Forecasting VaR and Risk Mangement under Basel Accords Michael McAleer

1. Forecasting VaR and Risk Mangement under Basel Accords Michael McAleer Erasmus University Rotterdam/ Tinbergen Institute The Netherlands / Institute of Economic Research, Kyoto University Japan Juan-Angel Jimenez-Martin Teodosio Perez-Amaral Complutense University of Madrid, Spain

2. Paper 1.- Has the Basel II Accord Encouraged Risk Management During the 2008-09 Financial Crisis? (January 24, 2010). Available at SSRN: http://ssrn.com/abstract=1397239 Paper 2.- GFC-Robust Risk Management Strategies under the Basel Accord (October 6, 2010). Available at SSRN: http://ssrn.com/abstract=1688385 Paper 3.- International Evidence on GFC-Robust Forecasts for Risk Management Under the Basel Accord (January 16, 2011). Available at SSRN: http://ssrn.com/abstract=1741565

3. Motivation 1. The Basel II Accord requires that banks and other Authorized Deposit-taking Institutions (ADIs) communicate their daily risk forecasts to the appropriate monetary authorities at the beginning of each trading day to determine regulatory capital requirements 2. There are different types of risk: • Credit Risk • Operational risk • Market risk • Interest rate • Equity risk • Exchange rate risk… 3.- There are several measures of Risk (standard deviation, β, VaR) • VaR is a measure of risk based on a probability of loss and a specific time horizon. Value at Risk is an estimate of the worst possible loss an investment could realize over a given time horizon, under normal market conditions (defined by a given level of confidence). • VaR is denominated in units of a currency or as a percentage of portfolio holdings. For e.g., a set of portfolio having a current value of say € 1 million- can be described to have a daily value at risk of €0.1 million- at a 99% confidence level, which means there is a 1/100 chance of the loss exceeding €0.1 million, considering no great paradigm shifts in the underlying factors. • Measure of Total Risk rather than Systematic (or Non-Diversifiable Risk) measured by Beta.

4. Advantages of VaR • VaR provides an measure of total risk. • VaR is an easy number to understand and explain to clients. • VaR translates portfolio volatility into a dollar value. A one day VAR of € 0.1 million using a probability of 5% means that there is a 5% chance that the portfolio could lose more than € 0.1 million the next trading day. • …Additionally • VaR is useful for monitoring and controlling risk within the portfolio. • VaR can measure the risk of many types of financial securities (i.e., stocks, bonds, commodities, foreign exchange, off-balance-sheet derivatives such as futures, forwards, swaps, and options, and etc.) • It is easy to implement a Back testing procedure… Calculate 1-Day 95% VAR for a (changing) portfolio each day for some substantial period of time (e.g., 100 Days) Compare the P/L on the succeeding trading day with the previous close of business day’s VAR Count the number of times the loss exceeds the VAR

5. Motivation • …Therefore the Basel II Accord requires that banks communicate their VaR forecasts to the appropriate monetary authorities at the beginning of each trading day to determine regulatory capital requirements. • Basel II accord was designed to reward institutions with superior risk management systems. • Financial Institutions are permitted to use Internal Models to calculate VaR. • Historical Simulation • Variance and covariance • Monte Carlo simulation …. But the model has to work correctly…

6. Motivation • … More than 10 violations in any financial year may required to adopt “standardized” approach. • When internal models lead to a greater number of violations than could reasonably be expected the bank is required: • To hold a higher level of capital. • An the monetary authority can impose a external model to forecast VaR . • … These are the reasons why bank managers may prefer risk management strategies that are passive and conservative rather than active and aggressive. But we know that excessive Conservatism has a negative impact on the profitability

7. Maximizing Profits, VaR and DCC A necessary condition to maximize (1) in a given period of (say) 250 days, is to minimize the total CRqt (3) for the period. The standard approach in the literature is to report in the estimate of VaR obtained from a given model. In this paper, we propose a robust strategy to communicate VaR. • Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions

8. Model for Forecasting VaR and Daily Capital Charges We assume that daily returns follow: • Risk management • Models for Forecasting VaR and Daily Capital Charges • Combining alternative Models to forecast VaR • Conclusions  is the critical value from the distribution of  to obtain the appropriate confidence level. It is possible for to be replaced by alternative estimates of the conditional variance. The VaR is: GARCH GJR (Glosten, Jaganathan and Runkle) Problems with GARCH(p,q) Models: - Non-negativity constraints may still be violated - GARCH models cannot account for leverage effects With both normal and t distribution errors EGARCH Exponentially Weighted Moving Average (EWMA) - RiskmetricsTM (1996) Daily Capital Charges are :

9. Risk management • Models for Forecasting VaR and Daily Capital Charges • Combining alternative Model to forecast VaR • Conclusions Data DATA DESCRIPTION Closing daily prices for Standard and Poor’s Composite 500 Index. SOURCE: Ecowin Financial Database SAMPLE: 3 January 2000 to 12 February 2009 Figure 1. Daily Returns on the S&P500 Index, 3 January 2000 – 12 February 2009 Figure 2. Daily Volatility in S&P500 Returns 3 January 2000 – 12 February 2009

10. Risk management • Models for Forecasting VaR and Daily Capital Charges • Combining alternative Models to forecast VaR • Conclusions Combining alternative Risk Model to Forecast VaR Banks need not restrict themselves to using only one of the available risk models. We propose a risk management strategy that consists in choosing from among different combinations of alternative risk models to forecast VaR. One of them can can be characterized as an aggressive strategy and another that can be regarded as a conservative strategy Figure 3. VaR for S&P500 Returns 2 January 2008 – 12 February 2009 Note: The upper blue line represents daily returns for the S&P500 index. The upper red line represents the infinum of the VaR forecasts for the different models described above. The lower green line corresponds to the supremum of the forecasts of the VaR for the same models.

11. Risk management • Models for Forecasting VaR and Daily Capital Charges • Combining alternative Models to forecast VaR • Conclusions Forecasting VaR and Calculating DCC – Performance of the Proposed models Table 3. Percentage of Days Minimizing Daily Capital Charges, Mean Daily Capital Charges, and Number of Violations for Alternative Models of Volatility Figure 5. Number of Violations Accumulated Over 260 Days 3 January 2008-12 February 2009

12. Risk management • Models for Forecasting VaR and Daily Capital Charges • Combining alternative Models to forecast VaR • Conclusions Forecasting VaR and Calculating DCC – Performance of the Proposed models Figure 4. VaR and Mean VaR for the Previous 60 Days to Calculate Daily Capital Charges for S&P500 Returns It can be observed from Figure 4 that daily capital charges always exceed VaR (in absolute terms). Moreover, immediately after the financial crisis had started, a significant amount of capital was set aside to cover likely financial losses. This is a positive feature of the Basel II Accord, since it can have the effect of shielding banks from possible significant financial losses.

13. Risk management • Models for Forecasting VaR and Daily Capital Charges • Combining alternative Models to forecast VaR • Conclusions Forecasting VaR and Calculating DCC – Performance of the Proposed models Figure 6. Duration of Minimum Daily Capital Charges for Alternative Models of Volatility Alternative risk models were found to be optimal before and during the financial crisis. (1) Before the global financial crisis from 3 January 2008 to 6 June 2008, the best model for minimizing daily capital charges is GARCH (coinciding with the Upperbound). For the period 6 June 2008 to 16 July 2008, GJR was best and, for only 5 days, EGARCH was the best. This is a period with relatively low volatility and few extreme values. (2) Riskmetrics is the best model from 16 July 2008 to 15 September 2008. The S&P500 reached a peak on 12 August 2008, after which it started to decrease. In the second half of September 2008, the volatility on returns began to increase considerably. (3) During most of the global financial crisis, from 24 September 2008 to the end of the sample, the best model was EGARCH_T. This is a period with considerably high volatility and a large number of extreme values of returns. EGARCH can capture asymmetric volatility, thereby providing a more accurate measure of risk during large financial turbulence. A Decision Rule to Minimize Daily Capital Charges in Forecasting Value-at-Risk (February 26, 2009). Available at SSRN: http://ssrn.com/abstract=1349844

14. Risk management • Models for Forecasting VaR and Daily Capital Charges • Combining alternative Models to forecast VaR • Conclusions Conclusions • Under the Basel II Accord, • Banks have to communicate their risk estimates to the monetary authorities • They can use a variety of VaR models to estimate risks. • Banks are subject to a back-test • Daily capital charges as protection against market risk must be set at the higher of the previous day’s VaR or the average VaR over the last 60 business days, multiplied by a factor k. • Banks objective is to maximize profits, so they wish to minimize their capital charges while restricting the number of violations in a given year below the maximum of 10 allowed by the Basel II Accord. From this target it follows naturally that ADI’s have to choose an optimal reporting policy that may strategically under-report or over-report their forecast of VaR in order to minimize the daily capital requirement. • We define risk management in terms of choosing sensibly from a variety of conditional volatility (risk) models, considering combining alternatives risk models. We propose both a conservative and an aggressive strategies. • We found optimal strategies using different combinations or alternatives risk models for predicting VaR and minimizing daily capital changes. • In this paper we analyzed the performance of existing state-of-the-art, as well as a novel, risk management strategies permitted under the Basel II framework, as applied to the S&P500 index. Such risk management strategies could well have provided adequate coverage against market risk during the 2008-09 period, which included the global financial crisis. • The area between the bounds provided by the aggressive and the conservative strategy can be seen to be fertile area for future research..

15. Agenda • Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • Robust Strategy for Market Risk Disclosures • Results • Conclusions

16. GFC-Robust Risk Management Strategy • Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-RobustRisk Management Strategy • Results • Conclusions What is a GFC-Robust Risk Management Strategy? A crisis robust strategy IS an optimal risk management strategy that remains unchanged regardless of whether it is used before, during or after a significant financial crisis. Parametric methods for forecasting VaR are typically fitted to historical returns assuming specific conditional distributions of returns, such as normality, Student-t, or generalized normal distribution. The VaR forecast depends on the parametric model, the conditional distribution and can be heavily affected by a few large observations. Some models provide many violations, but low daily capital charges. Additionally, these results can change drastically from tranquil to turbulent periods. Therefore , regardless of economic turbulence, is there a model to forecast VaR that provides a reasonable number of violations and daily capital charges? Why a GFC-Robust Risk Management Strategy is Needed?

17. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions GFC-Robust Risk Management Strategy Duration of Minimum Daily Capital Charges for Alternative Models of Volatility McAleer, Jimenez-Martin, Perez-Amaral (2010)

18. Data DATA Closing daily prices for Standard and Poor’s Composite 500 Index. From Reuters-Ecowin Financial Database SAMPLE: 3 January 2000 to 16 March 2010 • Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Daily Returns on S&P500 Index 3 January 2000 – 16 March 2010 Daily Returns volatitlity on S&P500 Index 3 January 2000 – 16 March 2010

19. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Data BEFORE CRISISDURING CRISIS AFTER CRISIS

20. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions • Proposal: use median of the point forecasts of the usual VaR models. Or in general, quantiles. • Associated with robust statistics, turns out to be the best in our case.

21. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Data VaR for S&P500 Returns 2 January 2008 – 16 March 2010

22. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Results

23. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Results Criteria for Comparing Percentile Strategies BEFORE CRISISDURING CRISIS AFTER CRISIS

24. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Conclusions • Under the Basel II Accord, • ADIs have to communicate their risk estimates to the monetary authorities • They can use a variety of VaR models to estimate risks. • ADIs are subject to back-testing • Daily capital charges as protection against market risk must be set at the higher of the previous day’s VaR or the average VaR over the last 60 business days, multiplied by a factor k. • VaR models currently in use can lead to high daily capital requirements or an excessive number of violations. • ADI’s objective is to maximize profits, so they wish to minimize their capital charges while restricting the number of violations in a given year below the maximum of 10 allowed by the Basel II Accord. From this target it follows naturally that ADI’s have to choose an optimal reporting policy that may strategically under-report or over-report their forecast of VaR in order to minimize the daily capital requirement.

25. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Conclusions II • In McAleer et al. (2010), the VaR model minimizing DCC before, during and after the GFC changed frequently. • In this paper we propose robust risk forecasts that use combinations of several conditional volatility models for forecasting VaR: eg: the median. • The median is robust, in that it yields reasonable daily capital charges, number of violations that do not jeopardize institutions that might use it, and more importantly, is invariant before, during and after the 2008-09 GFC. • The median is a model that balances daily capital charges and violation penalties in minimizing DCC.  

26. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Conclusions III • Combining forecasting models is within the spirit of the Basel II Accord, although its use would require approval by the regulatory authorities, as for any forecasting model. • This approach is not computationally demanding, even though several models have to be specified and estimated over time. • Further research is being carried out using a variety of different indexes from different countries. Temptative results confirm that the median is global financial crisis robust and clearly preferred in most cases to single models.

27. International evidence on GFC-robust forecasts for risk management under the Basel Accord Michael McAleer Erasmus University Rotterdam/ Tinbergen Institute The Netherlands / Institute of Economic Research, Kyoto University Japan Juan-Angel Jimenez-Martin Teodosio Perez-Amaral Complutense University of Madrid, Spain

28. Results I DATA Closing daily prices. From Reuters-Ecowin Financial Database SAMPLE: 3 January 2000 to 14 October 2010 • Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions IBEX35 Madrid CAC 40 Paris FTSE 100 London S&P500 New York DAX 30 Frankfurt NIKKEI TOKYO Dow Jones100 New York HSI Hong Kong SMI Zurich

29. Results II Daily Returns 3 January 2000 – 14 October 2010 • Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions

30. Results III Volatility of Daily Returns 3 January 2000 – 14 October 2010 • Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions

31. Results IV • Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Daily Returns Correlations 3 January 2008 – 14 October 2010 Correlations between Daily Returns Volatilities 3 January 2008 – 14 October 2010

32. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Results V BEFORE CRISISDURING CRISIS AFTER CRISIS

33. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Results VI • Proposal: use median of the point forecasts of the usual VaR models. Or in general, quantiles. • Associated with statistics, turns out to be the best in our case. • Mean of Daily Capital Charges • Number of Violations • Accumulated losses • Tick loss function

34. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Results VII Daily Capital charges Increase

35. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Results VIII Forecast Model Number of Violations Accumulated Losses

36. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Results IX Daily Capital charges Increase

37. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Results X Daily Capital charges Increase

38. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Results XI Daily Capital charges Increase

39. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Results XII Tick loss Function Increase

40. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Results XIII Tick loss Function Increase

41. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Results XIV Tick loss Function Increase

42. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Conclusions • Under the Basel II Accord, • ADIs have to communicate their risk estimates to the monetary authorities • They can use a variety of VaR models to estimate risks. • ADIs are subject to back-testing • Daily capital charges as protection against market risk must be set at the higher of the previous day’s VaR or the average VaR over the last 60 business days, multiplied by a factor k. • VaR models currently in use can lead to high daily capital requirements or an excessive number of violations. • ADI’s objective is to maximize profits, so they wish to minimize their capital charges while restricting the number of violations in a given year below the maximum of 10 allowed by the Basel II Accord. From this target it follows naturally that ADI’s have to choose an optimal reporting policy that may strategically under-report or over-report their forecast of VaR in order to minimize the daily capital requirement.

43. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Conclusions II • In McAleer et al. (2010), the VaR model minimizing DCC before, during and after the GFC changed frequently. • In this paper we propose robust risk forecasts that use combinations of several conditional volatility models for forecasting VaR, eg: the median. • The median is robust, in that it yields reasonable daily capital charges, number of violations that do not jeopardize institutions that might use it, and more importantly, is invariant before, during and after the 2008-09 GFC. • The median is a model that balances daily capital charges and violation penalties in minimizing DCC.  

44. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Conclusions III • Combining forecasting models is within the spirit of the Basel II Accord, although its use would require approval by the regulatory authorities, as for any forecasting model. • This approach is not computationally demanding, even though several models have to be specified and estimated over time. • Research is being carried out using a variety of different indexes from different countries. Results confirm that the median is global financial crisis robust and clearly preferred in most cases to single models.

45. Maximizing Profits, VaR and Daily Capital Charges • Models for Forecasting VaR • GFC-Robust Risk Management Strategy • Results • Conclusions Conclusions IV • Before the GFC, the best strategy for minimizing DCC and staying below 8 violations is the SUPREMUM, in 6 out of 9 indices. The second best is the EGARCH in 3 out of 9 indices. RISKMETRICS also beats the MEDIAN in 8 out of 9 indices. However, the best strategy for staying in the green zone (up to 4 violations) is the MEDIAN (8 out of 9 indices). • During the GFC, the SUPREMUM violates more than 8 times in 7 out of 9 indices while RISKMETRICS violates more than 8 times in 5 out of 9 indices. However, the MEDIAN beats RISKMETRICS in 5 indices while it keeps you in less than 8 violations, for 8 out of 9 indices. • After the GFC, the SUPREMUM is best in 5 out of 9 indices but violates heavily in the rest. In second place, in 2 out of 9 cases, comes EGARCH, but it also tends to violate in the other indices. The MEDIAN strategy keeps you green or with less than 8 violations for all indices, while it beats RISKMETRICS in 5 out of 9 indices.