Applying stress-testing on value at risk (VaR) methodologies

“Applying Stress-Testing On Value at Risk (VaR) Methodologies” AUTHORS José Manuel Feria Domínguez María Dolores Oliver Alfonso ARTICLE INFO José Manuel Feria Domínguez and María Dolores Oliver Alfonso (2004). Applying Stress-Testing On Value at Risk (VaR) Methodologies. Investment Management and Financial Innovations, 1(4) JOURNAL "Investment Management and Financial Innovations" FOUNDER LLC “Consulting Publishing Company “Business Perspectives” NUMBER OF REFERENCES 0 NUMBER OF FIGURES 0 NUMBER OF TABLES 0 © The author(s) 2018. This publication is an open access article. businessperspectives.org Investment Management and Financial Innovations, 4/2004 62 Applying Stress-Testing On Value at Risk (VaR) Methodologies José Manuel Feria Domínguez1, María Dolores Oliver Alfonso2 Abstract In recent years, Value at Risk (VaR) methodologies, i. e., Parametric VaR, Historical Simulation and the Monte Carlo Simulation have experienced spectacular growth within the new regulatory framework which is Basle II. Moreover, complementary analyses such a Stress-testing and Back-testing have also demonstrated their usefulness for financial risk managers. In this paper, we develop an empirical Stress-Testing exercise by using two historical scenarios of crisis. In particular, we analyze the impact of the 11-S attacks (2001) and the Latin America crisis (2002) on the level of risk, previously calculated by different statistical methods. Consequently, we have selected a Spanish stock portfolio in order to focus on market risk. Key words: Stress-Testing, Value at Risk, Market Risk Management. I. Introduction From a conceptual point of view, Value at Risk (VaR) needs to be defined previously in terms of certain parameters (time horizon, level of confidence and currency in reference), as well as some theoretical hypotheses. One of them has to do with stability which supposes that the VaR estimate is obtained for normal market conditions. This principle implies the exclusion of extreme scenarios characterized by high volatility levels that are defined by Jorion (1997) as Event Risk. Stress-Testing is a useful tool for financial risk managers because it gives us a clear idea of the vulnerability of a defined portfolio. By applying Stress-testing techniques we measure the potential loss we could suffer in a hypothetical scenario of crisis. In the words of William McDonough, the president of the New York Federal Commission Bank, “One of the most important functions of Stress-testing is to identify hidden vulnerabilities, often the result of hidden assumptions, and make clear to trading managers and senior management the consequences of being wrong in their assumptions”. II. Scenario Analysis Broadly speaking, there are different ways to develop the Stress-Testing exercise. Dowd (1998) distinguishes three main approaches: xHistorical Scenarios of Crisis: Scenarios are chosen from historical disasters such as the US stock market crash of October 1987, the bond price falls of 1994, the Mexican crisis of 1994, the Asian crisis of 1997, the Argentinean crisis of 2001, etc. xStylized Scenarios: Simulations of the effects of some market movements in interest rates, exchange rates, stock prices and commodity prices on the portfolio. These movements are expressed in terms of both absolute and relative changes. As the Derivatives Policy Group (1995) suggests: oParallel yield curve in ±100 basis points. oYield curve shifts of ±25 basis points. oStock index changes of ±10%. oCurrency changes of ±6%. oVolatility changes of ±20%. 1 Profesor Asociado, Finance Department, Pablo de Olavide University, Spain. 2 Profesora Titular, Finance Department, University of Seville, Spain. Investment Management and Financial Innovations, 4/2004 63 xHypothetical Events: A reflection process in which we have to think about the potential consequences of certain hypothetical situations such as an earthquake, an international war, a terrorist attack, etc. Scenario Analysis Historical Scenarios of Crisis Stylized Scenarios Hipothetical Events Fig. 1. Types of Scenario Analysis (Dowd, 1998) III. Methodological Issues Main Assumptions In this paper, we want to evaluate the response of Value at Risk methodologies to the Stress-testing exercise based on historical scenarios of crisis. The first step is to calculate VaR estimates by three alternative methods: Parametric VaR, Historical Simulation and the Monte Carlo Simulation. In a second part, we put press on those estimates by introducing both the stressed volatility and the correlation observed in two scenarios of crisis; in particular, the impact of the 11-S attack in New York (2001) and the Latin American Crisis of July 2002. Portfolio The selected portfolio consists of five common Spanish stocks, such as: TELEFÓNICA (TEF), BBVA (BBVA), BSCH (SAN), ENDESA (ELE), REPSOL (REP). Those shares are the blue chips of the Spanish Market and they represent more than 50% of the IBEX-35. It is also important to define the initial value of the position (portfolio), as well as the particular weights of each stock. In that sense, we are going to invest 100.000 € equally divided among the shares (Table 1). Moreover, the date used to calculate VaR has been set on 30 August 2002. If we want to asses the global position, we only have to multiply respective prices and number of shares. In that particular case, we have chosen the same weight for each stock, i. e., 20%. Table 1 Initial position (euros) Fecha VeR 30/08/2002 TEF ELE BBVA SAN REP TOTAL Nodetitulos 2.182 1.653 1.998 2.937 1.504 10.273 Cotización 9.17 € 12.10 € 10.01 € 6.81 € 13.30 € Valor 20.000 € 20.000 € 20.000 € 20.000 € 20.000 € 100.000 € Peso 20% 20% 20% 20% 20% 100% Time Horizon In this paper, we have selected a time window from 28 January 2000 to 30 August 2002 and it consists of 651 days of trading. For this period, we have transformed daily price series into logarithmic return series by using the following formula: Investment Management and Financial Innovations, 4/2004 64 »¼º«¬ª 1 ln t t tP P R. (1) In other words, our sample data is composed of 650 historical daily returns. Secondly, we have calculated the historical volatility for each return series as the following equation illustrates: 1 )( 1 2  ¦ T R T i i PV i=1,2...650 , (2) where V – sample standard deviation, T – total number of observations, P – medium return of the series, Ri– return of individual asset. Finally, in order to build up the stress-testing exercise, we have chosen two historical scenarios which are characterized for their respective high level of volatility: x11-S terrorist attacks in New York (2001) xBrazilian crisis (July 2002). The daily volatilities for each particular common stock in our portfolio have been calculated by using a mobile monthly window (20 days of market trading) as Figure 1 illustrates. We also plot (Figure 3) the daily volatility observed for the Spanish Stock Market Index (IBEX-35). Both charts reflect how risk, in terms of volatility, increases after these international events occur. 0,00% 1,00% 2,00% 3,00% 4,00% 5,00% 6,00% 7,00% 25/02/2000 25/03/2000 25/04/2000 25/05/2000 25/06/2000 25/07/2000 25/08/2000 25/09/2000 25/10/2000 25/11/2000 25/12/2000 25/01/2001 25/02/2001 25/03/2001 25/04/2001 25/05/2001 25/06/2001 25/07/2001 25/08/2001 25/09/2001 25/10/2001 25/11/2001 25/12/2001 25/01/2002 25/02/2002 25/03/2002 25/04/2002 25/05/2002 25/06/2002 25/07/2002 25/08/2002 horizonte temporal volatilidad diaria TEF ELE BBVA SAN REP Fig. 2. Daily volatility for individual stocks Scenario I Scenario II Investment Management and Financial Innovations, 4/2004 65 0,00% 0,50% 1,00% 1,50% 2,00% 2,50% 3,00% 3,50% 4,00% 25/02/2000 25/03/2000 25/04/2000 25/05/2000 25/06/2000 25/07/2000 25/08/2000 25/09/2000 25/10/2000 25/11/2000 25/12/2000 25/01/2001 25/02/2001 25/03/2001 25/04/2001 25/05/2001 25/06/2001 25/07/2001 25/08/2001 25/09/2001 25/10/2001 25/11/2001 25/12/2001 25/01/2002 25/02/2002 25/03/2002 25/04/2002 25/05/2002 25/06/2002 25/07/2002 25/08/2002 ibex Fig. 3. Daily volatility for IBEX-35 VaR Parameters Value at Risk (VaR) indicates the maximum loss which we can incur on a particular time horizon with a defined level of confidence. In other words, VaR, as a statistical estimate, requires the following parameters: xThe time horizon will be one day, i.e., we will estimate daily VaR, or DeaR (Daily Earnings at Risk). xThe level of confidence has been set at 95%. xThe currency used for reporting VaR figures is the Euro. Loss Profit ' Value (position) Probability VaR 95% pobability of P&L < VaR 5% probability of loss beyond VaR -C 0 Loss Profit ' Value (position) Probability VaR 95% pobability of P&L < VaR 5% probability of loss beyond VaR -C 0 Fig. 4. VaR Concept IV. Stress-Testing On VaR Methodologies In general, the basis of the Stress-Testing exercise is to recalculate the Value at Risk estimate by using a higher volatility than the observed one for the historical window selected, i.e., 651 trading days. For this purpose, we have computed the daily volatilities for each scenario of crisis. These are presented in Table 2. Scenario I Scenario II Investment Management and Financial Innovations, 4/2004 66 Table 2 Daily volatility for both scenario of crisis Fecha VeR TEF ELE BBVA SAN REP 30/08/2002 2.82% 1.81% 2.35% 2.52% 2.13% Fecha TEF ELE BBVA SAN REP Escenario I 11/10/2001 3.31% 1.96% 4.73% 4.90% 3.48% Fecha TEF ELE BBVA SAN REP Escenario II 09/08/2002 5.11% 4.51% 4.99% 5.63% 3.66% From an operational point of view, the main problem with Stress-Testing appears when incorporating correlation. Empirical evidence1 demonstrates that correlation is not constant over time; moreover, it fluctuates in periods of crisis. As Aragonés and Blanco (2000) point out, if we put pressure on correlation coefficients in an arbitrary way, probably, the newly calculated correlation matrix will not be positive defined and, as a consequence, its elements will not have internal consistency. For this reason, it is strongly recommended not only pressing volatilities up, but also the correlation matrix U . In practice, once we have calculated the correlation coefficients between pairs of stocks using a monthly mobile window, we can select the correlation observed for those days of maximum volatility levels, which corresponds to 11/10/2001 and 09/08/2002, respectively. From here, we have designed both stressed correlation matrices (Tables 3 and 4) whose determinants are positive: 0! U . (3) Table 3 Correlation matrix scenario I Matriz de correlación: Escenario I TEF ELE BBVA SAN REP TEF 100% 56.82% 66.24% 74.49% 45.88% ELE 56.82% 100% 74.07% 73.49% 65.36% BBVA 66.24% 74.07% 100% 93.68% 80.05% SAN 74.49% 73.49% 93.68% 100% 76.14% REP 45.88% 65.36% 80.05% 76.14% 100% Table 4 Correlation matrix scenario II Matriz de correlación: Escenario I TEF ELE BBVA SAN REP TEF 100% 74.87% 74.39% 76.97% 54.05% ELE 74.87% 100% 85.19% 84.50% 69.38% BBVA 74.39% 85.19% 100% 89.75% 76.99% SAN 76.97% 84.50% 89.75% 100% 59.24% REP 54.05% 69.38% 76.99% 59.24% 100% 1 Jackson (1996) and Mori, Ohsawa and Shimizu (1996) analysed such phenomena. Investment Management and Financial Innovations, 4/2004 67 According to Alexander y Leigh (1997), to ensure that the correlation matrix is positive defined, it must comply with the Cholesky mathematical property, that is: U T AA , (4) where, U – Correlation matrix, A – Cholesky matrix, T A– Transposed Cholesky matrix. We have also verified that stressed correlation matrices can be decomposed into Cholesky factors as Tables 5 and 6 illustrate. Table 5 Cholesky matrix scenario I Matriz de Cholesky: Escenario I TEF ELE BBVA SAN REP TEF 100% 0.00% 0.00% 0.00% 0.00% ELE 56.82% 82.29% 0.00% 0.00% 0.00% BBVA 66.24% 44.27% 60.43% 0.00% 0.00% SAN 74.49% 37.87% 45.63% 30.57% 0.00% REP 45.88% 47.75% 47.19% 7.67% 57.70% Table 6 Cholesky matrix scenario II Matriz de Cholesky: Escenario II TEF ELE BBVA SAN REP TEF 100% 0.00% 0.00% 0.00% 0.00% ELE 74.87% 66.29% 0.00% 0.00% 0.00% BBVA 74.39% 44.50% 49.86% 0.00% 0.00% SAN 76.97% 40.54% 28.98% 39.90% 0.00% REP 54.05% 43.61% 34.85% -25.41% 57.58% Stress-Testing and Parametric VaR Stress-Testing is very easy to apply when dealing with the parametric methodology because we only have to estimate on 30/08/2002 the stressed VaR for each scenario of crisis as formula 5 indicates: * , * 6449,1)( daily i Z i stressedVaR VZ  , (5) where i Z – initial value of the position maintained in stocki (20.000 Euros), * ,daily i V – daily volatility of the stocki associated to a stressed scenario, * Z – depends on the level of confidence; at 95% confidence its value is equal to -1,6449. Investment Management and Financial Innovations, 4/2004 68 In Tables 7 and 8 we present the individual VaR estimates associated with both scenarios of crisis. We can define a new magnitude which is raw VaR, with the aggregation of individual VaR’s, so it give us a global measure of risk without standing diversification benefits. If we want to have a more realistic idea of the risk exposure, it is necessary to introduce another estimate, which is diversified VaR or net VaR. For incorporating diversification effects, we apply the following formula: VVVaR T tportfolio  * , U (6) »»»»¼º««««¬ª tn t t VaR VaR VaR V , ,2 ,1 : Column vector of dimension (nx1) which represents non diversified individual VaR’s. It is calculated from the product of ZV  ** ZV >@ tntt TVaRVaRVaRV ,,2,1  : The transposed vector of V is calculated as ** VZ  ZV TT . Table 7 Individual VaR scenario I Escenario I TEF ELE BBVA SAN REP Valor inicial 20.000 € 20.000 € 20.000 € 20.000 € 20.000 € Volatilidad diaria 3,31% 1,96% 4,73% 4,90% 3,48% Z (95%) 1,6449 1,6449 1,6449 1,6449 1,6449 VeR individual 1.090,37 € 643.24 € 1.555,58 € 1.611,27 € 1.145,17 € Table 8 Individual VaR scenario II Escenario II TEF ELE BBVA SAN REP Valor inicial 20.000 € 20.000 € 20.000 € 20.000 € 20.000 € Volatilidad diaria 5.11% 4.51% 4.99% 5.63% 3,66% Z (95%) 1,6449 1,6449 1,6449 1,6449 1,6449 VeR individual 1.681,16 € 1.484,15 € 1.641,65 € 1.853,01 € 1.204,43 € In Tables 9 and 10 we have computed the diversified VaR for our portfolio in both stressed scenarios. Moreover, we have also calculated another interesting estimate, which is EaR (Earning at Risk). It is the maximum gain we can expect with a certain confidence level within a selected time period. In particular, we have estimated a 95% percentile. We notice that both figures, VaR and EaR, coincide because of the underlying assumption of normal distribution. Table 9 Correlated VaR scenario I Escenario I Nivel de confianza Horizonte temporal VeR correlacionado 5.391,11 € EaR correlacionado 5.391,11 € 95% 1 dia Ratio VeR/EaR 100% Ratio VeR/Valor de la cartera 5.39% Table 10 Investment Management and Financial Innovations, 4/2004 69 Correlated VaR scenario II Escenario II Nivel de confianza Horizonte temporal VeR correlacionado 7.061,43 € EaR correlacionado 7.061,43 € 95% 1 dia Ratio VeR/EaR 100% Ratio VeR/Valor de la cartera 7.06 % Stress-Testing and the Monte Carlo Simulation The Monte Carlo Simulation is based on the generation of random prices as follows: t tt ePP  HV 1, (7) where t P is the simulated price, 1t Pis the current price of the stock, H is a random variable which is distributed as a normal standardized, i.e., with P =0 and V =1, V is the daily volatility of the stock, tis an adjusted factor which transforms daily volatility into wider time horizons. In this paper, as VaR is estimated one day hence, its value is equal to one. In the case of a portfolio, composed by multiple assets, the previous formula cannot be applied because it is only valid for a single asset. Therefore, the process of generating random numbers is more complex; in other words, the historical correlation between shares should be incorporated in such a process. For this reason, and from a methodological point of view, the normal random numbers, H , should be transformed into correlated random numbers, Z, by using the Cholesky Matrix: >@ numbers random x matrixCholesky x numbers random Correlated x REP SAN BBVA ELE TEF A Z Z Z Z Z 15 5 4 3 2 1 55 * 15 »»»»»»¼º««««««¬ª »»»»»»¼º««««««¬ª HHHHH , (8) where Z is a vector of transformed normal variables which embodies the historical correlation, H is a vector of normal standardized variables, * Ais the stressed Cholesky Matrix for each scenario of crisis as Tables 5 and 6 show, respectively. For simulating 1.000 correlated and stressed prices from current prices (see Table 1) we should generate 1.000 Z vectors, as the subindex iindicates in the following equation: i REPREP i SANSAN i BBVABBVA i ELEELE i TEFTEF Z i tREP Z i tSAN Z i tBBVA Z i tELE Z i tTEF eP eP eP eP eP      * * * * * 30,13 81,6 01,10 10,12 17,9 , , , , , VVVVV i=1,2.....1.000. (9)