Measuring and Forecasting Portfolio Risk on the Romanian Capital Market Supervisor: Professor Moisa ALTAR MSc student: Stefania URSULEASA

 This paper describes, in the first part, various VaR methodologies ;  In the second part we implemented eight VaR models on the Romanian stock market  The last part of the paper highlights some conclusions Paper structure

Presentation contents  Paper objectives;  Value at Risk – practical aspects;  Theoretical aspects of the VaR models;  Data description;  VaR methods used;  Back testing VaR models;  Conclusions

1. Paper objectives  Apply eight types of VaR models on a portfolio of financial assets quoted on the Romanian Stock Exchange  Perform a back testing of the models in order to measure the performance of the methodologies used  Principal features of the models

2. Practical aspect of VaR models  Basle Accord Amendment (1996) for the calculation of market risk capital using VaR internal models  May 2004 – NBR issues a regulation project  May 2004 – NBR issues a regulation project regarding capital adequacy for credit institutions

3. Theoretical aspects of the VaR models  Categories of VaR models nonparametric models - Historical Simulation and the Hybrid Model; nonparametric models - Historical Simulation and the Hybrid Model; parametric models - Variance-Covariance Approaches; parametric models - Variance-Covariance Approaches; semiparametric models - Extreme Value theory, CAViaR. semiparametric models - Extreme Value theory, CAViaR.

4. Data description  daily closing quotations of the SIF1, SIF2 and SIF5 stocks  daily logarithmic returns from to  compute an equally weighted portfolio

Assumptions and Data Characteristics 1. Normality assumption  not normally distributed SIF1 SIF2

SIF5Portfolio

the positive skewness of the returns series for SIF1, SIF5, the portfolio and the negative one for the SIF2 returns series - showing that the returns distributions are asymmetric; the positive skewness of the returns series for SIF1, SIF5, the portfolio and the negative one for the SIF2 returns series - showing that the returns distributions are asymmetric; the excess kurtosis shows the “fat-tails” and higher probabilities for extreme events than for the normal distribution; the excess kurtosis shows the “fat-tails” and higher probabilities for extreme events than for the normal distribution; Jarque-Bera statistics are so large for all the four series as to show that the distribution is far from normal. The zero probability of the JB test also suggests that the normality null hypothesis is strongly rejected. Jarque-Bera statistics are so large for all the four series as to show that the distribution is far from normal. The zero probability of the JB test also suggests that the normality null hypothesis is strongly rejected.

2. Homoscedasticity assumption  Volatility clustering process (volatility mean reverts) 3. Stationarity assumption  Augmented Dickey-Fuller test - series stationary at 1% critical value 4. Serial independence assumption  Autocorrelation coefficients and the Liung-Box Q statistic test - strong autocorrelations for the first lags

5. VaR methods used 5.1. Nonparametric models: The historical simulation approach  estimates VaR directly by using the empirical percentiles of the historical returns distribution;  based on the concept of rolling windows;  portfolio returns within a 150 days window were sorted in ascending order and the 90th, 95th and 99th - quantiles of interest were given by the return that left 10%, 5% and 1% respectively of the observations on its left side. To compute the VaR the following day, the whole window was moved forward by one observation and the entire procedure was repeated.

Historical Simulation Chart

The antithetic historical simulation approach  Eliminates the trend in the sample series by reversing the sign of the positive observed returns and augmenting the original sample  From this point on the methodology is similar to the one from the historical simulation

Antithetic Historical Simulation Chart

The hybrid historical simulation approach  attributes exponentially declining weights to historical returns,where λ = 0.97  then the returns were ordered in ascending order and the corresponding weights are summed up until the desired confidence level is obtained

Hybrid historical simulation chart

5.2.Parametric Models The equally weighted moving average approach  based on the assumption that the variances and covariances of the returns are constant over the period of estimation and forecast;  estimated each element of the variance-covariance matrix by an equally weighted average of squared returns for the each variance and by an equally weighted average of cross products of returns for the covariances  The VaR measures were computed: Σ = variance-covariance matrix; δ = vector of sensitivities

The equally weighted moving average chart

Exponentially weighted moving average approach  promoted by Morgan – RiskMetrics  gives greater weights to recent returns

Orthogonal GARCH  tested the stationarity assumption of our system by confirming that no root of the characteristic polynomial lies outside the unit circle;  first is computed the correlation coefficients matrix for the normalized returns series of the three equities  once the eigenvectors are obtained and ordered accordingly with the decreasing magnitude of the eigenvalues the principal components can be written

 based on this relations the principal component series were computed and the modelled with the help of GARCH models  since the principal components are orthogonal their covariance matrix is simply the matrix of their variances  the variance of our portfolio returns series were determined by the following relation: where: A=matrix of normalized factor weights; ψ= the var-cov matrix of the principal components ψ= the var-cov matrix of the principal components

Orthogonal GARCH chart

5.3. Semiparametric models Extreme values theory models the tails of the return distribution rather than the whole distribution  first we ordered ascendingly the return series  we assumed that the distribution rank at which the tail starts is 5%  estimate the tail index  extreme quantile estimation was then computes using

CAViaR approach  moves the focus of attention from the distribution of returns directly to the behavior of the quantile  The autoregressive quantile specification that we chose was an Indirect G(1,1)  The first term from which the quantile’s autoregression begins was estimated using the Historical Simulation approach on a window of 150 days.

Reasons for choosing the I G(1,1)  Autocorrelation coefficients for squared portfolio returns -high degree of autocorrelation for the first lags =>mean equation to be an AR(1) *ARCH LM test no longer shows *ARCH LM test no longer shows evidence of GARCH effects

6.Back testing 6.1. Binary Loss Function  used by the supervision authorities in back testing procedures aiming to assess the quality of internal models  this statistic shows the number of exceptions obtained by the models

6.2. Multiple to obtain coverage

6.3. Mean Relative Bias  points out the extent to which the different VaR techniques produce risk estimate of similar average size

6.4. Root Mean Squared Relative Bias  reflects the degree to which the risk measures tend to vary around the all-model average risk measure ;  capture two effects:  the extent to which the average risk estimate provided by a given model systematically differs from the average of all the models and  the variability of each model’s risk estimate

7. Conclusions  Although it is not possible to formally test whether any particular VaR model outperforms others based on the back testing performed in this paper, it is possible to draw some broad conclusions regarding their features;  Apart from a few exceptions the performance of the models implemented in the paper is not greatly dissimilar across most of the performance criteria;

 the historical simulation attains the 90% and 95% confidence level, but not the 99% one. It replicates the sharpest losses registered by the portfolio, more accurately, of course for the 99th percentile, but the method has a great disadvantage: after a sharp decrease in the portfolio returns series - which the model generally reacts to by itself decreasing - followed by a quick return to normal levels, the VaR estimates remains biased downward, since it will take some time before the observation from the low return moment leaves the window;

 in the antithetic historical simulation the weak point described above is even more obvious. If after a massive loss the portfolio returns became highly positive - fact that frequently happens in our return series (remember that we have only 491 negative returns from a sample of 1081) - the historical simulation marks the high decrease in returns and then, due to the large variation of the series, returns to a normal level, although this level is lower then the one suggested by the underlying series. Because of the methodology of the antithetic historical simulation at a high desired degree of coverage the VaR model doesn’t return to a normal level after a situation like the one described above, and it remains very low. This fact is observed by the back testing methods we have used, which show a very large value of the MRB statistic and high variability for this model for the 99% confidence level;

 hybrid historical simulation show high correlation with the underlying series, especially for the 90% and 95% confidence level. For the highest level of confidence the model presents the same disadvantage as the previous two models;  the equally weighted moving average method underestimates risk for the 99th percentile although it seams to replicate smoothly the underlying return series behavior. Despite the fact that the number of exceptions returned by this model equaled the number of exceptions of other two models, the magnitude of those exceptions was greater for the hybrid approach. This is revealed by the multiple to obtain coverage that was the highest for this model;

 the exponentially weighted moving average model depends much of the underlying parameter specification. It can be seen that the model performed better then the equally weighted one because it obtained fewer exceptions for the 99th percentile, where the previous model failed. Even though this model didn’t attain its confidence level it only exceeded by one observation. More then that, the VaR estimates of this model seem to surprise better the behavior of the profits and losses series. The variability degree, the MRB and the RMSRB are almost the same for these last two models;

 the extreme values theory performs well only for the 90% confidence level. This could be because of the unfitted choice of the M parameter, which represent the point from which the tail starts. The model obtained the second highest values for the MRB and RMSRB proving higher conservatism then needed for the 99th percentile and underestimation of risk for the other two confidence levels;

 the orthogonal-GARCH replicated very accurately the behavior of the real returns series, but for the 99% confidence level it obtains two exceptions more then normal. The model has the lowest value for the mean relative bias statistic and for the root mean squared bias statistic, proving to perform very well;  the CAViaR method also performs very well, we could say even better then the orthogonal GARCH, except for the 95% confidence degree, where the model slightly exceeds the expected number of exceptions. The values obtained by this method for the MRB and RMSRB are comparable with the ones of the previous model. Also at the degrees of variability they are comparable;

 Despite our expectations seems that the models which best performed were a parametric model (O-GARCH) and a semiparametric model (CAViaR)  Basle Committee rules for back testing with the test confidence level to be 95%

 Apparently the fact that a model presents low variability or that it obtains fewer exceptions then its confidence level seem to be an appropriate model to be used by a bank. In reality if the underlying risk is not properly estimated, the financial institution will perform an inefficient allocation of financial resources