UMEÅ UNIVERSITY Department of physics
2013-02-25
The Use of Importance Sampling in
Bootstrap Simulations and in Moving
Block Bootstrap Simulations for
Efficient VaR Estimations
Gustav Bergström, gustav.bergstroem@gmail.com
Supervisor: Magnus Lundin, calu07@handelsbanken.se, Handelsbanken
Examiner: Markus Ådahl, markus.adahl@math.umu.se, Umeå University, Department of Mathematics and Mathematical statistics
Sammanfattning
Under 2008 chockades den finansiella sektorn av stora förändringar på den finansiella marknaden.
Detta startade en förändringsprocess av finansiella institutioners riskhantering och regelverkens utformning. Ett riskmått som flitigt används är Value at risk (). Det anses inte ge en fullgod bild av risken företagen är exponerade för och behöver därför kompletterande beräkningar. I sin enklaste form bygger på empiriskt data vilket inte ger en komplett bild av hela riskprofilen.
För att komplettera riskbilden används i detta arbete två metoder för att skatta konfidensintervall för
-måttet. Dessa intervall är tänkt att indikera inom vilket område med största sannolikhet ligger. Vanlig bootstrap och moving block bootstrap med tillhörande konfidensintervall för
studeras samt hur väl dessa modeller fångar in -måttet. För att minska kraven på datorkapacitet och reducera standardavvikelsen i simuleringarna används importance sampling i kombination med bootstrapmetoderna. Detta görs för -beräkningar med tre olika sannolikhetsnivåer 90, 95 och 99 procent. Dessa metoder studeras för att undersöka vilka värden på parametrarna i metoderna som ger lägst varians och mest effektiva skattningar.
En 1,1 -modell används, med skattade parametrar från en OMXS 30-tidserie, för att simulera fram nytt data då det inte finns något att jämföra med. Från 1,1 -modellen skattas ett -värde som anses motsvara det sanna värdet. Det förväntade antalet genomslag mellan beräknat med -modellen och predikterat data beräknas tillsammans med roten ur medelkvadratfelet, RMSE, mellan bootstrapintervallen och det -simulerade -värdet.
Antalet genomslag och RMSE antas motsvara bootstrap-metodernas prestation.
Vanlig bootstrap i kombination med importance sampling med skattade konfidensintervall för - siffran fånga in det förväntade antalet genomslagen mellan den skattade -siffran och förlusten.
RMSE-beräkningarna däremot resulterar i 0.004, 0.004 och 0.008 för de respektive ovannämnda sannolikhetsnivåerna. I och med att dess är noll skilda betyder det att intervallen inte rymmer den sanna -kurvan i alla tidssteg. Moving block bootstrap i kombination med importance sampling fångade in det förväntade antalet genomslagen. Resultaten från RMSE-beräkningarna visade att intervallen inte rymmer den sanna -kurvan för någon av sannolikhetsnivå. RMSE var dock närmare noll och beräknades till 0.003, 0.002 och 0.005 för respektive nivå. RMSE beräkningarna anses ge en mer rättvis bild av konfidensintervallens prestation och varken vanlig bootstrap eller moving block bootstrap anses fånga och hur det varierar.
Abstract
In 2008, change of the behavior on the financial market shocked the financial sector. This started a process of changes of financial institutions' risk management and regulatory framework. A common measure of risk is Value at Risk () and it do not provide an adequate picture of the risks that companies are exposed to. In its simplest form, -calculations are based on empirical data, which do not give a complete picture of the risk profile.
To complete the risk management two methods are examined in this work that both aim to estimate confidence intervals of the -curve. The intervals is supposed to indicate the range in which the correct most likely is located. The two methods are ordinary bootstrap and the moving block bootstrap. In order to reduce demands on computing power and reduce the standard deviation in each simulation importance sampling is combined with the bootstrap methods. The calculations are performed with three different probability levels 90, 95 and 99 percent.
When there is no true to compare with, data is simulated with a -model and used in the bootstrap methods. The parameters of the 1,1 -model are estimated from an OMXS30 time series. A is simulated with the 1,1 -model that represents the true . The expected number of exceedances is computed between the -simulated and values predicted from the -model. It is calculated along with the root mean square error, RMSE, between bootstrap intervals and the -simulated . These two measures are assumed to represent the bootstrap methods' performance.
The ordinary bootstrap in combination with importance sampling with corresponding confidence intervals capture the expected number of exceedances between the estimated and the loss.
RMSE calculations results in 0.004, 0.004 and 0.008 respectively for the different probability levels. The RMSE shows that the confidecne intervals do not hold the true -curve in each time step, since it is non-zero. Moving block bootstrap in combination with importance sampling also capture the expected number of exceedances shows that the intervals contain the true -curve for everey probability level. RMSE calculations results in 0.003, 0.002 and 0.005 for each probability level.
Table of content
1 Introduction ... 1
2 Theory ... 2
2.1 Price process model ... 2
2.2 Value at Risk ... 2
2.3 Backtesting ... 3
2.4 Autocorrelation ... 3
2.5 Bootstrap Methods ... 4
2.5.1 Ordinary bootstrap ... 4
2.5.2 Moving block bootstrap ... 4
2.5.3 Bootstrap confidence intervals ... 5
2.6 Importance Sampling ... 5
2.6.1 Designed distribution ... 7
2.6.2 Selection of the optimal ... 7
2.7 , ... 8
3 Methodology ... 9
3.1 Daily observations ... 9
3.2 The designed distributions ... 11
3.3 Stable variance ... 14
3.4 MBB with Importance sampling ... 17
3.5 Simulations and evaluations of the models ... 20
3.5.1 , -simulations ... 20
3.5.2 Summary of the bootstrap methods ... 21
3.5.3 Evaluation of the methods ... 21
4 Results ... 22
4.1 Bootstrap with importance sampling ... 22
4.2 MBB with importance sampling ... 24
5 Conclusions ... 27
5.1 Bootstrap with importance sampling ... 27
5.2 MBB with importance sampling ... 27
5.3 Summary... 28
6 Discussion ... 29
References ... 30
Appendix ... 31
Results ... 31
Ordinary bootstrap ... 31
Moving block bootstrap ... 33
1
1 Introduction
The financial crisis that struck the world in 2008 shocked the financial world. Although banks worldwide were strictly controlled by institutions comparable to the Swedish Financial Supervisory Authority, no financial institution expected the losses the financial sector would suffer. This led to proposed changes in how banks' risk management are performed to counteract losses in comparable financial turmoil.
Value at risk () is a widely used risk measure which plays an important part in regulations of banks all over the world. Even though is criticized for its lacks of coherence it can together with other risk measures give a fair view of the risk profile. The measure estimated with historical simulation takes the historical loss into account for a certain probability. Computing by empirical simulation is a relatively easy task but possibly misleading because of its retrospective approach.
Beside that might be misleading it does not indicate how much the measure may vary either.
The purpose is therefore to study together with an estimate of the possible range that will tell something about the measure’s accuracy.
One way of measuring the accuracy of the risk measure is to use bootstrap. It was introduced in 1979 and is used in mathematics for empirical estimations or in simulation of a distribution and its character (Efron, 1979). The basic idea is to randomly choose observations from the original sample and form new samples (bootstrap samples) which the accuracy can be estimated from. One important assumption when using bootstrap is independent identical distributed (i.i.d) data. One variation of the classical method, called moving block bootstrap (MBB) by Künsch (1989), is used with time dependent data. Instead of generate bootstrap samples by randomly choose single observations;
blocks of observations are randomly chosen. The data is often time dependent when dealing with financial time series and moving block bootstrap is therefore supposed to perform better. The benefit with all kinds of bootstrap methods is that no assumptions about the distribution are necessary.
Since the introduction of methods like bootstrap, which demands computational effort, efficient methods have been requested to obtain simulations with low variance and short computational time.
One method that has proven to reduce both of these parameters is importance sampling. Its main aim is to focus the computational effort on the more interesting parts of the distribution, in this thesis the extreme losses, with a distribution called designed distribution.
In this thesis a study of how to combine importance sampling with ordinary bootstrap and moving block bootstrap is carried out. The data that is used is generated from a -model with estimated parameters from an OMXS30 time series. By using data from a known model, an analytical
can be estimated. In order to carry out the study, the most efficient designed distribution, the optimal block length and the most efficient numbers of bootstrap samples are found.
The aim is ultimately to capture a -simulated -curve, considered the correct , with bootstrap confidence intervals.
2
2 Theory
In this section an overview of the basic theoretical concepts of this thesis is presented. The data and the model that generates the data is presented. The risk measure that is used is introduced. A way of evaluate the risk measure and a method of determine weather the data is time dependent is described. Two basic bootstrap methods are introduced together with a description of how it is combined with importance sampling. A -model is described in this section which is used in the final simulations to generate data and risk measures that are needed for evaluation of the model.
2.1 Price process model
The time series that will be studied in this thesis is the negative relative changes, i.e. the loss. It is given by the following equation
. (1)
is the actual loss and is the negative relative changes in every time step. This means that in the following figures, tables and other results a positive rate of return will represent a loss in value.
2.2 Value at Risk
Risk measures that are easy to understand and describe are few but desirable in the financial sector.
is a measure of such properties and is therefore widely used. is defined as the minimum loss of the percent worst losses. That is
! inf%& ' (: *+& , - (2)
where is the loss distribution of a portfolio’s time series, the confidence level and *+ is the cumulative distribution function (CDF) of the loss.
The ! calculated from the cumulative empirical distribution is illustrated in the figure below.
Note that this is an example with a normal distribution and not from the distribution that is used with the bootstrap methods.
3
Figure 1. An illustration of the ./0 calculated from the empirical distribution function.
The cumulative empirical distribution, *12& , is given by the following equation
*32& 42∑ 6728;4 8 9 &: (3)
where 8 is the rate of returns, 6 is the standard indicator function and < is the number of observations.
The is estimated by the empirical distribution, (Sun and Hong, 2009), and is defined as follows:
=>!2 *32 4 inf7&: *32& , : (4) where < is the number of observations and is the level of confidence.
If is calculated every day in a time period it creates a -curve that illustrates the change of
in different time steps.
2.3 Backtesting
One method to evaluate methods is called Backtesting. It is commonly used when evaluating
and describes the number of times the actual loss exceeds the . To evaluate the methods, the number of exceedances is calculated. It is calculated with the formula
? ∑ 628;4 8 @ ! (5)
Where 6 is the indicator function and is the losses.
2.4 Autocorrelation
The autocorrelation is the correlation of a time series between itself and a time lag and is used to determine the range of the dependency in a time series. There are different methods of calculating autocorrelation, in this thesis the one proposed by Box, Jenkins, and Reinsel is used. It follows the formula
4 AB CCD
E (6)
where
FB 42∑2 B;4G GH GIB GH J 0,1,2, … N. (7) and < is the number of observations in the sample, N is the number of time lags and FO is the lag 0 covariance i.e. the unconditional variance.
2.5 Bootstrap Methods
One method of estimating properties of a distribution without making assumptions of it is to make use of bootstrap. It comes in different varieties and the two types used in this thesis are ordinary bootstrap and moving block bootstrap (MBB).
An important assumption when using ordinary bootstrap is that the time series is independent. MBB however has proved to perform well with short-range dependent data. These two methods are presented in the sections below.
2.5.1 Ordinary bootstrap
Bootstrap methods are a variant of the Jackknife method (estimating by subset of empirical data) and were introduced by Efron (1979). The bootstrap method is a nonparametric method of determining the accuracy of the distribution statistics.
Given the observed sample P 4, Q, … , 2 where P~S& and is independent identical distributed. It is of interest to estimate the accuracy of TU*P& V, which is calculated from the empirical distribution.
The basic idea is to draw W new samples of size < with replacement from sample . The new samples are denoted PXY 4,XY , Q,XY , … , 2,XY where Z 1,2, … , W. TXY T [*1XPXY \ is computed from the bootstrap sample’s empirical distribution.
Different types of accuracy measures for TP are calculated with the bootstrap samples such as the bias of the measure
Z]^_X`` ab1UTPY TP V (8)
and the standard deviation
cdX`` eab1f[TPY ab1UTPY V\Qgh
i
. (9)
2.5.2 Moving block bootstrap
As mentioned earlier, most financial time series are not time independent and this independency is a necessary condition of the sample for good bootstrap estimations. Künsch (1989) introduced an alternated method called moving block bootstrap (MBB) that applies for dependent data.
The basic idea of MBB is the same as ordinary bootstrap with the difference that blocks are used instead of using single observations. Given the observed sample 4, Q, … , 2 where ~S& ,
5
we choose J </k overlapping blocks of length k with replacement from the < k l 1 blocks of observed data. < is the number of observations in the original sample.
By choosing blocks uniformly on 70,1, … , < k: and form the new samples with the blocks, an empirical distribution is constructed. The bootstrap sample will have the distribution
mnY 42∑ o2;4 pY (10)
where o is the Dirac measure and the blocks are i.i.d with probability distribution
4
2 qI4∑2 q;Oopr Y ,…,prsY (11)
and has the empirical distribution *1. We then form the bootstrap statistic
TY T [*1XPXY \. (12)
Properties, such as the accuracy, are estimated by using Equation 8 and 9.
2.5.3 Bootstrap confidence intervals
The bootstrap estimations are used to calculate confidence intervals. Calculations of bootstrap confidence intervals are specified in Davison and Hinkley (1997) and in Johns (1988). The simplest way of calculating the bootstrap 21 C confidence interval limits with significance level C are with the formula
[TUtI4 4 !Y u V\, [TUtI4 !Y uV\. (13) TY is calculated from the bootstrap samples with the corresponding quantiles specified in the parentheses.
2.6 Importance Sampling
Importance sampling is a way of estimating properties of a distribution by using samples from another distribution. This is done by increasing the probability of samples of interest so that they are simulated more often. Hesterberg (1988) describes how importance sampling is used in bootstrap methods that first was introduced by Johns (1988) and is given in a short version as follows:
If is a random variable with distribution S and w is a function of , we obtain a simple Monte Carlo estimate by generating numbers 8~S, x 1,2, … , <, and computing w8 w8 . The observed values of w then form an estimation of the distribution w .
The traditional view is that Monte Carlo simulation is a form of integration. The estimated expected value of w can be written in the following way
ayUw V z w S { z w |+ y+ } { . (14)
It is crucial that the distribution }& dominates the true distribution, that is }& @ 0 if S& @ 0 for all &.
The definition
6
n ~ w y+ |+ (15)
is estimated from quantiles and are obtained by the empirical distribution function. An interpretation of importance sampling called the sampling interpretation is used in this thesis and is appropriate when estimating from empirical distribution functions.
The applicable interpretation of quantiles estimations are the regression estimate and the ratio estimate (Hesterberg, 1988 and 1995). These estimates require that the distribution have a total probability equal to one.
Defining
~|+y+ (16)
where the expected value is
a| z & }& {& z S& {& 1 . (17) One way of ensuring that the ratio adds to one is to use the ratio weights where each bootstrap observations has the following weight
8`8 ~∑ U+U+ V
V
. (18)
This gives the ratio estimate
̂8`~ ∑ 28;4 8`8 w88 p. (19) Other weights called the regression weights will also be used. These weights are defined in the following way
|8`28 428U1 l F8 V (20)
where
F 4
∑ [ \i. (21)
Since the quantiles of the distribution are of interest, estimations of such statistics are calculated from by the empirical distribution. Y comes from another distribution and that is taken into account by calculating the CDF in the following way
*12 ∑ 28;4 868Y 9 (22)
where 8 is either the ratio or the regression weights. For the bootstrap samples this is calculated for every sample. The statistic is calculated from the bootstrap distributions with Equation 8 and 9 that correspond to variance and bias.
7 2.6.1 Designed distribution
There are different designed distributions when using importance sampling. However, four requirements should be fulfilled to obtain a designed distribution with satisfying results.
• The distribution is dominating the true distribution
• It is easy to generate random numbers from the distribution
• The likelihood ratio between the distributions is easy to calculate
• The designed distribution gives a low variance estimate.
One type of a designed distribution is called exponential tilting. This takes the form
}& ! S& . (23)
The function is the logarithm of the moment generating function, called the cumulant generating function, and is defined
logUz {*& V (24)
and where is the degree of tilting from the original distribution. The cumulant generating function registers important information about the distribution }& such as a|
and | (Glasserman, 2004).
Another type of designed distributions is a mixture of distributions. There is no standard formation of mixing distributions but normally the original distribution is mixed with some other distribution.
2.6.2 Selection of the optimal
When tilting the original distribution with exponential distributions the choice of the parameter , which controls the variance and determines the degree of tilting, is important. In Sun and Hong (2009) and Glasserman (2004), a description of how to select the optimal is presented. By using properties of the cumulant-generating function in Equation 24, the optimal (Y) that satisfies the following equation can be found:
=! . (25)
To obtain Y, define
: z {*& aUV, (26)
replace in Equation 24 and differentiate . It yields
UlogU VV UUVV. (27)
In the discrete case, Equation 27 becomes
∑∑
. (28)
Given the above formula the optimal can be found numerically by combining Equation 28 with Equation 25.
8
2.7 ,
One problem with financial time series is that the variance is time dependent. Simulating or predicting the movement of such series is therefore problematic. Tsay (2002) describes a way of simulating this type of time series by a model called generalized autoregressive conditional heteroscedasticity model (-model).
In order to test the bootstrap models a -model is estimated and realized over a selected period. The realization results in new data (-data) with which the different methods are applied.
Let A – be the mean-corrected logreturn. Then follows a , ^ -model if
c ¡ (29)
where
cQ Ol ∑ ¢ 8 8Q
8;4 l ∑ £c £Q
£;4 . (30)
¡ is a sequence of i.i.d random numbers with mean 0 and variance 1. The parameters are limited by O@ 0, 8 , 0, £, 0 and ∑¤¥¦¢, 8;4 8l 8 § 1 where x 1,2, … , . These parameters are found by using maximum likelihood method with the function
¨w ∏ Sw ª;4 (31)
where Sw is the normal density function and where w depends on the parameters from Equation 30.
9
3 Methodology
The two different samples that are used in this thesis is a OMXS30 time series and a FTSE100 time series. The OMXS30 is used in all simulations and FTSE100 is used in the first part when the most efficient designed distribution is deterimened.
An overweiv of the two different samples that are used and their distributions are presented in the following section. This illustrates the difference in their distirbution and which of the samples that contain the most extreme losses. These illustrations of the distribution are also interesting since comparing the original distributions and the bootstrap distributions tells how much the original distirbution is tilted. The autocorrelation of OMXS30 is illustrated that time series is used in the final simulations.
The designed distribution is presented in a section below and illustrated in a figures together with the original distribution. The different weights from Equation 18 and 20 are tested with the different designed distirbutions in order to determine which weight togheter with which desgined dstribution that give estimations with lowest standard deviation. The number of bootstraps and the bootstrap sample size is tested with OMXS30 data to obtain optimal values of these parameters. Three different designed distribution are evaluated with the entire dataset. The above is first tested for ordinary bootstrap. The results for ordinary bootstrap is assumed to hold for MBB aswell instead the optimal block length is tested instead.
The methodology section finishes with a summary of the final simulations for both ordinary bootstrap and moving block bootstrap.
3.1 Daily observations
In the figure below the distribution of the OMXS30 sample is illustrated in a histogram. It shows the range of the sample and in which region observations are most common.
Figure 2. A smooth approximation of the OMXS30 time series´ probability density function.
The independence of the time series is an important assumption for ordinary bootstrap. In order to determine whether there exists any dependency the series is tested for autocorrelation. If there exits
-0.150 -0.1 -0.05 0 0.05 0.1
50 100 150 200 250 300 350 400
Rate of return
Number of observation
10
autocorrelation in the first moment of the time series it means that there exists time dependence.
Existence of autocorrelation of the second moment means that the volatility of the time series is time dependent, often referred to as heteroscedasticity. The autocorrelation of the first and the second moment is calculated by Equation 6 and 7 and are illustrated in the following figure.
Figure 3. Autocorrelation for all time lags of the OMXS30 time series (A). Autocorrelation for all time lags for the second moment of the time series (B). The 95 % confidence intervals (blue).
The blue lines in Figure 3 correspond to a 95 percent confidence interval. If the dots (red) in the figure exceed the lines corresponding to the confidence intervals, autocorrelation exists. Figure 3A corresponds to the autocorrelation of the first moment and such autocorrelation exists. Figure 3B shows that there is autocorrelation i.e. time dependent variance.
In the figure below the FTSE 100 sample is illustrated in a histogram that shows its distribution.
Figure 4. A smooth approximation of the probability density function for the FTSE time series.
0 500 1000 1500 2000 2500 3000 3500 4000 4500
-0.2 0 0.2 0.4 0.6 0.8
Lag
Sample Autocorrelation
0 500 1000 1500 2000 2500 3000 3500 4000 4500
-0.2 0 0.2 0.4 0.6 0.8
Lag
Sample Autocorrelation
Sample Autocorrelation Function
-0.10 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
50 100 150 200 250 300 350 400 450
Rate of return
Number of observation
(A) (B)
11
The time series of FTSE100 (Figure 4) contains less extreme losses than the time series of OMXS30 (Figure 3). Because the fact that the time series differ from each other it is valid to keep both when trying to find the best designed distribution since they should give different results. Since the FTSE100 time series is not used in the final simulations it is not tested for autocorrelation.
3.2 The designed distributions
In order to determine which type of designed distribution the results in -estimations with low standard deviation, low bias and that are easy to implement, three distributions are evaluated.
The distributions tested and described below are; the exponential tilted distribution, a mixture distribution of two constant functions and a mixture distribution of a constant- and a second order function.
The exponential tilted distribution which have been stated to be the most efficient (Hesterberg, 1988), and as mentioned in a previous section has the form
}& ! S& (32)
where normalizes the function to unit mass and is the cumulant generating function from Equation 24. is the degree of tilting, & is the observed loss and S& is the empirical density distribution.
The mixture distribution of two constant functions which is a distribution generated with the formula }Q& « l O¬O® (33)
where § and normalizes the distribution. Since corresponds to the probability that the observation of gain are drawn it is set to a low probability. The constant is therefore set to 0.00002.
The mixture distribution of a constant and a second order function given by the formula
}¯& « l &Q ¬OO®. (34)
where normalizes the distribution, & is the losses and a constant set to 0.00002 as the for the above designed distribution.
The three different designed distributions are calculated from the empirical data with normalized constant. They are illustrated together with the original PDF in the figure below.
12
Figure 5. The PDF for the original sample (A) and the sample distributions °± with (B), °²± (C) and °³± (D).
Depending on the choice of the tilting constant Figure 5B varies. It is created with the optimal Y that will give the lowest variance of the bootstrap estimate. The choice of Y is described in section 2.6.2 Select optimal of beta. Figure 5 illustrates with which probability the different observations are drawn when bootstrapping. Since the interesting observations are the extreme losses (to the right in each figure) they are drawn with higher probability.
To evaluate the performance of the designed distribution a bootstrap sample from each distribution are illustrated. An example of one bootstrap sample drawn with the different probabilities is shown in the following figure.
-0.150 -0.1 -0.05 0 0.05 0.1 0.005
0.01 0.015 0.02
Rate of Return
Probability
(A)
-0.150 -0.1 -0.05 0 0.05 0.1 0.02
0.04 0.06 0.08 0.1 0.12
Rate of Return
Probability
(B)
-0.150 -0.1 -0.05 0 0.05 0.1 1
2 3 4 5 6x 10-4
Rate of Return
Probability
(C)
-0.150 -0.1 -0.05 0 0.05 0.1 0.002
0.004 0.006 0.008 0.01 0.012
Rate of Return
Probability
(D)
13
Figure 6. The upper right and the lower part of the figure are samples from the different designed distribution. The upper left is a bootstrap sample without any tilting.
The designed distribution that generates most samples with extreme losses is the exponential tilted distribution (Figure 6B). This is an indication that exponential tilting is successful when estimations of quantiles are done.
The weights described in Equation 18 and 20 are compared to determine which of them that is preferable in bootstrap. The ratio and the regression weights are evaluated by studying the bootstraps’ CDFs and which are presented in the following two figures. The CDFs are calculated with 100 bootstrap samples in both figures. The first figure presents the CDFs calculated with the regression weights.
-0.150 -0.1 -0.05 0 0.05 0.1 100
200 300 400
Loss
Number of observation (A)
-0.050 0 0.05 0.1 0.15
200 400 600
Loss
Number of observation (B)
-0.10 -0.05 0 0.05 0.1
100 200 300 400
Loss
Number of observation (C)
-0.10 -0.05 0 0.05 0.1
50 100 150 200
Loss
Number of observation (D)
-0.150 -0.1 -0.05 0 0.05 0.1
0.5 1
(A)
Rate of Return
Probability
-0.15 -0.1 -0.05 0 0.05 0.1
-0.5 0 0.5 1
(B)
(D) (C)
1.5
Rate of Return
Probability
-0.150 -0.1 -0.05 0 0.05 0.1
0.5 1 1.5
Rate of Return
Probability
-0.150 -0.1 -0.05 0 0.05 0.1
0.5 1 1.5
Rate of Return
Probability
14
Figure 7. The CDFs for the different designed distribution (B, C, and D) and for the ordinary bootstrap (A) calculated with the regression weight.
It is seen in Figure 7B that the Y axis differs from the rest and shows values below zero. This indicates that the CDF goes below zero at some value and this weight is therefore not used in the simulations.
The following figure shows the corresponding CDFs for the ratio weights.
Figure 8. The empirical distribution function for the different designed distribution (B, C, and D) and for the ordinary bootstrap (A) calculated with the ratio weights.
Although both of the estimates (Figure 7 and 8) result in CDFs with a maximum slightly larger than one, they are used. This phenomenon appears because of round off errors of the relative changes in the implementation. As mentioned above, the regression weights result in values below zero and instead the ratio weights are applied.
3.3 Stable variance
In order to determine which designed distribution that results in estimations with lowest variance and least computational time, the different distributions are tested with different bootstrap sizes and different number of bootstraps. The sample sizes of the bootstrap samples are therefore limited to 252 observations. In the following figures, BS means bootstrap and BS IS means bootstrap with importance sampling followed by the designed distribution. The OMXS30 sample is used for studies of stable variance and low computational time. The distribution of the time series OMXS30 and FTSE100 in Figure 2 and 4 are similar and the distributions are therefore assumed to perform equally well.
-0.150 -0.1 -0.05 0 0.05 0.1
0.2 0.4 0.6 0.8 1
(A)
Rate of Return
Probability
-0.150 -0.1 -0.05 0 0.05 0.1
0.5 1 1.5
(B)
Rate of Return
Probability
-0.150 -0.1 -0.05 0 0.05 0.1
0.5 1 1.5
(C)
Rate of Return
Probability
-0.150 -0.1 -0.05 0 0.05 0.1
0.5 1 1.5
(D)
Rate of Return
Probability
15
The number of bootstraps that yields the lowest variance is determined. Due to the fact that the simulation time increases with the number of bootstrap the maximum number of bootstraps is set to 200. The standard deviation with different number of bootstraps are calculated and illustrated in the figure below.
Figure 9. The variance dependency of the number of bootstrap samples.
It is not necessary to sate which of the designed distributions }Q& red and }¯& (cyan) in Figure 9 that generates samples with lowest standard deviation since the other two distributions perform better. Ordinary bootstrap (blue) results however in lower standard deviation than the two previous mentioned. The lowest standard deviation is obtained by the exponential tilted distribution }4&
(green). The figure also illustrates that a stable variance is hard to obtain with a limit of 200 bootstraps. Increasing the number of bootstraps stabilizes the standard deviation without reducing it.
The number of bootstrap is set to 100 since it fluctuate a bit less with 100 bootstrap samples or higher.
The following figure illustrates the standard deviations dependency of the bootstrap sample size. The suitable number of bootstrap samples from Figure 9 is used and the following figure presents simulations with 100 bootstrap samples. Note that the horizontal axis in this figure corresponds to the size of each bootstrap sample and not the number of bootstrap samples.
0 20 40 60 80 100 120 140 160 180 200
1 2 3 4 5 6 7 8 9 10 11x 10-3
Number of bootstrap samples
Standard deviation
BS BS IS, g1(x) BS IS, g2(x) BS IS, g3(x)
16
Figure 10. The standard deviation with different bootstrap sample sizes.
Figure 10 illustrates that a sample size of 245 observations do not give a stable standard deviation. It is set to 245 not 252, a whole year, because an uneven number is not equally convenient to implement in simulations to obtain Figure 10. As mentioned in previous section the computational time is measured and it rises quickly with the bootstrap sample size. In the simulations in the final part of the thesis a bootstrap sample size of 252 observations is however equally convenient and it is therefore used.
The following two tables contain results that determine which of ordinary bootstrap and bootstrap with importance sampling that results in -estimations with lowest variance. These results are calculated from bootstrap samples drawn from the entire dataset. The is set to 0.99. The expected value of is calculated as the mean of the estimations obtained by the bootstrap samples. The standard deviation and the bias are calculated for all bootstrap samples with Equation 8 and 9 respectively and the time is measured. The number of bootstrap samples is set to 100 and data from OMXS30 and FTSE100 is used.
Table 1. The different methods of calculating ./¸.¹¹ for 100 numbers of bootstraps and sample size set to 252.
Bootstrap Bootstrap with importance sampling
°± - }4& }Q& }¯&
º%./¸.¹¹- 0.0422 0.0418 0.0415 0.0421
»¼%./¸.¹¹- 0.0064 0.0032 0.0043 0.0025
½¾/%./¸.¹¹- 0.0005 0.0002 -0.0001 0.0004
»¾¿ 2.22 2.05 2.06 2.05
The O.ÀÀ of the entire OMXS30 time series is estimated to 0.0416 by historical simulation. A corresponding is estimated with ordinary bootstrap and bootstrap with importance sampling and is presented in the Table 1. Since the bootstrap sample size only stabilizes the standard deviation it is set to nine times the sample size of the importance sampling methods for the ordinary bootstrap.
This gives comparable simulation times and a suitable comparison.
150 160 170 180 190 200 210 220 230 240 250
0 0.002 0.004 0.006 0.008 0.01 0.012
Bootstrap sample size
Standard deviation
BS BS IS, g1(x) BS IS, g2(x) BS IS, g3(x)
17
The O.ÀÀ is computed for the FTSE100 time series with the different methods and the results are presented in the table below. The with probability 0.99 is calculated from the entire sample and is estimated to 0.0316.
Table 2. The different methods of calculating ./¹¹ for 100 numbers of bootstraps and sample size 252.
Bootstrap Bootstrap with importance sampling
°± - }4& }Q& }¯&
º%./¸.¹¹- 0.0340 0.0322 0.0326 0.0322
»¼%./¸.¹¹- 0.0072 0.0022 0.0059 0.0027
½¾/%./¸.¹¹- 0.0023 0.0005 0.0010 0.0006
»¾¿ 2.16 2.08 2.01 2.07
It can be concluded from the above table that the method that gives the least standard deviation and bias is bootstrap tilted with the design distribution }4& . Note however that the designed distribution }4& has the longer simulation times. The simulation time varies from simulation to simulation and is only an indicator of the computational effort needed for the simulations. This method is used in simulations to test how the estimations of the accuracy with bootstrap are performing.
3.4 MBB with Importance sampling
There is no established standard of implying importance sampling in combination with moving block bootstrap. One problem is that the blocks do not correspond to a specific value but J-values. There is however one probability corresponding to each block. By assigning a value to each block that matches the block´s extreme value a tilted distribution can be calculated.
There are four different alternatives tested for what value that represents the extreme value of the blocks. These are the mean, the maximum, the minimum and the first value of the block. In order to determine which alternative that generates most samples from the right side of the distribution a sample of the different alternatives is illustrated in the figure below.
18
Figure 11. A sample of the bootstraps for the three different ways of ranking the blocks extremeness; the maximum (A), the minimum (B), the mean (C) and the first observation (D).
The different alternatives to rank the blocks are presented in Figure 11 with the maximum (A), the minimum (B), the mean (C) and the first observation in each block (D). The tilting method that that generates most extreme losses is used. No alternative stands out as the best but the maximum (A) and the first value of each block (D) is chosen.
One obstacle arises in the need to tilt the distribution. That is how to assign the ratio weight (Equation 18) to the different values in the blocks. This is partially solved due to the fact that relative changes are used, and is calculated by the following equation
Á f1 l ∑ H Y
, Y
D g &H8 . (35)
where Á8 is the vector containing the new weights for the blocks, &H8 is the xth bootstrapped block, is the weight calculated with tilting and à corresponds to the Ãth observation in the block.
The standard deviation with respect to the number of bootstraps for the different designed distributions is calculated. This is executed for the two best alternatives from the figure below. The left part is tilted as before by }4& , }Q& and }¯& using the maximum of each block and the right part in the same way using the first observation of each block.
(A) (B)
(C) (D)
19
Figure 12. The standard deviation of the different design distribution when used with importance sampling for different bootstrap sample sizes.
It is seen from the Figure 12 that the standard deviation decreases as the bootstrap sample size increases. The method that gives the lowest variance is tilting with respect to the maximum value in each block and it is therefore used in the final simulations. The figure shows that larger bootstrap sample size is preferable and a study of sample sizes larger than 600 is therefore interesting.
However, a sample size larger than 300 is time consuming and the sample size is therefore set to 300 in the final simulations.
One way of deciding the block length is to set it to the range of the time series´ time dependency.
The OMXS30 and FTSE100 time series´ dependence is however too long to include in blocks. A study of how the block length affects the standard deviation is illustrated in the figure below.
Figure 13. The standard deviation of the different MBB methods and its dependence of the block length.
Figure 13 shows that the two most efficient methods are MBB with the design distribution }4 and }¯. Since }4 is proven to be the more efficient distribution in previous work (Hesterberg, 1988) it is used in the final simulations. The block length did not significantly affect the standard deviation and it is set to 10 observations since Figure 13 indicates that it gives a stable standard deviation.
200 400 600 800
2 3 4 5 6 7 8 9 10
x 10-3
Bootstrap sample size
Standard deviation
MBB MBB IS, g1(x) MBB IS, g2(x) MBB IS, g3(x)
200 400 600 800
2 3 4 5 6 7 8 9 10
x 10-3
Bootstrap sample size
Standard deviation
MBB MBB IS, g1(x) MBB IS, g2(x) MBB IS, g3(x)
10 20 30 40 50 60 70 80 90 100 110
1.5 2 2.5 3 3.5 4 4.5 5 5.5x 10-3
Block length
Standard deviation
MBB MBB IS, g1(x) MBB IS, g2(x) MBB IS, g3(x)
(A) (B)
20
The expected value, the standard deviation, the bias and the computational time of the different methods are presented in the table below.
Table 3. The results of the different design distributions when tilting with respect to the maximum value in each block.
MBB MBB with importance sampling
°± - }4& }Q& }¯&
º%./¸.¹¹- 0.0412 0.0422 0.0420 0.0420
»¼%./¸.¹¹- 0.0053 0.0019 0.0038 0.0019
½¾/%./¸.¹¹- -0.0003 0.0006 0.0003 0.0004
»¾¿ 7.0 6.9 6.9 6.9
The method that gives the lowest variance and bias is MBB with importance sampling with }¯ (Table 3). MBB with importance sampling with }4 is however used since the difference in performance between }4m and }¯ is negligible. Another advantage using }4 is the possibility to minimize the standard deviation in each time lag by the optimal choice of beta with the method described in section 2.6.2.
3.5 Simulations and evaluations of the models
The methods of the final simulations are presented in this section. The method of obtaining data for the simulations, a summary of the parameters of the bootstrap methods and how the model is evaluated is described.
3.5.1 , -simulations
The notation , ^ is the general notation of the -model. In this thesis the
1,1 -model is used. Equations 29 and 30 from section 2.7 is expressed as
c ¡ (36)
where
cQ Ol 4 4Q l 4c 4Q . (37)
By estimating the parameters of the -model on the time interval %0, T- and with that model simulate new data, data from a known model is obtained. A predicted variance cªI4 is simulated and used to predict Ä number of returns. The is calculated using these predicted values and considered as the reference in comparison with other models.
The initial variance is set to cO4 !!E
(38)
in the first time step. In the rest of the simulations the variance from the previous time step is used.
A root-mean-square-error (RMSE) between the -prediction and the historical data is calculated to determine the accuracy of the -model. This is presented in the table below.
21
Table 4. The RMSE of the -model in comparison with the historical data.
RMSE Historical data 0.0012
The result from Table 4 tells that a 1,1 -model predicts daily changes with short-range dependency with good accuracy.
3.5.2 Summary of the bootstrap methods
From sections 3.3 and 3.4 it is concluded that the design distribution that generates lowest variance is }4. To estimate bootstrap confidence intervals both ordinary bootstrap and MBB are in this case used with daily data simulated by a -model as described in the previous section. The simulations are done with three different choices of for the -calculations, since it is of interest to remark what happens when increasing . The bootstrap confidence intervals are computed with a significance level set to 95 percent.
3.5.3 Evaluation of the methods
The number of exceedances is calculated for the upper and lower limit of the bootstrap interval of
and for the computed with 100000 -predicted values for evaluation of the methods.
The number of exceedances of these curves compared with the -simulated data in each time interval and calculated with Equation 5. The estimated values in each time step creates curves that are compared with a predicted curve from the -model (Appendix 1, Figure 18).
The expected number of exceedances is a concept used in the results. It is explained in the following example. Consider a -curve calculated from a time series of 1252 observation, a set to 0.99 and the based on a sliding interval of 252 observations. This result in a -curve based on 1000 number of -values. By comparing the -curve and the 1000 corresponding observations the number of exceedances can be calculated with Equation 5. This is statistically expected to 1000 1 0.99 10 exceedances.
To evaluate the bootstrap models a comparison of the -simulated and the bootstrap-
with confidence intervals is done. The comparison reflects the performance of the bootstrap confidence interval estimations. Because of this the RMSE is calculated between the -
and closest bootstrap- confidence interval. The error is set to zero if the intervals capture the
-. This follows the formula
ÅÆatÇ ÈÉ Ê∑ [¤ËÌ[XU , ÍÎV,XUi, ÍÎV\\iÉUi, ÏÍÎÏ , V 2 2
8;4 (39)
where &4 and &Q is the upper and lower confidence interval respectively.
The RMSE between the - and the bootstrap is calculated by the standard formula.
22
4 Results
The result of the thesis is presented in this section. Both the results of ordinary bootstrap and MBB are presented with the number of exceedances, the RMSE and the number of times the bootstrap intervals succeed to capture the . A selection from figures in the appendix that illustrates both success and failures of the intervals are also included.
4.1 Bootstrap with importance sampling
The results from the RMSE-calculations are presented below.
Table 5. The RMSE calculated between different ./-curves simulated with ordinary bootstrap with importance sampling compared with the -simulated ./-curve.
Root-mean-square-error of the bootstrap estimations
0 0.90 0.95 0.99
./- and -./-curve 0.006 0.007 0.011
Confidence interval- and -./-curve 0.004 0.004 0.008
The RMSE is non-zero for all significance levels when comparing the - and the bootstrap (Table 5). The RMSE-calculation between the - and the bootstrap confidence intervals that is closest to zero is calculated with O.ÀO and O.ÀÐ.
An example that illustrates scenarios where the bootstrap intervals fail to capture the -curve and therefore contribute to non-zero errors in Table 5 is seen in the following figure
Figure 14. A section from Figure 21 in Appendix 1 that illustrates a section where the bootstrap confidence intervals lack in performance.
-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Date
Rate of Return
BS IS VaR 95 BS IS VaR CI GARCH(1,1) VaR 95
23
Figure 14 comes from a period around 1997, which is a more volatile period (see Appendix 1, section Ordinary bootstrap, Figure 21). As can be seen in Figure 14 the bootstrap intervals fail to contain the
-curve in some time steps, which contribute to a non-zero RMSE in Table 5.
The following figure illustrates an example where the bootstrap confidence intervals succed to capture the .
Figure 15. A section from Figure 21 in Appendix 1 that shows where the bootstrap confidence intervals succed to capture the ./-curve.
Figure 15 comes from a period around 2007, which is a less volatile period (see Appendix 1, section Ordinary bootstrap, Figure 21). The section in Figure 15 illustrates a period where the bootstrap intervals succeed to contain the -curve. It is not contributing to any RMSE in the bootstrap confidence interval part of Table 5. Since the different -curves in the figure are not overlapping the RMSE -curve in Table 5 becomes non-zero.
The following table shows the result of the bootstrap confidence intervals and their ability to capture the . The results are presented as the number of times the bootstrap confidence intervals succeed to capture the expressed in percent.
Table 6. The number of times when the bootstrap confidence intervals fails to capture the ./.
0 0.90 0.95 0.99
Percentage of successful prediction 61 % 62 % 56 %
As can be seen in Table 6 ordinary bootstrap succeed to predict how O.ÀÐ varies in 62 percent of the time steps. Bootstrap predictions of `.Àдs variation succeed the most number of times and of O.ÀÀ´s variation the least.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
Date
Rate of Return
BS IS VaR 95 BS IS VaR CI GARCH(1,1) VaR 95
