THE CONSEQUENCES OF BASEL III REQUIREMENTS FOR LIQUIDITY HORIZON AND IT’S IMPLICATION ON OPTIMAL TRADING STRATEGY
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Table of contents
Introduction....................................................................................................... 3
Calculating the ES according to the Basel standards........................................... 6
Literature review................................................................................................ 9
Methodology and data..................................................................................... 14
Results............................................................................................................. 26
Conclusion...................................................................................................... 41
Bibliography.................................................................................................... 42
Calculating the ES according to the Basel standards.
In this subsection, we present the methodology for calculating the ES outlined in Basel III and provide a simple example to give the basic idea of how the inclusion of the LH parameter affects the total ES. The minimum capital requirements for market risk were revised in January 14th, 2019, which replaced the earlier version from January 2016 and will come into effect on January 1st, 2022. The standards introduced some notable changes in terms of banks’ internal risk modelling but kept the old method of calculating expected shortfall intact. According to the standards, the banks have a discretion in terms of modelling their expected shortfall, but they also must fulfill the minimum requirements imposed by Basel III. The banks are required to use 97.5th percentile one-tailed confidence level for estimating the 10-day ES and scale the estimates by the respective liquidity horizons of risk factors:
2
( − −1)
= √( ( ))2 + ∑ ( ( , )√)(1)
≥2
Yuanyuan Chen, Xuefeng Gao, Duan Li. 2018. "Optimal Execution using hidden orders." Journal of Economic Dynamic and Control 89-116.
Zhao, Jingdong, Hongliang Zhu, and Xindan Li. 2018. "Optimal execution with price impact under Cumulative ProspectTheory." Physica A 490 1228-1237.
Ziegel, Johanna F., Fabian Kruger, Alexander Jordan, and Fernando Fasciati. 2019. "Robust Forecast Evaluation of Expected Shortfall." Journal of Financial Econometrics 18 (1): 95-120.
Оригинальность работы 93%
Table of contents
Introduction....................................................................................................... 3
Calculating the ES according to the Basel standards........................................... 6
Literature review................................................................................................ 9
Methodology and data..................................................................................... 14
Results............................................................................................................. 26
Conclusion...................................................................................................... 41
Bibliography.................................................................................................... 42
Introduction.
In this research we present a formula-based approach to derive the optimal levels of liquidity horizon for banks’ market risk capital requirements in accordance with Basel III standards. Our topic mainly concerns bank’s market risk, since the liquidity horizon (LH) is an important parameter for computing the total expected shortfall (ES), which in turn replaced the value at risk (VaR) as a risk measure for bank’s internal modelling. The LH, according to Basel (2019), is defined as “the time assumed to be required to exit or hedge a risk position without materially affecting market prices in stressed market conditions”, and is used in banks’ internal model approach to calculate the acceptable level of capital safety – specifically, it figures in the computation of ES – the expected loss in case if VaR is exceeded.
Our proposition of a formula-based approach to determine the LH is motivated by at least two factors. Firstly, investments still make up a substantial portion of banks assets. As shown in Figure 1, the total investments in tradable securities among global systemically important banks (G-SIBs) was around 20.93 bln. USD for the last FY in 2019. Although the share of investments has been trending downward during the last decade, the amount of assets that directly fall under the market risk supervision is still considerable. Secondly, in our opinion, the absence of formulas or any objective methodology that justifies the use of the current LH values in ES calculation leaves a huge space for its further improvements. As far as we are concerned, the current Basel recommendation in calculating the expected shortfall has two major disadvantages. Firstly, the choice of market-risk factors and their respective time horizons is set arbitrarily without any objective foundation and is subject to constant discussions. The choice of specific LH values is not based on strong evidence and thus, represent the opinion of the authority as for the most appropriate time frame to eliminate the respective risks in case of market turbulence. Secondly, the provided methodology of calculating the ES heavily
3
In this research we present a formula-based approach to derive the optimal levels of liquidity horizon for banks’ market risk capital requirements in accordance with Basel III standards. Our topic mainly concerns bank’s market risk, since the liquidity horizon (LH) is an important parameter for computing the total expected shortfall (ES), which in turn replaced the value at risk (VaR) as a risk measure for bank’s internal modelling. The LH, according to Basel (2019), is defined as “the time assumed to be required to exit or hedge a risk position without materially affecting market prices in stressed market conditions”, and is used in banks’ internal model approach to calculate the acceptable level of capital safety – specifically, it figures in the computation of ES – the expected loss in case if VaR is exceeded.
Our proposition of a formula-based approach to determine the LH is motivated by at least two factors. Firstly, investments still make up a substantial portion of banks assets. As shown in Figure 1, the total investments in tradable securities among global systemically important banks (G-SIBs) was around 20.93 bln. USD for the last FY in 2019. Although the share of investments has been trending downward during the last decade, the amount of assets that directly fall under the market risk supervision is still considerable. Secondly, in our opinion, the absence of formulas or any objective methodology that justifies the use of the current LH values in ES calculation leaves a huge space for its further improvements. As far as we are concerned, the current Basel recommendation in calculating the expected shortfall has two major disadvantages. Firstly, the choice of market-risk factors and their respective time horizons is set arbitrarily without any objective foundation and is subject to constant discussions. The choice of specific LH values is not based on strong evidence and thus, represent the opinion of the authority as for the most appropriate time frame to eliminate the respective risks in case of market turbulence. Secondly, the provided methodology of calculating the ES heavily
3
Figure 1.
The amount and the share of investments in G-SIB book from 2009-2019.
bln.60.0044.82%45.14%45.30%
54.2654.3146.00%
51.0450.73
52.03
50.95
49.77
USD
48.60
48.24
50.00
44.00%
44.71
42.64% 42.52%
41.25
42.20%
40.00
42.00%
40.37%39.69%
30.00
40.00%
38.27% 38.41% 38.55%
20.00
38.00%
10.0018.4920.1823.1221.6320.6621.9519.4820.2219.0520.8420.9336.00%
0.00
34.00%
20092010201120122013201420152016201720182019
Years
Total assetsTotal investmentsShare of investments
Source: created by the author from data in Financial Stability Board, TR Eikon. aggregates the market risk factors to only 5 types of market instruments (see Table 2), leaving behind other individual security-specific types of risks that also affect portfolio value. As we know, each market instrument has its own intrinsic market risk which cannot be simply diversified away, so the current methodology may fall short in case when the bank’s portfolio consists of not diversified portfolio.
To resolve this issue, we propose a different approach. Our methodology is based on a seminal contribution from Almgren & Chriss (2000), who studied the optimal portfolio execution problem in presence of uncertainty and derived an explicit formula for calculating the “half-life” – the optimal time horizon for the execution of an individual security. We modified their formula of “half-life” to accommodate for practical estimation procedures based on available data and incorporate it directly into the computation of optimal expected shortfall (OES), which solves the issue of
4
The amount and the share of investments in G-SIB book from 2009-2019.
bln.60.0044.82%45.14%45.30%
54.2654.3146.00%
51.0450.73
52.03
50.95
49.77
USD
48.60
48.24
50.00
44.00%
44.71
42.64% 42.52%
41.25
42.20%
40.00
42.00%
40.37%39.69%
30.00
40.00%
38.27% 38.41% 38.55%
20.00
38.00%
10.0018.4920.1823.1221.6320.6621.9519.4820.2219.0520.8420.9336.00%
0.00
34.00%
20092010201120122013201420152016201720182019
Years
Total assetsTotal investmentsShare of investments
Source: created by the author from data in Financial Stability Board, TR Eikon. aggregates the market risk factors to only 5 types of market instruments (see Table 2), leaving behind other individual security-specific types of risks that also affect portfolio value. As we know, each market instrument has its own intrinsic market risk which cannot be simply diversified away, so the current methodology may fall short in case when the bank’s portfolio consists of not diversified portfolio.
To resolve this issue, we propose a different approach. Our methodology is based on a seminal contribution from Almgren & Chriss (2000), who studied the optimal portfolio execution problem in presence of uncertainty and derived an explicit formula for calculating the “half-life” – the optimal time horizon for the execution of an individual security. We modified their formula of “half-life” to accommodate for practical estimation procedures based on available data and incorporate it directly into the computation of optimal expected shortfall (OES), which solves the issue of
4
aggregated risk factors by scaling each security’s ES individually inside a portfolio with the optimal LH specific to each security at different times. We then backtested the behavior of both regulatory ES (RES) and optimal ES (OES) estimates on time series of two major US stock indices: S&P500 and Dow Jones Industrial by using a novel regression-based approach suggested by Bayer, S., & Dimitriadis, T. (2018). Specifically, we employed the strict ES regression (S-ESR) and the intercept ES regression (I-ESR) forms of the test.
We obtained two interesting results. Firstly, throughout the period of our study, the optimal LH of both SP500 and DJIA stocks did not exceed 1, which meant that index portfolios could be depleted in less than a day. Secondly, we found that due to low LH values, our OES model slightly underestimated the ES compared to the RES, but further check with both S-ESR and I-ESR tests revealed that while OES model gives correct estimates of ES, the RES instead strongly overestimates it.
The structure of our thesis is as follows. In the next section we present the current methodology of calculating the ES as described in Basel (2019). Then in the literature review, we provide a detailed review of the current state of research in terms of the general framework of estimating and backtesting techniques of the ES, the portfolio optimal execution framework and studies that included the LH concept in their studies. In our methodology, we present the base model of Almgren and Chriss (2000) and outline the key points in implementing our strategy to obtain the ES with optimal LH values. In our results, we present some statistics of the optimal LH and ES calculated by using the Basel III and our own formula and provide the results of the ES backtests. Finally, we finish our thesis with concluding remarks.
5
We obtained two interesting results. Firstly, throughout the period of our study, the optimal LH of both SP500 and DJIA stocks did not exceed 1, which meant that index portfolios could be depleted in less than a day. Secondly, we found that due to low LH values, our OES model slightly underestimated the ES compared to the RES, but further check with both S-ESR and I-ESR tests revealed that while OES model gives correct estimates of ES, the RES instead strongly overestimates it.
The structure of our thesis is as follows. In the next section we present the current methodology of calculating the ES as described in Basel (2019). Then in the literature review, we provide a detailed review of the current state of research in terms of the general framework of estimating and backtesting techniques of the ES, the portfolio optimal execution framework and studies that included the LH concept in their studies. In our methodology, we present the base model of Almgren and Chriss (2000) and outline the key points in implementing our strategy to obtain the ES with optimal LH values. In our results, we present some statistics of the optimal LH and ES calculated by using the Basel III and our own formula and provide the results of the ES backtests. Finally, we finish our thesis with concluding remarks.
5
Calculating the ES according to the Basel standards.
In this subsection, we present the methodology for calculating the ES outlined in Basel III and provide a simple example to give the basic idea of how the inclusion of the LH parameter affects the total ES. The minimum capital requirements for market risk were revised in January 14th, 2019, which replaced the earlier version from January 2016 and will come into effect on January 1st, 2022. The standards introduced some notable changes in terms of banks’ internal risk modelling but kept the old method of calculating expected shortfall intact. According to the standards, the banks have a discretion in terms of modelling their expected shortfall, but they also must fulfill the minimum requirements imposed by Basel III. The banks are required to use 97.5th percentile one-tailed confidence level for estimating the 10-day ES and scale the estimates by the respective liquidity horizons of risk factors:
2
( − −1)
= √( ( ))2 + ∑ ( ( , )√)(1)
≥2
- where:
-
- RES is the regulatory liquidity-adjusted ES.
-
- T is the length of the base horizon, i.e. 10 days.
-
- ( ) is ES at horizon T of a portfolio with positions P = (pi) with respect to shocks to all risk factors that the positions P are exposed to.
-
- ( , ) is ES at horizon T of a portfolio with positions P = (pi) with respect to shocks for each position pi in the subset of risk factors Q (pi, j) and all other risk factors held constant.
-
- LHj is the liquidity horizon j as shown in Table 1:
Bibliography
Almgren, Robert, and Neil Chriss. 2000. "Optimal Execution of portfolio transactions." Journal of Risk 3 (2): 5-39.
Andreassen, Paul B. 1988. "Explaining the Price-Volume Relationship: The Difference between Price Changes and Changing Prices." Organizational Behavior and Human Deciion Processes 41: 371-389.
Artzner, Philippe, Freddy Delbaen, Eber Jean-Marc, and David Heath. 2001.
"Coherent measure of risk." Mathematical Finance 9 (3): 203-228.
Basel. 2019. "Minimum capital requirements for market risk." Basel Committee on Banking Supervision. https://www.bis.org/bcbs/publ/d457.htm.
Bayer, and Dimitriadis. 2019b. "esreg: Joint Quantile and Expected Shortfall Regression. R package." https://cran.r-project.org/package=esreg.
Bayer, Sebastian, and Timo Dimitriadis. 2018. "Regression based expected shrtfall backtesting." arXiv preprint arXiv:1801.04112.
Bollerslev, Tim. 1986. "Generalized autoregressive conditional heteroskedasticity." Journal of Econometrics 31 (3): 307-327.
Brunovsky, Pavol, Ales Cerny, and Jan Komadel. 2017. "Optimal trade execution under endogenous presure to liquidate: Theory and numerical solutions." European Journal of Operational Research 1159-1171.
Chang, Chia-Lin, Juan-Angel Jimenez-Martin, Esfandiar Maasoumi, Michael McAleer, and Teodosio Pérez-Amaral. 2019. "Choosing expected shortfall over VaR in Basel III using stochastic dominance." International Review of Economics & Finance 60: 95-113.
Colldeforns-Papiol, Gemma, and Luis Ortiz-Gracia. 2018. "Computation of market risk measures with stochastic liquidity horizon." Journal of Computational and Applied Mathematics 342: 431-450.
42
Almgren, Robert, and Neil Chriss. 2000. "Optimal Execution of portfolio transactions." Journal of Risk 3 (2): 5-39.
Andreassen, Paul B. 1988. "Explaining the Price-Volume Relationship: The Difference between Price Changes and Changing Prices." Organizational Behavior and Human Deciion Processes 41: 371-389.
Artzner, Philippe, Freddy Delbaen, Eber Jean-Marc, and David Heath. 2001.
"Coherent measure of risk." Mathematical Finance 9 (3): 203-228.
Basel. 2019. "Minimum capital requirements for market risk." Basel Committee on Banking Supervision. https://www.bis.org/bcbs/publ/d457.htm.
Bayer, and Dimitriadis. 2019b. "esreg: Joint Quantile and Expected Shortfall Regression. R package." https://cran.r-project.org/package=esreg.
Bayer, Sebastian, and Timo Dimitriadis. 2018. "Regression based expected shrtfall backtesting." arXiv preprint arXiv:1801.04112.
Bollerslev, Tim. 1986. "Generalized autoregressive conditional heteroskedasticity." Journal of Econometrics 31 (3): 307-327.
Brunovsky, Pavol, Ales Cerny, and Jan Komadel. 2017. "Optimal trade execution under endogenous presure to liquidate: Theory and numerical solutions." European Journal of Operational Research 1159-1171.
Chang, Chia-Lin, Juan-Angel Jimenez-Martin, Esfandiar Maasoumi, Michael McAleer, and Teodosio Pérez-Amaral. 2019. "Choosing expected shortfall over VaR in Basel III using stochastic dominance." International Review of Economics & Finance 60: 95-113.
Colldeforns-Papiol, Gemma, and Luis Ortiz-Gracia. 2018. "Computation of market risk measures with stochastic liquidity horizon." Journal of Computational and Applied Mathematics 342: 431-450.
42
Dimitris Bertsimas, Andrew W. Lo. 1998. "Optimal control of execution costs." Journal of Financial Markets 1-50.
Du, Bian, Hongliang Zhu, and Jingdong Zhao. 2016. "Optimal execution in high-frequency trading with Bayesian learning." Physica A 461 767-777.
Firoozi, Dena, and Peter E Caines. 2017. "An Optimal Execution Problem in Finance with Acquisition and Liquidation Objectives: an MFG Formulation." IFAC PapersOnLine 49460-4967.
Forsyth, P.A, J.S. Kennedy, T.S. Tse, and H. Winchiff. 2012. "Optimal trade execution: a mean quadratic variation approach." Journal of economic dynamic and control 1971-1991.
Forsyth, PA. 2011. "A hamilton jacobi bellman approach to optimal trade execution." Applied numerical mathematics 241-265.
Ghalanos, A. 2020. "rugarch: Univariate GARCH models. R package version 1.4-2." https://cran.r-project.org/package=rugarch.
Karpoff, Jonathan M. 1987. "The Relation Between Price Changes and Trading Volume: A Survey." The Journal of Financial and Quantitative Analysis 22
(1): 109-126.
Kellner, Ralf, and Daniel Rösch. 2016. "Quantifying market risk with Value-at-Risk or Expected Shortfall? – Consequences for capital requirements and model risk." Journal of Economic Dynamic adn Control 68: 45-63.
Lazar, Emese, and Ning Zhang. 2019. "Model risk of expected shortfall." Journal of Banking and Finance 105: 74-93.
Maa, Guiyuan, Chi Chung Siu, and Song-Ping Zhu. 2019. "Optimal investment and consumption with return predictability and execution costs." Economic Modelling XXX.
43
Du, Bian, Hongliang Zhu, and Jingdong Zhao. 2016. "Optimal execution in high-frequency trading with Bayesian learning." Physica A 461 767-777.
Firoozi, Dena, and Peter E Caines. 2017. "An Optimal Execution Problem in Finance with Acquisition and Liquidation Objectives: an MFG Formulation." IFAC PapersOnLine 49460-4967.
Forsyth, P.A, J.S. Kennedy, T.S. Tse, and H. Winchiff. 2012. "Optimal trade execution: a mean quadratic variation approach." Journal of economic dynamic and control 1971-1991.
Forsyth, PA. 2011. "A hamilton jacobi bellman approach to optimal trade execution." Applied numerical mathematics 241-265.
Ghalanos, A. 2020. "rugarch: Univariate GARCH models. R package version 1.4-2." https://cran.r-project.org/package=rugarch.
Karpoff, Jonathan M. 1987. "The Relation Between Price Changes and Trading Volume: A Survey." The Journal of Financial and Quantitative Analysis 22
(1): 109-126.
Kellner, Ralf, and Daniel Rösch. 2016. "Quantifying market risk with Value-at-Risk or Expected Shortfall? – Consequences for capital requirements and model risk." Journal of Economic Dynamic adn Control 68: 45-63.
Lazar, Emese, and Ning Zhang. 2019. "Model risk of expected shortfall." Journal of Banking and Finance 105: 74-93.
Maa, Guiyuan, Chi Chung Siu, and Song-Ping Zhu. 2019. "Optimal investment and consumption with return predictability and execution costs." Economic Modelling XXX.
43
Martins-Filho, Carlos, Feng Yao, and Maximo Torero. 2018. "Nonparametric estimation of conditional value-at-risk and expected shortfall based on extreme value theory." Econometric Theory 34 (1): 23-67.
Meng, Xiaochun, and James W. Taylor. 2020. "Estimating Value-at-Risk and Expected Shortfall using the intraday low and range data." European Journal of Operational Research 280 (1): 191-202.
Novales, Alfonso, and Laura Garcia-Jorcano. 2019. "Backtesting extreme value theory models of expected shortfall." Quantitative Finance 19 (5): 799-825.
Oded, Jacob. 2009. "Optimal Execution of open market stock repurchase programs." Journal of Financial Markets 832-869.
Ortiz-Gracia, Luis. 2019. "Expected shortfall computation with multiple control variates." Applied Mathematics and Computation 373: 125018.
Siu, Chi Chung, Ivan Guo, Song-Ping Zhu, and Robert J.Elliot. 2019. "Optimal execution with regime-switching market resilience." Journal of Economic Dynamic and Control 17-40.
Skoglund, Jimmy, and Wei Chen. 2011. "On the choice of liquidity horizon for incremental risk charges: are the incentives of banks and regulators aligned?" Journal of Risk Model Validation 5 (3): 37-57.
Supervision, Basel Committee on Banking. 2019. "Minimum capital requirements
for market risk." www.bis.org. January.
https://www.bis.org/bcbs/publ/d457.htm.
Taylor, James W. 2019. "Forecasting Value at Risk and Expected Shortfall Using a Semiparametric Approach Based on the Asymmetric Laplace Distribution." Journal of Business & Economic Statistics 37 (1): 121-133.
Xun, Li, Zhou Yangzhi, and Zhou Yong. 2019. "A generalization of Expected Shortfall based capital allocation." Statistics and Probability Letters 146: 193-199.
44
Meng, Xiaochun, and James W. Taylor. 2020. "Estimating Value-at-Risk and Expected Shortfall using the intraday low and range data." European Journal of Operational Research 280 (1): 191-202.
Novales, Alfonso, and Laura Garcia-Jorcano. 2019. "Backtesting extreme value theory models of expected shortfall." Quantitative Finance 19 (5): 799-825.
Oded, Jacob. 2009. "Optimal Execution of open market stock repurchase programs." Journal of Financial Markets 832-869.
Ortiz-Gracia, Luis. 2019. "Expected shortfall computation with multiple control variates." Applied Mathematics and Computation 373: 125018.
Siu, Chi Chung, Ivan Guo, Song-Ping Zhu, and Robert J.Elliot. 2019. "Optimal execution with regime-switching market resilience." Journal of Economic Dynamic and Control 17-40.
Skoglund, Jimmy, and Wei Chen. 2011. "On the choice of liquidity horizon for incremental risk charges: are the incentives of banks and regulators aligned?" Journal of Risk Model Validation 5 (3): 37-57.
Supervision, Basel Committee on Banking. 2019. "Minimum capital requirements
for market risk." www.bis.org. January.
https://www.bis.org/bcbs/publ/d457.htm.
Taylor, James W. 2019. "Forecasting Value at Risk and Expected Shortfall Using a Semiparametric Approach Based on the Asymmetric Laplace Distribution." Journal of Business & Economic Statistics 37 (1): 121-133.
Xun, Li, Zhou Yangzhi, and Zhou Yong. 2019. "A generalization of Expected Shortfall based capital allocation." Statistics and Probability Letters 146: 193-199.
44
Yuanyuan Chen, Xuefeng Gao, Duan Li. 2018. "Optimal Execution using hidden orders." Journal of Economic Dynamic and Control 89-116.
Zhao, Jingdong, Hongliang Zhu, and Xindan Li. 2018. "Optimal execution with price impact under Cumulative ProspectTheory." Physica A 490 1228-1237.
Ziegel, Johanna F., Fabian Kruger, Alexander Jordan, and Fernando Fasciati. 2019. "Robust Forecast Evaluation of Expected Shortfall." Journal of Financial Econometrics 18 (1): 95-120.
