Using Patterns of Volatility in Calculating VaR

Transcription

1 Using Patterns of Volatility in Calculating VaR Professor Radu Titus MARINESCU, Ph.D. Artifex University of Bucharest Lecturer Mădălina Gabriela ANGHEL, Ph.D. Artifex University of Bucharest Daniel DUMITRESCU, Ph.D. Student Academy of Economic Studies Bucharest Adina Mihaela DINU, Ph.D. Student Academy of Economic Studies Bucharest Abstract Based on the above, it can be concluded that the use of Value at Risk method allows a more efficient allocation of financial resources available, thus eliminating the overexposure to a single risk source. Also, the VaR model allows capital investors to properly assess their activity and position in the capital market, depending on the level of risk they are willing to and take. Key words: value at risk, EWMA model, GARCG model, capital market Determination of value Value at Risk can be carried out using models EWMA (Exponentially Weighted Moving Average), or using the GARCH model. Following are presented the main theoretical aspects regarding the use of the two types of models mentioned above to determine the Value at Risk indicator. 1. VaR calculation using EWMA According to the model EWMA (Exponentially Weighted Moving Average) volatility depends on the previous yield and previous volatility : (current volatility) λ = weighting constant rt -1 = yield in previous period Revista Română de Statistică Trim III/2013- Supliment 143

2 λ is a parameter that indicates the financial asset volatility persistence. If its value is higher, the shock occurred at one time in the market is more persistent. Parameter λ 1 shows how quickly the asset volatility responds to a shock in any direction. The greater the value of this parameter is, the greater the reaction of volatility at shock is. RiskMetrics uses a λ value of 0.94 for daily data. Ways to incorporate volatility calculated by EWMA models in VaR models a) The first way is to incorporate volatility is historical simulation with weighting of data according to volatility. In this case, past performance is standardized based on conditional volatility. b ) In the case of Monte Carlo simulation using EWMA, yields can be simulated considering that a normal distribution follows, but the covariance matrix is created using EWMA. c) Analytical VaR using EWMA. In generating the covariance matrix we use an equation that is analogous to a variance equation: (covariance between active 1 and 2) r 1, t - 1 or r 2, t -1 = yields of both active in the preceding period Once defined, the covariance matrix can be used to calculate VaR using, in this sense, the analytical method (which is recommended for simple portfolios ) or the Monte Carlo simulation (shown for portfolios that include options). For analytical simulation, the VaR for h days, with level of relevance α is: Zα = the critical value of normal standard distribution for α level of relevance; P = the current value of the portfolio; σ = standard deviation forecast for a horizon of h days. The standard deviation is calculated using a covariance matrix of yields for h days. 144 Revista Română de Statistică Trim. III/ Supliment

3 When represented at assets level, it shall take the following form: w = ( w1, w2,..., wn ) is the share of assets in the portfolio ; V = forecast, for a horizon of h days, of the covariance matrix for the yields of the assets included in the portfolio. When represented at risk factor level, it takes the following form: β = ( β 1, β 2,..., β n) represents the sensitivity factors of the portfolio; V = forecast, for a horizon of h days, of the covariance matrix for risk factors yelds. The forecast of the covariance matrix on a horizon of h days, for simple portfolios, is obtained by applying the rule, multiplying the covariance matrix for a one-day horizon with. This methodology is not suitable for portfolios that include options because it will lead to incorrect results, in this case beeing recommended to use a covariance matrix for horizon h 2. Calculating VaR using GARCH models In constructing an ARCH model we have to consider two separate equations : one for media contingent - the equation of evolution of asset returns; one for the conditional variance - the volatility equation. In 1986, Tim Bollerslev propose the following form of the GARCH model (p,q ) for the determination of the indicator value Value at Risk : rt= a ARMA process (m, n) or a Random Walk model (if β1, i = 0, where i = 1, m and β2, j = 0, where i = 1, n ) ; ht ( volatility) = an ARCH process (q ) and GARCH (p) ; Revista Română de Statistică Trim III/2013- Supliment 145

4 α1 = persistent volatility; α2 = speed of response to shocks in market volatility. The following condition must be met in order to have an not explosive process (explosive volatility): The ARCH and GARCH coefficients must be a subunit and positive. This model, interpreted in a financial context, describes how an agent is trying to forecast volatility for the next period based on the average long-term ( α0 ) of the variance, of the prior variance (GARCH term) and of information on the volatility observed in the previous period (ARCH term). If the asset return in the previous period was unexpectedly high, the agent will increase the variance expected in the future. The model accepts the situation where large changes in the course of financial assets is likely to continue to follow its variations, a phenomenon that is called "clustering volatility ". Tests on mature financial markets showed a response of exchange rate volatility generally lower threshold of 0.25 and a degree of permanence to its upper limit over 0.7. Subsequently GARCH model was extended to relax certain assumptions or to incorporate asymmetry of financial assets rate impact or to separate the trend volatility of short-term volatility. The inclusion of GARCH models in VaR calculation can be done in several ways, namely: Analytical VaR by using a covariance matrix based on GARCH models; historical simulation when data are weighted according to the volatility - the data are standardized according to their volatility estimated by GARCH models; Monte Carlo Simulation - Evolution of the yields can be simulated on the basis of the covariance matrix calculated on the GARCH models basis, which enables both simulation of volatility evolution and simulation of asset returns evolution - which is an advantage if the portfolio contains options. To calculate the covariance matrix, the correlation coefficient can be considered constant, in which case the covariance is determined by correlation coefficients and variables: σij, t +1 = covariance between the two assets i and j ; ρij = correlation coefficient between the two assets; σi, t +1 and σj, t +1 = represents the variances of the two assets. 146 Revista Română de Statistică Trim. III/ Supliment

5 Using GARCH models has been suggested in practice for modeling direct P/L portfolio and calculate VaR by its conditional volatility, thus avoiding the calculation of covariance matrices. 3. Using VaR (value at risk) model to analyze capital market In papers published in our country or abroad, Value at Risk model is widely used to determine the risk associated with investment activity. Such a model is shown by Anghelache Gabriela - Victoria and Radu Alina Nicoleta in the article " VAR method used in market risk ". In it is analyzed, with the VAR model, the evolution of market risk for the BET- C index. In this case, volatility is estimated using a GARCH model. Based on the above methodology elements, it is analized the evolution of BET C index, resulting in a GARCH model (1,1) with innovations normally distributed. Estimated GARCH model is stationary, but has a high persistence in volatility. In conclusion this study provides a graphical presentation of actual loss over a period of ten days for Bet Index and Value at Risk values determined both on the 1% quantile of the normal distribution and on the distribution of the 1% t-students. Based on the above, it can be concluded that the use of Value at Risk method allows a more efficient allocation of financial resources available, thus eliminating the overexposure to a single risk source. Also, the VaR model allows capital investors to properly assess their activity and position in the capital market, depending on the level of risk they are willing to and take. In this part I decided to apply the model to determine the VaR (value at risk) to estimate the Var level under different scenarios for the portfolio consisting of 10 titles. For this it is required to present the daily returns vector and covariance matrix of the 10 titles analyzed in the portfolio management. Daily return vector hopes of 10 titles: Share Daily return ELMA SNP BRM TGN TUFE SIF SIF Revista Română de Statistică Trim III/2013- Supliment 147

6 Share Daily return TEL BIO TLV Source: own calculations Covariance matrix for the daily returns of securities: Source: own calculations Historical simulation method involves determining the continuous daily returns of the securities using for this purpose the following formula: = Continuous daily yield title "x" = Stock price at time "t" = Stock price at time " t -1" The values obtained are presented in appendices. Estimating the risk by applying the variance - covariance method requires to determine the level of trust to be took in consideration for the portfolio variance estimation for which the calculation is made. By applying this method, the risk is determined by the following relationship: W = market value of the portfolio c = trust coefficient for distribution of probability considered (tabulated values in table) σp =portfolio volatility 148 Revista Română de Statistică Trim. III/ Supliment

7 From the tables on repartition function of normal standard distribution result the following values for the parameter "c" corresponding probabilities : Probability (P) c 90% 1,28 95% 1,65 97,5% 1,96 99% 2,33 Next, we estimate several scenarios of VaR using the variance covariance method.to begin, we consider the case of echiponderate portfolio containing the 10 shares (titles). Also we have to made some clarifications as: We believe that the investment is RON 1,000,000, σp for this echiponderate portfolio is % (calculated in a previous chapter). Trust level Confidence Daily VaR coeficient c (historical) 90% 1, lei 95% 1, lei 99% 2, lei Portfolio that has the highest value of return ( ) ( P12 ) corresponds σp equals The vector of structure is the following: X ELMA X SNP X BRM X TGN X TUFE X SIF5 X SIF4 X TEL X BIO X TLV 0.00% 10.00% 60.00% 5.00% 0.00% 5.00% 5.00% 0.00% 5.00% 10.00% Appropriate Value at Risk is : Trust level Confidence Daily VaR coeficient c (historical) 90% 1, % 1, % 2, The VaR is calculated similarly for the other portfolios. P σ P P P P P P Revista Română de Statistică Trim III/2013- Supliment 149

8 P σ P P P P P P P P P P P P P P Trust level c P1 P2 P3 P4 P5 P6 90% 1, % 1, % 2, Trust level c P7 P8 P9 P10 P11 P12 90% 1, % 1, % 2, Trust level c P13 P14 P15 P16 P17 P18 90% 1, % 1, % 2, Bibliography Codirlaşu, A. ; Chidesciuc N.Al. (2008 ) - "Banking Econometrics. Applied Econometrics using EViews Course Notes" Miclaus, PG ; Bobirca, A., Lupu, R., Ungureanu, st. (2009 ) - "A Garch dynamic conditional correlation model for the computation of dynamic VAR on the Romanian capital market", Economic Journal, Lucian Blaga University of Sibiu Anghelache, Gabriela Victoria, Radu Alina Nicoleta - "VaR method used in market risk", International Scientific Symposium "Effects and solutions to economic crisis financial Artifex Publishing, Revista Română de Statistică Trim. III/ Supliment