an application is developed to measure expected shortfall (ES) and value at risk (VaR) impacts under FRTB versus conventional VaR in a currency option position by considering stress scenarios from the 2007–9 and 2020–1 crises and back-testing procedures.
Findings
The suggested algorithm satisfactorily captures impacts via the sensitivities-based method, and higher risk capital demands are expected for emerging economies. Also, the planned FRTB methodology to measure ES and VaR is appropriate; in particular, historical metrics perform well. Astonishingly, their revealed impacts are more significant under the 2020–1 pandemic crisis than the 2007–9 financial crisis.
Originality/value
The proposals developed weave a communication bridge between the standardised and internal approaches of FRTB regulation, which can be scaled up technologically and institutionally.
Keywords
Citation
Grajales, C.A. and Medina Hurtado, S. (2023), "Sensitivities-based method and expected shortfall for market risk under FRTB: its impact on options risk capital", Journal of Economics, Finance and Administrative Science, Vol. 28 No. 55, pp. 96-115. https://doi.org/10.1108/JEFAS-12-2021-0268
Publisher
:Emerald Publishing Limited
Copyright © 2023, Carlos Alexander Grajales and Santiago Medina Hurtado
License
Published in Journal of Economics, Finance and Administrative Science. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence maybe seen at http://creativecommons.org/licences/by/4.0/legalcode
1. 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 (
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,
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,
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
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
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.
2. Literature review
2.1 Market risk measures
2.1.1 Value at risk, V a R
The monetary measure (in dollars)
If the loss is assumed to be
2.1.2 Expected shortfall, E S
The internal models approach in FRTB has chosen to follow the
More precisely, following Rockafellar and Uryasev (2002), let us consider that
Alternatively, other expressions that define
Additionally, from the works of Rockafellar and Uryasev (2002) and McNeil et al. (2005),
2.2 Basel FRTB regulation
The following body of concepts and relations is taken from BIS (2019).
2.2.1 IMA – E S
Now let us consider some features of IMA regarding the
The time horizons for liquidity are established as 10, 20, 40, 60 and 120 days. However, all calculations are based on changes in risk factors across overlapping 10-day periods, which is crucial since the samples produced are conceptually dependent.
Symbolically, if the risk factors are classified in bucket
Next, to obtain the capital requirement, it is necessary to estimate the so-called weighted expected shortfall,
The regulation agrees upon two performance tests (back-testing and profit and loss). The first must be carried out in the banking book, while both tests must be carried out in the trading book. Furthermore, in the trading book, the back-testing is based on the 1-day
2.2.2 SA – SbM
FRTB regulation in the standardised approach includes the sensitivities-based method, where the market risk capital is estimated through delta, vega and curvature risks for the portfolio risk factors. Calculating such capital involves various tasks in which risk classes are identified, and according to objective sensitivity, risk factors are grouped into buckets or categories. From here, the capital requirement (
Symbolically, using the same notation and formulation as that established in FRTB, if
Consequently, the aggregation
For other complementary formulations, as well as those related to estimations of capital requirement associated with curvature risk, see the regulation directly in BIS (2019).
2.3 Back-testing in FRTB
A back-testing proof is a procedure, usually of a statistical nature, which evaluates the accuracy of a risk measure from historical data. The
2.3.1 Validation proofs
A set of eight statistical back-testing tests is referenced below to validate a
Frequency tests
Traffic light (TL), see BCBS (1996, 2006, 2011).
Binomial test (Bin), see an adaptation in Hull (2018).
Unconditional coverage test (UC), see Kupiec (1995).
Time until first failure (TUFF), see Kupiec (1995).
Independence tests
Conditional coverage independence (CCI), see Christoffersen (2012, 1998).
Time between failures independence (TBFI), see Haas (2001).
Mixed frequency and independence tests.
Conditional coverage mixed (CC), see Christoffersen (2012, 1998).
Time between failures mixed (TBF), see Haas (2001).
3. Method
3.1 FRTB standardised approach – sensitivities-based method
The different portfolio instruments in the trading book are related to risk factors and determine their price. In turn, risk factors are made to correspond to seven risk classes: interest rate (GIRR), three types of credit spread (CSR), equity (stock exchange), commodities and foreign exchange rate (FX).
Additionally, the different risk factors are grouped into buckets or categories in each risk class. For example, in the equity class, such buckets can be defined based on asset capitalisation and membership to a particular industry sector and a developed or emerging market. Buckets can also be specified according to a trading currency, the credit quality of an institution, the type of a commodity and the exchange rate between two currencies, among others.
Subsequently, SbM requires estimating three risk exposures via delta, vega and curvature for each factor. Regarding vega sensitivity, its risk-weighting considers the liquidity horizon of the respective risk class to which the factor belongs.
According to FRTB regulation, sensitivities must be aggregated under prescribed correlations, at bucket level and across buckets of the same class. Finally, the capital requirement
Therefore, the logical structure of the processes involved in calculating capital requirements under the FRTB sensitivities-based method is complex. Next, we suggest Algorithm 1, which represents a preliminary version for a banking industry implementation of the SbM programme. The algorithm partly extends the work of Orgeldinger (2018) and is aligned with the need to automate modern standardised processes in market risk management from 2023. The implementation could also facilitate experimentation in calculating risk capital for hypothetical portfolios via FRTB SA SbM regulation.
The lines in Algorithm 1, except those containing loop processes, would lead to their own algorithms. The amount of “for” cycles reflects the entire process's high complexity and computational demand. Comments in grey in lines 1, 2, 3 and 25 indicate specific items of the FRTB regulation to appeal to carry out the corresponding process. The remaining lines are the consequence of an integrated vision of Chapter MAR21 in FRTB.
Finally, lines 17, 19 and 25 involve, respectively, calculations for risk aggregation within and across buckets, and the establishment of capital requirement by scenario analysis according to correlations between buckets. Technical formulations were described and referenced in Section 2.2.2.
3.2 FRTB internal models approach – expected shortfall
We propose an adaptation of the traditional historical simulation method to estimate the
(Scenarios for option pricing). Consider a portfolio with initial value
(
Proof. Without loss of generality, assume that
Now assume that
(
Second, an adaptation to the historical simulation method is proposed to estimate
(
Proof. The estimation is achieved by appealing to equation (7), making the losses
(
(
(stress scenarios to consider in FRTB). The stress scenarios involved in definition 3.1 can be achieved from the 2007–9 global financial crisis and the 2020–1 pandemic crisis. The resulting measures,
4. Results
4.1 Illustration: risk capital by the sensitivities-based method in Basel FRTB
BCBS documents in BCBS (2019) two simple examples to estimate capital requirements through the sensitivities-based method in the standardised approach, endeavouring to promote a level of understanding for its industrial implementation. However, the examples are remarkably brief and not very explicit in their calculations.
Therefore, this illustration looks to widen the scope of the example applied to an option position. To achieve this, the following procedure follows the steps of Algorithm 1 in Section 3.1, complements some estimations to be made in spreadsheets and explicitly indicates the FRTB articles to be used. These additional characteristics thus serve as an auxiliary guide towards studying and implementing the sensitivities-based method.
The banking portfolio in question consists of a short position put option on the hypothetical Telco asset, which reports in Canadian currency (CAD). Table 1 shows the essential data of the portfolio and reproduces Table A2.1 in BCBS (2019).
Consequently, Figures 1–3 estimate the delta, vega and curvature risk capital in spreadsheets following the guidelines of Algorithm 1 under medium correlation scenarios. The notation used in the previous Tables was introduced in Section 2.2.2.
The right column of each spreadsheet references the specific FRTB items that support the calculations. When the calculations for delta and vega are repeated under low and high correlation scenarios, generating similar spreadsheets, delta risk capital remains at CAD 0.21, while vega capital increases to CAD 0.9567. As in BCBS (2019), it is assumed that the risk capital for curvature given the correlation stress is kept at CAD 1.75. In the worksheet represented in Figure 3, the calculations in the cells marked with # are ignored. However, the corresponding magnitudes are calculated as usual in derivatives valuation. The risk capital under the sensitivities-based method turns out to be CAD 2.9167. As a comparison, the results shown coincide with those reported in BCBS (2019) for vega and curvature risks in medium correlation scenarios. The remaining calculations shown in our study do not appear there.
To conclude, it is noted that total capital requirement is significantly greater than the option's value (by a factor of 7.6). Therefore, the bucket associated with the option is b6 (cell B3 in Figure 1), which means that the underlying asset, affiliated with the telecommunications sector, has a high capitalisation (
The development of Algorithm 1 on a large scale would allow access to a platform to measure market risk and its possible impacts on a portfolio via the SbM schema. Such endeavour entails significant challenges regarding the instantiation of Algorithm 1. First, starting from a given portfolio, linguistic routines must be written to identify factors, buckets and risk classes. In addition, it is necessary to identify weighting factors and correlation matrices, generate sensitivities calibration routines from market data and create expert reports from text summaries. Finally, implementing each line within the algorithm loops should be processed in separate modules.
From a research point of view, analysing the algorithm's scope from computer science theories, like algorithm engineering, and evaluating its computational complexity using parallel processors, are open inquiries (see, e.g. Kliemann and Sanders (2016)).
4.2 Application: E S V a R
A portfolio composed of an at-the-money (ATM) European call option on GBP/USD having an annual maturity
On the other hand, the estimated risk measures are compared with conventional
Finally, according to Section 2.3, back-testing tests are executed for the
4.2.1 Market data and stress scenarios
In the following, dates are written in dd/mm/yyyy format. The data used correspond to the GBP/USD exchange rate and the treasury curves for the zero-coupon rate in the USA and the UK corresponding to the annual tenors
Period i: 2007–9 financial crisis. Date range considered: 02/03/2009–01/18/2010.
Period ii: 2020–1 covid-19 crisis. Date range considered: 02/10/2020–02/22/2021.
4.2.2 Information for the estimation of V a R E S
Table 2 summarises the information needed to measure the risk metrics in question. The valuation date
The estimated
4.2.3 Risk metrics, comparisons and impacts
To begin with, the following notation is established:
Subsequently, with the information given in Table 2 and the methodology defined in Section 3.2, the agreed risk metrics are estimated in relative terms as shown in Table 3, where risk metrics are computed traditionally for each date
For comparison purposes, the results in Table 3 can be represented in area diagrams according to the pair
The following approximate pattern is observed in Table 3.
Expiration
Crisis stress scenarios – FRTB: The 2020–1 crisis has a more significant impact on the historical
Horizon
4.2.4 Back-testing
Figure 5 shows call option returns vs
In the first column of Table 4, there are two valuation dates denoted by
The overall performance of all risk measures under back-testing is good. However, we believe that comparatively the performance of the historical
In sum, the previous observations and the results reported in Section 4.2.3 suggest that historical
Severity validation tests for
5. Discussion
This work develops two proposals for market risk estimations and their associated impacts under FRTB regulation, which comes into effect in 2023. First, it proposes an algorithm to estimate risk capital via the SA-SbM framework. An illustration of a hypothetical position in options shows a possible implementation of the algorithm in spreadsheets to evaluate impacts, yielding that the risk capital to position value ratio for an advanced economy is 7.6. Second, we design a methodology to assess
The above illustration of the algorithm suggests that, for emerging economies, the risk capital to position value ratio would be more drastic than previously indicated. Financial institutions must join efforts or restructure their portfolios to meet SA SBM requirements. On the other hand, the proposed methodology for evaluating
Implementing the two proposals can become a technological tool for the banking industry or an experimentation platform to explore possible marginal impacts concerning SbM or
6. Conclusions
With the new Basel FRTB regulation for market risk in 2023, banking institutions must prepare for future impacts and complex challenges in knowledge management, technological and human resources implementation, ethical and accounting frameworks, portfolio restructuring and institutional capital. The two proposals presented in this work draw communication bridges between the sensitivities-based method in the standardised approach (SA-SbM) and the expected shortfall in the internal models approach (IMA-ES).
The proposals aim to provide methodologies whose implementations assist in measuring market risk and the associated impacts on risk capital. The bridges seek to analyse the impacts of risk capital in different economies and make the associated risk capitals more convergent and comparable. All the above will support decision-making processes concerning risk-based capital requirements and strengthen institutional risk management processes.
Our research shows that under the SA-SbM schema and from a stylised option portfolio, the risk capital to position value ratio is higher than 7 and would impact emerging economies. Also, under the IMA-ES framework and from a currency options portfolio, we present evidence that the methodology defined for measuring market risk is satisfactory, considering FRTB requirements and a statistical point of view.
The technical and managerial complexity levels involved in the Basel FRTB regulation are today a focus of attention for institutions and academics. This work focuses on the FRTB-SA-SbM and FRTB-IMA-ES components of the FRTB regulation for market risk in options. The effects of SA-DRC, SA-RRAO, IMA-NMRF and IMA-DRC components remain to be measured.
Figures
Figure 4
VaR and ES under FRTB for a call option on GBP/USD (see Table 3)
Option portfolio in Basel FRTB
| Instrument | Currency | Maturity | Industry | Exercise type | Market value (CAD) |
|---|---|---|---|---|---|
| put option @ Telco | EUR | 2 years | Telecommunications | European | −0.38 |
Source(s): Own elaboration adapted from BCBS (2019), Table A2.1
| Risk metric valuation dates (t) | 22/01/2021, 03/06/2021 | |
|---|---|---|
| Stress scenarios | Period i | 03/02/2009–18/01/2010 |
| Period ii | 10/02/2020–22/02/2021 | |
| Call on GBP/USD | Pricing model | Black Scholes |
| Underlying value | Exchange rate at | |
| Annual maturity | ||
| Strike | Underlying value at | |
| Volatility GBP/USD | Over 250 days prior to | |
| Local and foreign rates | Zero curves in the USA and the UK | |
| Risk metrics | Conventional | Delta, delta-gamma |
| Normal, | ||
| Confidence level and risk horizon |
Source(s): Own elaboration
VaR and ES at N days under FRTB and traditional VaR
| Greeks | −0.2249 | −0.1601 | −0.7112 | −0.5064 | −1.0058 | −0.7161 | |
| −0.2101 | −0.1528 | −0.6375 | −0.4700 | −0.8785 | −0.6533 | ||
| Crisis 2007–9 | −0.2421 | −0.1726 | −0.7475 | −0.5144 | −1.1062 | −0.7467 | |
| −0.2470 | −0.1760 | −0.7640 | −0.5257 | −1.1308 | −0.7633 | ||
| −0.2621 | −0.1911 | −0.4821 | −0.3546 | −0.6432 | −0.4812 | ||
| −0.2908 | −0.2072 | −0.9113 | −0.6271 | −1.3515 | −0.9126 | ||
| −0.3321 | −0.2367 | −1.0503 | −0.7228 | −1.5597 | −1.0534 | ||
| −0.3032 | −0.2225 | −0.5837 | −0.4414 | −0.7055 | −0.5423 | ||
| Crisis 2020–1 | −0.2174 | −0.1554 | −0.6560 | −0.4795 | −0.8455 | −0.6068 | |
| −0.2216 | −0.1584 | −0.6696 | −0.4894 | −0.8630 | −0.6193 | ||
| −0.2149 | −0.1556 | −0.6977 | −0.5454 | −0.8296 | −0.6696 | ||
| −0.2601 | −0.1859 | −0.7908 | −0.5776 | −1.0196 | −0.7318 | ||
| −0.2964 | −0.2118 | −0.9053 | −0.6608 | −1.1673 | −0.8379 | ||
| −0.3133 | −0.2313 | −0.9037 | −0.7826 | −0.9241 | −0.7978 | ||
| Greeks | −0.2238 | −0.1594 | −0.7077 | −0.5041 | −1.0009 | −0.7129 | |
| −0.2088 | −0.1519 | −0.6332 | −0.4669 | −0.8722 | −0.6487 | ||
| Crisis 2007–9 | −0.3344 | −0.2392 | −1.0525 | −0.7239 | −1.5624 | −1.0574 | |
| −0.3413 | −0.2441 | −1.0765 | −0.7403 | −1.5988 | −1.0818 | ||
| −0.3508 | −0.2592 | −0.6086 | −0.4621 | −0.7731 | −0.6072 | ||
| −0.4026 | −0.2877 | −1.2912 | −0.8868 | −1.9245 | −1.3006 | ||
| −0.4605 | −0.3289 | −1.4939 | −1.0251 | −2.2320 | −1.5071 | ||
| −0.4027 | −0.3003 | −0.7172 | −0.5649 | −0.8282 | −0.6725 | ||
| Crisis 2020–1 | −0.2989 | −0.2146 | −0.8730 | −0.6354 | −1.1183 | −0.8050 | |
| −0.3049 | −0.2189 | −0.8918 | −0.6489 | −1.1429 | −0.8224 | ||
| −0.2898 | −0.2122 | −0.8271 | −0.6809 | −0.9300 | −0.8036 | ||
| −0.3585 | −0.2572 | −1.0606 | −0.7700 | −1.3632 | −0.9788 | ||
| −0.4091 | −0.2933 | −1.2199 | −0.8842 | −1.5710 | −1.1264 | ||
| −0.4126 | −0.3103 | −0.9686 | −0.8948 | −0.9800 | −0.9080 | ||
Note(s): Call option on GBP/USD
Source(s): Own elaboration
Back-testing on different VaR metrics: FRTB and conventional frameworks
|
Conflict of interest: The authors declare that they have no conflict of interest.
References
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999), “Coherent measures of risk”, Mathematical Finance, Vol. 9 No. 3, pp. 203-228.
BCBS (1996), “Supervisory framework for the use of ‘backtesting’ in conjunction with the internal models approach to market risk capital requirements”, available at: https://www.bis.org/publ/bcbs22.htm
BCBS (2006), “Basel II: international convergence of capital measurement and capital standards: a revised framework – comprehensive version”, available at: https://www.bis.org/publ/bcbs128.htm
BCBS (2011), “Revisions to the Basel II market risk framework”, available at: https://www.bis.org/publ/bcbs193.pdf
BCBS (2019), “Explanatory note on the minimum capital requirements for market risk”, available at: https://www.bis.org/bcbs/publ/d457_note.pdf
BIS (2019), “Minimum capital requirements for market risk”, available at: https://www.bis.org/bcbs/publ/d457.htm
Chen, S.X. (2008), “Nonparametric estimation of expected shortfall”, Journal of Financial Econometrics, Vol. 6 No. 1, pp. 87-107.
Christoffersen, P.F. (1998), “Evaluating interval forecasts”, International Economic Review, Vol. 39 No. 4, pp. 841-862.
Christoffersen, P. (2012), Elements of Financial Risk Management, 2nd ed., Academic Press - Elsevier, doi: 10.1016/C2009-0-22827-3.
Colletaz, G., Hurlin, C. and Pérignon, C. (2013), “The risk map: a new tool for validating risk models”, Journal of Banking and Finance, Vol. 37 No. 10, pp. 3843-3854.
Daníelsson, J. and Zigrand, J.-P. (2006), “On time-scaling of risk and the square-root-of-time rule”, Journal of Banking and Finance, Vol. 30 No. 10, pp. 2701-2713.
Deng, K. and Qiu, J. (2021), “Backtesting expected shortfall and beyond”, Quantitative Finance, Vol. 21 No. 7, pp. 1109-1125.
Embrechts, P., Liu, H. and Wang, R. (2018), “Quantile-based risk sharing”, Operations Research, Vol. 66 No. 4, pp. 936-949.
Embrechts, P., Liu, H., Mao, T. and Wang, R. (2020), “Quantile-based risk sharing with heterogeneous beliefs”, Mathematical Programming, Vol. 181 No. 2, pp. 319-347.
Haas, M. (2001), New Methods in Backtesting, Financial Engineering Research Center Caesar, Bonn.
Hull, J. (2018), Risk Management and Financial Institutions, 5th ed., Wiley Finance, Hoboken, NJ.
Kliemann, L. and Sanders, P. (2016), Algorithm Engineering – Selected Results and Surveys, Springer.
Kratz, M., Lok, Y.H. and McNeil, A.J. (2018), “Multinomial VaR backtests: a simple implicit approach to backtesting expected shortfall”, Journal of Banking and Finance, Vol. 88, pp. 393-407.
Kupiec, P. (1995), “Techniques for verifying the accuracy of risk management models”, Journal of Derivatives, Vol. 3, pp. 73-84.
Laurent, J.-P., Sestier, M. and Thomas, S. (2016), “Trading book and credit risk: how fundamental is the Basel review?”, Journal of Banking and Finance, Vol. 73, pp. 211-223.
Majumder, M.T.H. and Li, X. (2018), “Bank risk and performance in an emerging market setting: the case of Bangladesh”, Journal of Economics, Finance and Administrative Science, Vol. 23 No. 46, pp. 199-229, doi: 10.1108/JEFAS-07-2017-0084.
McNeil, A.J., Frey, R. and Embrechts, P. (2005), Quantitative Risk Management: Concepts, Techniques, and Tools, Princeton University Press, Princeton, NJ.
Menéndez, S.C. and Hassani, B.K. (2021), “Expected shortfall reliability—added value of traditional statistics and advanced artificial intelligence for market risk measurement purposes”, Mathematics, Vol. 9 No. 17, doi: 10.3390/math9172142.
Nadarajah, S., Zhang, B. and Chan, S. (2014), “Estimation methods for expected shortfall”, Quantitative Finance, Vol. 14 No. 2, pp. 271-291.
Nieto, M.R. and Ruiz, E. (2016), “Frontiers in VaR forecasting and backtesting”, International Journal of Forecasting, Vol. 32 No. 2, pp. 475-501.
Orgeldinger, J. (2018), “Recent issues in the implementation of the new Basel minimum capital requirements for market risk”, Emerging Science Journal, Vol. 2 No. 2, pp. 65-77.
Patton, A.J., Ziegel, J.F. and Chen, R. (2019), “Dynamic semiparametric models for expected shortfall (and Value-at-Risk)”, Journal of Econometrics, Vol. 211 No. 2, pp. 388-413.
Pederzoli, C. and Torricelli, C. (2021), “An assessment of the fundamental review of the trading book: the capital requirement impact on a stylised financial portfolio”, International Journal of Banking, Accounting and Finance, Vol. 12 No. 4, pp. 389-403.
Porretta, P. and Agnese, P. (2021), “The fundamental review of trading book: new standard approach and risk management impacts”, Journal of Risk Management in Financial Institutions, Vol. 14 No. 2, pp. 209-219.
Rockafellar, R. and Uryasev, S. (2000), “Optimization of conditional value-at-risk”, The Journal of Risk, Vol. 2 No. 3, pp. 21-41.
Rockafellar, R. and Uryasev, S. (2002), “Conditional value-at-risk for general loss distributions”, Journal of Banking and Finance, Vol. 26 No. 7, pp. 1443-1471.
Rockafellar, R. and Uryasev, S. (2013), “The fundamental risk quadrangle in risk management, optimization and statistical estimation”, Surveys in Operations Research and Management Science, Vol. 18 Nos 1-2, pp. 33-53.
Scaillet, O. (2004), “Nonparametric estimation and sensitivity analysis of expected shortfall”, Mathematical Finance, Vol. 14 No. 1, pp. 115-129.
Serrano Bautista, R. and Núñez Mora, J.A. (2021), “Value-at-risk predictive performance: a comparison between the CaViaR and GARCH models for the MILA and ASEAN-5 stock markets”, Journal of Economics, Finance and Administrative Science, Vol. 26 No. 52, pp. 197-221, doi: 10.1108/JEFAS-03-2021-0009.
Valerio Roncagliolo, F.C. and Villamonte Blas, R.N. (2022), “Impact of financial stress in advanced and emerging economies”, Journal of Economics, Finance and Administrative Science, Vol. 27 No. 53, pp. 68-85, doi: 10.1108/JEFAS-05-2021-0063.
Acknowledgements
The authors thank the reviewers for their careful observations and suggestions.