Portfolio Risk Analysis - The Effectiveness of different VaR Models in times of Crisis

Portfolio Risk Analysis

FI6012: Assignment

Barry Sheehan

MSc. in Computational Finance

2015

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Title Page

University: University of Limerick

Program: MSc. in Computational Finance

Year of Submission: 2015

Author: Barry Sheehan

Student Number: 0854867

Title: Portfolio Risk Analysis

Word count:

Lecturer: Orla McCullagh

This assignment is solely the work of the author and submitted in partial fulfilment of the

requirements of the MSc. in Computational Finance.

XBarry Sheehan

MSc. in Computational Finance Candidate

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Table of Contents

Title Page ................................................................................................................................... 1

1.1 Introduction .......................................................................................................................... 3

1.2 Research Aims and Hypotheses ....................................................................................... 3

1.3 Hang Seng Index .............................................................................................................. 4

1.4 Minimum Variance Portfolio ........................................................................................... 4

2.1 Preliminary Testing of Returns ............................................................................................ 6

2.2 Descriptive Statistics ........................................................................................................ 6

2.3 Normality of Returns ........................................................................................................ 7

2.4 Autocorrelation of Returns and Stationarity .................................................................... 8

3.1 Value-at-Risk and ETL Models ......................................................................................... 10

3.2 Parametric Normal Linear VaR...................................................................................... 10

3.3 Student-T Linear VaR .................................................................................................... 11

3.4 Generalised Pareto Distribution VaR ............................................................................. 11

3.5 Back-testing over 2008 Global Financial Crisis ............................................................ 13

6.1 Research Findings and Conclusion .................................................................................... 14

Bibliography ............................................................................................................................ 16

Appendix A: Constituent Members of the Hang Seng Index 2004-2008 ............................... 18

.................................................................................................................................................. 18

Appendix B: Minimum Variance Portfolio Weightings and Descriptive Statistics ............... 19

.................................................................................................................................................. 19

Appendix C: Matlab Script File “FI6012_Portfolio_Risk_Analysis_Assignment.m” ............ 20

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1.1 Introduction

Value at Risk (VaR) modelling is a statistical methodology in which financial institutions can

quantify their potential risk of capital loss over a certain risk horizon (Alexander, 2008)

(Doric, 2011), (Kourouma, et al., 2011). Under the Basel II Accord for minimum market risk

capital requirements in financial institutions, the VaR approach was the standard risk

management tool utilized to calculate the level of capital reserves in order to survive financial

crises, specifically, “regulatory capital is based on … the 1% 10-day VaR if the value-at-risk

approach is used. (Alexander, 2008, p. 401)”.

The simplicity of the VaR methodology provides risk managers the ability to provide a

succinct and non-technical estimate of potential downside risk over a future time horizon to a

specified degree of confidence. However, it is the accuracy of its measurement that

incorporates a “challenging statistical problem” (Doric, 2011), with assumptions lending to

significant model risk.

One of the many common criticisms of the use of VaR as a measurement for significant

financial downside risk is its inability to estimate the magnitude of the exceptional losses. For

this reason, this paper includes the Expected Tail Loss (ETL) measure proposed by Arzner et.

al. implying that it provides the “worst conditional expectation measure of risk” (Artzner, et

al., 1999). ETL statistical modelling provides an estimation of the expected loss given that

the loss exceeds the VaR (Alexander, 2008, p. 128).

This paper compares the three Value at Risk and Expected Tail Loss modelling techniques on

the Hang Seng index (HSI) and a Minimum Variance Portfolio (MVP) of the 27 consistent

members of the HSI during the period 2004 – 2008. The models, namely Systematic Normal

Linear, Student-t Linear and Generalized Pareto Distribution, were chosen following a

preliminary empirical analysis of the HSI and MVP returns over the 2004 – 2007 period and

tested over 2008 to highlight the competency of the models when exposed to the Global

Financial Crisis. Integrated into the analysis is a discussion on the advancements in modern

risk modelling, and how the Basel committee intend to correct the shortcomings implicit to

the Basel II Accord which many commentators believe it could have and has contributed to

economic instability (Sollis, 2009), (Danielsson, et al., 2001), (Rebonato, 2010) , (Jorion,

2009)

1.2 Research Aims and Hypotheses

The aim of the paper is two-fold. First, within the scope of the research paper is to determine

the most accurate VaR model of both the HSI and MVP, and what assumptions and

characteristics of each model make them more or less suitable. Second, this author would like

to prove the hypothesis that the MVP created through application of the standard mean-

variance portfolio selection framework developed by Harry Markowitz will outperform the

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Hang Seng Index. (Markowitz, 1952) Outperformance will be measured in terms of an out-

of-sample test through the 2008 financial crisis where the respective daily VaR estimates and

returns will be measured and compared against one another. The optimal weightings of the

MVP will be calculated at 31/12/2007, with the expected daily return set equal to the HSI

equivalent, and weightings held static over the course of the 2008 trading year.

1.3 Hang Seng Index

The Hang Seng Index (HSI) is a capitalization-weighted index of a selection of companies

from the Stock Exchange of Hong Kong. The core elements of the index are divided into four

sub-indices: Commerce and Industry, Finance, Utilities, and Properties (Bloomberg, 2015).

The HSI is included in the analysis as a benchmark of comparison to the Minimum Variance

Portfolio derived from its members. Appendix A shows the 27 consistent members of the

index throughout the window of analysis 2004 – 2008, from which the MVP will be selected.

Figure 1 shows that the number of observations in the sample for portfolio selection and for

preliminary testing and VaR/ETL initial estimation is 989, allowing for the 245 observations

in 2008 to be used in an out-of-sample test for the VaR models.

Figure 1: Trading days per year Hang Seng Index 2004-2008

1.4 Minimum Variance Portfolio

Modern Portfolio Theory (MPT) was first introduced by Harry Markowitz in his paper

‘Portfolio Selection’, in which he introduced the concept of combining the investor’s utility

to maximise expected return whilst minimise the variance on an investment portfolio

(Markowitz, 1952). Prior to this seminal paper, the investment banking industry focused on

finding and investing in what Pachamanova describes as “winners", i.e. assets that appeared

to provide high expected returns irrespective of risk (Pachamanova & Fabozzi, 2010). This

framework of individual asset selection ignored the combined effect each singular asset

would have on the total portfolio return, an aspect identified by Markowitz’s mean-variance

model stating “to reduce risk, it is necessary to avoid a portfolio whose securities are all

highly correlated with each other” (Markowitz, 1959).

Year Count Cumulative

2004 249 249

2005 247 496

2006 247 743

2007 246 989

2008 245 1234

No. of Observations by Year

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Figure 2: The Efficient Frontier of Optimal Portfolios (Alexander, 2008, p. 247)

The mean-variance portfolio optimization model suggested by Markowitz, can be adapted to

suit our analysis in its ability to identify the unique optimal portfolio selection from a large

collection of assets with the lowest risk (variance of return) for any given expected rate of

return, or with the highest expected return for any given risk as illustrated in Figure 2.

Therefore, the portfolio selection was constrained to provide the Minimum Variance Portfolio

(MVT) with an expected daily return greater than or equal to that of the HSI mean return

over the 2004 – 2007 horizon.

The simplifications and assumptions of the standard optimization model proposed by

Markowitz have been subject to similar criticisms to that of traditional VaR models,

implicitly within the assumption that the each of the assets’ returns are normally distributed

and the input parameters are thereby set to the arithmetic mean for expected return and

variance of return for risk. Empirical research has shown that often the distribution of equity

returns deviates from normality exhibiting “excess kurtosis” (Kon, 1984). Non-normality of

returns indicates that the use of the variance of returns is inappropriate (Sharpe, et al., 1991).

Figure 3: Framework of Portfolio Optimisation Methodology (Radlak, 2008) adapted

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The standard mean-variance optimisation model is implemented in Matlab utilizing a

quadratic programming solver i.e. an optimisation problem with a quadratic objective

function (variance-covariance matrix of returns) and linear constraints (expected return

equal to the HSI mean return, no-shorting constraint and portfolio weights/allocations must

sum to one). Several iterative and search algorithms exist to solve for quadratic programming

problems such as the Active Set method, Augmented Lagrangian method and the Conjugate

Gradient method (Nocedal & Wright, 2006)

Appendix B shows the weightings derived from the mean-variance portfolio optimization

constrained to have the same expected return as the Hang Seng Index over the 4 year period

2004 to 2007. It also shows that the standard deviation of daily returns for the MVT is

significantly lower that of the HSI over the 4 year period, implying less risk for the same

return.

2.1 Preliminary Testing of Returns

For the purposes of VaR model selection, and perhaps a view on how the application of

incorrect models could lead to substantial model risk, the distribution of the HSI and MVP

returns over 2004 – 2007 will be analysed by means of a battery of standardised tests for

normality, stationarity and autocorrelation.

2.2 Descriptive Statistics

The distributions of the daily returns for the HSI and MVP show signs of non-normality, with

high levels of excess kurtosis and slight negative skew. The evidence of leptokurtic

distributions indicate that the portfolio and index have a higher probability of realising large

negative outlier returns than the normal distribution would predict.

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Figure 4: Histograms of MPT (left) and HSI (right) Returns 2004 - 2007

.

2.3 Normality of Returns

In order the prove the assumption of the returns following non-normal distributions, the same

model validation process as (Alexander, 2008), (Kourouma, et al., 2011) was followed, where

Jarque-Bera test for normality was conducted on the HSI and MVP daily returns. Alexander

states that “the Jarque–Bera test applies to any random variable whenever we need to justify

an assumption of normality” (Alexander, 2008, p. 158) i.e. the test is conducted under the

null hypothesis that the daily returns are normally distributed.

The results of the tests are incorporated into the QQ-plot (Figure 5), signifying to reject the

null hypothesis that the daily returns for both MVP and HSI follow a normal distribution.

Figure 5 also depicts an abundance of both negative and positive outliers. These “fat-tail”

attributes of the daily returns provide an indication for the suitability of a Student-t

distribution being fitted to both the index and portfolio, as suggested by (Sollis, 2009). Sollis

warned of the implications of assuming normality when a fatter tailed distribution should be

utilized stating it could “lead to an underestimate of the true VaR” (Sollis, 2009).

Furthermore, the prevalence of negative outlier returns encourage the application of Extreme

Value Theory (EVT) for estimation of negative tail risk, focusing solely on the “tail

behaviour of asset returns” (Kourouma, et al., 2011).

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.040

20

40

60

80

100

120

140Histogram of Optimized Portfolio Daily Returns 2004-2007

Optimized Portfolio Daily Returns

Fre

quency

-0.06 -0.04 -0.02 0 0.02 0.04 0.060

20

40

60

80

100

120

140

160

180

200Histogram of Hang Seng Index Daily Returns 2004-2007

Hang Seng Index Daily Returns

Fre

quency

Mean

Return:

0.00078535

Excess

Kurtosis:

1.8595

Skewness:

-0.26298

Mean

Return

0.00078535

Excess

Kurtosis:

2.7252

Skewness:

-0.16533

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Figure 5: Quantile-Quantile Plot of MVT (top) and HSI (bottom)

2.4 Autocorrelation of Returns and Stationarity

The Basel II regulatory framework sets the VaR estimate based on a 10 day risk horizon, and

accepts the use of scaling up the 1-day VaR by multiplying by the square root of 10

(Danielsson, et al., 2001). However, this assumption assumes that the returns are

independently and identically distributed (i.i.d.) i.e. no autocorrelation in returns (Alexander,

2008, p. 21).

The autocorrelation of daily returns were examined qualitatively using correlelograms and

Partial-ACF plots and quantitatively using the “Ljung-Box Q-test”. This popular technique

for testing autocorrelation tests “the null hypothesis for this test is that the first m

autocorrelations are jointly zero” (Ljung & Box, 1978). Figure 6 merges both the illustration

of the ACF and PACF plots indicating statistically significant lags at lag 3 for both the MVP

and HSI. The results of the Ljung-Box Q-tests are also provided in Figure 6, rejecting the

null hypothesis in both cases.

This result implies that the use of the scaling method globally used and accepted under the

Basel II framework will result in under-estimation or over-estimation of the VaR, thereby

increasing the model risk unless measures are taken to transform the time series.

Compounded in the result of significant autocorrelation is the inference that the daily returns

-4 -3 -2 -1 0 1 2 3 4-0.04

-0.02

0

0.02

0.04

Standard Normal Quantiles

Quantile

s o

f O

ptim

ized P

ort

folio D

aily R

etu

rns QQ-Plot of Optimized Portfolio Daily Returns 2004-2007

-4 -3 -2 -1 0 1 2 3 4-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Standard Normal Quantiles

Quantile

s o

f H

SI

Index D

aily R

etu

rns

QQ-Plot of Hang Seng Index Daily Returns 2004-2007

Jarque-Bera p-value:

0.001

Null Hypothesis of

Gaussian returns:

Reject with 95%

Confidence

Jarque-Bera p-value:

0.001

Null Hypothesis of

Gaussian returns:

Reject with 95%

Confidence

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data may be non-stationary, given that “for stationary processes, autocorrelation between

any two observations only depends on the time lag h between them” (Mathworks, 2014).

Figure 6: ACF and PACF plots for MVP (top row) and HSI (bottom row)

This result was investigated using the Augmented Dickey-Fuller and KPSS tests for a unit

root in the daily returns i.e. a time series with a stationary first difference. Both tests reject the

null hypothesis that the MVP and HSI daily returns follow a trend stationary process over the

period 2004 – 2007. By definition, it can be assumed that the mean and variance of both

variables are not finite constants (Alexander, 2008, p. 203).

In summary, the preliminary testing resulted in almost identical attributes for both the MVP

and HSI daily returns 4 year time series’. Both showed leptokurtic distributions with slight

negative skewness, statistically significant autocorrelation, and followed a non-stationary

process.

In terms of VaR model selection, the Parametric Normal Linear VaR and ETL will be

implemented as a benchmark model which will provide a good indication of the

consequences implicit to model risk and the use of risk models which do not accurately

reflect the distribution of returns. As an extension, the Parametric Student-T Linear VaR

model will be applied with a view to improving on the Parametric Normal Linear VaR model

in its ability to capture the “fat tails” (Sollis, 2009). Finally, deviating from the traditional

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

Lag

Sam

ple

Auto

corr

ela

tion

ACF of Optimized Portfolio Daily Returns 2004-2007

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

Lag

Sam

ple

Part

ial A

uto

corr

ela

tions

PACF of Optimized Portfolio Daily Returns 2004-2007

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

Lag

Sam

ple

Auto

corr

ela

tion

ACF of Hang Seng Index Daily Returns 2004-2007

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

Lag

Sam

ple

Part

ial A

uto

corr

ela

tions

PACF of Hang Seng Index Daily Returns 2004-2007

Ljung-Box Q-test p-value:

0.031944

Null Hypothesis of no

Autocorrelation:

Reject with 95%

Confidence

Ljung-Box Q-test p-value:

0.013336

Null Hypothesis of no

Autocorrelation:

Reject with 95%

Confidence

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VaR methodology, the Generalised Pareto Distribution VaR model will be used in order to

view replicate the distribution explicitly at the fat tails.

3.1 Value-at-Risk and ETL Models

In this section, the three VaR and ETL models will be implemented, the assumptions inherent

to each model will be described and related to the attributes considered in Section 2.1 of the

MVP and HSI daily returns. The VaR and ETL model estimates will be reported as at

31/12/2007 (Figure 7 ) and each VaR model will be back-tested and compared over 2008, the

Global Financial Crisis (Figures 8 and 9 ). All VaR and ETL estimates are reported as a

percentage of the portfolio value as per Alexander and parametric formulae may be seen in

Matlab implementation. Discounted mean returns are excluded from the Parametric Normal

and Student-T Linear VaR and ETL estimates as these are assumed to be nil as per the Basel

II Directives.

3.2 Parametric Normal Linear VaR

As summarised in Section 2.4, the Parametric Normal Linear VaR was implemented as a tool

for benchmarking the improved risk model against. The two fundamental assumptions of the

model, “i.i.d. and normally distributed” (Alexander, 2008, p. 18) daily returns are directly

violated by that of the MVP and HSI. With large excess kurtosis exhibited by both sets of

returns, it is expected that this model will underestimate the VaR level.

The equally weighted moving average of the constant variance (implied by stationarity)

results in what Alexander describes as “ghost features”, where the Parametric Normal Linear

VaR will remain high or low for exactly the number of days in the sample period following

an extreme outlier return (Alexander, 2008, p. 20). The effect of “ghost features” can be de-

sensitized by increasing the observations in the sample. Therefore, the back-test implemented

utilized 4 years of historical returns at each iteration in order to re-calculate the covariance

matrix for the updated VaR estimate over the 245 trading days in 2008.

The Parametric Normal Linear VaR performed very poorly, as expected, in the back-tests

with 27 exceedances for the MVP and 34 exceedances for the HSI. Therefore, the model

rejected the null hypothesis of Kupiec’s unconditional coverage test, that the indicator

function has a constant “success” or “exceedance” probability equal to the significance level

chosen, 1%.

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3.3 Student-T Linear VaR

In order to fit the leptokurtic daily returns of both the HSI and MVP, the same methodology

deployed in Section 3.2 was followed, replacing the normal distribution with the Student-T

distribution per Alexander “the normal linear VaR model can be extended to the case where

the portfolio’s returns, or the risk factor returns, have leptokurtic and skewed distributions …

Student t distributed VaR” (Alexander, 2008, p. 129)

This required the additional parameter, degrees of freedom ν, to be estimated by using the

maximum likelihood estimation tool in Matlab to fit the daily returns distribution. This

caused problems in the back-testing capability of the model, as the value of ν contracted

during November 2008 during times of high volatility and outlier negative returns and the

value dipping below 2. This resulted in the Student-T VaR formula providing a complex

number due to the calculation of the square root of a negative number.

Despite this set-back, the Student-T Linear VaR model outperformed the Parametric Normal

Linear VaR in terms of the back-test over the 2008 Financial Crisis, with 27 exceedances for

the HSI model and only 9 exceedances for the MVP model. Despite this increment in

performance relative to its predecessor, the Student-T Linear VaR model was rejected when

tested for unconditional coverage with the Kupiec Test.

Figure 7 highlights a stark increase in both VaR and ETL estimates with the Student-T

Linear model, highlighting its ability to forecast large negative outlier returns with more

precision.

3.4 Generalised Pareto Distribution VaR

As identified in the preliminary testing, the non-normal distribution of the HSI and MVP

daily returns characterised by significant excess kurtosis requires a risk estimation technique

which deviates from the restrictive model assumptions as per the traditional linear VaR and

ETL methodology. Extreme Value Theory is a risk management technique which explicitly

“focuses on the tail behaviour of asset returns” (Kourouma, et al., 2011).

Specifically, the “Generalised Pareto Distribution” (GPD) risk modelling framework

“applies to only those returns above some pre-defined threshold u” (Alexander, 2008, p.

167). The choice of u is highlighted as a potential source of model risk is highlighted by both

Kourouma and Alexander, specifying that it is important to have enough observations below

the specified threshold in order to “obtain a reasonably accurate estimate of the GPD scale

and tail index parameters, β and ξ” (Alexander, 2008, p. 168). Furthermore, Kourouma

points out that there is no satisfactory program which chooses the optimal value for the

threshold u, leaving it to the discretion of the risk manager.

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Given that the sample size of daily returns for both the HSI and MPT only extends 998 days,

the choice of a high threshold u was imperative to the success of this risk model. The

threshold was set at 15%, incorporating 148 left-tail returns in order to make accurate

estimates of the GPD distribution parameters.

On implementation of GPD VaR and ETL models the daily returns are first normalized by

subtracting the sample mean and divided by the standard deviation. The GPD parameters are

then estimated using maximum likelihood estimation in Matlab (Alexander, 2008, p. 169)

Once the GPD VaR and ETL estimates are computed, they “are then de-normalised by

multiplying by the return standard deviation and adding the mean” (Chopping, 2014)

Figure 7 shows the high VaR and ETL estimates produced by the GPD model relative to both

the Student-T Linear model and Parametric Normal Linear Model. In the 2008 back-test the

GPD VaR model dominates the other models for both HSI and MVP, with only 2 and 3

exceedances observed respectively. The GPD Model was also the only model failing to reject

the null hypothesis of unconditional convergence per the Kupiec test performed in Matlab.

Figure 7: VaR and ETL Model Estimates as at 02/01/2008 and Back-Test Results

02/01/2008 - 31/12/2008

Parametric Normal Linear 1% 1-day VaR 1% 1-day ETL

In-Sample Test

Exceedances

Hang Seng Index 2.64% 3.02% 34

Mean Variance Portfolio 2.04% 2.33% 27

Student-T Linear 1% 1-day VaR 1% 1-day ETL

2008 In-Sample Test

Exceedances



Generalized Pareto

Distribution

1% 1-day VaR

(u=15%)

1% 1-day ETL

(u=15%)

2008 In-Sample Test

Exceedances



02/01/2008 Only

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3.5 Back-testing over 2008 Global Financial Crisis

The Basel Directive identifies the importance of backtesting“The competent authorities shall

examine the institution capability to perform back-testing on both actual and hypothetical

changes in the portfolio value.” (Directive 2010/76/EU of the European Parliament and of the

Council, 2010) The Bernoulli back-testing approach was used for the daily VaR estimates on

the three models and applied to the Hang Seng Index and the static Minimum Variance

Portfolio. Bernoulli “successes” or VaR exceedances are highlighted with markers in Figure

8 and Figure 9. This method is analogous to that outlined by Alexander, “Regulators

recommend a very simple type of backtest, which is based on a 1% daily VaR estimate and

which covers a period of only 250 days” (Alexander, 2008).

Figure 8: Back-test of HSI VaR Models over 2008 Global Financial Crisis

Figure 8 illustrates the performance of the three VaR Model estimates over the year of the

2008 Financial Crisis. It is clear that the GPD VaR model was the only methodology accurate

in its forecasting ability in financial downturn. Perhaps this validates Rebonato’s theory

proposed in that the modern risk management framework should be able to utilize “plural

models”, deviating on the traditional methods employed by market participants searching for

the “unique, true model” (Rebonato, 2010).

0 50 100 150 200 250-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.151% Daily Normal Linear VaR, Student-t Linear and GPD VaR vs. Daily Returns of HSI 2008

HSI Daily Returns

HSI Daily 1% Normal Linear VaR

HSI Daily 1% Student-t Linear VaR

HSI Daily 1% GPD VaR

Normal Linear VaR Exceedance

Student-t Linear VaR Exceedance

GPD VaR Exceedance

Number of Normal Linear VaR

Exceedances :

34

Number of Student-t Linear VaR

Exceedances :

28

Number of GPD VaR

Exceedances :

2

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Figure 9: Back-test of MVP VaR Models over 2008 Global Financial Crisis

Figure 9 again depicts the prevalence of the GPD VaR model for MVP back-tested over 2008

over the alternative linear methodology.

6.1 Research Findings and Conclusion

At the outset of this research paper the aims were outlined as follows:

1. Determine the most accurate VaR model of both the HSI and MVP.

It is clear form the back-testing results that the GPD VaR outperforms the alternatives in

terms of providing an accurate assessment of tail risk with a degree of confidence which

holds over the 2008 financial crisis.

2. Prove the hypothesis that the MVP created through application of the standard mean-

variance portfolio selection framework developed by Harry Markowitz will outperform

the Hang Seng Index.

0 50 100 150 200 250-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.151% Daily Normal Linear VaR, Student-t Linear and GPD VaR vs. Daily Returns of Minimum Variance Portfolio 2008

Port Daily Returns

Port Daily 1% Normal Linear VaR

Port Daily 1% Student-t Linear VaR

Port Daily 1% GPD VaR

Normal Linear VaR Exceedance

Student-t Linear VaR Exceedance

GPD VaR Exceedance

Number of Normal Linear VaR

Exceedances :

27

Number of Student-t Linear VaR

Exceedances :

9

Number of GPD VaR

Exceedances :

3

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Figure 10 and Figure 11 implies that this hypothesis has been proven.

Relative Performance 02/01/2008 - 31/12/2008

Equity Group Mean Standard Deviation

Minimum Variance Portfolio -0.14% 2.4%

Hang Seng Index -0.26% 3.2% Figure 10: HSI vs. MVP 2008

Figure 1:HSI vs MVP - VaR and diff returns

-12.00%

-10.00%

-8.00%

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

Minimum Variance Portfolio Versus Hang Seng Index: Net Daily Returns and GPD Value-at-Risk

Portfolio Daily Returns - HIS Daily Returns HSI GPD VaR Portfolio GPD VaR

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MATLAB,@ RISK, or VBA (Vol. 173). s.l.:John Wiley & Sons.

Rebonato, R., 2010. Post-crisis financial risk management: some suggestions. Journal of Risk

Management in Financial Institutions, 3(2), pp. 148-155.

Sharpe, G. J., Alexander, J. & Bailey, V., 1991. Investments. 6 ed. New York: John Wiley.

Sollis, R., 2009. Value at Risk: a critical overview. Journal of Financial Regulation and

Compliance, 17(4), pp. 398 - 414.

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Appendix A: Constituent Members of the Hang Seng Index 2004-2008

Hang Seng Index Members GICS Sector

02/01/2004 31/12/200802/01/2004 -

31/12/2008 101 HK Equity Financials

5 HK Equity 5 HK Equity 101 HK Equity 1038 HK Equity Utilities

941 HK Equity 941 HK Equity 1038 HK Equity 11 HK Equity Financials

13 HK Equity 2628 HK Equity 11 HK Equity 1199 HK Equity Industrials


2388 HK Equity 388 HK Equity 12 HK Equity 1249772D HK Equity Financials

16 HK Equity 16 HK Equity 1249772D HK Equity 13 HK Equity Industrials

1249772D HK Equity 939 HK Equity 13 HK Equity 16 HK Equity Financials

883 HK Equity 1249772D HK Equity 16 HK Equity 19 HK Equity Financials





12 HK Equity 3328 HK Equity 267 HK Equity 291 HK Equity Consumer Staples



293 HK Equity 330 HK Equity 3 HK Equity 330 HK Equity Consumer Discretionary







551 HK Equity 23 HK Equity 66 HK Equity 762 HK Equity Telecommunication Services


101 HK Equity 101 HK Equity 8 HK Equity 883 HK Equity Energy


97 HK Equity 17 HK Equity 941 HK Equity

992 HK Equity 83 HK Equity GICS Sector Count

1199 HK Equity 688 HK Equity Financials 10

20 HK Equity 6 HK Equity Utilities 4

291 HK Equity 4 HK Equity Industrials 5

511 HK Equity 762 HK Equity Consumer Staples 1

363 HK Equity 267 HK Equity Consumer Discretionary 3

144 HK Equity Telecommunication Services 3

CHNETQ HK Equity Energy 1

66 HK Equity Total Members 27

700 HK Equity

291 HK Equity

2038 HK Equity

293 HK Equity

8 HK Equity

1199 HK Equity

551 HK Equity

1038 HK Equity

Hang Seng Index Members by Date

*Consistent Members

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Appendix B: Minimum Variance Portfolio Weightings and Descriptive Statistics

Portfolio optimisation obtained the following:

EquityMinimum Variance

Portfolio Weightings

101 HK Equity 0.00% Equity Optimized Weightings Mean Std Dev Skewness Excess Kurtosis

1038 HK Equity 0.00% 11 HK Equity 2.94% 0.05% 1.09% 142.25% 13.06

11 HK Equity 2.94% 2 HK Equity 30.80% 0.04% 0.88% -2.55% 2.84



1249772D HK Equity 0.00% 330 HK Equity 6.39% 0.17% 2.17% 55.47% 6.05

13 HK Equity 0.00% 6 HK Equity 12.71% 0.04% 1.02% -39.06% 4.06




23 HK Equity 3.22% Total Weight 100.00%

2388 HK Equity 0.00%

267 HK Equity 0.00%

291 HK Equity 0.00% ** Figures Based on returns in date range 02/01/2004 - 31/12/2007

293 HK Equity 0.00%

3 HK Equity 16.62%

330 HK Equity 6.39% Index Mean Std Dev Skewness Excess Kurtosis

4 HK Equity 0.00% Hang Seng Index 0.079% 1.13% -16.53% 2.73

494 HK Equity 0.00% ** Figures Based on returns in date range 02/01/2004 - 31/12/2007

5 HK Equity 0.00%

551 HK Equity 0.00%

6 HK Equity 12.71%

66 HK Equity 14.92%

762 HK Equity 0.00%

8 HK Equity 0.00%

883 HK Equity 3.16%

941 HK Equity 9.22%

Hang Seng Index Descriptive Statistics

*Portfolio Mean Daily Return = 0.079%

Minimum Variance Portfolio Descriptive Statistics

* Portfolio Standard Deviation = 0.876%

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Appendix C: Matlab Script File “FI6012_Portfolio_Risk_Analysis_Assignment.m”

% FI6012: Portfolio Risk Analysis % Barry Sheehan 0854867 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 1) Initialise Share Price Data: % All data taken from Bloomberg on 30/03/2015 % Daily HSI Share and Index values from 02/01/2004 - 31/12/2008 HSI_Index = xlsread('Hang_Seng_Data.xlsx','HSI_Index');% 1234 by 1 matrix HSI_Share = xlsread('Hang_Seng_Data.xlsx','HSI_Share');% 1234 by 27 matrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 2) Calculate Daily Returns, Expected Returns, Covariance: HSI_Index_Ret = price2ret(HSI_Index); HSI_Share_Ret = price2ret(HSI_Share); % Mean/Expected Return over first 4 years: Exp_Daily_Return_HSI_Index = mean(HSI_Index_Ret(1:988,1)); Exp_Daily_Return_HSI_Share = mean(HSI_Share_Ret(1:988,:)); % Covariance Matrix Return over first 4 years: Share_Cov = cov(HSI_Share_Ret(1:988,:)); % Test Share Covariance Matrix for Symmetry: if norm(Share_Cov-Share_Cov')>0 errmsg = 'Error: Covariance Matrix is not symmetric'; else disp('Covariance Matrix is Symmetric') end Eigenvalues=eig(Share_Cov); if min(Eigenvalues)<0 errmsg = 'Covariance Matrix is not Positive Definite'; else disp('Covariance Matrix is Positive Definite') end % Result: Covariance Matrix is Positive Defnite and Symmetric. % Therefore, solution to portfolio optimisation will be a global minimum. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 3) Create Optimal Mean-Variance Portfolio Weightings with same % Expected Daily Return as Hang Seng Index % Objective Function: 0.5(x'.Cov.x) Cov=2*Share_Cov; % Optimisation Constraints: % i) Return Constraint: -Mean.x<=-R (inequality format accepted in quadprog % functionality) Mean=-Exp_Daily_Return_HSI_Share; R=-Exp_Daily_Return_HSI_Index; % ii) Sum of Weights must equal one: e'.x=1 Aeq=ones(1,length(Exp_Daily_Return_HSI_Share)); beq=1; % iii) No Short-selling Allowed lb=zeros(length(Exp_Daily_Return_HSI_Share),1); % Mean-variance Portfolio Optimisation: % Minimize Variance s.t. % i) Expected Return = HSI Index annualised Ret % ii) Sum of Portfolio Weights = 1 % iii)No Short-Selling Allowed i.e. Portfolio Weights >=0 [PortfolioWeights]=quadprog(Cov,[],Mean,R,Aeq,beq,lb,[],[],[]); % Theoretical Min Variance Portfolio Daily Returns: (2004 - 2007): Hist_Port_DailyRet = PortfolioWeights'*HSI_Share_Ret(1:988,:)'; % Min Variance Portfolio Expected Daily Returns and Std Dev: Portfolio_Exp_SD = sqrt(PortfolioWeights'*Share_Cov*PortfolioWeights); Portfolio_Exp_Return = PortfolioWeights'*Exp_Daily_Return_HSI_Share'; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 4) Create Histograms for Returns Data % i) Plot histogram of daily Optimized Portfolio Daily Returns 2004 - 2007: figure subplot(1,2,1) histfit(Hist_Port_DailyRet) title('Histogram of Optimized Portfolio Daily Returns 2004-2007') xlabel('Optimized Portfolio Daily Returns') ylabel('Frequency')

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str1 = {'Mean Return:' ,num2str(mean(Hist_Port_DailyRet)),... 'Excess Kurtosis:' ,num2str(kurtosis(Hist_Port_DailyRet)-3),... 'Skewness: ' ,num2str(skewness(Hist_Port_DailyRet))}; annotation('textbox', [0.36,0.7,0.055,0.21],'String', str1); % ii) Plot histogram of daily Hang Seng Index Daily Returns 2004 - 2007: subplot(1,2,2) histfit(HSI_Index_Ret(1:988,1)) title('Histogram of Hang Seng Index Daily Returns 2004-2007') xlabel('Hang Seng Index Daily Returns') ylabel('Frequency') str2 = {'Mean Return' ,num2str(mean(HSI_Index_Ret(1:988,1))),... 'Excess Kurtosis:' ,num2str(kurtosis(HSI_Index_Ret(1:988,1))-3),... 'Skewness: ' ,num2str(skewness(HSI_Index_Ret(1:988,1)))}; annotation('textbox', [0.8,0.7,0.055,0.21],'String', str2); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 5) Time-series Testing: % i) Jarque-Bera Test for Normality [h1,p1] = jbtest(Hist_Port_DailyRet); if h1==1 h1_String='Reject with 95% Confidence'; else h1_String='Fail to reject with 95% Confidence'; end [h2,p2] = jbtest(HSI_Index_Ret(1:988,1)); if h2==1 h2_String='Reject with 95% Confidence'; else h2_String='Fail to reject with 95% Confidence'; end % ii) QQ-Plots for Visual illustration of Normality figure subplot(2,1,1) qqplot(Hist_Port_DailyRet) title('QQ-Plot of Optimized Portfolio Daily Returns 2004-2007') ylabel('Quantiles of Optimized Portfolio Daily Returns') str1 = {'Jarque-Bera p-value: ' ,num2str(p1),... 'Null Hypothesis of Gaussian returns: ',h1_String}; annotation('textbox', [0.75,0.65,0.10,0.154],'String', str1); subplot(2,1,2) qqplot(HSI_Index_Ret(1:988,1)) title('QQ-Plot of Hang Seng Index Daily Returns 2004-2007') ylabel('Quantiles of HSI Index Daily Returns') str2 = {'Jarque-Bera p-value: ' ,num2str(p2),... 'Null Hypothesis of Gaussian returns: ',h2_String}; annotation('textbox', [0.75,0.15,0.1,0.154],'String', str2); % Result of i) and ii): Both Hang Seng Index and Optimised Portfolio % Returns Non-Normally Distributed with 95% confidence %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % iii) Test for Autocorrelation % % a) Ljung-Box Q-test for residual autocorrelation Portfolio_Residual = Hist_Port_DailyRet - mean(Hist_Port_DailyRet); [h3,p3] = lbqtest(Portfolio_Residual); if h3==1 h3_String='Reject with 95% Confidence'; else h3_String='Fail to reject with 95% Confidence'; end Index_Residual = HSI_Index_Ret(1:988,1) - mean(HSI_Index_Ret(1:988,1)); [h4,p4] = lbqtest(Index_Residual); if h4==1 h4_String='Reject with 95% Confidence'; else h4_String='Fail to reject with 95% Confidence'; end % b) ACF and PACF plots: figure subplot(2,2,1) autocorr(Hist_Port_DailyRet) title('ACF of Optimized Portfolio Daily Returns 2004-2007') str3 = {'Ljung-Box Q-test p-value: ' ,num2str(p3),... 'Null Hypothesis of no Autocorrelation: ',h3_String}; annotation('textbox', [0.3,0.745,0.125,0.16],'String', str3, 'color','b');

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subplot(2,2,2) parcorr(Hist_Port_DailyRet) title('PACF of Optimized Portfolio Daily Returns 2004-2007') subplot(2,2,3) autocorr(HSI_Index_Ret(1:988,1)) title('ACF of Hang Seng Index Daily Returns 2004-2007') str4 = {'Ljung-Box Q-test p-value: ' ,num2str(p4),... 'Null Hypothesis of no Autocorrelation: ',h4_String}; annotation('textbox', [0.3,0.27,0.125,0.16],'String', str4, 'color','b'); subplot(2,2,4) parcorr(HSI_Index_Ret(1:988,1)) title('PACF of Hang Seng Index Daily Returns 2004-2007') % Result of iii): % Reject Null Hypothesis that both "Hang Seng Index" and "Min Variance % Portfolio are not autocorrelated with 95% confidence %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % iv) Test for Stationarity % Dickey-Fuller test for stationarity h5 = adftest(Hist_Port_DailyRet); h6 = adftest(HSI_Index_Ret(1:988,1)); % KPSS test for stationarity h7 = kpsstest(Hist_Port_DailyRet); h8 = kpsstest(diff(HSI_Index_Ret(1:988,1))); % Result of iv): Reject the null hypothesis of a unit root % for both Hang Seng Index and Optimised Portfolio Returns with 95% % confidence. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 6) Estimate Systematic Normal and Student t Linear VaR on Hang Seng Index % and Minimum Variance Portfolio: Alpha=0.01; % Significance Level: as specified by Basel II % i) Hang Seng Index Systematic Normal Linear 1-day 1% VaR VaR_HSI_NormLinear_1day = norminv((1-Alpha))*std(HSI_Index_Ret(1:988,1)); % Hang Seng Index Systematic Normal Linear 1-day 1% ETL % (Eqn: Ref. Alecander, C. Vol.4 pg 129) ETL_HSI_NormLinear_1day = (normpdf(norminv(Alpha,0,1))... *std(HSI_Index_Ret(1:988,1)))/Alpha; % ii) Minimum Variance Normal Linear 1-day 1% VaR VaR_Port_NormLinear_1day = norminv((1-Alpha))*Portfolio_Exp_SD; % Minimum Variance Normal Linear 1-day 1% ETL ETL_Port_NormLinear_1day = (normpdf(norminv(Alpha,0,1))... *Portfolio_Exp_SD)/Alpha; % iii) HSI Index Systematic Student-t 1-day 1% VaR %Degrees of Freedom parameter estimated by fitting distribution using MLE V1=mle(HSI_Index_Ret(1:988,:),'distribution','tlocationscale'); VaR_HSI_StudentT_1day=sqrt((V1(3)-2)/V1(3))*tinv(1-Alpha,V1(3))*... std(HSI_Index_Ret(1:988,1)); % HSI Index Systematic Student-t 1-day 1% ETL % (Eqn: Ref. Alecander, C., Vol.4 pg 130) ETL_HSI_StudentT_1day=((V1(3)- 2 +tinv(Alpha,V1(3))^2)* ... tpdf(tinv(Alpha,V1(3)),V1(3))*std(HSI_Index_Ret(1:988,1)))/... (Alpha*(V1(3)-1)); % iv) Minimum Variance Systematic Student-t 1-day 1% VaR %Degrees of Freedom parameter estimated by fitting distribution using MLE V2=mle(Hist_Port_DailyRet,'distribution','tlocationscale'); VaR_Portfolio_StudentT_1day=sqrt((V2(3)-2)/V2(3))*tinv(1-Alpha,V2(3))...

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*Portfolio_Exp_SD; % Minimum Variance Systematic Student-t 1-day 1% ETL ETL_Portfolio_StudentT_1day=((V2(3)- 2 +tinv(Alpha,V2(3))^2)* ... tpdf(tinv(Alpha,V2(3)),V2(3))*Portfolio_Exp_SD)/... (Alpha*(V2(3)-1)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 7) GPD VaR: %i) Hang Seng Index GPD VaR: NormalisedRet_HSI=(HSI_Index_Ret(1:988,1)-mean(HSI_Index_Ret(1:988,1)))/... std(HSI_Index_Ret(1:988,1)); % Normalised Returns: Mean=0, Variance=1 %set threshold u=15% u_15perc_HSI=prctile(NormalisedRet_HSI,15); ThresholdU1=NormalisedRet_HSI(NormalisedRet_HSI<u_15perc_HSI); % Maximum Likelihood Parameters for GPD Parameters GPD_HSI_Param = gpfit(abs(ThresholdU1)); VaR_HSI_GPDNorm_1day = u_15perc_HSI+(GPD_HSI_Param(2)/(GPD_HSI_Param(1)*... (length(NormalisedRet_HSI))^GPD_HSI_Param(1)))*((length(ThresholdU1)... /(1-Alpha))^GPD_HSI_Param(1) - length(NormalisedRet_HSI)... ^GPD_HSI_Param(1)); % De-normalise VaR Estimate VaR_HSI_GPD_1day=VaR_HSI_GPDNorm_1day* std(HSI_Index_Ret(1:988,1))+... mean(HSI_Index_Ret(1:988,1)); % Hang Seng Index GPD ETL: ETL_HSI_GPDNorm_1day = VaR_HSI_GPDNorm_1day+(GPD_HSI_Param(2)... -GPD_HSI_Param(1)*(VaR_HSI_GPDNorm_1day))... /(1-GPD_HSI_Param(1)); % De-normalise ETL Estimate ETL_HSI_GPD_1day=ETL_HSI_GPDNorm_1day* std(HSI_Index_Ret(1:988,1))+... mean(HSI_Index_Ret(1:988,1)); %ii) Minimum Variance Portfolio GPD VaR: %set threshold u=15% NormalisedRet_Port=(Hist_Port_DailyRet(:)-mean(Hist_Port_DailyRet))/... std(Hist_Port_DailyRet); % Normalised Returns: Mean=0, Variance=1 u_15perc_Port=prctile(NormalisedRet_Port,15); ThresholdU2=NormalisedRet_Port(NormalisedRet_Port<u_15perc_Port); % Maximum Likelihood Parameters for GPD Parameters GPD_Port_Param = gpfit(abs(ThresholdU2)); VaR_Port_GPDNorm_1day = u_15perc_Port+(GPD_Port_Param(2)/(GPD_Port_Param(1)*... (length(Hist_Port_DailyRet))^GPD_Port_Param(1)))*((length(ThresholdU2)... /(1-Alpha))^GPD_Port_Param(1) - length(Hist_Port_DailyRet)... ^GPD_Port_Param(1)); % De-normalise VaR Estimate VaR_Port_GPD_1day=VaR_Port_GPDNorm_1day* std(Hist_Port_DailyRet)+... mean(Hist_Port_DailyRet); % Minimum Variance Portfolio GPD ETL: ETL_Port_GPDNorm_1day = VaR_Port_GPDNorm_1day+(GPD_Port_Param(2)... -GPD_Port_Param(1)*(VaR_Port_GPDNorm_1day))... /(1-GPD_Port_Param(1)); % De-normalise ETL Estimate ETL_Port_GPD_1day=ETL_Port_GPDNorm_1day* std(Hist_Port_DailyRet)+... mean(Hist_Port_DailyRet); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 7) Bernoulli VaR In-Sample Backtesting: % Test VaR Models over 2008 period and most accurate VaR model Test_Period=length(HSI_Index_Ret); Estimation_Window=Test_Period-245;

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% i) Hang Seng Index Systematic Normal Linear 1-day 1% VaR VaR_HSI_NormLinear_2008TS=zeros(Test_Period,1); for t= Estimation_Window+1:Test_Period; t1=t-Estimation_Window; t2=t-1; Window=(HSI_Index_Ret(t1:t2)); VaR_HSI_NormLinear_2008TS(t)=norminv(1-Alpha)*std(Window); end % Points where daily returns exceeds daily 1% VaR Estimate Bernoulli_Exceed_Point1=find(HSI_Index_Ret(Estimation_Window+1:... Test_Period)<-VaR_HSI_NormLinear_2008TS(Estimation_Window+1:... Test_Period)); Bernoulli_Exceed_Returns1=NaN(245,1); Bernoulli_Exceed_Returns1(Bernoulli_Exceed_Point1)=HSI_Index_Ret... (Bernoulli_Exceed_Point1+Estimation_Window); % Bernoulli Exceedance Counting formula Exceedance_HSI_NL=length(Bernoulli_Exceed_Point1); % Kupiec’s (1995) unconditional coverage test n=Test_Period-Estimation_Window; n1=Exceedance_HSI_NL; n0=n-n1; MUexpected=Alpha; MUobserved=n1/n; LR1= ((MUexpected^n1)*((1-MUexpected)^n0))/((MUobserved^n1)*... ((1-MUobserved)^n0)); %Likelihood Ratio if -2*log(LR1)>6.6349 Kupiec1 = 1;% Reject, at 1% Confidence level, null hypothesis of a % constant "success" probapility equal to Alpha else Kupiec1 = 0; % Fail to reject Null Hypothesis end % ii) Hang Seng Index Systematic Student-t Linear 1-day 1% VaR VaR_HSI_StudentT_2008TS=zeros(Test_Period,1); for t= Estimation_Window+1:Test_Period; t1=t-Estimation_Window; t2=t-1; Window=(HSI_Index_Ret(t1:t2)); VaR_HSI_StudentT_2008TS(t)=sqrt((V1(3)-2)/V1(3))*tinv(1-Alpha,V1(3))*... std(Window); end Bernoulli_Exceed_Point2=find(HSI_Index_Ret(Estimation_Window+1:... Test_Period)<-VaR_HSI_StudentT_2008TS(Estimation_Window+1:... Test_Period)); Bernoulli_Exceed_Returns2=NaN(245,1); Bernoulli_Exceed_Returns2(Bernoulli_Exceed_Point2)=HSI_Index_Ret... (Bernoulli_Exceed_Point2+Estimation_Window); % Bernoulli Exceedance Counting formula Exceedance_HSI_ST=length(Bernoulli_Exceed_Point2); % Kupiec’s (1995) unconditional coverage test n=Test_Period-Estimation_Window; n1=Exceedance_HSI_ST; n0=n-n1; MUexpected=Alpha; MUobserved=n1/n; LR2= ((MUexpected^n1)*((1-MUexpected)^n0))/((MUobserved^n1)*... ((1-MUobserved)^n0)); %Likelihood Ratio if -2*log(LR2)>6.6349 Kupiec2 = 1;% Reject, at 1% Confidence level, null hypothesis of a % constant "success" probapility equal to Alpha else Kupiec2 = 0; % Fail to reject Null Hypothesis end

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% iii) Hang Seng Index GPD 1-day 1% VaR,threshold=15% VaR_HSI_GPD_2008TS=zeros(Test_Period,1); for t= Estimation_Window+1:Test_Period; t1=t-Estimation_Window; t2=t-1; Window=(HSI_Index_Ret(t1:t2,1)-mean(HSI_Index_Ret(t1:t2,1)))/... std(HSI_Index_Ret(t1:t2,1)); u_15perc_HSI=prctile(Window,15); ThresholdU=Window(Window<u_15perc_HSI); GPD_HSI_Param = gpfit(abs(ThresholdU)); VaR_HSI_GPDNorm_2008TS=u_15perc_HSI+(GPD_HSI_Param(2)/(GPD_HSI_Param(1)*... (length(Window))^GPD_HSI_Param(1)))*((length(ThresholdU)... /(1-Alpha))^GPD_HSI_Param(1) - length(Window)... ^GPD_HSI_Param(1)); VaR_HSI_GPD_2008TS(t)=VaR_HSI_GPDNorm_2008TS*... std(HSI_Index_Ret(t1:t2,1))+mean(HSI_Index_Ret(t1:t2,1)); end % Points where daily returns exceeds daily 1% VaR Estimate Bernoulli_Exceed_Point3=find(HSI_Index_Ret(Estimation_Window+1:... Test_Period)<VaR_HSI_GPD_2008TS(Estimation_Window+1:... Test_Period)); Bernoulli_Exceed_Returns3=NaN(245,1); Bernoulli_Exceed_Returns3(Bernoulli_Exceed_Point3)=HSI_Index_Ret... (Bernoulli_Exceed_Point3+Estimation_Window); % Bernoulli Exceedance Counting formula Exceedance_HSI_GPD=length(Bernoulli_Exceed_Point3); % Kupiec’s (1995) unconditional coverage test n=Test_Period-Estimation_Window; n1=Exceedance_HSI_GPD; n0=n-n1; MUexpected=Alpha; MUobserved=n1/n; LR3= ((MUexpected^n1)*((1-MUexpected)^n0))/((MUobserved^n1)*... ((1-MUobserved)^n0)); %Likelihood Ratio if -2*log(LR3)>6.6349 Kupiec3 = 1;% Reject, at 1% Confidence level, null hypothesis of a % constant "success" probapility equal to Alpha else Kupiec3 = 0; % Fail to reject Null Hypothesis end % Plot results of Bernoulli exceedance backtest figure plot(HSI_Index_Ret(Estimation_Window+1:Test_Period)) hold on plot(-VaR_HSI_NormLinear_2008TS(Estimation_Window+1:Test_Period),'--r') plot(-VaR_HSI_StudentT_2008TS(Estimation_Window+1:Test_Period),'--g') plot(VaR_HSI_GPD_2008TS(Estimation_Window+1:Test_Period),'--m') plot(Bernoulli_Exceed_Returns1,'rv','MarkerSize',10) hold on plot(Bernoulli_Exceed_Returns2,'g^','MarkerSize',10) hold on plot(Bernoulli_Exceed_Returns3,'m+','MarkerSize',20) title('1% Daily Normal Linear VaR, Student-t Linear and GPD VaR vs. Daily Returns of HSI 2008') legend('HSI Daily Returns','HSI Daily 1% Normal Linear VaR',... 'HSI Daily 1% Student-t Linear VaR','HSI Daily 1% GPD VaR',... 'Normal Linear VaR Exceedance','Student-t Linear VaR Exceedance', ... 'GPD VaR Exceedance') str3 = {'Number of Normal Linear VaR Exceedances : ' ... ,num2str(Exceedance_HSI_NL),'Number of Student-t Linear VaR Exceedances :'... ,num2str(Exceedance_HSI_ST),'Number of GPD VaR Exceedances :'... ,num2str(Exceedance_HSI_GPD)}; annotation('textbox', [0.2,0.15,0.15,0.24],'String', str3); % iv) Minimum Variance Portfolio Systematic Normal Linear 1-day 1% VaR Sample_Port_DailyRet = (PortfolioWeights'*HSI_Share_Ret(:,:)')'; VaR_Port_NormLinear_2008TS=zeros(Test_Period,1);

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for t= Estimation_Window+1:Test_Period; t1=t-Estimation_Window; t2=t-1; Portfolio_SD = sqrt(PortfolioWeights'*cov(HSI_Share_Ret(t1:t2,:))... *PortfolioWeights); VaR_Port_NormLinear_2008TS(t)=norminv(1-Alpha)*Portfolio_SD; end % Points where daily returns exceeds daily 1% VaR Estimate Bernoulli_Exceed_Point4=find(Sample_Port_DailyRet(Estimation_Window+1:... Test_Period)<-VaR_Port_NormLinear_2008TS(Estimation_Window+1:... Test_Period)); Bernoulli_Exceed_Returns4=NaN(245,1); Bernoulli_Exceed_Returns4(Bernoulli_Exceed_Point4)=Sample_Port_DailyRet... (Bernoulli_Exceed_Point4+Estimation_Window); % Bernoulli Exceedance Counting formula Exceedance_Port_NL=length(Bernoulli_Exceed_Point4); % Kupiec’s (1995) unconditional coverage test n=Test_Period-Estimation_Window; n1=Exceedance_Port_NL; n0=n-n1; MUexpected=Alpha; MUobserved=n1/n; LR4= ((MUexpected^n1)*((1-MUexpected)^n0))/((MUobserved^n1)*... ((1-MUobserved)^n0)); %Likelihood Ratio if -2*log(LR4)>6.6349 Kupiec4 = 1;% Reject, at 1% Confidence level, null hypothesis of a % constant "success" probapility equal to Alpha else Kupiec4 = 0; % Fail to reject Null Hypothesis end % v) Minimum Variance Portfolio Systematic Student-t Linear 1-day 1% VaR VaR_Port_StudentT_2008TS=zeros(Test_Period,1); for t= Estimation_Window+1:Test_Period; t1=t-Estimation_Window; t2=t-1; Portfolio_SD = sqrt(PortfolioWeights'*cov(HSI_Share_Ret(t1:t2,:))... *PortfolioWeights); VaR_Port_StudentT_2008TS(t)=sqrt((V2(3)-2)/V2(3))*tinv(1-Alpha,V1(3))*... Portfolio_SD; end % Points where Portfolio daily returns exceeds daily 1% VaR Estimate Bernoulli_Exceed_Point5=find(Sample_Port_DailyRet(Estimation_Window+1:... Test_Period)<-VaR_Port_StudentT_2008TS(Estimation_Window+1:... Test_Period)); Bernoulli_Exceed_Returns5=NaN(245,1); Bernoulli_Exceed_Returns5(Bernoulli_Exceed_Point5)=... Sample_Port_DailyRet(Bernoulli_Exceed_Point5+Estimation_Window); % Bernoulli Exceedance Counting formula Exceedance_Port_ST=length(Bernoulli_Exceed_Point5); % Kupiec’s (1995) unconditional coverage test n=Test_Period-Estimation_Window; n1=Exceedance_Port_ST; n0=n-n1; MUexpected=Alpha; MUobserved=n1/n; LR5= ((MUexpected^n1)*((1-MUexpected)^n0))/((MUobserved^n1)*... ((1-MUobserved)^n0)); %Likelihood Ratio if -2*log(LR5)>6.6349 Kupiec5 = 1;% Reject, at 1% Confidence level, null hypothesis of a % constant "success" probapility equal to Alpha else Kupiec5 = 0; % Fail to reject Null Hypothesis end % vi) Minimum Variance Portfolio GPD 1-day 1% VaR,threshold=15%

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VaR_Port_GPD_2008TS=zeros(Test_Period,1); %set threshold u=15% for t= Estimation_Window+1:Test_Period; t1=t-Estimation_Window; t2=t-1; Window1=(PortfolioWeights'*HSI_Share_Ret(t1:t2,:)')'; % Normalise Returns over Estimation Window Window2=(Window1(:)-mean(Window1))/std(Window1); u_15perc_Port=prctile(Window2,15); ThresholdU=Window2(Window2<u_15perc_Port); GPD_Port_Param = gpfit(abs(ThresholdU)); VaR_Port_GPDNorm_2008TS=u_15perc_Port+(GPD_Port_Param(2)/... (GPD_Port_Param(1)*(length(Window2))^GPD_Port_Param(1)))*... ((length(ThresholdU)/(1-Alpha))^GPD_Port_Param(1) - length(Window2)... ^GPD_Port_Param(1)); VaR_Port_GPD_2008TS(t)=VaR_Port_GPDNorm_2008TS* std(Window1)+... mean(Window1); end % Points where daily returns exceeds daily 1% VaR Estimate Bernoulli_Exceed_Point6=find(Sample_Port_DailyRet(Estimation_Window+1:... Test_Period)<VaR_Port_GPD_2008TS(Estimation_Window+1:... Test_Period)); Bernoulli_Exceed_Returns6=NaN(245,1); Bernoulli_Exceed_Returns6(Bernoulli_Exceed_Point6)=Sample_Port_DailyRet... (Bernoulli_Exceed_Point6+Estimation_Window); % Bernoulli Exceedance Counting formula Exceedance_Port_GPD=length(Bernoulli_Exceed_Point6); % Kupiec’s (1995) unconditional coverage test n=Test_Period-Estimation_Window; n1=Exceedance_Port_GPD; n0=n-n1; MUexpected=Alpha; MUobserved=n1/n; LR6= ((MUexpected^n1)*((1-MUexpected)^n0))/((MUobserved^n1)*... ((1-MUobserved)^n0)); %Likelihood Ratio if -2*log(LR6)>6.6349 Kupiec6 = 1;% Reject, at 1% Confidence level, null hypothesis of a % constant "success" probapility equal to Alpha else Kupiec6 = 0; % Fail to reject Null Hypothesis end % Plot results of Bernoulli exceedance backtest figure plot(Sample_Port_DailyRet(Estimation_Window+1:Test_Period)) hold on plot(-VaR_Port_NormLinear_2008TS(Estimation_Window+1:Test_Period),'--r') plot(-VaR_Port_StudentT_2008TS(Estimation_Window+1:Test_Period),'--g') plot(VaR_Port_GPD_2008TS(Estimation_Window+1:Test_Period),'--m') plot(Bernoulli_Exceed_Returns4,'rv','MarkerSize',10) hold on plot(Bernoulli_Exceed_Returns5,'g^','MarkerSize',10) hold on hold on plot(Bernoulli_Exceed_Returns6,'m+','MarkerSize',20) title('1% Daily Normal Linear VaR, Student-t Linear and GPD VaR vs. Daily Returns of Minimum Variance Portfolio 2008') legend('Port Daily Returns','Port Daily 1% Normal Linear VaR',... 'Port Daily 1% Student-t Linear VaR','Port Daily 1% GPD VaR',... 'Normal Linear VaR Exceedance','Student-t Linear VaR Exceedance', ... 'GPD VaR Exceedance') str4 = {'Number of Normal Linear VaR Exceedances : ' ... ,num2str(Exceedance_Port_NL),'Number of Student-t Linear VaR Exceedances :'... ,num2str(Exceedance_Port_ST),'Number of GPD VaR Exceedances :'... ,num2str(Exceedance_Port_GPD)}; annotation('textbox', [0.2,0.15,0.15,0.24],'String', str4);

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Covariance Matrix is Symmetric Covariance Matrix is Positive Definite Warning: P is less than the smallest tabulated value, returning 0.001. Warning: P is less than the smallest tabulated value, returning 0.001. Warning: Maximum likelihood has converged to an estimate of K < -1/2. Confidence intervals and standard errors can not be computed reliably. Warning: Maximum likelihood has converged to an estimate of K < -1/2. Confidence intervals and standard errors can not be computed reliably. Warning: Maximum likelihood has converged to an estimate of K < -1/2.

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.040

20

40

60

80

100

120

140Histogram of Optimized Portfolio Daily Returns 2004-2007

Optimized Portfolio Daily Returns

Fre

quency

-0.06 -0.04 -0.02 0 0.02 0.04 0.060

20

40

60

80

100

120

140

160

180

200Histogram of Hang Seng Index Daily Returns 2004-2007

Hang Seng Index Daily Returns

Fre

quency

Mean

Return:

0.00078535

Excess

Kurtosis:

1.8595

Skewness:

-0.26298

Mean

Return

0.00078535

Excess

Kurtosis:

2.7252

Skewness:

-0.16533

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Published with MATLAB® 7.14

Date post:	17-Aug-2015
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Portfolio Risk Analysis - The Effectiveness of different VaR Models in times of Crisis

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