Measuring and Forecasting Measuring and Forecasting Portfolio Risk on the Portfolio Risk on the

Romanian Capital MarketRomanian Capital Market

Supervisor: Professor Moisa ALTARSupervisor: Professor Moisa ALTARMSc student: Stefania URSULEASA MSc student: Stefania URSULEASA

This paper describes, in the first part, This paper describes, in the first part, various VaR methodologiesvarious VaR methodologies;;

In the second part we implemented eight In the second part we implemented eight VaR models on the Romanian stock VaR models on the Romanian stock marketmarket

The last part of the paper highlights some The last part of the paper highlights some conclusionsconclusions

Paper structure

Presentation contentsPresentation contents

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

1. Paper objectives1. Paper objectives

Apply eight types of VaR models on a Apply eight types of VaR models on a portfolio of financial assets quoted on portfolio of financial assets quoted on the Romanian Stock Exchange the Romanian Stock Exchange

Perform a back testing of the models Perform a back testing of the models in order to measure the performance in order to measure the performance of the methodologies usedof the methodologies used

Principal features of the modelsPrincipal features of the models

2. Practical aspect of VaR 2. Practical aspect of VaR modelsmodels

Basle Accord Amendment (1996) for Basle Accord Amendment (1996) for the calculation of market risk capital the calculation of market risk capital using VaR internal models using VaR internal models

May 2004 – NBR issues a regulation May 2004 – NBR issues a regulation project project regarding capital adequacy for credit institutions

3. Theoretical aspects of the 3. Theoretical aspects of the VaR modelsVaR models

Categories of VaR modelsCategories of VaR models nonparametric modelsnonparametric models - Historical Simulation - Historical Simulation

and the Hybrid and the Hybrid Model;Model;

parametric modelsparametric models - Variance-Covariance - Variance-Covariance Approaches; Approaches;

semiparametric modelssemiparametric models - Extreme Value theory, - Extreme Value theory, CAViaR.CAViaR.

4. Data description4. Data description

daily closing quotations of the SIF1, daily closing quotations of the SIF1, SIF2 and SIF5 stocksSIF2 and SIF5 stocks

daily logarithmic returns from daily logarithmic returns from 01.11.1999 to 30.04.2004 01.11.1999 to 30.04.2004

compute an equally weighted compute an equally weighted portfolio portfolio

Assumptions and Data Assumptions and Data CharacteristicsCharacteristics

1.1. Normality assumptionNormality assumption

not normally distributednot normally distributed

SIF1 SIF1 SIF2SIF2

SIF5SIF5 PortfolioPortfolio

• the positive skewness of the returns series the positive skewness of the returns series for SIF1, SIF5, the portfolio and the for SIF1, SIF5, the portfolio and the negative one for the SIF2 returns series - negative one for the SIF2 returns series - showing that the returns distributions are showing that the returns distributions are asymmetric;asymmetric;

• the excess kurtosis shows the “fat-tails” the excess kurtosis shows the “fat-tails” and higher probabilities for extreme and higher probabilities for extreme events than for the normal distribution;events than for the normal distribution;

• Jarque-Bera statistics are so large for all the Jarque-Bera statistics are so large for all the four series as to show that the distribution four series as to show that the distribution is far from normal. The zero probability of is far from normal. The zero probability of the JB test also suggests that the normality the JB test also suggests that the normality null hypothesis is strongly rejected.null hypothesis is strongly rejected.

2. Homoscedasticity assumption2. Homoscedasticity assumption

Volatility clustering process (volatility Volatility clustering process (volatility mean reverts)mean reverts)

3. Stationarity assumption3. Stationarity assumption

Augmented Dickey-Fuller test - series Augmented Dickey-Fuller test - series stationary at 1% critical valuestationary at 1% critical value

4. Serial independence assumption4. Serial independence assumption

Autocorrelation coefficients and the Liung-Autocorrelation coefficients and the Liung-Box Q statistic test - strong Box Q statistic test - strong autocorrelations for the first lags autocorrelations for the first lags

5. VaR methods used5. VaR methods used5.1. Nonparametric models5.1. Nonparametric models::

5.1.1.The 5.1.1.The historical simulation approachhistorical simulation approach estimates VaR directly by using the empirical estimates VaR directly by using the empirical

percentiles of the historical returns percentiles of the historical returns distribution;distribution;

based on the concept of rolling windows;based on the concept of rolling windows; portfolio returns within a 150 days window portfolio returns within a 150 days window

were sorted in ascending order and the 90th, were sorted in ascending order and the 90th, 95th and 99th - quantiles of interest were 95th and 99th - quantiles of interest were given by the return that left 10%, 5% and given by the return that left 10%, 5% and 1% respectively of the observations on its 1% respectively of the observations on its left side. To compute the VaR the following left side. To compute the VaR the following day, the whole window was moved forward day, the whole window was moved forward by one observation and the entire procedure by one observation and the entire procedure was repeated.was repeated.

Historical Simulation ChartHistorical Simulation Chart

200 300 400 500 600 700 800 900 1000-0.2

-0.15

-0.1

-0.05

0Historical Simulation

Time Measure

Ret

urn

Return VaR0.99VaR0.95VaR0.90

5.1.2. The 5.1.2. The antithetic antithetic historical simulation historical simulation

approachapproach EliminatesEliminates the trend in the sample the trend in the sample

series by reversing the sign of the series by reversing the sign of the positive observed returns and positive observed returns and augmenting the original sample augmenting the original sample

From this point on the methodology From this point on the methodology is similar to the one from the is similar to the one from the historical simulation historical simulation

Antithetic Historical Simulation Antithetic Historical Simulation ChartChart

200 300 400 500 600 700 800 900 1000-0.2

-0.15

-0.1

-0.05

0Antithetic Historical Simulation

Time Measure

Ret

urn

Return VaR0.99VaR0.95VaR0.90

5.1.3. The 5.1.3. The hybrid historical hybrid historical simulation approachsimulation approach

attributes exponentially declining weights to attributes exponentially declining weights to historical returns historical returns

,where λ = 0.97,where λ = 0.97 then the returns were ordered in ascending then the returns were ordered in ascending

order and the corresponding weights are order and the corresponding weights are summed up until the desired confidence level summed up until the desired confidence level is obtained is obtained

Tkktw 1/11

1

Hybrid historical simulation Hybrid historical simulation chartchart

200 300 400 500 600 700 800 900 1000-0.2

-0.15

-0.1

-0.05

0Hybrid Historical Simulation for = 0.97

Time Measure

Ret

urn

Return VaR0.99VaR0.95VaR0.90

5.2.Parametric Models5.2.Parametric Models5.2.1. The 5.2.1. The equally weighted moving average equally weighted moving average

approachapproach

based on the assumption that the variances and based on the assumption that the variances and covariances of the returns are constant over the period covariances of the returns are constant over the period of estimation and forecast;of estimation and forecast;

estimated each element of the variance-covariance estimated each element of the variance-covariance matrix by an equally weighted average of squared matrix by an equally weighted average of squared returns for the each variance and by an equally returns for the each variance and by an equally weighted average of cross products of returns for the weighted average of cross products of returns for the covariancescovariances

The VaR measures were computed:The VaR measures were computed:

Σ = variance-covariance matrix; δ = vector of Σ = variance-covariance matrix; δ = vector of sensitivities sensitivities

Σδδ'ZVaR

The equally weighted moving The equally weighted moving average chartaverage chart

200 300 400 500 600 700 800 900 1000-0.2

-0.15

-0.1

-0.05

0Equally Weighted Moving Average

Time Measure

Ret

urn

Return VaR0.99VaR0.95VaR0.90

5.2.2. 5.2.2. Exponentially weighted moving Exponentially weighted moving

average approachaverage approach promoted by Morgan – RiskMetricspromoted by Morgan – RiskMetrics gives greater weights to recent returns gives greater weights to recent returns

1

0

11 ')1(ˆ

T

sstst

st rr

200 300 400 500 600 700 800 900 1000-0.2

-0.15

-0.1

-0.05

0Exponentially Weighted Moving Average for = 0.97

Time Measure

Retu

rn

Return VaR0.99VaR0.95VaR0.90

5.2.3. Orthogonal GARCH5.2.3. Orthogonal GARCH tested the stationarity assumption of our tested the stationarity assumption of our

system by confirming that no root of the system by confirming that no root of the characteristic polynomial lies outside the unit characteristic polynomial lies outside the unit circle;circle;

first is computed the correlation coefficients first is computed the correlation coefficients matrix for the normalized returns series of matrix for the normalized returns series of the three equities the three equities

once the eigenvectors are obtained and once the eigenvectors are obtained and ordered accordingly with the decreasing ordered accordingly with the decreasing magnitude of the eigenvalues the principal magnitude of the eigenvalues the principal components can be written components can be written

3

1iiii RwP

based on this relations the principal based on this relations the principal component series were computed and the component series were computed and the modelled with the help of GARCH models modelled with the help of GARCH models

since the principal components are since the principal components are orthogonal their covariance matrix is simply orthogonal their covariance matrix is simply the matrix of their variances the matrix of their variances

the variance of our portfolio returns series the variance of our portfolio returns series were determined by the following relation:were determined by the following relation:

where: A=matrix of normalized factor weights;where: A=matrix of normalized factor weights;

ψ= the var-cov matrix of the principal ψ= the var-cov matrix of the principal componentscomponents

')var( AARP

Orthogonal GARCH chartOrthogonal GARCH chart

0 100 200 300 400 500 600 700 800 900 1000-0.2

-0.15

-0.1

-0.05

0Orthogonal GARCH

Time Measure

Ret

urn

Return VaR0.90VaR0.95VaR0.99

5.3. Semiparametric models5.3. Semiparametric models5.3.1. Extreme values theory5.3.1. Extreme values theory

models the tails of the return distribution models the tails of the return distribution rather than the whole distribution rather than the whole distribution

first we ordered ascendingly the return first we ordered ascendingly the return series series

we assumed that the distribution rank at we assumed that the distribution rank at which the tail starts is 5% which the tail starts is 5%

estimate the tail indexestimate the tail index

extreme quantile estimation was then extreme quantile estimation was then computes using computes using

1

1

)ln()ln(1

1ˆ/1

m

kmk rr

m

/11 /ˆ pTmrr mp

5.3.2. CAViaR approach5.3.2. CAViaR approach moves the focus of attention from the moves the focus of attention from the

distribution of returns directly to the distribution of returns directly to the behavior of the quantilebehavior of the quantile

The autoregressive quantile specification The autoregressive quantile specification that we chose was an Indirect G(1,1)that we chose was an Indirect G(1,1)

The first term from which the quantile’s The first term from which the quantile’s autoregression begins was estimated autoregression begins was estimated using the Historical Simulation approach using the Historical Simulation approach on a window of 150 days. on a window of 150 days.

2/1213

2121 ttt yVarVaR

Reasons for choosing the I Reasons for choosing the I G(1,1)G(1,1) Autocorrelation coefficients for squared Autocorrelation coefficients for squared

portfolio returnsportfolio returns

-high degree of autocorrelation -high degree of autocorrelation

for the first lagsfor the first lags

=>mean equation to be an AR(1)=>mean equation to be an AR(1)

*ARCH LM test no longer shows *ARCH LM test no longer shows

evidence of GARCH effects evidence of GARCH effects

6.Back testing6.Back testing6.1. 6.1. Binary Loss FunctionBinary Loss Function used by the supervision authorities in used by the supervision authorities in

back testing procedures aiming to back testing procedures aiming to assess the quality of internal models assess the quality of internal models

this statistic shows the number of this statistic shows the number of exceptions obtained by the models exceptions obtained by the models

Exceptions Confidence levelHS99% 12 9HS95% 42 47HS90% 85 93HS_a99% 4 9HS_a95% 16 47HS_a90% 32 93HS_h99% 12 9HS_h95% 46 47HS_h90% 87 93EqWMA99% 12 9EqWMA95% 32 47EqWMA90% 50 93ExWMA99% 10 9ExWMA95% 33 47ExWMA90% 52 93EVT99% 11 9EVT95% 55 47EVT90% 91 93O-G99% 11 9O-G95% 31 47O-G90% 49 93CAViaR99% 9 9CAViaR95% 50 47CAViaR90% 76 93

HS

Extr Val

O-GARCH

CAViaR

HS_a

HS_h

EqWMA

ExWMA

6.2. 6.2. Multiple to obtain Multiple to obtain coveragecoverage

Multiple to Obtain Coverage

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

HS HS_a HS_h EqWMA ExWMA Extr Val O-GARCH CAViaR

99% 95%

6.3. 6.3. Mean Relative BiasMean Relative Bias points out the extent to which the different points out the extent to which the different

VaR techniques produce risk estimate of VaR techniques produce risk estimate of similar average size similar average size

Mean Relative Bias

-3%

-2%

-1%

0%

1%

2%

3%

4%

5%

6%

HS HS_a HS_h EqWMA ExWMA Extr Val O-GARCH

CAViaR

90%VaR 95%VaR 99%VaR

6.4. 6.4. Root Mean Squared Root Mean Squared Relative BiasRelative Bias

reflects the degree to which the risk reflects the degree to which the risk measures tend to vary around the all-measures tend to vary around the all-model average risk measure ;model average risk measure ;

capture two effects: capture two effects: the extent to which the average risk the extent to which the average risk

estimate provided by a given model estimate provided by a given model systematically differs from the average systematically differs from the average of all the models andof all the models and

the variability of each model’s risk the variability of each model’s risk estimate estimate

Variability of the VaR models

0

0,005

0,01

0,015

0,02

0,025

HS HS_a HS_h EqWMA ExWMA Extr Val O-GARCH

CAViaR

99% 95% 90%

7. Conclusions7. Conclusions Although it is not possible to formally Although it is not possible to formally

test whether any particular VaR model test whether any particular VaR model outperforms others based on the back outperforms others based on the back testing performed in this paper, it is testing performed in this paper, it is possible to draw some broad possible to draw some broad conclusions regarding their features; conclusions regarding their features;

Apart from a few exceptions the Apart from a few exceptions the performance of the models performance of the models implemented in the paper is not greatly implemented in the paper is not greatly dissimilar across most of the dissimilar across most of the performance criteria; performance criteria;

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

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

hybrid historical simulationhybrid historical simulation show high show high correlation with the underlying series, correlation with the underlying series, especially for the 90% and 95% confidence especially for the 90% and 95% confidence level. For the highest level of confidence the level. For the highest level of confidence the model presents the same disadvantage as model presents the same disadvantage as the previous two models;the previous two models;

the the equally weighted moving averageequally weighted moving average methodmethod underestimates risk for the 99th underestimates risk for the 99th percentile although it seams to replicate percentile although it seams to replicate smoothly the underlying return series smoothly the underlying return series behavior. Despite the fact that the number of behavior. Despite the fact that the number of exceptions returned by this model equaled exceptions returned by this model equaled the number of exceptions of other two the number of exceptions of other two models, the magnitude of those exceptions models, the magnitude of those exceptions was greater for the hybrid approach. This is was greater for the hybrid approach. This is revealed by the multiple to obtain coverage revealed by the multiple to obtain coverage that was the highest for this model;that was the highest for this model;

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

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

the the orthogonal-GARCHorthogonal-GARCH replicated very replicated very accurately the behavior of the real returns accurately the behavior of the real returns series, but for the 99% confidence level it series, but for the 99% confidence level it obtains two exceptions more then normal. obtains two exceptions more then normal. The model has the lowest value for the The model has the lowest value for the mean relative bias statistic and for the mean relative bias statistic and for the root mean squared bias statistic, proving root mean squared bias statistic, proving to perform very well;to perform very well;

the the CAViaRCAViaR method also performs very method also performs very well, we could say even better then the well, we could say even better then the orthogonal GARCH, except for the 95% orthogonal GARCH, except for the 95% confidence degree, where the model confidence degree, where the model slightly exceeds the expected number of slightly exceeds the expected number of exceptions. The values obtained by this exceptions. The values obtained by this method for the MRB and RMSRB are method for the MRB and RMSRB are comparable with the ones of the previous comparable with the ones of the previous model. Also at the degrees of variability model. Also at the degrees of variability they are comparable;they are comparable;

Despite our expectations seems that the Despite our expectations seems that the models which best performed were a models which best performed were a parametric model (O-GARCH) and a parametric model (O-GARCH) and a semiparametric model (CAViaR)semiparametric model (CAViaR)

Basle Committee rules for back testing with Basle Committee rules for back testing with the test confidence level to be 95%the test confidence level to be 95%

Nonrejection region for number of exceptions N VaR confidence level T=255days T=510days T=1000days

99% N<7 1<N<11 4<N<17 95% 6<N<21 16<N<36 37<N<65 90% 16<N<36 38<N<65 81<N<120

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