Hypothetical Scenario Analysis in Modeling of Market Risk

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Hypothetical Scenario Analysis in

Modeling of Market Risk

Emma Berglund

emma.berglund87@gmail.com Supervisor

Carl Magnus Lundin calu07@handelsbanken.se Handelsbanken Examiner Lars-‐Daniel Öhman lars-‐daniel.ohman@math.umu.se

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Sammanfattning

Det senaste decenniets kriser har skapat ett behov av att hantera risker på ett mer omfattade sätt. Nya regelverk och riktlinjer syftar till att ge banker och andra finansiella institut varningsignaler vid ett tidigt skede i finansiell turbulens för att undvika stora förluster. Ett sätt är att använda sig av stressat Value at Risk, 𝑉𝑎𝑅, då marknadens beteende tydligt skiljer sig åt i finansiell stress.

En vanlig metod skatta 𝑉𝑎𝑅 är att använda sig av empiriska data från finansiellt stressade perioder. Nackdelen med detta är att det ger ett bakåtblickande mått som endast tar hänsyn till kriser som redan uppstått. Användningen av riskmodellerna förutsätter att historiska data är ett bra sätt att mäta risk och att det skulle följa en känd statistisk process. Detta har dock visat sig vara en överskattad metod och därmed har ett behov att använda sig av hypotetiska scenarion uppstått. I detta arbete undersöks vad som händer i korrelationerna mellan en portföljs tillgångar i en stressad tid i jämförelse med en lugn. Sedan undersöks om dessa korrelationer kan användas för att stressa 𝑉𝑎𝑅-‐siffran och varna för eventuella förluster.

Modellerna som används är FHS Unconditional Student’s t-‐model och MV-‐GARCH (1,1). Den förstnämna går ut på att generera slumptal från en t-‐fördelning och korrelera dessa enligt korrelationer som beräknats med hjälp av det historiska datamaterialet. Det historiska datamaterialet skalas sedan med variansen i varje tidssteg och ytterligare korrelationer beräknas. Detta är en del i filtered historical simulation, FHS. Den multivariata garch-‐modellen använder sig av observationen och variansen i föregående tidssteg, samt tillgångarnas standardiserade residualers korrelation vid en beräkning av nästa tidsstegs avkastning. De simulerade tidsserierna från de ovannämnda modellerna används för att beräkna 𝑉𝑎𝑅 -‐siffror som syftar till att efterlikna motsvarande siffror med användning av det historiska datamaterialet.

Vid skapandet av hypotetiska scenarier modifieras de beräknade korrelationerna. Det som används för att modifiera korrelationerna är de relativa förändringarna i korrelationerna mellan en lugn period och efterföljande kris. Dessa multipliceras sedan med korrelationerna under hela tidsperioden. Även korrelationerna i kriserna används samt en konstant relativ förändring i korrelationen utan empirisk förankring. För att sedan utföra tester om korrelationen kan användas för att stressa 𝑉𝑎𝑅-‐siffran appliceras de modifierade korrelationerna i modellerna och simuleringar genererar nya 𝑉𝑎𝑅-‐siffror. Dessa jämförs sedan med de ordinära 𝑉𝑎𝑅-‐siffrorna från simuleringarna och det historiska datamaterialet.

Den modifiering som mest effektivt stressar 𝑉𝑎𝑅 är när korrelationen vid en turbulent tid från mitten av 2002 till mitten av 2003 används. Resultatet visar dock en för liten förändring av 𝑉𝑎𝑅-‐siffran för att helt förlita sig på korrelation som parameter att stressa 𝑉𝑎𝑅-‐siffran.

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Abstract

The last decade's crises have created a need to manage risk in a more comprehensive way. New guidelines and regulations aim to give banks and other financial institutions warning signals at an early stage of a financial turmoil to avoid big losses. One way is to use Stressed Value at Risk, VaR, as the market behavior clearly differs in financial stress.

A common method to estimate 𝑉𝑎𝑅 is to use empirical data from financial stressed periods. The downside of this is that it gives a retrospective measure that only takes already occurred crises into account. The use of risk management models assumes that historical data is a good basis for risk measurement and that the history will follow a known statistical process. This has been proven to be an overrated method and a need to use the hypothetical scenarios has emerged.

This work examines what happens in the correlations between a portfolio's assets in a stressful time in comparison to the correlations in a calm period of time. It also examines whether these correlations can be used to stress the ordinary 𝑉𝑎𝑅 to alert for potential losses.

The models used are FHS Unconditional Student's t-‐model and MV-‐GARCH (1,1). The first mentioned model generates random numbers from a t-‐distribution and correlates them according to the historical data set. The data set is scaled with the variance of each time step before the correlation is calculated. This is called filtered historical simulation, FHS. The multivariate GARCH model uses the observation and the variance in the previous time step, and also the asset correlation is used in the calculation of the return in the next time step.

The correlations are modified in order to create the hypothetical scenarios. The modifications that are used are the relative changes in the correlations between a calm and turbulent time period. These are then multiplied by the correlations for the entire period. Also the correlations in the crises are used and applied on the whole time period as well as a constant relative change in the correlation without empirical support. To test if the correlation can be used to stress the 𝑉𝑎𝑅 the modified correlations is applied in the models and simulations generate new 𝑉𝑎𝑅 numbers. These are compared with both the ordinary 𝑉𝑎𝑅 from the simulation and the historical data set.

The modification that most effective stresses the 𝑉𝑎𝑅 is when the correlation at a turbulent time from mid-‐2002 to mid-‐2003 are used. The results show, however, that the small change of 𝑉𝑎𝑅 is too small to completely rely on the correlation parameter to stress the 𝑉𝑎𝑅.

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1 Introduction

The demands of banking authorities resulting from the financial crisis over the last decade have created a need to manage risk in a more distinct and comprehensive way. New guidelines and regulations have recently been developed to meet the needs in today’s risk management. These regulations aim to alert banks and other financial institutions for possible losses at an early stage in order to reduce them.

The new regulations include the requirement to produce better methods to test stressed measurements such as stressed Value at Risk, 𝑉𝑎𝑅. (EBA European Banking Authority, 2012). A well-‐ used method to manage stressed 𝑉𝑎𝑅 is to use historical data from financial stressed times. However, there are drawbacks to using historical data. One of the problems is that historical data gives a retrospective measure and only considers events that have already occurred. This means that most risk management models have relied on the assumptions that historical data is a good basis for risk measurement and that it will follow a known statistical process. This has however been proven to be overestimated in times of financial turmoil. Assumptions about the market in stable conditions for a period of time will indicate good conditions in the near future and will not foresee vulnerability or possible shocks. Correlation and other relationships have also been proven unreliable in times of turmoil. In fact, history has shown that the market’s behavior in financially stressed times is clearly different from its behavior in non-‐stressed times. The assumptions, that the history would follow a known statistical process, led to an underestimation of the market developments and extreme events at the end of the last decade resulting in losses far higher than expected, (Basel Committee of Banking Supervision., 2009).

New stress testing techniques have been developed since the crisis. One basic method is to change one parameter in the model while keeping the others constant to test the sensitivity. A way to implement this is to make use of hypothetical analysis of events that are unlikely to occur but plausible. Hypothetical analysis means that calculations are made on data that do not contain the true history, but rather a modified variant.

The purpose of this thesis is to provide a complement to ordinary 𝑉𝑎𝑅-‐calculations and stress tests by analyzing hypothetical scenarios. The hypothetical data is inspired by earlier stressed events in the historical data. The correlation is studied to determine if it can be used to generate hypothetical scenarios with fluctuations similar to the historical fluctuations of a portfolio’s time series.

The correlation between the assets in a fixed portfolio will be studied to demonstrate how different categories of assets are related to each other. This analysis is also performed in order to observe if differences in the correlation occur between financially stressed and calm periods of time, different stressed periods and different calm periods. Market circumstances changes over time which creates a problem with using data that span over a long historical period. One of the reasons is fluctuations in volatility. Therefore, it is necessary to find a way to filter the historical events by the volatility and study what happens with the correlations and compare those with the results before the filtering. The models FHS Unconditional Student’s t-‐model and MV-‐GARCH (1,1) are used to give a simulation of the history and enable changes in the correlations. The methodological questions are if the correlations during a crisis or if the relative changes in correlations between a calm period and a crisis can be used to stress the ordinary 𝑉𝑎𝑅. It is also of interest to study if there is some general

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way to modify the correlations and stress the 𝑉𝑎𝑅 without knowing what the correlation looked like or how it behaved earlier in time. To test this, the correlation changes from a calm period to a stressed are multiplied with the ordinary correlation, and the new modified simulated 𝑉𝑎𝑅-‐curve can be obtained and analyzed. To test if a correlation during a crisis can be used, this is simply applied on the whole time period.

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2 Theoretical background

This section describes a rough theory of the models and other mathematical concepts. A description of the risk measurement used is found in the first section, followed by a derivation of how to make use of the Monte Carlo simulation in making estimates. Finally, the models used in the simulations are described.

2.1 Value at Risk

Value at Risk, 𝑉𝑎𝑅 is a common risk measure and is widely used in the market risk departments of financial institutions. It is a simple and intuitive way of measuring a position’s risk exposure. For the institution, 𝑉𝑎𝑅 represents a position’s maximum loss during a certain time period for a given probability 𝛼. An illustration of the 𝑉𝑎𝑅 for a curve from a normal distribution is shown in Figure 1.

Figure 1. Shows the 𝑽𝒂𝑹 at the significant level 𝟏 − 𝜶. The yellow area has the value 𝜶. The formula for calculating the 𝑉𝑎𝑅 is the following:

𝑉𝑎𝑅_!!! 𝑋 = sup 𝑥 ∈ 𝑅 𝑃 𝑋 ≤ 𝑥 ≤ 𝛼 (1)

where 𝑋 is a random variable and 𝛼 is the probability.

2.2 Monte Carlo Simulation

Simulations utilizing Monte Carlo methods are a commonly used technique of estimating mathematical systems. When it is difficult to analytically calculate the value of a parameter this method can be used. The method relies on generating random numbers from a certain probability distribution.

To illustrate the method, assume one wants to calculate the expected value of the function 𝑓 𝑥 with a given probability distribution function 𝜓 𝑥 where 𝑥 ∈ ℝ. The expected value is given by

𝜇 = 𝐸! ! 𝑓 𝑥 = ∫ 𝑓 𝑥 𝜓 𝑥 𝑑𝑥 (2)

This can be simulated by drawing values 𝑥 from the given distribution 𝜓 𝑥 and using 𝑥!, where 𝑖 = 1,2 … 𝑁, to calculate 𝑓! = 𝑓 𝑥! . This gives the Monte Carlo estimator

𝜇 ≔ !

! 𝑓 𝑥!

!

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2.3 Unconditional Student’s t-‐distribution

It has been suggested that many of the financial time series can be accurately modeled by random numbers from a student’s t-‐distribution and such simulations has been proven successful in earlier 𝑉𝑎𝑅 estimations, (Tsay, R., 2002), (Alexander, C., Sheedy, E., 2008). The simulation makes use of the mean adjusted returns, i.e. the mean of the time series is calculated and subtracted from each value in the time series. The mean-‐adjusted returns 𝑎! have the following distribution

𝑎!~𝑇 𝜈 𝜎 _!!!! !!/!

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where 𝑇 𝜈 is the standardized Student’s t distribution with 𝜈 degrees of freedom. The mean is zero and 𝜎 is the standard deviation of the time series. In the one dimensional case, the 𝑉𝑎𝑅 is calculated as

𝑉𝑎𝑅_!!! = 𝑇_! 𝛼 !! !!! !

!/!

𝜎. (5)

2.4 Filtered Historical Simulation

A problem with simulations based on historical events is that the volatility seems to change over time. To get observations that are drawn from the standardized empirical return distribution the returns 𝑟 are filtered in the following way

𝑎! =_!!!

!,!!" (6)

where 𝜎!,!"# is estimated with a GARCH(1,1) which is described in the next section, see section 2.5.

2.5 GARCH (1,1)

The autoregressive conditional heteroskedastic (ARCH) model takes returns in previous time step into account in predictions. One drawback to this method is that it ignores changes in variance which could be advantageous to take into account. The Generalized Autoregressive Conditional Heteroskedasticity model (GARCH) extends the simpler ARCH-‐model by taking the conditional variance into consideration.

Although the distribution of the observations in a time series is unknown, GARCH gives a reliable simulation of financial time series. This due to the capturing of the conditional variance described above.

Consider a process

𝑋!= 𝐶 + 𝑎! (7)

where 𝑎! is the mean-‐adjusted log return such that 𝑎! = 𝑟!− 𝜇! and 𝐶 is the mean of the historical observations. The GARCH (1,1) is described as follows

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Maximum likelihood is used to estimate the parameters. The approach is to maximize the logarithm of 𝐿 𝜃 which is defined as

𝐿 𝜃 = ! 𝑓 𝜃

!!! (9)

where 𝑓 𝜃 is the density function, in this case the normal density function

𝑓 𝜃 = !

!!!_!!𝑒

!!!!

!!!!_. ₍₁₀₎

The logarithm of both sides gives

log 𝐿 𝜃 = ! log 𝑓 𝜃

!!! (11)

and the log likelihood function log 𝐿 𝜃 that will be maximized for the normal probability function is rewritten as 𝐿 𝜃 = −!_!log 2𝜋 −!_! !!!!log 𝜎!!−!_! !! ! !!! ! !!! (12)

where 𝜃 denotes all the unknown parameters in 𝜎!! and 𝑎!.

2.6 Multivariate GARCH (1,1)

The constant correlation multivariate GARCH was proposed by Tim Bollerslev (1990). It provides the possibility of simulating a mixture of time series with respect to correlations.

Let 𝑎! be the returns with mean zero and

𝑎!|ℱ!!!~𝑁 0, 𝐻! (13)

where ℱ!!! is all the available information up through time 𝑡 − 1 and

𝐻! = 𝐷!𝑅!𝐷! (14)

and 𝑅! is the correlation matrix which, in the constant correlation case, is 𝑅! = 𝑅. Note that Equation 14 requires that the correlation matrix is positive definite. 𝐷! in this equation is the diagonal of standard deviations ℎ!" for the 𝑖!!_{time
series
composed
in
a
univariate
GARCH(1,1)
as
follows}

ℎ!" = 𝛼!!+ 𝛼!!𝑟!"!!! + 𝛽!!ℎ!"!! (15) The coefficients 𝛼!!, 𝛼!! and 𝛽!!for the respective time series have the same restrictions as the corresponding coefficients in Equation 8 and are estimated by maximum likelihood. The log-‐ likelihood in the multivariate case is defined as

𝐿 𝜃 = −!_! !!!! 𝑘log 2𝜋 + 2log|𝐷! + log 𝑅! + 𝜖!!𝑅!!!𝜖!) (16) where 𝜃 denotes all the unknown parameters in 𝐷! and 𝜖! and 𝜖!~𝑁 0, 𝑅! is the standardized residuals in the univariate GARCH model.

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2.7 Cholesky’s decomposition

In a desire to create a linear relationship between a set of independent variables, Cholesky’s decomposition can be used. Let 𝜌 be a positive and definite 𝑛×𝑛 matrix. If an upper triangular matrix 𝑈 exists such that 𝜌 = 𝑈!_{𝑈
then
the
matrix
𝑈
is
called
the
Cholesky
factor
of 𝜌.}

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3 Methodology

This section describes the portfolio that is used in the thesis. That is a distribution and an autocorrelation analysis of the different time series in the portfolio and also how the time period is divided into financially calm and turbulent subintervals. The models used in the simulations are FHS Unconditional t-‐model and MV-‐GARCH(1,1). The models together with an evaluation of them are presented in this section. The end of the section describes how the hypothetical data is created and how the stressed 𝑉𝑎𝑅 is defined.

3.1 Portfolio

The portfolio used in this study contains of a single share in each of the indices OMXS30, FTSE100, S&P500, and one American dollar expressed in SEK and also a British pound expressed in SEK. The time series used to know the change of those will thus be the foreign exchange rates USDSEK and GBPSEK.

The asset OMXS30 is a market value-‐weighted index for the Stockholm Stock Exchange and consists of the 30 most traded stocks. FTSE100 is a share index which consists of the 100 companies with the highest capitalization of the London Stock Exchange Group. S&P500 is a stock market index with the 500 top publicly American traded companies. These three indices, as well as the foreign exchange rates corresponding to the same countries as the indices, are chosen because banks commonly invest in them.

The time series consist of the daily relative changes from 1994-‐01-‐03 to 2011-‐07-‐11 where OMXS30, USDSEK and GBPSEK are calculated from the prices in SEK. However, the relative changes in the time series S&P500 and FTSE100 are expressed in relative changes calculated with the countries’ currencies respectively, see their prices over time in Figure 3. To obtain the series corresponding to a relative change calculated with SEK the exchange rate USDSEK has to be used. If this conversion is used, the correlation between S&P500 and USDSEK will be strengthened in a way that does not only include the market conditions. Therefore, the fact that the international indices’ relative changes depend on another currency than SEK will be ignored and treated as if they were calculated from SEK.

As the 𝑉𝑎𝑅 of interest is on a one day horizon, the change in the portfolio can be consider as the relative change from the day before if 1 SEK had been invested in each asset. To illustrate how the value of the different assets has changed over time, the price for every asset is observed from Bloomberg from 2011-‐07-‐03 and the daily prices are calculated backwards with the daily relative changes

𝑃!!!= _!!!!!

! (17)

where 𝑃! is the price on day 𝑡 and 𝑝! is the daily relative change from day 𝑡 − 1 to 𝑡. The daily prices from 1994-‐01-‐03 to 2011-‐07-‐03 are obtained for currencies and shown in Figure 2. Note that the indices prices in Figure 3 are expressed in different currencies.

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Figure 2. The daily prices for USD and GBP from 1994-‐01-‐03 to 2011-‐07-‐11.

Figure 3. The daily prices for OMXS30, FTSE100 and S&P500 from 1994-‐01-‐03 to 2011-‐07-‐11.

The daily relative changes, also denoted as returns, of the currencies and indices over time are shown in Figure 4 and 5.

1995 2000 2005 2010 4 6 8 10 12 14 16 Date Pr ic e i n SEK USD GBP 1995 2000 2005 2010 0 1000 2000 3000 4000 5000 6000 7000 Date Pr ic e i n t h e i n d ic e s c o u n tr ie s ' c u rr e n c ie s r e s p e c ti v e ly (S E K , U S D , G B P ) OMXS30 S&P500 FTSE100

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Figure 4. The daily relative changes of the currencies from 1994-‐01-‐03 to 2011-‐07-‐11.

Figure 5. Daily relative changes of the indices from 1994-‐01-‐03 to 2011-‐07-‐11.

The daily relative changes of the assets are added together for each day and represent the portfolio, see Figure 6. 1995 2000 2005 2010 -0.05 0 0.05 Date Re tu rn s USD 1995 2000 2005 2010 -0.05 0 0.05 Date Re tu rn s GBP 1995 2000 2005 2010 -0.1 0 0.1 Date Re tu rn s OMXS30 1995 2000 2005 2010 -0.1 0 0.1 Date Re tu rn s S&P500 1995 2000 2005 2010 -0.05 0 0.05 0.1 Date Re tu rn s FTSE100

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Figure 6. The daily relative changes of each asset added together for each day represents the portfolio. 3.1.1 Distribution analysis

A simulation of the portfolio requires understanding of its behavior as well as the behavior for all assets respectively. In this section an analysis of the distribution is made for the different assets and the portfolio with histogram fitted with normal distribution and t-‐distribution. An estimation of the distributions parameters is done and the fitting are obtained by the density functions. There is also a quantile-‐quantile plot, qq-‐plot, for the different time series quantiles versus the quantiles of a normal distribution and of a t-‐distribution. As mentioned before, the standardized Student’s t-‐ distribution is commonly used in modeling financial time series. Therefore, a limit is set to only include tests for this distribution and for normal distribution.

To get a quantitative comparison between the two tested distributions a chi-‐square goodness-‐of-‐fit test is done. The chi-‐square statistic

𝜒!₌ !!!!!!

!!

!

!!! (18)

is computed where the data is grouped into bins, 𝐸! is the expected value of the counts and 𝑂! are the observed counts from the time series. The number of bins, 𝑁, is set to ten and the test statistic is compared with the chi-‐square distribution. The number of the degrees of freedom is set to 𝑁 − 3. The parameters needed for the respective distributions are estimated by maximum likelihood. The null hypothesis in the test is that the data comes from the tested distribution. The results of the test are presented in the end of this section.

1995 2000 2005 2010 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Date Da ily p e rc e n ta g e c h a n g e s

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Figure 7. Histogram of the assets and the portfolio’s distributions vs. the fitted normal distribution (red line).

Figure 8. The assets and the portfolio observations’ quantiles (vertical axis) vs. the normal distribution (horizontal axis).

As can be seen in Figure 7, all six time series distributions have more extended peaks than the fitted line from the normal distributions. This is an indication that the distributions probably have heavier tails. It is also clear in the qq-‐plots in Figure 8 that the ends of the curves are shifted from the red line

-0.20 0 0.2 200 400 600 OMXS30 -0.20 0 0.2 200 400 600 S&P500 -0.20 0 0.2 200 400 600 FTSE100 -0.20 0 0.2 200 400 600 USD -0.20 0 0.2 200 400 600 GBP -0.20 0 0.2 200 400 600 Portfolio -5 0 5 -0.1 0 0.1 0.2 0.3 OMXS30 -5 0 5 -0.2 -0.1 0 0.1 0.2 S&P500 -5 0 5 -0.1 -0.05 0 0.05 0.1 FTSE100 -5 0 5 -0.05 0 0.05 USD -5 0 5 -0.05 0 0.05 GBP -5 0 5 -0.4 -0.2 0 0.2 0.4 Portfolio

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corresponding to the normal distribution. If the top of the curve is shifted to the left it is an indication that the distribution of the time series has heavier tails than the reference distribution. To obtain the student’s t-‐distribution parameter, degrees of freedom, the maximum likelihood approach described in section 2.5 is used. However, the student’s t density function is used and defined as follows

𝑓 𝜃 = ! !!! ! ! !"! !_! !! !!!_! ! ! ! !!!_! (19)

where 𝑥 are the observations in the time series, 𝜇 is the mean, 𝜎 is the standard deviation and 𝜈 is the degrees of freedom. 𝜃 represents the unknown parameters 𝜈 and 𝜇, and 𝜎 is given by

𝜎 = !

!!!, for 𝜈 > 2 (20)

Figure 9. Histogram of the assets and portfolios distributions vs. student’s t distribution (red line).

-0.20 0 0.2 200 400 600 OMXS30 -0.20 0 0.2 200 400 600 S&P500 -0.20 0 0.2 200 400 600 FTSE100 -0.20 0 0.2 200 400 600 USD -0.20 0 0.2 200 400 600 GBP -0.20 0 0.2 200 400 600 Portfolio

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Figure 10. The assets and the portfolios observations’ quantiles (vertical axis) vs. the student’s t distribution (horizontal axis) with 4 degrees of freedom for the currencies and 3 for indices.

A comparison between Figure 7 and 9 indicates clearly that the series distributions follow the reference distribution more closely in Figure 9. The qq-‐plots in Figure 10 also show signs that the sample’s quantiles follow the quantiles of a student’s t distribution more closely than those from a normal distribution in Figure 8.

In the following table some general information of the time series is presented. Table 1. General information of the time series.

OMXS30 S&P500 FTSE100 USD GBP

Mean 4.1 ⋅ 10!! _{3.0 ⋅ 10}!! _{1.8 ⋅ 10}!! _{−3.7 ⋅ 10}!!_{−2.8 ⋅ 10}!!

Standard deviation 0.015 0.012 0.011 0.0071 0.0060

Skewness 0.24 0.21 0.055 −0.15 −0.083

Kurtosis 6.9 15.0 9.7 6.0 6.4

Skewness is a measure of the asymmetry of the distribution. The value is positive, negative or undefined and indicates whether the right, the left or none of the tails are longer than the other. The skewness is defined as

𝑠 =! !!!_!! ! (21)

The skewness of a sample without correcting for bias can be estimated as follows:

𝑠!= ! ! !!!!!!!!! ! ! !!!!!!!! ! ! (22) -0.5 0 0.5 -0.4 -0.2 0 0.2 0.4 OMXS30 -0.5 0 0.5 -0.2 -0.1 0 0.1 0.2 S&P500 -0.5 0 0.5 -0.2 -0.1 0 0.1 0.2 FTSE100 -0.1 0 0.1 -0.1 -0.05 0 0.05 0.1 USD -0.1 0 0.1 -0.1 -0.05 0 0.05 0.1 GBP -0.5 0 0.5 -0.5 0 0.5 Portfolio

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Kurtosis is a measure of the shape of the distribution and more specifically, the peak. It indicates how much the distribution deviates from a normal distribution which has a kurtosis of three. A kurtosis more than three is called leptokurtic and characterizes a thin peak and fat tails. The kurtosis is defined as

𝑘 =! !!! !

!! (23)

and to obtain the uncorrected bias measure of a sample, the following Equation applies

𝑘!= ! ! !!!!!!!! ! ! ! !!!!!!!!! ! (24)

The kurtosis value of all assets in Table 1 indicates that they come from a distribution with heavy tails, for example from the student’s t-‐distribution. This was expected, as mentioned before, as it is well know that distributions of financial returns often have heavy tails.

The chi-‐square statistic corresponds to a value 1 − 𝑝 where a 𝑝-‐value less that 0.05 strengthens a rejection of the null hypothesis for a confidence level of 95 percent. The 𝑝-‐value of the null hypothesis stating that the time series of the portfolio is normally distributed is 1.37 ⋅ 10!!!_{and
the} corresponding 𝑝-‐value for student’s t-‐distribution is 0.302.

3.1.2 Autocorrelation and Heteroskedasticity

To determine the dependency between returns, the autocorrelation function, ACF, is calculated. If the return 𝑎! is correlated with 𝑎!!!, where 𝑘 is the lag, the correlation is called autocorrelation. The estimates used are of Box, Jenkins and Reinsel, specifically

𝑧! =!_!!

! (25)

where 𝑧! is the autocorrelation of lag 𝑘 and 𝑐_!= !

! 𝑎!− 𝑎

!!!

!!! 𝑎!!!− 𝑎 𝑘 = 0,1,2, … , 𝐾 (26) 𝑎! is the return sequence and 𝑎 is the sample mean. If the observations in the sequence 𝑎! are squared and 𝑎 is its mean the test is for heteroskedasticity instead.

In figures 11 through 15 the ACF of all the time series are illustrated. The blue lines represent a 95 percentage confidence interval.

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Figure 11. The autocorrelation function of the time series for OMXS30 and its heteroskedasticity to the right.

Figure 12. The autocorrelation function of the time series for S&P500 and its heteroskedasticity to the right.

0 1000 2000 3000 4000 -0.2 0 0.2 0.4 0.6 0.8 Lag Sa m p le Au to c o rr e la ti o n

Sample Autocorrelation Function

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Figure 13. The autocorrelation function of the time series for FTSE100 and its heteroskedasticity to the right.

Figure 14. The autocorrelation function of the time series for USD and its heteroskedasticity to the right.

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Figure 15. The autocorrelation function of the time series for GBP and its heteroskedasticity to the right.

It is illustrated that there is autocorrelation in all assets and clearly heteroskedasticity as well. This suggests that it is inappropriate to simulate the returns without considering the autocorrelation. A way to get around this is to filter the innovations by, for example, the conditional variances. The filtered innovations are then assumed to be independent and identically distributed.

3.1.3 Subintervals of the time series

It is of interest to determine what happens with the correlation between a financial non-‐stressed and stressed time. To simplify, the whole period of data is divided into subintervals. The intervals are chosen by delimiting the financial shocks in time, using the time around them as stressed time periods and using the intervals before and after these as calm periods. The different calm periods are not necessarily equally calm but rather relatively calm in comparison with its environment.

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In Figure 16 the prices of OMXS30 are illustrated.

Figure 16. The prices of OMXS30 in SEK over time.

It is possible to discern some great losses over time in the figure above and it is also possible to see that these periods in Figure 16 are more volatile than their environment. The cluster of the time series shown in Figure 6 does not necessarily occur exactly at the same time as the price fall and it is therefore difficult to decide when the shock starts. The shocks are defined roughly by selecting time periods where the loss is more than 20 percent within 50 days for the index OMXS30 (shown in Figure 16), and where the portfolio variant described in section 3.2 has a daily relative change less than −0.09 at least four times during the same 50 day period. Those periods that meet these two conditions are considered to be financially stressed times. The exact dates are picked by arbitrarily selecting a date where the relative changes start to increase and after the upturn, when the relative changes have decreased, see Figure 6. The subintervals are presented in Table 2.

Table 2. The dates of the subintervals.

Subinterval Period from Period to

Period 1 1994 − 01 − 04 1998 − 07 − 15 Period 2* 1998 − 07 − 16 1998 − 10 − 08 Period 3 1998 − 10 − 09 2002 − 06 − 14 Period 4* 2002 − 06 − 17 2003 − 04 − 08 Period 5 2003 − 04 − 09 2008 − 05 − 16 Period 6* 2008 − 05 − 19 2009 − 04 − 28 Period 7 2009 − 04 − 29 2011 − 07 − 11

*Period 2, 4 and 6 are considered to be stressed.

3.2 Simulation with Unconditional Student’s t

As argued in previous sections, the assets return series seem to have heavy tails and using student’s

1994 1996 1998 2001 2003 2006 2008 2011 200 400 600 800 1000 1200 1400 1600 Date Va lu e o f O M XS3 0 i n SEK

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To make use of the model, one generates random numbers from a student’s t-‐distribution such that a matrix of the same size as the portfolio is obtained. It is generated with three degrees of freedom for the indices and four degrees for the currencies. These numbers were obtained by the maximum likelihood method. The correlation is calculated by the mean-‐adjusted returns in the different time periods of the historical data. To take the correlation into account, Cholesky’s decomposition is used to correlate the generated random numbers.

Let a vector 𝑟!,! be the generated numbers for time period 𝑖 and asset number 𝑗 where 𝑖 = 1,2, … ,7 and 𝑗 = 1,2, … ,5. To obtain generated numbers with the same variance as the historical data a vector 𝑎_!,! is created as follows

𝑎!,! = 𝜎!,!𝑟!,! (27)

where 𝜎!,! = 𝜎!,! _!!!

!!!

!!/!

is a constant with the standard deviation of asset 𝑗 in time period 𝑖. A matrix can now be obtained as follows

𝒂𝒊 = 𝑎!,! 𝑎!,! 𝑎!,! 𝑎!,! 𝑎!,! (28) A correlated matrix corresponding to the historical data is given by

𝑨𝒊 = 𝒂𝒊𝑈! (29)

where 𝑈! is the Cholesky factor of the correlation matrix calculated by the historical data in time period 𝑖. To obtain the whole portfolio estimation, for the entire time period, the calculations in Equation 29 are done for all time periods and then simply put together in the correct time sequence. To obtain the simulated time series corresponding to the portfolios’ time series, the mean of the time series is added to the simulated mean adjusted returns.

3.2.1 FHS Unconditional t-‐model

The problem with simulations based on historical events, such as in this case, is that volatility may influence the correlations and therefore provide misleading information. To get around this problem Filtered Historical Simulation, FHS, is used.

Consider the matrix 𝑟 which represents the historical data for all five assets in the portfolio. To get observations that are drawn from a standardized empirical return distribution 𝑟 is filtered. Hence, a sequence 𝑎!,! is obtained where 𝑖 and 𝑗 is the same as above. A description of filtered historical simulation is found in section 2.4.

The correlations for the different time periods are now calculated by the filtered events the same way as described above. One might argue that the estimated standard deviations by GARCH(1,1) could be used after filtering the historical innovations, but for simplicity the same standard deviation as in Equation 27 are used in the simulations. The FHS Unconditional t-‐model is given by Equation 27, 28 and by

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where 𝑈! is the Cholesky factor of the correlation matrix calculated by the filtered historical data in time period 𝑖.

The 𝑉𝑎𝑅 is calculated as in Equation 1 every day by the latest 252 days. The 252 first days in the historical data will therefore not be compared with any 𝑉𝑎𝑅 estimation.

3.3 Simulation with MV-‐GARCH (1,1)

Simulation of a portfolio containing different assets may require taking into consideration that the returns vary differently to get a satisfying and realistic result. The MV-‐GARCH uses a constant correlation approach, but although the correlations remain constant, it will still allow the conditional heteroskedasticity to be time-‐variant.

In this work the interpretation of Kevin Sheppard (2001, 2003), based on the theory of Tim Bollerslev (1990) is used.

The whole time period is used to estimate the coefficients for the assets’ time series respectively. This means that the coefficients in Equation 15 will be determined for the same time period as the prediction time period. Predictions with GARCH normally use the coefficients estimated by an appropriate calibration time, but in this case, when the aim is to answer the question of whether the data can be stressed by correlation, no calibration time is used. Also in these estimations the mean-‐ adjusted returns are used as in the Unconditional Student’s t-‐model. By the covariance matrix obtained in Equation 14 the observation in the time step 𝑡 is obtained as follows

𝑟_! = 𝐻_!!/!𝜂_! (31)

where 𝐸 𝜂!𝜂!! = 𝐼, the identity matrix.

One can argue that the coefficient estimations should be based on the fact that the innovations are student’s t-‐distributed, but it is arguable that the model based on an assumption of normal distributed innovations in the coefficients estimations performs well (Gouriéroux, C., 1997).

The time series corresponding to the portfolio that are simulated by this model are used to calculate a 𝑉𝑎𝑅 which is calculated the same way as described in section 3.2.1.

3.4 Evaluation of the models

This section describes two tests performed on the models with the objective to evaluate them. The results of the test are also presented.

3.4.1 Root mean square error

To test how well the model estimates the historical data a root mean square error (RMSE) is calculated for both models as follows

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Table 3. The RMSE between the historical data and the simulations. The RMSE values for FHS Unconditional Student’s t is a mean of 100 iterations. The RMSE for MV-‐GARCH(1,1) is obtained by one iteration.

Model RMSE )

Returns FHS Unconditional Student’s t 0.0470

MV-‐GARCH(1,1) 0.0478

Model _{(𝜶 = 𝟎. 𝟎𝟏)}RMSE _{(𝜶 = 𝟎. 𝟎𝟓)}RMSE _{(𝜶 = 𝟎. 𝟏)}RMSE _{(𝜶 = 𝟎. 𝟐)}RMSE

𝑽𝒂𝑹

FHS Unconditional

Student’s t 0.0141 0.0103 0.0083 0.0067

MV-‐GARCH(1,1) 0.0146 0.0085 0.0068 0.0050

The information in Table 3 indicates that the models perform almost equally. The unconditional model shows a slightly better RMSE of both returns and 𝑉𝑎𝑅 for 𝛼 = 0.01. However, for other probabilities, the RMSE calculated of the 𝑉𝑎𝑅 calculations from the MV-‐GARCH model is noticeable smaller than for the other model. Since the 𝑉𝑎𝑅-‐curve is analyzed in the tests of the stress effects by the correlation, the MV-‐GARCH seems to be more accurate.

3.4.2 Backtesting

A backtest to test the models performance is used. First the number of historical events that exceeds their 𝑉𝑎𝑅-‐curve is calculated. By using the historical data and using the 𝑉𝑎𝑅-‐curves simulated by the models, the exceedances can be compared for the different 𝑉𝑎𝑅 curves. It is also of interest to compare the outcome of this with the statistical expected number of exceedances. This is done for 𝛼 = 0.01, 𝛼 = 0.05, 𝛼 = 0.1, and 𝛼 = 0.2 to see if the stability varies for different probabilities. In section Backtesting in Appendix 1 a backtest is presented where the simulated innovations exceedances of the historical 𝑉𝑎𝑅 curve are calculated and compared with the statistical exceedances.

For 𝛼 = 0.01

Figure 17. The difference in exceedances by the historical data on the 𝑽𝒂𝑹-‐curves. The diagram to the left is a comparison between the 𝑽𝒂𝑹-‐curves of the models and the historical 𝑽𝒂𝑹-‐curve. The right diagram is a comparison between the 𝑽𝒂𝑹-‐curves of the models and the statistical expected number of exceedances for the given probability.

-‐40 -‐20 0 20 40 60 1 2 3 4 5 6 7 -‐40 -‐20 0 20 40 60 1 2 3 4 5 6 7 FHS Uncond. t MV-‐GARCH(1,1)

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Table 4. The number of exceedances in each period for the given probability.

Period 1 Period 2 Period 3 Period 4 Period 5 Period 6 Period 7

FHS Uncon. t 11 7 5 4 15 6 5

MV-‐GARCH(1,1) 26 7 9 6 32 6 7

Historical 13 4 7 7 13 5 6

Statistical 9,17 0,61 9,46 2,05 12,83 2,37 5,52

The number of exceedances between the 𝑉𝑎𝑅-‐curve generated by the FHS Unconditional Student’s t-‐model and the historical 𝑉𝑎𝑅-‐curve is more consistent than in a comparison with MV-‐GARCH (1,1). The number of exceedances by the unconditional model also seems to be more consistent with the statistical number of exceedances. The diagram to the right in Figure 17 supports the fact that FHS Unconditional Student’s t-‐model performs better for 𝛼 = 0.01 which was also the conclusion by the RMSE values in Table 3.

For 𝛼 = 0.05

Figure 18. The difference in exceedances by the historical data on the 𝑽𝒂𝑹-‐curves. The diagram to the left is a comparison between the 𝑽𝒂𝑹-‐curves of the models and the historical 𝑽𝒂𝑹-‐curve. The right diagram is a comparison between the 𝑽𝒂𝑹-‐curves of the models and the statistical expected number of exceedances for the given probability. Table 5. The number of exceedances in each period for the given probability.

Period 1 Period 2 Period 3 Period 4 Period 5 Period 6 Period 7

FHS Uncon. t 48 22 65 28 98 18 26

MV-‐GARCH(1,1) 51 15 39 24 85 30 16

Historical 52 12 47 22 75 15 18

Statistical 45,85 3,05 47,3 10,25 64,15 11,85 27,6

With this probability of the 𝑉𝑎𝑅 the MV-‐GARCH (1,1) models 𝑉𝑎𝑅 curve performs better as it is closer to both the historical and statistical number of exceedances.

-‐40 -‐20 0 20 40 60 1 2 3 4 5 6 7 -‐40 -‐20 0 20 40 60 1 2 3 4 5 6 7 FHS Uncond. t MV-‐GARCH(1,1)

Hypothetical Scenario Analysis in Modeling of Market Risk