applying hybrid approach to calculating var in croatia

“APPLYING HYBRID APPROACH TO CALCULATING VAR IN CROATIA“

M.Sc. Saša Žiković Faculty of Economics University of Rijeka/Assistant

Ivana Filipovića 4, Rijeka, Croatia

Phone: +385 51 355 111 Fax: +385 51 212 268 E-mail: [email protected]

Key words: Value at Risk, historical simulation, hybrid approach, Croatia

1. INTRODUCTION

The most prominent of risks present in trading operations of a bank is the market risk, since it

reflects the potential economic loss caused by the decrease in the market value of a bank’s

portfolio of securities.

According to the 1996 Market Risk Amendment to the Basel Accord, besides using the

standardized approach, banks can set their capital requirements for market risk of their trading

positions based on the ten-day 1-percent VaR. The 1996 Amendment allows ten-day 1-

percent VaR to be measured as a multiple of one-day 1-percent VaR by using a simplistic

square root of time rule1. Although VaR is a conceptually simple measure of risk, computing

VaR in practice can be very difficult due to a simple reason that position risk of a bank's

portfolio depends on the joint distribution of all of the securities composing that particular

portfolio. Fortunately, portfolio level risk measurement requires only a univariate, portfolio-

level model2 thus drastically reducing the computational burden of multivariate models. If

interest centres on the distribution of the portfolio returns, then this distribution can be

modelled directly from portfolio returns rather than via aggregation based on a larger and

almost inevitably less-well-specified multivariate model.

In this paper the author examines the theoretical background of two nonparametric

approaches to calculating VaR, historical simulation and hybrid approach developed by

Boudoukh, Richardson and Whitelaw, and examines their performance in a transitional capital

market such as Republic of Croatia. The paper evaluates and analyses the out-of-sample

forecasting accuracy of both methods on two Croatian indexes, CROBEX – the official index of

Zagreb Stock Exchange and VIN - the official index of Varazdin Stock Exchange. In section 2 of

the paper, an overview of Value at Risk as a methodology for measuring market risk is

presented. Section 3, present the methodology of calculating VaR via historical simulation. In

section 4, a hybrid nonparametric approach to calculating VaR is presented and its advantages

and disadvantages are discussed. Section 5 analyses the characteristics of Croatian financial

market. In section 6, out-of-the-sample performance evaluation of historical simulation and

hybrid BRW approach, with different observation windows and decay factors is performed on

two Croatian stock indexes. The conclusions are summarized in Section 7.

1 Basel Committee on Banking Supervision (1996): ‘Amendment to the Capital Accord to incorporate Market

Risks. Bank for International settlements’, BIS, January 1996 2 Benson, P. and Zangari, P. (1997): ‘A general approach to calculating VaR without volatilities and

correlations’, RiskMetrics Monitor, (second quarter 1997): 19-23

2. MEASURING MARKET RISK USING VAR METHODOLOGY

Value at Risk (VaR) has become the standard measure that financial analysts use to quantify

market risk. It is defined as the maximum potential loss in value of a portfolio of financial

instruments with a given probability over a certain horizon. It indicates how much can a bank

expect to lose with probability C for a given time horizon3. The main advantage of VaR as a

risk measure over other risk measures is that it is theoretically simple. VaR can be used to

summarize the risk of individual positions, or a risk of large portfolio of an internationally

active bank. VaR reduces the risk associated with any portfolio to just one number, the

expected loss associated with a given probability over a defined holding period.

VaR for a given probability C can be expressed as:

VaRc = F-1(C)

where F–1(C) denotes the inverse of cumulative probability distribution of the changes in the

market value of a portfolio. Thus, losses greater than the estimated VaR should only occur

with the probability 1-C. For example, if a VaR calculated at the C% confidence level is

accurate, then losses greater than the VaR measure, so-called “tail events”, should on average

only occur C*N times in every N trading days.

While VaR is a very easy and intuitive concept, its measurement is a very challenging

statistical problem. Although the existing models for calculating VaR employ different

methodologies, some are entire parametric, while other are semi parametric or nonparametric

in their nature, they all follow a common general structure:

1) Marking to market of a portfolio of securities,

2) Estimation of distribution of portfolio returns (using either parametric or

nonparametric methods),

3) Calculating the VaR of the portfolio.

The main difference among numerous VaR methods is related to the estimation of distribution

that adequately describes the portfolio returns, i.e. the way the problem of how to estimate the

possible changes in the value of the portfolio is dealt with.

Research papers dealing with VaR calculation or volatility forecasting in the financial markets

of Croatia or EU new member states are very rare. Croatia as an EU member candidate state

offers large profits for investors, and represents a very interesting opportunity for foreign

investors, primarily banks. Most of the banking sector in Croatia, about 92%, is foreign

owned, and banks that operate in Croatia employ the same risk measurement models in

forming of provisions as they do in developed markets. This means that risk managers in

Croatian banks de facto presume similar or even equal market characteristics and behaviour in

Croatian market, as they would expect in developed markets. This is a dangerous assumption,

which is not realistic. Employing VaR models in forming of bank’s provisions that are not

suited to financial markets that they are used on, can have serious consequences for banks,

resulting in significant losses in trading portfolio that could pass undetected by the employed

risk measurement models, leaving the banks unprepared for such events.

3 Jorion, P. (2001): Value at Risk, The New Benchmark for Managing Financial Risk, Second Edition, New

York: McGraw Hill

Most of the widespread VaR methods used in the financial industry do not capture the

following characteristics of financial markets that are known since the pioneering work of

Mandelbrot (1963)4:

1. Equity returns are usually skewed to the left.

2. Distribution of financial return is leptokurtotic, i.e. it has fatter tails and a higher peak than

described under Gaussian distribution.

3. There is significant autocorrelation in financial time series, i.e. periods of high and low

volatility tend to cluster.

The last assumption is a very important characteristic of financial returns, since it means that

market volatility can be consider as being quasistable, subject to change in longer periods, but

stable in the short run. This gives some justification to the assumption that is the basis for

most of the VaR models (historical simulation among them), that recent past will be similar to

near future.

3. MEASURING VAR USING HISTORICAL SIMULATION

One of the most common methods for VaR estimation is the historical simulation (HS VaR).

This approach drastically simplifies the procedure for computing VaR, since it doesn’t make

any direct distributional assumption about portfolio returns.

Banks often rely on VaR figures calculated by historical simulations. The value of VaR is

calculated as the 100C’th percentile or the (T+1)C’th order statistic of the set of pseudo

portfolio returns.

In principle it is easy to construct a time series of historical portfolio returns using current

portfolio holdings and historical asset returns. The returns on the indexes in this paper are

calculated as:

1

ln)1ln(−

=+=t

ttt

P

PRr

The problem is that historical asset prices for the assets held at some point in time may not be

available. In such a case “pseudo” historical prices must be constructed using either pricing

models, factor models or some ad hoc consideration. The current assets without historical

prices can for example be matched to “similar” assets by capitalization, industry, leverage,

and duration. Historical pseudo asset prices and returns can then be constructed using the

historical prices on these substitute assets:

tT

N

i

tiTitw RWrwr '1

,,, ≡=∑=

, t = 1, 2,…, T

Historical simulation VaR can than be expressed as:

))1((|1 CTrVaRHS w

C

TT +≡− +

where ))1(( CTrw + is taken from the set of ordered pseudo returns { })(),...,2(),1( Trrr www . If

(T+1)C is not an integer value then the two adjacent observations can be interpolated to

4 Mandelbrot, B. (1963): 'The Variation of Certain Speculative Prices', The Journal of Business, Vol. 36, No. 4,

(October 1963): 394-419

calculate the VaR. Historical simulation has some serious problems, which have been well

documented5. Perhaps most importantly, it does not properly incorporate conditionality into

the VaR forecast. The only source of dynamics in the HS VaR is the fact that the observation

window is updated with the passing of time. This source of conditionality is minor in practice.

Historical Simulation is based on the concept of rolling windows. The process of calculating

VaR by historical simulation begins by choosing a length of the window of observations,

which usually ranges from two months to two years. Calculated portfolio returns within the

observation window are sorted and the desired quantile is given by the return xi that satisfies

the condition that j out of n observations do not exceed it. The probability that j out of n

observations do not exceed some fixed value of observed variable x follows a binomial

distribution:

jnj xFxFnjxnsobservatiojP −−

=≤ )}(1{)}({}_{

It follows that the probability of at least i observations in the selected sample do not exceed x

also follows a binomial distribution:

∑=

−−

=

n

ij

jnj

i xFxFnjxG )}(1{)}({)(

Gi(x) is the distribution function of the order statistic and thus also of the VaR.

To compute the VaR the following day, the whole window is moved forward by one

observation and the entire procedure is repeated.

Historical simulation method assigns equal probability weight of 1/N to each observation.

This means that the historical simulation estimate of VaR at the C confidence level

corresponds to the N(1-C) lowest return in the N period rolling sample. Because a crash is the

lowest return in the N period sample, the N(1-C) lowest return after the crash, turns out to be

the (N(1-C)-1) lowest return before the crash. If the N(1-C) and (N(1-C)-1) lowest returns

happen to be very close in magnitude, the crash actually has almost no impact on the

historical simulation estimate of VaR for the long positions in a portfolio of securities. From

the equation for historical simulation it can be seen that HS VaR changes significantly only if

the observations around the order statistic ))1(( CTrw + change significantly.

Although historical simulation makes no explicit assumptions about the distribution of

portfolio returns, an implicit assumption is hidden behind the procedure: the distribution of

portfolio returns doesn’t change within the window. From this implicit assumption several

problems may arise in using this method in practice. From the assumption that all the returns

within the observation window used in historical simulation have the same distribution, it

follows that all the returns of the time series also have the same distribution: if yt-window,...,yt

and yt+1-window,...,yt+1 are independently and identically distributed (IID), then also yt+1 and yt-

window has to be IID, by the transitive property. Another serious problem of the historical

simulation is the fact that for the empirical quantile estimator to be consistent, the size of

observation window must go to infinity. The length of the observation window hides another

serious problem. Forecasts of VaR under historical simulation are meaningful only if the

5 Pritsker, M. (2001): ‘The Hidden Dangers of Historical Simulation’, Board of Governors of the Federal

Reserve System, Economics Discussion Series, Working paper, No. 27, April 2001.

historical data used in the calculations have the same distribution. The length of the window

must satisfy two contradictory properties: it must be large enough, in order to make statistical

inference significant, and it must not be too large, to avoid the risk of taking observations

outside of the current volatility cluster. Clearly, there is no easy solution to this problem.6

If the market is moving from a period of low volatility to a period of high volatility, VaR

forecasts based on the historical simulation will under predict the true risk of a position since

it will take some time before the observations from the low volatility period leave the

observation window.

Finally, VaR forecasts based on historical simulation may present predictable jumps, due to

the discreteness of extreme returns. If VaR of a portfolio is computed using a rolling window

of N days and that today’s return is a large negative number, it is easy to predict that the VaR

estimate will jump upward, because of today’s observation. The same effect (reversed) will

reappear exactly after N days, when the large observation drops out of the observation

window.

4. EXPONENTIAL MODIFICATION OF HISTORICAL SIMULATION

When relaxing the assumption that returns are IID, it might be reasonable to assume that

simulated returns from the recent past better represent today portfolio's risk than returns from

the distant past. Boudoukh, Richardson, and Whitelaw, BRW hereafter, used this idea to

introduce a generalization of the historical simulation and assign a relatively higher amount of

probability weight to returns from the more recent past.7

The BRW approach combines RiskMetrics and historical simulation methodologies, by

applying exponentially declining weights to past returns of the portfolio. Each of the most

recent N returns of the portfolio, yt, yt-1, ..., yt-N+1, is associated a weight,

1

1

1,...,

1

1,

1

1 −

−−

−−

−− N

NNNλ

λλ

λλλ

λλ

respectively8. After the probability weights are assigned,

VaR is calculated based on the empirical cumulative distribution function of returns with the

modified probability weights. The basic historical simulation method can be considered as a

special case of the more general BRW method in which the decay factor - λ is set equal to 1.

The BRW method involves a simple modification of the historical simulation. However, the

modification makes a large difference. The most recent return in the BRW methods receives

probability weights of just over 1% for λ = 0,99 and of just over 3% for λ = 0,97. In both

cases, this means that if the most recent observation is the worst loss of the N days, then it will

be the VaR estimate at the 1% confidence level. Hence, the BRW methods appear to remedy

the main problems with the historical simulation methods because very large losses are

immediately reflected. The simplest way to implement BRW's approach is to construct a

history of N hypothetical returns that the portfolio would have earned if held for each of the

previous N days, rt-1,…, rt-N and then assign exponentially declining probability weights wt-

6 Manganelli, S. and Engle F.R. (2001): ‘Value at Risk models in Finance’, ECB working paper series, Working

paper, No. 75, August 2001 7 Boudoukh, J., Richardson, M. and Whitelaw, R. (1998): ‘The best of both worlds’, Risk, Vol. 11, No. 5, (May

1998): 64-67

8 The role of the term

Nλ

λ

−

−

1

1 is to ensure that the weights sum to 1.

1,…, wt-N to the return series9. Given the probability weights, VaR at the C percent confidence

level can be approximated from G(.; t;N), the empirical cumulative distribution function of r

based on return observations rt-1,…, rt-N.

∑=

−≤−=

N

i

itxr wNtxGit

1

}{1),;(

Because the empirical cumulative distribution function, unless smoothed, for example via

kernel smoothing as suggested by Butler and Schachter (1998)10, is discrete, the solution for

VaR at the C percent confidence level will typically not correspond to a particular return from

the return history. Instead, the BRW solution for VaR at the C percent confidence level can be

between a return that has a cumulative distribution that is less than C, and one which has a

cumulative distribution that is more than C. These returns can be used as estimates of the

BRW method's VaR estimates at confidence level C. The estimate that understates the BRW

estimate of VaR at the C percent confidence level (upper limit) is given by11:

)),;(|},...{inf(),,|( 11 CNtrGrrrCNtBRW Ntt

u ≥∈= −−−λ

and the estimator of lower limit is given by:

)),;(|},...{sup(),,|( 11 CNtrGrrrCNtBRW Ntt

o ≤∈= −−−λ

where λ is the exponential weight factor, N is the length of the history of returns used to

compute VaR, and C is the VaR confidence level.

),,|( CNtBRW u λ is the lowest return of the N observations whose empirical cumulative

probability is greater than C, and ),,|( CNtBRW o λ is the highest return whose empirical

cumulative probability is less than C.

The main issue in evaluation of BRW based VaR, as a risk measure, is the extent to which

VaR forecasts based on the BRW method respond to changes in the underlying risk factors. It

is important to know under what circumstances risk estimates increase when using the

),,|( CNtBRW u λ estimator. The result is provided in the following proposition:

If ),,( NtBRWr u

t λ> then ),,(),,1( NtBRWNtBRW uu λλ ≥+ .

When the VaR estimate using the BRW method is estimated for returns during time period

t+1, the return at time t−N is dropped from the sample, the return at time t receives weight

Nλλ

−−

1

1 and the weight on all other returns are λ times their earlier values.

Consequently, r(C) is defined as:

9 The weights sum to 1 and are exponentially declining at rate λ (0 < λ ≤ 1)

∑=

− =N

i

itw1

1

itit ww −−− = λ1 10 Butler, J.S. and Schachter, B. (1998): ‘Estimating Value-at-Risk with a Precision Measure by Combining

Kernel Estimation with Historical Simulation’, Review of Derivatives Research, No. 1, (1998): 371-390 11 Pritsker, M. (2001)

}),;(|,...1,{)( 11 CNtrGNirCr tt ≤== −−

To verify the proposition, it suffices to examine how much probability weight the VaR

estimate at time t+1 places below ),,( NtBRW u λ . There are two cases to consider:

Case 1: )(Crr Nt ∉− In this case, since by assumption, )(Crrt ∉ then

)),,((),,1);,,(( NtBRWGNtNtBRWG uu λλλλ <+ . Therefore,

),,()),,1;(|},...{inf(),,1( 1 NtBRWCNtrGrrrNtBRW u

Ntt

u λλλ ≥≥+∈=+ −−

Case 2: )(Crr Nt ∈− . In this case, since )(Crrt ∈ by assumption, then

)),,((),,1);,,(( NtBRWGNtNtBRWG oo λλλλ <+

Therefore,

),,()),,1;(|},...{sup(),,1( 1 NtBRWCNtrGrrrNtBRW o

Ntt

o λλλ ≤≤+∈=+ −−

The proposition shows that when losses at time t are bounded below the BRW VaR estimate

at time t, the BRW VaR estimate for time t+1 will indicate that risk at time t+1 is no greater

than it was at time t. To understand the importance of this proposition, it suffices to examine

the case when today's BRW VaR estimate for tomorrow's return is conditionally correct, but

that risk changes with returns, so that tomorrow's return will influence risk for the day after

tomorrow. Under these circumstances, an important question is what is the probability that a

VaR estimate that is correct today will increase tomorrow. The answer provided by the

proposition is that tomorrow's VaR estimate will not increase with probability 1−C. So, for

example, if C is equal to 1%, then a VaR estimate that is correct today will not increase

tomorrow with probability 99%.

Although the BRW approach suffers from the explained logical inconsistency, this approach

still represents a significant improvement over the historical simulation, since it drastically

simplifies the assumptions needed in the parametric models and it incorporates a more

flexible specification than the historical simulation approach. To better understand the

assumptions behind the BRW approach and its connection to historical simulation, BRW

quantile estimator can be expressed as12:

( )∑ ∑+−=

= −++ =≤=t

Ntj

N

i jitijCt CyyINfIyq1

1 1,1 )();(ˆ λ

where );( Nfi λ are the weights associated with return yi and I(●) is the indicator function. If

NNfi /1);( =λ BRW quantile estimator equals the historical simulation estimator. The main

difference between BRW approach and historical simulation is in the specification of the

quantile process. With historical simulation each return is given the same weight, while with

the BRW approach returns have different weights, depending on how old the observations are.

Strictly speaking, none of these models is nonparametric, since a parametric specification is

proposed for the quantile. Boudoukh, Richardson and Whitelaw in their original paper set λ

12 Manganelli, S. and Engle F.R. (2001)

equal to 0,97 and 0,99, as in their framework no statistical method is available to estimate this

unknown parameter.

5. ANALYSIS OF CROATIAN FINANCIAL MARKET

For a transitional economy, with a short history of market economy such as Croatia the main

problem for a statistically significant analysis is the short history of active trading in the

financial markets and their low liquidity. Because of the short time series of returns of

individual stock and their highly variable liquidity it is practical to analyse the stock indexes

of the two stock exchanges in Croatia. Although a small country, Croatia has two stock

exchanges, Zagreb Stock Exchange (ZSE) and Varazdin Stock Exchange (VSE). This

situation reduces further the liquidity of an already illiquid market, so choosing to analyse the

portfolio of securities that form the official stock indexes of Zagreb Stock Exchange

(CROBEX) and Varazdin Stock Exchange (VIN), and by definition present the most liquid

stocks on these stock exchanges presents itself as a logical choice.

The analysis of the two stock indexes is performed in period 04.01.2000. – 04.01.2006. In this

observation period the obtained sample of CROBEX index consists of 1469 observations, and

for VIN index, the sample consists of 1483 observations. The data is sourced from ZSE and

VSE. Since the period from the beginning of year 2000 onward was very turbulent for

Croatian financial market, the analysis of the selected indexes consists of three parts (entire

period, first and second half of the period). This procedure is implemented to detect any

structural changes in the characteristics of selected indexes, i.e. test if the assumption of

stationarity of time series can be applied, while at the same time no generality is lost due to

the statistically significant number of observations in analysed half-periods.

In case of CROBEX index, half-periods consist of 734 observations, and the first half-period

covers the period from 04.01.2000 to 19.12.2002, for VIN index half periods consist of 741

observations, and the first half period, covers the same period as CROBEX index.

The two analysed indexes show a strong positive trend. This was expected since Croatian

securities are currently trading at a discount compared to other surrounding market, especially

EU new member states. Due to the process of accession of Republic of Croatia to the

European Union, and the adjustment of legislation and business climate to European

standards, the growth of stock indexes is a natural consequence. Croatia is subject to an ever-

growing inflow of foreign direct and portfolio investments that is further boosting the

appreciation of Croatian securities. From figures 1 to 4 it is visible that there is significant

volatility clustering and presence of extreme positive and negative returns.

Looking at a measure of linear dependence between two variables i.e. correlation coefficient

which equals 0,9714 for the entire analysed period, suggest that the CROBEX index and VIN

index are strongly positively correlated. First half-period sample correlation coefficient

between CROBEX index and VIN index equals 0,8954. Second half period sample

correlation coefficient between CROBEX index and VIN index equals 0,9782. These results

indicate that in both half-periods there is significant linear dependence between these two

indexes, as could be expected because although there are composed of completely different

securities, they both represent the same market. The fact that there is a strong linear

dependence between these two indexes, that has strengthened even further in the last three

years, and the fact that the presence of two stock exchanges in such a small country reduces

the transparency and liquidity of the whole financial market gives a strong argument to those

that advocates the merger of two Croatian stock exchanges. In spite of the strong linear

dependence of the two indexes, their descriptive statistics, in particular higher moments

around the mean, tell a different story. Descriptive statistics for the CROBEX and VIN index

are presented in table 1.

Table 1 - Summary descriptive statistics for CROBEX and VIN index in the period 04.01.2000 - 04.01.2006.

Both CROBEX and VIN index have significant positive means and medians in the entire

analysed period as well in both half-periods. This clearly points to the conclusion that

securities composing these two indexes had a steady positive mean, resulting in considerable

capital gains for the investors. For both indexes the highest positive means are present in the

second half-period which covers the period from 20.12.2002 to 04.01.2006, and can be easily

explained by the positive influence of the accession negotiations between Croatia and EU.

Mean, median and mode of the stock indexes are not equal for individual indexes, which is

assumed under normality of the distribution. Standard deviation of both indexes is quite high

during the entire analysed period, and is equal to 13,59% for CROBEX index, and 12,80% for

VIN index. Both indexes were more volatile in the first analysed half-period, equalling

15,96% for CROBEX index, and 14,93% for VIN index. Based on the standard deviation of

the two indexes, it turns out that CROBEX index was the more volatile index, and thus

riskier. Combining this conclusion with the realized daily mean return, during the entire

period, for CROBEX index (0,067%) and for VIN index (0,128%), it turns out that VIN

index, in the entire observed period, was at the same time less volatile and more profitable,

especially in the second half period when its daily mean return reached 0,160%.

Third moment around the mean (skewness) for both indexes, and all the analysed periods, is

significantly different from the zero, which is assumed under normal distribution. In the entire

analysed period VIN index had negative skewness of –0,68183, while CROBEX index had

positive skewness of 0,2392. This fact is very important for the investors meaning that the

probability of positive returns occurring is far greater when investing in CROBEX index than

in VIN index. The interesting phenomenon is the significant positive skewness in the second

half period of the VIN index and negative skewness of CROBEX index. This shows that in

the last three years the skewness of the analysed indexes has completely changed, VIN index

has become positively skewed, while CROBEX index has become negatively skewed. The

fourth moment around the mean (kurtosis) for both indexes, and all the analysed periods is

different from the zero, as presumed under normality. Both indexes experienced extreme daily

returns in the observed period. The high values of kurtosis for these indexes, especially VIN

index, indicate to the investors investing in either stock exchange that they can expect

unusually high, both positive and negative returns on their investments.

Combining the third and fourth moment of the VIN index with the previously described mean

and standard deviation, one may conclude that, although in the entire observation period,

negative returns were more frequent than positive returns, the magnitude of the positive

Descriptive

statistics

CROBEX

index (whole

period)

CROBEX

index (first

half)

CROBEX

index

(second half)

VIN index

(whole

period)

VIN index

(first half)

VIN index

(second

half)

Mean 0,00067 0,00059 0,00074 0,00128 0,00096 0,00160

Median 0,00023 0,00046 0,00009 0,00053 0,00 0,00100

Mode 0,00 0,00 0,00 0,00 0,00 0,00

Minimum -0,09032 -0,08854 -0,09032 -0,15670 -0,15670 -0,06547

Maximum 0,11399 0,11399 0,07355 0,10186 0,10186 0,05837

St.Dev. 0,01359 0,01596 0,01071 0,01280 0,01493 0,01025

Skewness 0,23920 0,39708 -0,30973 -0,68183 -0,94255 0,35245

Kurtosis 9,86525 7,59346 11,71601 20,76386 20,83433 5,59279

returns was significantly higher than the magnitude of loses resulting in a strong positive

trend and continually growing index. In case of the CROBEX index, for the whole period, the

standard deviation was higher compared to the VIN index, and also the mean return was

lower, meaning that from the classical portfolio theory perspective CROBEX index can be

viewed as an inferior portfolio compared to VIN index. Skewness and kurtosis, during the

entire period, of CROBEX index indicate that from higher moments perspective was in fact

not as inferior to the VIN index as could be expected from looking at just the first two

moments of the distribution of the returns of these two indexes.

To determine if the daily returns of CROBEX and VIN indexes are normally distributed,

normality of distribution is tested in several ways. For CROBEX index, the tests and analysis

are performed on 1469 observations for the entire period and 734 observations for half-

periods. For VIN index, the tests and analysis are performed on 1483 observations for the

entire period and 741 observations for half-periods. The simplest test of normality is the

analysis of the third and fourth moment around the mean of the distribution. Third moment

around the mean, asymmetry, in the case of normal distribution should be zero. Negative

asymmetry means that the distribution is skewed to the left, which implies that there is a

greater chance of experiencing negative returns, and vice versa. Fourth moment around the

mean, kurtosis, in the case of normal distribution, when modified, should also equal zero.

Kurtosis higher than zero means that the distribution has fatter tails than normal distribution,

meaning that more extreme events occur more frequently in the sample data than can be

expected under normal distribution. More sophisticated tests of normality of distribution used

in this paper are Lilliefors test, Shapiro-Wilks test and Jarque-Bera test. Normality tests for

the CROBEX and VIN index are presented in table 2.

Table 2 – Tests of normality of distribution for CROBEX and VIN index returns in the period 04.01.2000 -

04.01.2006.

All normality tests show that the hypothesis of the normality of returns for CROBEX and

VIN indexes, over all the analysed periods, should be rejected at 5% significance level.

Probability values of distribution of returns being normal, for both indexes, under all of the

normality test are zero, strongly indicating that there is no possibility that the returns on these

indexes are normally distributed. The distribution of CROBEX and VIN index returns are

leptokurtotic and are not symmetrical i.e. they skew to the left and to the right, as can be seen

from figure 5.

Since the assumption of IID returns underlies the logic behind the historical simulation it is

necessary to test whether returns of the analysed time series are indeed IID. Returns on

financial assets themselves are usually not dependent (correlated), otherwise traders could

forecast daily returns. Returns squared are usually dependent; meaning that volatility of the

returns can be forecasted, but not the direction of the change of a variable.

Normality

tests

CROBEX

index (whole

period)

CROBEX

index (first

half)

CROBEX

index

(second half)

VIN index

(whole

period)

VIN index

(first half)

VIN index

(second

half)

Lilliefors 0,09406 0,10067 0,08178 0,10510 0,11095 0,08961(p value) 0,00 0,00 0,00 0,00 0,00 0,00

Shapiro Wilk 0,89222 0,90728 0,89983 0,86270 0,84574 0,92738

(p value) 0,00 0,00 0,00 0,00 0,00 0,00

Jarque-Bera 5.900,70 1.741,80 4.118,80 26.470,00 13.241,00 957,33(p value) 0,00 0,00 0,00 0,00 0,00 0,00

Critical value for Lilliefors test (whole period = 0,023124, half period = 0,032703)

Critical value for Jarque-Bera test = 5,9915

A widely accepted approach to detecting volatility clusters, which is actually autoregression

in the data, is the Ljung-Box Q-statistic calculated on the squared returns and Engle’s

Archtest. Ljung-Box Q-statistic is the lth autocorrelation of the T-squared returns, and

calculates whether the size of the movement at time t has any useful information to predict the

size of the movement at time t+l. Engle's Archtest for the presence of autoregressive

conditional heteroskedasticity (ARCH) effects tests the hypothesis that a time series of sample

residuals consists of IID Gaussian disturbances, i.e., that no ARCH effects exist. Ljung-Box

Q-statistic and Archtest for CROBEX and VIN index are presented in table 3.

The Ljung-Box Q-statistic and Engle’s Archtest confirm that there is significant

autocorrelation and ARCH effects present in both indexes i.e. that the volatilities tend to

cluster together (periods of low volatility are followed by further periods of low volatility and

vice versa), and for all the analysed periods, with the exception of CROBEX index in the

second-half period, meaning that the returns on CROBEX and VIN index are not IID. The

results are that much more indicative when considering that the hypothesis of IID was

rejected for all the tested time lags (5, 10, 15 and 20 days) and all of the indexes, with the

exception of CROBEX index in the second half-period. This is very indicative for risk

managers, because these tests prove that the elementary assumption of historical simulation is

not satisfied, and that the VaR figures obtained from it cannot be trusted and can at best

provide only unconditional coverage. Because of the existence of volatility clustering BRW

approach could provide a better alternative to historical simulation, due to the fact it assigns

more probability weight to recent events, thus providing an updating scheme that is more

responsive to changes in the market.

6. PERFORMANCE EVALUATION OF HISTORICAL SIMULATION AND BRW APPROACH ON CROATIAN STOCK INDEXES

The performance of the historical simulation and BRW approach is tested in two ways. First

test is the Kupiec test13, a simple expansion of the failure rate, which is also prescribed by

Basel Committee on Banking Supervision. The set-up for this test is the classic framework for

a sequence of successes and failures, also known as Bernoulli trials. The second test used is

the test of temporal unpredictability of extreme returns i.e. tail events. While extreme returns

do happen, a good VaR estimator should not allow them to cluster together. An efficient VaR

estimator should react in such a way that once a tail event occurs, it increases in such a way

that, given this new estimate, the next tail event has the same probability of occurring i.e. it is

temporally independent. The testing of temporal independence of tail events is done by

examining the sample autocorrelation function of tail events, which in the case of temporal

independence should not be significant.

For the period of 1.000 days realized daily returns of CROBEX and VIN index are compared

with the VaR forecasts based on historical simulation and BRW approach at 95%, 97,5% and

99% confidence level. Four historical simulation models with rolling windows of (50, 100,

250 and 400 days), and two BRW models with decay factors of 0,99 and 0,97, are shown in

figures 6 to 17. Backtesting results of Kupiec test are presented in table 5, and backtesting

results of test of temporal independence of tail events is presented in table 4.

Backtesting results based on Kupiec procedure, for both indexes indicate that historical

simulation based on shorter rolling windows (50 and 100 days) performs poorly at both 95%,

13 Jorion, P. (2001)

97,5% and at 99% confidence level. Historical simulation based on longer observation periods

(250 and 400 days) was accepted at 5% significance level, as being unconditionally correct

for both indexes at 95% and 97,5% confidence level but failed at 99% confidence level, with

the exception of historical simulation based on 400 day rolling window which provided

unconditionally correct coverage for VIN index at all the tested confidence levels. A fair

performance of historical simulation models based on longer observation periods can be

attributed to the fact that more extreme losses were present in the observation period, and that

is why the binomial test accepted these models as being unconditionally correct, but

unfortunately these models are also the slowest to react to the changes. Backtesting results for

historical simulation based on test of temporal independence of tail events show that there

exists temporal dependence in the tail events for all of the tested historical simulation models.

When comparing the results of test for temporal independence between the indexes, they are

mixed, and show that for CROBEX index historical simulation performed quite well while for

VIN index it performed poorly. This results show that historical simulation is not an efficient

VaR estimator and does not react adequately to the changes in the market and occurrence of

extreme events i.e. it does not adapt adequately and timely to the changes, and thus does not

reflect the true risk of a position.

Backtesting results for BRW approach based on Kupiec procedure, for both indexes show that

BRW approach performed quite good, with BRW model with decay factor set at 0,97

provided unconditional coverage for 95% and 97,5% confidence levels for both indexes but

failed at 5% significance level for 99% confidence level. BRW model with decay factor set at

0,99 provided unconditional coverage for both indexes, at all of the tested confidence levels.

Backtesting results for BRW approach based on test of temporal independence of tail events

show mixed results. The existence of temporal dependence in the tail events under BRW

approach for VIN index can be confirmed for both BRW models. In case of the CROBEX

index, the existence of temporal dependence in the tail events under BRW approach can be

rejected. This shows that when it comes to temporal independence, BRW approach should not

be taken for granted since it also showed that it to does not always react adequately to the

changes in the market and can also be misleading about the true level of risk.

On a sample of two Croatian stock indexes, CROBEX and VIN, BRW approach proved to be

a far better performer according to both the Kupiec test and test of temporal independence.

BRW model with decay factor set at 0,99, according to the Kupiec test provided

unconditional coverage for both indexes, at all of the tested confidence levels and as such was

the best performer. According to the test of temporal independence BRW models were also

better performers than any historical simulation models.

The results point to the conclusion that even though historical simulation with longer

observation periods provided correct unconditional coverage for 95% and 97,5% confidence

levels banks and other investors should be very careful when using it. Historical simulation

should not be used for high confidence level estimates (above 95%), especially models based

on shorter rolling windows. The obtained results show that although BRW approach also has

its flaws, especially when testing for temporal dependence in the tail events, it brings

significant improvement to historical simulation with minimal additional computational effort

and could prove to be an interesting alternative to historical simulation.

7. CONCLUSION

Banks operating in Croatia and other EU new member and member candidate states employ

the same risk measurement models in forming of provisions as they do in developed markets.

This means that risk managers de facto presume similar or even equal market characteristics

and behaviour in these developing markets, as they would expect in developed markets. This

is a dangerous assumption, which is not realistic. Employing VaR models in forming of

bank’s provisions that are not suited to financial markets that they are used on, can have

serious consequences for banks, resulting in significant losses in trading portfolio that could

pass undetected by the employed risk measurement models, leaving the banks unprepared for

such events. In this paper the author evaluates the performance of a widely spread VaR risk

measure – historical simulation against a hybrid model developed by Boudoukh, Richardson,

and Whitelaw, in transitional surroundings. The performance of the VaR models is evaluated

out-of-the-sample for two Croatian stock indexes based on a time series of 1.000 observations

by using Kupiec test and test of temporal independence of tail events.

The author finds that the basic assumption underlying the implementation of historical

simulation is violated in case of two tested indexes i.e. the assumption that analysed returns

are IID was rejected. Based on the performed tests it can be concluded that historical

simulation should not be used for high confidence level estimates (above 95%), especially

models based on shorter rolling windows. The obtained results show that although BRW

approach also has its flaws, especially when testing for temporal dependence in the tail

events, it brings significant improvement to historical simulation with minimal additional

computational effort. BRW approach should be further studied and tested in other transitional

and emerging economies, because based on these obtained results it prove to be a far better

alternative to historical simulation.

BIBLIOGRAPHY:

1. Basel Committee on Banking Supervision (1996): ‘Amendment to the Capital Accord to incorporate

Market Risks. Bank for International settlements’, BIS, January 1996

2. Benson, P. and Zangari, P. (1997): ‘A general approach to calculating VaR without volatilities and

correlations’, RiskMetrics Monitor, (second quarter 1997): 19-23

3. Boudoukh, J., Richardson, M. and Whitelaw, R. (1998): ‘The best of both worlds’, Risk, Vol. 11, No. 5,

(May 1998): 64-67

4. Butler, J.S. and Schachter, B. (1998): ‘Estimating Value-at-Risk with a Precision Measure by

Combining Kernel Estimation with Historical Simulation’, Review of Derivatives Research, No. 1,

(1998): 371-390

5. Jorion, P. (2001): Value at Risk, The New Benchmark for Managing Financial Risk, Second Edition,

New York: McGraw Hill

6. Mandelbrot, B. (1963): 'The Variation of Certain Speculative Prices', The Journal of Business, Vol. 36,

No. 4, (October 1963): 394-419

7. Manganelli, S. and Engle F.R. (2001): ‘Value at Risk models in Finance’, ECB working paper series,

Working paper, No. 75, August 2001

8. Pritsker, M. (2001): ‘The Hidden Dangers of Historical Simulation’, Board of Governors of the Federal

Reserve System, Economics Discussion Series, Working paper, No. 27, April 2001.

Table 3 - Test of independency for CROBEX and VIN index returns in the period 04.01.2000 - 04.01.2006.

Table 4 - Test of independency for CROBEX and VIN index tail events in the period 04.01.2000 - 04.01.2006.

CROBEX index

Ljung-Box-Pierce Q-test (CROBEX index - whole period) Ljung-Box-Pierce Q-test (CROBEX index - first half)

Period

(days)H p-value Statistic

Critical

value

Period


Critical

value

5 1 0 156,830 11,070 5 1 0 105,460 11,070

10 1 0 169,500 18,307 10 1 0 109,870 18,307

15 1 0 184,490 24,996 15 1 0 119,710 24,996

20 1 0 199,270 31,410 20 1 0 128,680 31,410

ARCH test (CROBEX index - whole period) ARCH test (CROBEX index - first half)

Period


Critical

value

Period


Critical

value

5 1 0 122,500 11,070 5 1 0 86,947 11,070

10 1 0 129,660 18,307 10 1 0 91,913 18,307

15 1 0 144,310 24,996 15 1 0 103,550 24,996

20 1 0 144,370 31,410 20 1 0 104,120 31,410

Ljung-Box-Pierce Q-test (CROBEX index - second half) Ljung-Box-Pierce Q-test (VIN index - whole period)

Period


Critical

value

Period


Critical

value

5 0 0,55914 3,933 11,070 5 1 0 140,200 11,070

10 0 0,68312 7,442 18,307 10 1 0 140,860 18,307

15 0 0,88390 8,876 24,996 15 1 0 150,860 24,996

20 0 0,97845 9,353 31,410 20 1 0 156,510 31,410

ARCH test (CROBEX index - second half) ARCH test (VIN index - whole period)

Period


Critical

value

Period


Critical

value

5 0 0,62209 3,509 11,070 5 1 0 147,560 11,070

10 0 0,77846 6,424 18,307 10 1 0 148,560 18,307

15 0 0,93332 7,752 24,996 15 1 0 157,560 24,996

20 0 0,98110 9,150 31,410 20 1 0 159,130 31,410

Ljung-Box-Pierce Q-test (VIN index - first half) Ljung-Box-Pierce Q-test (VIN index - second half)

Period


Critical

value

Period


Critical

value

5 1 0 74,249 11,070 5 1 0 104,120 11,070

10 1 0 74,516 18,307 10 1 0 118,710 18,307

15 1 0 78,853 24,996 15 1 0 140,520 24,996

20 1 0 80,807 31,410 20 1 0 163,490 31,410

ARCH test (VIN index - first half) ARCH test (VIN index - second half)

Period


Critical

value

Period


Critical

value

5 1 0 81,225 11,070 5 1 0 72,142 11,070

10 1 0 81,460 18,307 10 1 0 75,961 18,307

15 1 0 85,560 24,996 15 1 0 94,324 24,996

20 1 0 85,970 31,410 20 1 0 99,334 31,410

Ljung-Box-Pierce Q-test (Tail events for HS 50 95%) Ljung-Box-Pierce Q-test (Tail events for HS 250 95%)

Period


Critical

value

Period


Critical

value

5 0 0,35268 5,548 11,070 5 1 0,01962 13,435 11,070

10 0 0,39100 10,582 18,307 10 0 0,10062 15,965 18,307

15 0 0,20365 19,227 24,996 15 0 0,11886 21,595 24,996

20 0 0,43852 20,313 31,410 20 0 0,32669 22,258 31,410


Period


Critical

value

Period


Critical

value

5 1 0,03837 11,751 11,070 5 1 0,01108 14,837 11,070

10 0 0,09236 16,263 18,307 10 0 0,05531 17,979 18,307

15 1 0,04948 25,035 24,996 15 0 0,05376 24,725 24,996

20 0 0,14492 26,666 31,410 20 0 0,18561 25,426 31,410

VIN index

Table 5 – Backtesting results for 1.000 historical simulation and BRW VaR forecasts for CROBEX and VIN

index

CROBEX index

Ljung-Box-Pierce Q-test (Tail events for BRW λ=0,97 95%)

Period


Critical

value

5 0 0,68598 3,091 11,070

10 0 0,91650 4,596 18,307

15 0 0,93993 7,568 24,996

20 0 0,98885 8,402 31,410


Period


Critical

value

5 1 0,04230 11,501 11,070

10 0 0,16225 14,242 18,307

15 0 0,17775 19,850 24,996

20 0 0,42272 20,572 31,410


Period


Critical

value

Period


Critical

value

5 1 1,14E-05 30,571 11,070 5 1 0,00142 19,710 11,070

10 1 0,00024 33,354 18,307 10 1 0,02010 21,146 18,307

15 1 1,41E-05 49,582 24,996 15 1 0,00203 35,589 24,996

20 1 6,33E-07 66,661 31,410 20 1 1,50E-05 57,905 31,410


Period


Critical

value

Period


Critical

value

5 1 0,01018 15,0430 11,070 5 1 2,12E-06 34,254 11,070

10 0 0,08009 16,7500 18,307 10 1 0,00012 35,074 18,307

15 1 0,00346 33,9510 24,996 15 1 1,77E-05 48,978 24,996

20 1 1,26E-06 64,7860 31,410 20 1 4,60E-08 73,633 31,410


Period


Critical

value

5 0 0,16974 77,639 11,070

10 0 0,46721 96,999 18,307

15 1 0,019947 28,269 24,996

20 1 0,0009445 45,497 31,410


Period


Critical

value

5 1 0,009488 15,214 11,070

10 0 0,080374 16,738 18,307

15 1 0,0046158 33,053 24,996

20 1 1,45E-05 57,984 31,410

Number of failures 65 41 22 53 35 22 46 26 14

Probability value 0,01493 0,00101 0,00027 0,30017 0,02104 0,00027 0,68847 0,36964 0,08241

Frequency of failures 0,065 0,041 0,022 0,053 0,035 0,022 0,046 0,026 0,014

Confidence interval0,05052

0,0821

0,02958

0,05521

0,01384

0,03312

0,03995

0,06876

0,0245

0,04834

0,01384

0,03312

0,03387

0,06088

0,01705

0,03787

0,00767

0,02338

Accept model NO NO NO YES NO NO YES YES NO

HS 250

(99%)

HS 250

(97,5%)

HS 100

(95%)

HS 100

(97,5%)

HS 100

(99%)

HS 250

(95%)Model

HS 50

(95%)

HS 50

(97,5%)

HS 50

(99%)

VIN index

Figure 1 – Daily values of CROBEX index in the period 04.01.2000 - 04.01.2006. (1469 observations)

J a n 2 0 0 0 J a n 2 0 0 1 J a n 2 0 0 2 J a n 2 0 0 3 J a n 2 0 0 4 J a n 2 0 0 5 J a n 2 0 0 66 0 0

8 0 0

1 0 0 0

1 2 0 0

1 4 0 0

1 6 0 0

1 8 0 0

2 0 0 0

2 2 0 0

Figure 2 – Daily returns on CROBEX index in the period 04.01.2000 - 04.01.2006. (1468 observations)

J a n 2 0 0 0 J a n 2 0 0 1 J a n 2 0 0 2 J a n 2 0 0 3 J a n 2 0 0 4 J a n 2 0 0 5 J a n 2 0 0 6- 0 . 1

- 0 . 0 5

0

0 . 0 5

0 . 1

0 . 1 5

Return





0,05862

0,01544

0,0355

0,00767

0,02338

0,03995

0,06876

0,01787

0,03904

0,00767

0,02338

0,03301

0,05975

0,01787

0,03904

0,0055

0,0196

Accept model YES YES NO YES YES NO YES YES YES

BRW

λ=0,99

(99%)

ModelHS 400

(95%)

HS 400

(97,5%)

HS 400

(99%)

BRW

λ=0,97

(95%)

BRW

λ=0,97

(97,5%)

BRW

λ=0,97

(99%)

BRW

λ=0,99

(95%)

BRW

λ=0,99

(97,5%)


Probability value 0,11056 0,00514 5,57E-06 0,46247 0,09729 0,08241 0,46247 0,13381 0,08241



0,07434

0,02703

0,05179

0,017053

0,037865

0,03734

0,06539

0,02116

0,04372

0,00767

0,02338

0,03734

0,06539

0,02033

0,04255

0,00767

0,02338

Accept model YES NO NO YES NO NO YES YES NO

HS 250

(97,5%)

HS 250

(99%)

HS 100

(95%)

HS 100

(97,5%)

HS 100

(99%)

HS 250

(95%)Model

HS 50

(95%)

HS 50

(97,5%)

HS 50

(99%)





0,06427

0,01624

0,03669

0,0055

0,0196

0,03387

0,06088

0,01951

0,04139

0,00917

0,02585

0,03129

0,05749

0,01464

0,03431

0,00412

0,01702

Accept model YES YES YES YES YES NO YES YES YES

BRW

λ=0,99

(97,5%)

BRW

λ=0,99

(99%)

ModelHS 400

(95%)

HS 400

(97,5%)

HS 400

(99%)

BRW

λ=0,97

(95%)

BRW

λ=0,97

(97,5%)

BRW

λ=0,97

(99%)

BRW

λ=0,99

(95%)

Figure 3 – Daily values of VIN index in the period 04.01.2000 - 04.01.2006. (1483 observations)

J a n 2 0 0 0 J a n 2 0 0 1 J a n 2 0 0 2 J a n 2 0 0 3 J a n 2 0 0 4 J a n 2 0 0 5 J a n 2 0 0 62 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

1 4 0 0

1 6 0 0

1 8 0 0

2 0 0 0

2 2 0 0

Figure 4 – Daily returns on VIN index in the period 04.01.2000 - 04.01.2006. (1482 observations)

J a n 2 0 0 0 J a n 2 0 0 1 J a n 2 0 0 2 J a n 2 0 0 3 J a n 2 0 0 4 J a n 2 0 0 5 J a n 2 0 0 6- 0 . 2

- 0 . 1 5

- 0 . 1

- 0 . 0 5

0

0 . 0 5

0 . 1

0 . 1 5

Return

Figure 5 - Probability plot for CROBEX and VIN index, period 04.01.2000 - 04.01.2006

CROBEX index VIN index

Figure 6 – Historical simulation VaR for 1.000 observations at confidence level of 1, 2.5 and 5 percent with 50

days observation window for CROBEX index

J a n 2 0 0 2 J a n 2 0 0 3 J a n 2 0 0 4 J a n 2 0 0 5 J a n 2 0 0 6- 0 . 1

- 0 . 0 5

0

0 . 0 5

0 . 1

0 . 1 5

Return

C R O B E X

H S 5 0 9 5 %

H S 5 0 9 7 , 5 %

H S 5 0 9 9 %

-0.05 0 0.05 0.1

0.01 0.05 0.1 0.25 0.5 0.75 0.9 0.95 0.99

Return

Probability

-0.15 -0.1 -0.05 0 0.05 0.1

0.01 0.05 0.1 0.25 0.5 0.75 0.9 0.95 0.99

Return

Probability




- 0 . 0 5

0

0 . 0 5

0 . 1

0 . 1 5

Return

C R O B E X

H S 1 0 0 9 5 %

H S 1 0 0 9 7 , 5 %

H S 1 0 0 9 9 %




- 0 . 0 5

0

0 . 0 5

0 . 1

0 . 1 5

Return

C R O B E X

H S 2 5 0 9 5 %

H S 2 5 0 9 7 , 5 %

H S 2 5 0 9 9 %




- 0 . 0 5

0

0 . 0 5

0 . 1

0 . 1 5

Return

C R O B E X

H S 4 0 0 9 5 %

H S 4 0 0 9 7 , 5 %

H S 4 0 0 9 9 %

Figure 10 – BRW VaR (λ=0,97) for 1.000 observations at confidence level of 1, 2.5 and 5 percent for CROBEX

index


- 0 . 0 5

0

0 . 0 5

0 . 1

0 . 1 5

Return

C R O B E X

B R W H S 9 5 %

B R W H S 9 7 , 5 %

B R W H S 9 9 %

Figure 11 – BRW VaR (λ=0,99) for 1.000 observations at confidence level of 1, 2.5 and 5 percent for CROBEX

index


- 0 . 0 5

0

0 . 0 5

0 . 1

0 . 1 5

Return

C R O B E X

B R W H S 9 5 %

B R W H S 9 7 , 5 %

B R W H S 9 9 %


days observation window for VIN index

J a n 2 0 0 2 J a n 2 0 0 3 J a n 2 0 0 4 J a n 2 0 0 5 J a n 2 0 0 6- 0 . 0 8

- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

0

0 . 0 2

0 . 0 4

0 . 0 6

Return

V I N

H S 5 0 9 5 %

H S 5 0 9 7 , 5 %

H S 5 0 9 9 %

Figure 13 – Historical simulation VaR for 1.000 observations at confidence level of 1, 2.5 and 5 percent with

100 days observation window for VIN index


- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

0

0 . 0 2

0 . 0 4

0 . 0 6

Return

V I N

H S 1 0 0 9 5 %

H S 1 0 0 9 7 , 5 %

H S 1 0 0 9 9 %




- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

0

0 . 0 2

0 . 0 4

0 . 0 6

Return

V I N

H S 2 5 0 9 5 %

H S 2 5 0 9 7 , 5 %

H S 2 5 0 9 9 %




- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

0

0 . 0 2

0 . 0 4

0 . 0 6

Return

V I N

H S 4 0 0 9 5 %

H S 4 0 0 9 7 , 5 %

H S 4 0 0 9 9 %

Figure 16 – BRW VaR (λ=0,97) for 1.000 observations at confidence level of 1, 2.5 and 5 percent for VIN index


- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

0

0 . 0 2

0 . 0 4

0 . 0 6

Return

V IN

B R W H S 9 5 %

B R W H S 9 7 , 5 %

B R W H S 9 9 %

Figure 17 – BRW VaR (λ=0,99) for 1.000 observations at confidence level of 1, 2.5 and 5 percent for VIN index


- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

0

0 . 0 2

0 . 0 4

0 . 0 6

Return

V I N

B R W H S 9 5 %

B R W H S 9 7 , 5 %

B R W H S 9 9 %

applying hybrid approach to calculating var in croatia

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