COMPARING ESTIMATORS OF VAR AND CVAR UNDER

THE ASYMMETRIC LAPLACE DISTRIBUTION

By

HSIAO-HSIANG HSU

A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN STATISTICS

UNIVERSITY OF FLORIDA

2005

Copyright 2005

by

Hsiao-Hsiang Hsu

To my parents, brother, and husband

ACKNOWLEDGMENTS

I would like to especially thank my advisor, Dr. Alexandre Trindade. He guided

me through all the research, and gave me invaluable advice, suggestions and comments.

This thesis could never have been done without his help. I also want to thank Dr. Ramon

C. Littell and Dr. Ronald Randles for serving on my committee and providing valuable

comments.

I am also grateful to Yun Zhu for her support and patience.

Finally, I would like to thank my families for their unwaving affection and

encouragement.

iv

TABLE OF CONTENTS page

ACKNOWLEDGMENTS ................................................................................................. iv

LIST OF TABLES............................................................................................................ vii

LIST OF FIGURES ......................................................................................................... viii

ABSTRACT.........................................................................................................................x

CHAPTER

1 INTRODUCTION ........................................................................................................1

2 DEFINITIONS, PROPERTIES AND ESTIMATIONS...............................................4

Definitions ....................................................................................................................4 Definition 2.1: VaR (Value at Risk).....................................................................4 Definition 2.2: CVaR (Conditional Value at Risk) ...............................................4 Definition 2.3: AL distribution (The Asymmetric Laplace distribution) ..............5

Basic Properties ............................................................................................................7 Proposition 2.1: The Coefficient of Skewness ......................................................7 Proposition 2.2: The Coefficient of (excess) Kurtosis .........................................7 Proposition 2.3: The Quantiles ..............................................................................7 Proposition 2.4: VaR and CVaR for the AL distribution ......................................8

Estimations ...................................................................................................................8 Parametric Estimation ...........................................................................................8

Maximum Likelihood Estimation ( MLE ) ....................................................9 Method of Moments Estimation ( MME ) ...................................................11

Semi-parametric Estimation ................................................................................12 Nonparametric Estimation...................................................................................15

3 SIMULATION STUDY .............................................................................................16

Simulation...................................................................................................................16 Comparison.................................................................................................................16

Y~AL(1, 0.8, 1) ...................................................................................................17 Y~AL(0, 1, 1) ......................................................................................................20 Y~AL(1, 1.2, 1) ...................................................................................................21

v

4 EMPIRICAL APPLICATIONS .................................................................................26

Data.............................................................................................................................26 Interest Rates .......................................................................................................26 Exchange Rates ...................................................................................................31

Comparison.................................................................................................................34

5 CONCLUSION...........................................................................................................36

APPENDIX

THE PROBABILITIES OF THE MLES OF VAR AND CVAR BEING DEGENERATE..........................................................................................................37

LIST OF REFERENCES...................................................................................................38

BIOGRAPHICAL SKETCH .............................................................................................40

vi

LIST OF TABLES

Table page 3-1. Summary of related parameters of Y~AL(0,0.8,1). [n=200, reps=500].....................17

3-2. The MSEs of VaR and CVaR at different confidence levels: α=0.9, 0.95 and 0.99 of Y~AL(0, 0.8,1). [n=200, reps=500].............................................................18

3-3. Summary of related parameters ofY~AL(0,1,1). [n=200, reps=500].........................20

3-4. The MSEs of VaR and CVaR at different confidence levels: α=0.9, 0.95 and 0.99 of Y~AL(0,1,1). [n=200, reps=500].................................................................21

3-5. Summary of related parameters of Y~(0,1.2,1). [n=200, reps=500] ..........................22

3-6. The MSEs of VaR and CVaR at different confidence levels: α=0.9, 0.95 and 0.99 of Y~AL(0,1.2,1). [n=200, reps=500]..............................................................22

4-1. Summary statistics for the interest rates, after taking logarithm conversion..............26

4-2. Summary statistics for the exchange rates, after taking logarithm conversion. .........31

vii

LIST OF FIGURES

Figure page 2-1. VaR and CVaR for the possible losses of a portfolio...................................................5

2-2. Asymmetric Laplace densities with θ=0, τ=1, and κ =2, 1.25, 1, 0.8,0.5.....................6

2-3. The probabilities of the MLEs of VaR and CVaR being degenerate at different confidence levels ..........................................................................................10

2-4. Estimating tail index by plotting ( )log , kn k y

n⎧ ⎫−⎛ ⎞⎨ ⎬⎜ ⎟

⎝ ⎠⎩ ⎭, where n=200 in this case.

The largest value of m, 98, gives a roughly straight line, and the slope of the line is –1.865421. 2R =0.96....................................................................14

3-1. Histogram of simulated AL (0,0.8,1) data. [n=200, reps=500] ..................................18

3-2. The comparisons of three estimators of Y~AL(0, 0.8,1) at different confidence levels: α=0.9, 0.95 and 0.99. [n=200, reps=500] .........................................19

3-3. Histogram of simulated AL (0,1,1) data. [n=200, reps=500] .....................................20

3-4. The comparisons of three estimators of Y~AL(0, 1, 1) at different confidence levels: α=0.9, 0.95 and 0.99. [n=200, reps=500] .........................................21

3-5. Histogram of simulated AL (0,1.2,1) data. [n=200, reps=500] ..................................22

3-6. The comparisons of three estimators of Y~AL(0, 1.2, 1) at different confidence levels: α=0.9, 0.95 and 0.99. [n=200, reps=500] .........................................23

3-7. Relationships between κ, the skewness parameter, and MSEs at different confidence levels, α=0.9, 0.95 and 0.99. [n=200, reps=500] .......................24

3-8. Relationships among MSEs of VaR and CVaR, the skewness parameter, κ, and confidence level, α for the three different estimators (parametric, semiparametric, and nonparametric) ............................................................25

4-1. Histogram and normal quantile plot of interest rates on 30-year Treasury bonds, sample size = 202.. .......................................................................................28

viii

4-2. The Q-Q plot of interest rates on 30-year Treasury bonds vs. fitted AL distributions..................................................................................................30

4-3. The difference between CVaR and VaR for 99 quantiles of interest rates on 30 year Treasury bonds (top) and and that of simulated AL distributions (with different simulation seeds)(bottom)....................................................31

4-4. Histogram and normal quantile plot of Taiwan Dollar daily exchange rates, 6/1/00 to 6/7/05, sample size = 1833.......................................................................32

4-5. The Q-Q plot of Taiwan Dollar daily exchange rates, 6/1/00 to 6/7/05 vs. fitted AL distribution .............................................................................................33

4-6. The difference between CVaR and VaR for 99 quantiles of Taiwan Dollar daily exchange rates from 6/1/00 to 6/7/05 and and that of simulated AL distributions (with different simulation seeds).. ...........................................34

4-7. The comparison of three estimators of interest rates on 30-year Treasury bonds, from February 1977 to December 1993, at different confidence levels:α=0.9, 0.95 and 0.99. [n=200, reps=500] ..........................................34

4-8. The comparison of three estimators of Taiwan Dollar daily exchange rates,from 6/1/00 to 6/7/05, at different confidence levels: α =0.9, 0.95 and 0.99. [n=200, reps=500] ........................................................................................35

ix

Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the

Requirements for the Master of Science in Statistics

COMPARING ESTIMATORS OF VAR AND CVAR UNDER THE ASYMMETRIC LAPLACE DISTRIBUTION

By

Hsiao-Hsiang Hsu

December 2005 Chair: Alexandre Trindade Major Department: Statistics

Assessing the risk of losses in financial markets is an issue of paramount

importance. In this thesis, we compare two common estimators of risk, VaR and CVaR,

in terms of their mean squared errors (MSEs). Three types of estimators are considered:

parametric, under the asymmetric laplace (AL) law; semiparametric by assuming Pareto

tails; and ordinary nonparametric estimators, which can be expressed as L-statistics.

Parametric and nonparametric estimators have respectively the lowest and highest MSEs.

By assessing two types of quantile plots on interest rate and exchange rate data, we

determine that the AL distribution provides a plausible fit to these types of data.

x

CHAPTER 1 INTRODUCTION

Risk management has been an integral part of corporate finance, banking, and

financial investment for a long time. Indeed, the idea has been dated to at least four

decades ago, with Markowitz’s pioneering work on portfolio selection [1]. However, the

paper did not attract interest until twenty years after it was published. It was the financial

crash of 1973-1974 that proved that past good performance was simply a result of bull

market and that risk also had to be considered. This resulted in the increasing popularity

of Markowitz’s ideas on risk, portfolio performance and the benefits of diversification.

In the past few years, the growth of financial market and trading activities has

prompted new studies investigating reliable risk measurement techniques. The Value-at-

Risk (VaR) is a most popular measure of risk in either academic research or industry

application. This is a dollar measure of the minimum loss that would be expected over a

period of time with a given probability. For example, a VaR of one thousand dollars for

one day at a probability of 0.05 means that the firm would expect to lose at least $1

thousand in one day 5 percent of the time. Or we can also express this as a probability of

0.95 that a loss will not exceed one thousand dollars. In this way, the VaR becomes a

maximum loss with a given confidence level. The most influential contribution in this

field has been J.P Morgan’s RiskMetrics methodology, within which a multivariate

normal distribution is employed to model the joint distribution of the assets in a portfolio

[2]. However, the VaR approach suffers problems when the return and losses are not

normally distributed which is often the case. It underestimates the losses since extreme

1

2

events should happen with equally chance at each day. Obvious explanations for this

finding are negative skewness and excess kurtosis in the true distribution of market

returns, which cannot be accounted for by using a normal density model as in

RiskMetrics.

Another risk measure that avoids the problem is Conditional Value at Risk

(CVaR). The concept of CVaR was first introduced by Artzuer, Delbaen, Eber, and

Heath [3], and formulated as an optimization problem by Rockefellar and Uryasev [4].

CVaR is the conditional mean value of the loss exceeding VaR. It is a straightforward

way to avoid serial dependency in the predicted events and thus base one’s forecast on

the conditional distribution of the portfolio returns given past information. Although

CVaR has not become a standard in the finance industry, it is likely to play a major role

as it currently does in the insurance industry. Therefore, in the thesis, we consider both

of those two measurements for broader application.

A correct statistical distribution of financial data is needed first before any proper

predicative analysis can be conducted. Although the normal distribution is widely used, it

has several disadvantages when applied to financial data. The first potential problem is

one of statistical plausibility. The normal assumption is often justified by reference to the

central limit theory, but the central limit theory applies only to the central mass of the

density function, and not to its extremes. It follows that we can justify normality by

reference to the central limit theory only when dealing with more central quantiles and

probabilities. When dealing with the extremes, which are often the case in financial data,

we should therefore not use the normal to model. Second, most financial returns have

excess kurtosis. The empirical fact that the return distributions have fatter tails than

3

normal distribution has been researched since early 1960s when Mandelbrot reported his

first findings on stable (Parentian) distributions in finance [5]. Since then, several

researchers have observed that practically all financial data have excess kurtosis, which is

the leptokurtic phenomena. Thus, using the statistics of normal distributions to

characterize the financial market is potentially very hazardous. Since Laplace

distributions can account for leptokurtic and skewed data, they are natural candidates to

replace normal models and processes.

In this thesis, the aim is to compare parametric, semiparametric, and nonparametric

estimators of VaR and CVaR random sampling from the Asymmetric Laplace

distribution. To do so, we calculate their mean squared error (MSE), a popular criterion

for measuring the accuracy of estimators. Broadly speaking, the best estimator should

have smallest MSE.

The plan of this thesis is as follows. Chapter 2 provides some background to the

study by introducing some definitions and propositions related to VaR, CVaR and the

Asymmetric Laplace distribution. Chapter 3 compares the parametric, semiparametric,

and nonparametric three different estimators of Asymmetric Laplace distribution.

Chapter 4 provides empirical analysis by using interest rates and currency exchange rates

data. Chapter 5 concludes the article. Additional tables are included in the Appendix A.

CHAPTER 2 DEFINITIONS, PROPERTIES AND ESTIMATIONS

Let Y be a continuous real-valued random variable defined on some probability

space (Ω, Α, Ρ), with distribution function and density function f(.). Both the first

and second moments of Y are finite.

F .b g

Definitions

Definition 2.1: VaR (Value at Risk)

The VaR refers to a particular amount of money, the maximum amount we are likely to

lose over a period of time, at a specific confidence level. If positive values of Y represent

losses, the VaR of Y at probability level α is defined to be the αth quantile of Y.

VaRα Y( )≡ ζα Y( )= F−1 α( ) (2.1)

Definition 2.2: CVaR (Conditional Value at Risk)

The CVaR of Y at probability level α,is the mean of the random variable that results by

turncasting Y at ζα and discarding its lower tail.

CVaRα Y( )≡ φα Y( )= E Y Y ≥ζα( ) . (2.2)

Expanding on the definition, we obtain

φζ

ζ α ααα

αζ α α

YE YI Y

P YydF y yf y dyb g b gc h

b g b g b g=≥

≥=

−=

−

∞

ζ

∞z z11

11

(2.3)

or we can have an equivalent definition of CVaR in terms of the quantile function of Y:

φαα α

Y Fb g b g=−

− u duz11

11 (2.4)

4

5

Figure 2-1. VaR and CVaR for the possible losses of a portfolio

Definition 2.3: AL distribution (The Asymmetric Laplace distribution)

Random variable Y is said to follow an Asymmetric Laplace distribution if there exist

location parameter θ∈ℜ, scale parameter τ≥0, and skewness parameter κ>0, such that the

probability density function of Y is of the form

f y( )=κ 2

τ (1+ κ 2)

exp −κ 2

τy −θ

⎛

⎝ ⎜

⎞

⎠ ⎟ ,if y ≥ θ

exp −2

κτy −θ

⎛

⎝ ⎜

⎞

⎠ ⎟ ,if y < θ

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

(2.5)

or, the distribution function of Y is of the form

F y( )=1−

11+ κ 2 exp −

κ 2τ

y −θ⎛

⎝ ⎜

⎞

⎠ ⎟ ,if y ≥ θ

κ 2

1+ κ 2 exp −2

κτy −θ

⎛

⎝ ⎜

⎞

⎠ ⎟ ,if y < θ

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

. (2.6)

6

We denote the distribution of Y by AL(θ, κ, τ) and write Y~ AL(θ, κ, τ). The

mean of the distribution is given by

µ = θ +

τ2

1κ

−κ⎛ ⎝ ⎜

⎞ ⎠ ⎟ (2.7)

Its variance is

σ τκ

κ µ22

22 2

21

= + τ 2FHG

IKJ = + . (2.8)

The value of the skewness parameter κ is related to µ and τ as follows,

κ τµ τ µ

τ µτ

=+ +

=+ −2

2

222 2

2 2 µ, (2.9)

and it controls the probability assigned to each side of θ. If κ=1, the two probabilities are

equal and the distribution is symmetric about θ. This is the standard Laplace distribution

Figure 2-2. Asymmetric Laplace densities with θ=0, τ=1, and κ =0.5, 1, and 2.

7

Basic Properties

Proposition 2.1: The Coefficient of Skewness

For a distribution of an random variable Y with a finite third moment and standard

deviation greater then zero, the coefficient of skewness is a measure of symmetry that is

independent of scale. If Y~ AL(θ, κ, τ), the coefficient of skewness, γ1, is defined by

γ κκ

κκ

1

33

22

32

2

1

1= ×

−

+FHGIKJ

. (2.10)

The coefficient of skewness is nonzero for an AL distribution. As κ increases

within the interval , then the corresponding value of 0,∞b g γ 1 decreases from 2 to –2.

Thus, the absolute value of γ 1 is bounded by two.

Proposition 2.2: The Coefficient of (excess) Kurtosis

For a random variable Y with a finite fourth moment, the coefficient of (excess) kurtosis

can be defined as

γ 2 = 6 −

12(1 κ 2 + κ 2)2

. (2.11)

It is a measure of peakness and of heaviness of the tails. If γ 2>0, the distribution

is said to be leptokurtic (heavy-tailed). Otherwise, it is said to be platykurtic (light-tailed).

The skewness coefficient of the AL distribution is between 3 ( the least value for

asymmetric Laplace distribution when κ=1) and 6 (the largest value attained for the

limiting exponential distribution when κ→0).

Proposition 2.3: The Quantiles

If Y~ AL(θ, κ, τ), then the qth quantile of an AL random variable is,

8

ξq =θ +

τκ2

log 1+ κ 2

κ 2 q⎧ ⎨ ⎩

⎫ ⎬ ⎭

for q ∈ 0,1+ κ 2

κ 2

⎛

⎝ ⎜

⎤

⎦ ⎥ ,

θ −τ2κ

log (1+ κ 2)(1− q) for q ∈1+ κ 2

κ 2 ,1⎛

⎝ ⎜

⎞

⎠ ⎟ .

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

(2.12)

Proposition 2.4: VaR and CVaR for the AL distribution

If Y~ AL(θ, κ, τ), for α ≥ 0.5, then its standardization X Y AL= −θ τ κb g b g~ ,0 1, . Since

both VaR and CVaR are translation invariant and positively homogenous [7],

VaRα Y( )= θ + τVaRα X( ) , (2.13)

and

CVaRα Y( )= θ + τCVaRα X( ). (2.14)

Therefore, no generality is lost by focusing on the standard case X ~ AL (0, κ, 1),

provided θ and τ are known. VaR and CVaR are then easily obtained.

ζκ α

κα Xb g c hb g= −

+ −log 1 1

2

2

, (2.15)

and

φ ζκα αX Xb g b g= +

12

. (2.16)

Estimations

We now look at some of the most popular approaches to the estimation of VaR and

CVaR.

Parametric Estimation

The parametric approach estimates the risk by fitting probability curves to the data

and then inferring the VaR from the fitted curve.

9

Maximum Likelihood Estimation ( MLE )

Consider now the most general case of estimating all three parameters. If Y ~

AL(θ, κ, τ), the maximum likelihood estimators (MLEs) are available in closed form [6].

Define first the functions,

δ1 θ( )=

1n

Yi −θ( )+

i=1

n

∑, (2.17)

δ2 θ( )=1n

Yi −θ( )−

i=1

n

∑ , (2.18)

and

h θ( )= 2log δ1 θ( ) + δ2 θ( )[ ]+ δ1 θ( )δ2 θ( )

(2.19)

Letting the index 1 ≤ r ≤ n be such that

h Y r( )( )≤ h Y i( )( ), for i =1,.....,n,

the MLE of θ is . Provided 1 < r < n, the MLEs of ( κ, τ ) are: ( )rΥ=θ

( )( ) ( )( )[ ] 41

12ˆ rrY Υ= δδκ (2.20)

/ /τ δ δ δ δ=

/+L

NMOQP2 1 2

1 4

1

1 2

2

1 2Y Y Y Yr r r rb g b g b g b ge j e j e j e j (2.21)

(If r = 1 or r = n, the MLEs of ( κ, τ ) do not exist.) Defining

ωα,κ ≡ log 1+ κ 2( )1−α( )[ ,] (2.22)

the MLEs of VaR and CVaR are then obtained by equivariance,

( )2ˆ

ˆˆˆ ˆ,

κ

ωτθζ κα

α −=Y (2.23)

( ) ( )2ˆ

ˆˆˆκ

τζφ αα += YY (2.24)

10

However, after doing some experiments, we found the MLEs of VaR and CVaR will

be degenerate most of the time when all three parameters are unknown (Appendix).

Figure 2-3. shows that the probabilities of the MLEs being degenerate rise with both the

sample size, n, and the skewness parameter,κ .

α = 05. α = 0 625.

5

25

0.25 0.

5

0.75

1

0

0.5

1

prob.of

being

degenerate

n

kappa

5

25

0.25 0.

5

0.75

1

0

0.5

1

prob.of

being

degenerate

n

kappa

α = 0 75. α = 0875.

5

25

0.25 0.

5

0.75

1

0

0.5

1

prob.of

being

degenerate

n

kappa

5

25

0.25 0.

5

0.75

1

0

0.5

1

prob.of

being

degenerate

n

kappa

Figure 2-3. The probabilities of the MLEs of VaR and CVaR being degenerate at

different confidence levels

There is another way to estimate the parameters when all of them are unknown.

According to Ayebo and Kozubowski [8], in the case when all parameters are unknown,

one can estimate the mode (θ ) using one of the nonparametric methods (Bickel [9] and

11

Vieu [10]) for several estimation models). After getting θ , we can apply the following

formulas for κ and τ , assuming θ is known, ([6], Chapter 3), to get the maximum

likelihood estimates.

,κδ θδ θn = 2

1

4b gb g (2.25)

.τ δ θ δ θ δ θ δn = ×2 14

24

1 2b g b g b g b gc hθ+ (2.26)

Remark:

In our analysis, we assume that the location parameter,θ , is zero. This is a

reasonable assumption when the data consists of logarithmic growth rates such as interest

rates, stock returns, and exchange rates [8].

Method of Moments Estimation ( MME )

The method of moments approach is also considered in the thesis. Assuming that

the θ is known, which is set to be zero in the study, the method of moments estimators of

µ and τ are given by ([6], Chapter 3)

µ n ni

n

Yn

Y= ==∑1

1i , (2.27)

τ n ii

n

nnY Y= −

=∑1 22

1

2 . (2.28)

Then, we can compute k using relation (2.9).

Remark: After checking all the cases in the thesis, we found that the MLE and the

MME of each parameter are almost the same. Therefore, we only calculate the MLEs

for parametric estimation in the study.

12

Semi-parametric Estimation

When observing financial data, e.g. stock returns, interest rates, or exchange rates, a

much less restrictive assumption is to model the return distributions as having a Pareto

left tail, or equivalently that the loss distribution has a Pareto right tail. This allows for

the skewness and kurtosis of returns, while making no assumptions about the underlying

distribution away from the tails. We follow the development of Rupport ([11], Chapter

11. ) For y> 0,

P Y y L y y a> = −b g b g , (2.29)

where is slowly varying at infinity and a is the tail index. Therefore, if

and then

L yb gy1 0> y0 0>

P Y yP Y y

L yL y

yy

a>>

=FHGIKJ

−

1

0

1

0

1

0

b gb g

b gb g (2.30)

now suppose that and y VaR Y1 1= α ( ) y VaR Y0 0

= α ( ) , where 0 0 1< <α α .

Then, ( 2.29 ) becomes

11

1

0

1

0

1

0

1

0

−−

=>

>=

FHG

IKJ

−αα

α

α

α

α

α

α

P Y VaR Y

P Y VaR Y

L VaR Y

L VaR YVaR YVaR Y

ab gn sb gn s

b gn sb gn s

b gb g (2.31)

Because L is slowly varying at infinity and VaR and VaR are assumed to

be reasonably large, we make the approximation that

Yα1b g Yα 0

b g

L VaR X

L VaR Xα

α

1

0

1b gn sb gn s ≅ , (2.32)

so ( 2.32) simplifies to

13

VaR YVaR Y

aα

α

αα

1

0

11

0

1

1b gb g =

−−FHGIKJ . (2.33)

Now dropping the subscript “1” of α 1 , we have

VaR Y VaR Ya

α ααα

b g b g=−−FHGIKJ0

11

0

1

(2.34)

that is,

ζ ζ ααα αY Y

ab g b g=−−FHGIKJ

~0

11

0

1

(2.35)

We now extend this idea to CVaR similarly, giving

CVaR Y CVaR Ya

α ααα

b g b g=−−FHGIKJ0

11

0

1

, (2.36)

or we can write,

ϕ ϕ ααα αY Y

ab g b g=−−FHGIKJ

~0

11

0

1

. (2.37)

Equations (2.35) and (2.37) become semiparametric estimators of VaR and

when VaR and CVaR are replaced by nonparametric estimates

(2.41), (2.42) and the tail index a is estimated by the regression estimator.

Yα b gCVaR Yα b g Yα 0

b g Yα 0b g

To see this, note that by (2.29), we have

log[ ] log logP Y y L y a y> = −b g b g . (2.38)

If n is the sample size and 1 ≤ ≤k n , then

P Y y n knn> ≅−

b ge j (2.39)

14

⇒−FHGIKJ ≅ −log log log .n k

nL a y nb ge j (2.40)

One can then use the linearity of the plot of log ,n kn

y kk

m−FHGIKJ

FHG

IKJ

RSTUVW =

b ge j1

for different

to guide the choice of . m m

The value of m is selecting by plotting log ,n kn

y kk

m−FHGIKJ

FHG

IKJ

RSTUVW =

b ge j1

for various values

of m and choosing the largest value of m giving a roughly linear plot. If we fit a straight

line to these points by least squares then minus the slope estimates the tail index a.

For example, if a random sample Y Y is drawn from the AL (0, 0.8,1)

distribution, and denote the corresponding order statistics of the sample.

For getting the value of m, we first need to plot

Y Y1 2 3 200, , , ...

Y Y Y Y1 2 3 200b g b g b g b g, , , ...

log ,n kn

y kk

m−FHGIKJ

FHG

IKJ

RSTUVW =

b ge j1

, where n=200.

The plotted points and the least squares line can be seen in Figure 2-4.

Figure 2-4. Estimating tail index by plotting ( )⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ −

kyn

kn ,log , where n=200 in this case.

The largest value of m, 98, gives a roughly straight line, and the slope of the line is –1.865421. 2R =0.96

15

A least squares line was fit to these 98 points and R2 =0.96, indicating a good fit to

a straight line. The slope of the line is –1.865421, so a is 1.865421. After getting a, we

can obtain the semiparametric estimators by functions (2.35) and (2.37).

Nonparametric Estimation

This is the least restrictive approach to the estimation of VaR and CVaR. The

nonparametric approach seeks to estimate VaR or CVaR without making any

assumptions about the distribution of returns and losses. The essence of the approach is

that one can try to let the data speak for themselves as much as possible. (See for example

[7].)

When a random sample, Y1,...,Yn , from AL distribution is available, consistent

nonparametric estimators (NPEs) of VaR and CVaR taken the form of L-statistics. If

Y1( ) ≤ ... ≤ Y n( ) denote the corresponding order statistics of the sample, the estimator of

VaR is the αth empirical quantile,

~

,ζ α αY Y kb g b g= (2.41)

where k nα α= denotes the greatest integer less than or equal to nα . The

estimator of CVaR is the corresponding empirical tail mean,

~φαα α

Yn k

Y rr k

n

b g b g=− + =

∑11

(2.42)

Theoretically, parametric approaches are more powerful than nonparametric

approach, since they make use of additional information contained in the assumed density

or distribution function.

CHAPTER 3 SIMULATION STUDY

In this chapter, we compare the three types of estimators (parametric,

semiparametric, and nonparametric) of VaR and CVaR in terms of their bias, variance,

and MSE. The data is generated via Monte Carlo from AL distribution. The MSEs are

obtained empirically.

Simulation

There are several ways to generate random values from an AL distribution. Here

is an example of using two i.i.d standard exponential random variables.

We can generate the Y AL~ ( , , )θ κ τ by the following algorithm.

• Generate a standard exponential random variable W . 1

• Generate a standard exponential random variable W , independent of W . 2 1

• Set Y W← + −θ Wτκ

κ2

11 2( ) .

• RETURN Y. Comparison

In this section, we would like to compare three different approaches: parametric,

semiparimetric, and nonparametric to estimate VaR and CVaR. In order to measure the

goodness of those estimation procedures, using mean square error (MSE) to check their

goodness. MSE is a common criterion for comparing estimators and it is composed of

bias and variance. A better estimator should have smaller MSE. Besides checking the

goodness of those estimators, we would also like to know how different κ , the skewness

parameter, would affect the MSEs. Without loss of generality, here, we focus only on the

standard case, Y AL~ ( , , )0 1κ .

16

17

Before doing the following analysis, we need first to know how to estimate the

VaR and CVaR in the standard case. After some routine calculations from (2.23),(2.24),

the MLEs of VaR and CVaR in the standard case are:

,ζτωκα

α κYb g = −2

(3.1)

φ ζκα αY Yb g b g= +

12

(3.2)

Remark: Note that φ ζκα αY Yb g b g− =

12

, which is independent of α . Thos will

form the basis of a goodness-of-fit tool in Chapter 4.

Y~AL(1, 0.8, 1)

The skewness parameter, κ , controls the probability assigned to each side of θ .

Therefore, while κ = 0.8, the distribution would be moderately skewed to the right. The

histogram of simulated values from is shown in Figure 3-1, In Table below,

we summarize the corresponding estimated parameters and coefficients of skewness and

kurtosis for a random sample of size=200, reps=500, drawn fromY A .

AL( , . , )0 08 1

L~ , . ,0 08 1b gTable 3-1. Summary of related parameters of Y~AL(0,0.8,1). [n=200, reps=500]

κ τ skewness kurtosis(adjusted)

0.79954 0.99336 0.134882 3.52629

Since 90%, 95% and 99% are the most common quantiles when analyzing financial

data, we consider only those three confidence levels in this study.

As mentioned already, the VaR and CVaR are contingent on the choice of

confidence level, and will generally change when the confidence level changes. Thus,

the MSEs of VaR and CVaR of different quantiles will also change correspondingly.

18

This is illustrated in Table 3-2, which shows the corresponding MSEs of VaR and CVaR

at the 95%, 99%, 99.5% levels of confidence.

Figure 3-1. Histogram of simulated AL (0,0.8,1) data. [n=200, reps=500]

Table 3-2. The MSEs of VaR and CVaR at different confidence levels: α=0.9, 0.95 and 0.99 of Y~AL(0, 0.8,1). [n=200, reps=500]

MSEs of VaR

α =0.9 α =0.95 α =0.99

Parametric 0.0044458 0.0086326 0.023385

Semiparametric 0.014017 0.012534 0.20382

Nonparametric 0.82425 1.5001 3.9623

MSEs of CVaR

α =0.9 α =0.95 α =0.99

Parametric 0.020068 0.029502 0.054907

Semiparametric 0.025519 0.054298 0.35212

Nonparametric 0.78834 1.3716 3.1509

Figure 3-2 illustrates the comparisons of three different estimation approaches.

Because MSEs are composed of bias and variance, the comparisons of biases and

variances of VaR and CVaR are also shown in this figure. From Figure 3-2, not

19

surprisingly, one can find that the parametric approach is the best way for estimating the

VaR and CVaR according to its minimum MSE among those three approaches. The

parametric approach is more powerful than the others, because it makes use of most

information contained in the assumed density or distribution function.

-1

-0.5

0

0.5

1

1.5

2

2.5

0.9 0.95 0.99

Alpha

Bia

s (V

aR)

-1.5

-0.5

0.5

1.5

2.5

0.9 0.95 0.99

AlphaB

ias

(CV

aR)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.9 0.95 0.99

Alpha

Var

ianc

e (V

aR)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.9 0.95 0.99

Alpha

Var

ianc

e (C

VaR

)

0

1

2

3

4

5

0.9 0.95 0.99

Alpha

MS

E (

VaR

)

0

1

2

3

4

5

0.9 0.95 0.99

Alpha

MS

E (

CV

aR)

Figure 3-2. The comparisons of three estimators of Y~AL(0, 0.8,1) at different

confidence levels: α=0.9, 0.95 and 0.99. [n=200, reps=500]

20

Y~AL(0, 1, 1)

In this case, the skewness parameter, κ , is assumed to be 1, which means that the

two probabilities are equal and the distribution is symmetric about θ , which is assumed

to be 0 here. This is the standard Laplace distribution. In Figure 3-3, the symmetric

distribution was shown very clearly; therefore, one can expect that the coefficient of

skewness should be close to zero. As for the MLEs of parameters and other coefficients

are demonstrated in Table 3-3.

Table 3-3. Summary of related parameters ofY~AL(0,1,1). [n=200, reps=500] κ τ skewness kurtosis(adjusted)

1.0018 0.99435 0 3

Figure 3-3. Histogram of simulated AL (0,1,1) data. [n=200, reps=500]

A summary of MSEs of VaR and CVaR under different estimation approaches is

given in Table 3-4.

21

Table 3-4. The MSEs of VaR and CVaR at different confidence levels: α=0.9, 0.95 and 0.99 of Y~AL(0,1,1). [n=200, reps=500]

MSEs of VaR α =0.9 α =0.95 α =0.99

Parametric 0.0031112 0.0057482 0.01605

Semiparametric 0.0063773 0.012763 0.17371

Nonparametric 0.42144 0.79774 2.3073

MSEs of CVaR

α =0.9 α =0.95 α =0.99

Parametric 0.0031112 0.0057482 0.01605

Semiparametric 0.0063773 0.012763 0.17371

Nonparametric 0.42144 0.79774 2.3073

The comparison of different estimation approaches is shown in Figure 3-4.

0

0.5

1

1.5

2

2.5

0.9 0.95 0.99

Alpha

MS

E (

VaR

)

0

0.5

1

1.5

2

2.5

0.9 0.95 0.99

Alpha

MS

E (

CV

aR)

Figure 3-4. The comparisons of three estimators of Y~AL(0, 1, 1) at different confidence

levels: α=0.9, 0.95 and 0.99. [n=200, reps=500]

Remark: Since the results are similar to the previous one, we only demonstrate the comparison of MSEs in

the following two cases.

22

Y~AL(1, 1.2, 1)

Now, κ ,which is set to be 1.2. As the histogram of simulated AL numbers shown

in Figure 3-5, the distribution seems to be lightly skewed left. The coefficient of

skewness is therefore less than zero. Table3-5 illustrates some related parameters.

Table 3-5. Summary of related parameters of Y~(0,1.2,1). [n=200, reps=500] κ τ Skewness Kurtosis(adjusted)

1.2019 0.99386 -0.118189 3.36603

Figure 3-5. Histogram of simulated AL (0,1.2,1) data. [n=200, reps=500]

The comparison result of VaR and CVaR for those three approaches is illustrated in

Table 3-6, and is the same as previous two cases: the parametric one is the best one and

the nonparametric one is the worst. The comparison is also shown in Figure 3-6.

Table 3-6. The MSEs of VaR and CVaR at different confidence levels: α=0.9, 0.95 and 0.99 of Y~AL(0,1.2,1). [n=200, reps=500]

MSEs of VaR

α =0.9 α =0.95 α =0.99

Parametric 0.0026005 0.0046065 0.013319

Semiparametric 0.0075317 0.017683 0.15612

Nonparametric 0.2238 0.46272 1.4338

23

MSEs of CVaR α =0.9 α =0.95 α =0.99

Parametric 0.013104 0.016882 0.032377

Semiparametric 0.031763 0.056587 0.23916

Nonparametric 0.21668 0.42903 1.1449

0

0.5

1

1.5

2

0.9 0.95 0.99

Alpha

MS

E (

VaR

)

0

0.5

1

1.5

2

0.9 0.95 0.99

AlphaM

SE

(C

VaR

)

Figure 3-6. The comparisons of three estimators of Y~AL(0, 1.2, 1) at different

confidence levels: α=0.9, 0.95 and 0.99. [n=200, reps=500]

Before moving on, it might be a good idea to pause at this point to see the

relationships among the confidence level, α , the skewness parameter, κ , and the MSEs.

From Figure 3-7, we could recognize that the MSEs of VaR and CvaR might fall or

remain constant as κ rises.

α =0.9

0

0.2

0.4

0.6

0.8

1

0.8 1 1.2

Kappa

MS

E (

VaR

)

0

0.2

0.4

0.6

0.8

1

0.8 1 1.2

Kappa

MS

E (

CV

aR)

24

α =0.95

0

0.5

1

1.5

2

0.8 1 1.2

Kappa

MS

E (

VaR

)

0

0.5

1

1.5

2

0.8 1 1.2

Kappa

MS

E (

CV

aR)

α =0.99

0

1

2

3

4

5

0.8 1 1.2

Kappa

MS

E (

VaR

)

0

1

2

3

4

5

0.8 1 1.2

Kappa

MS

E (

CV

aR)

Figure 3-7. Relationships between κ, the skewness parameter, and MSEs at different

confidence levels, α=0.9, 0.95 and 0.99. [n=200, reps=500]

To form a more complete picture, we need to see how the MSEs change as we

allow both those two parameters to change under different estimation approaches. The

results are illustrated in Figure 3-8, which enables us to read off the value of the MSEs

for any given combination of these two parameters. Those histograms show how the

MSEs change as the underlying parameters change and convey information that the

MSEs rise with α but decline with κ .

25

MSEs of VaR for parametric estimator MSEs of CVaR for parametric estimator

0.8 1 1.2

0.9

0.99

0

0.01

0.02

0.03

0.04

0.05

0.06

mse

kappa

alpha

parametric

0.8 1 1.2

0.9

0.99

0

0.01

0.02

0.03

0.04

0.05

0.06

mse

kappa

alpha

parametric

MSEs of VaR for semiparametric estimator MSEs of CVaR for semiparametric

estimator

0.8 1 1.2

0.9

0.990

0.1

0.2

0.3

0.4

mse

kappa

alpha

Semiparametric (VaR)

0.8 1 1.2

0.9

0.99

0

0.1

0.2

0.3

0.4

mse

kappa

alpha

Semiparametric (CVaR)

MSEs of VaR for nonparametric estimator MSEs of CVaR for nonparametric

estimator

0.8 1 1.2

0.9

0.990

1

2

3

4

mse

kappa

alpha

nonparametric

0.8 1 1.2

0.9

0.990

1

2

3

4

mse

kappa

alpha

nonparametric

Figure 3-8. Relationships among MSEs of VaR and CVaR, the skewness parameter, κ,

and confidence level, α for the three different estimators (parametric, semiparametric, and nonparametric)

CHAPTER 4 EMPIRICAL APPLICATIONS

We present in this section the interest rates and exchange rates data sets along with

the quantitative analysis to determine if the AL distribution is an adequate model for the

data by using goodness-of-fit techniques. If the data sets do fit the AL distribution, we

would like to compare the MSEs of VaR and CVaR for the three estimation approaches

and to see which one is the best estimator.

Data

Interest Rates

Table 4-1 reports summary statistics, including estimates of the coefficients of

skewness and kurtosis. The data are the interest rates on 30-year Treasury bonds on the

last working days of the month. The database was downloaded from:

http://finance.yahoo.com. The variable of interest is the logarithm of the interest rate

ratio for two consecutive days. The data were transformed accordingly. This sample is

the same as that previously consider by Kozubowski and Podgorski [12], and it goes from

February 1977 through December 1993, yielding a sample size = 202.

Table 4-1. Summary statistics for the interest rates, after taking logarithm conversion. Mean S.D. Min Max Q1 Q3 Skewness KurtosisInterest rate

-0.00046 0.01492 -0.04994 0.05855 -0.00933 0.00761 -0.05706 1.9603Remark: The sample goes from February 1977 through December 1993, yielding a sample size =202

Figure 4-1 contains a histogram and a normal quantile plot. The normal quantile

plot is one of the most useful tools for assessing normality. The plot is to compare the

26

27

data values with the values one would predict for a standard normal distribution. The

comparison is based on the idea of quantiles. If the data came perfectly from a standard

normal distribution, the theoretical and empirical quantiles would be expected to be

similar. Thus, all the points would fall along a straight line. However, if the plot is

markedly nonlinear, then it is doubtful those data are normally distributed.

From the histogram, we can find that the data have a higher peak in the center and

heavier tails than normal distribution. Since it is quite symmetric, we could expect a

skewness near zero. Due to the heavier tails, we might expect the kurtosis to be larger

than for a normal distribution. In fact, from the summary statistics in Table 4-1, the

skewness is around zero and the kurtosis is near two, which indicate moderate kurtosis.

Furthermore, by looking at the normal quantile plot, it is quite clear that the data do not

follow normal distribution, since the dots do not quite fit to a straight line and have some

outliers.

Now, we have observed that the data set do not follow the normal distribution.

However, there is no agreement regarding the best theoretical model for fitting the

interest rates data. In a recent study, Kotz et al., [6], try to fit the skew Laplace

distribution to both interest rates and currency exchange rates data because of its fat-tail

and sharp peaks at the origin. In their experiment, they found except for a slight

discrepancy in skewness, the match between empirical and theoretical values is close.

Thus, in the study, we consider fitting the AL models to both interest rates and currency

exchange rates.

28

Figure 4-1. Histogram and normal quantile plot of interest rates on 30-year Treasury

bonds.

Remark: The sample goes from February 1977 through December 1993, yielding a sample size =202

To determine if the AL distribution functions describe the data well, we employ here

the most popularly used Quantile-Quantile plot or Q-Q plot graphical technique to

examine the data set. The idea of the Quantile-Quantile (Q-Q) plot is similar to the

29

normal quantile plot. It is a graphical technique for determining if two data sets come

from populations with a common distribution. A Q-Q plot is a plot of the quantiles of the

first data set against the quantiles of the second data set. By a quantile, it means the

fraction (or percent) of points below the given value. If the two datasets come from a

population with the same distribution, the points should fall approximately along this

reference line. The greater the departure from this reference line, the greater the evidence

for the conclusion that the two data sets have come from populations with different

distributions. The plot in Figure 4-2 is the Q-Q plot of interest rates data set and AL

distributions. To obtain the Q-Q plot, we need to fit an AL distribution to the interest

rates data. Estimate the κ and τ by formula (2.25), and (2.26), and then plot the

empirical quantiles, from the 1st to the 99th, against the corresponding quantiles

calculated from (2.12) with the MLEs for κ and τ substituted in.

From this plot, we can see that most of the data points fall on a straight line. It is

evident even to the naked eyes that AL distributions model these data more appropriately

than normal distributions.

30

Figure 4-2 The Q-Q plot of interest rates on 30-year Treasury bonds vs. fitted AL distributions

Remark1: We compute 99 quantiles for the Q-Q plot.

Remark2: The sample goes from February 1977 through December 1993, yielding a sample size =202

Now, we apply another goodness-of-fit test. The idea is similar to the Q_Q plot. If

the distance between the nonparametric estimators of CVaR and VaR for each quantile

from the interest rates data set is similar to that of AL distributions, then we can conclude

that the data might fit an AL distribution.

In Figure 4-3, we can observe that the difference between CVaR and VaR for each

quantile of interest rates on 30-year Treasury bond is similar to that of an AL

distribution. This provides additional evidence that the interest rates data set can be

plausibly explained by AL distributions.

0

0.01

0.02

0.03

0.04

0.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Quantiles

CV

aR-V

aR

0

0.01

0.02

0.03

0.04

0.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Quantiles

CV

aR-V

aR

0

0.01

0.02

0.03

0.04

0.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Quantiles

CV

aR-V

aR

31

Figure 4-3. The difference between CVaR and VaR for 99 quantiles of interest rates on

30-year Treasury bonds (top) and that of simulated AL distribution (with different simulation seeds) (bottom).

Remark1: We compute 99 quantiles for the dataset.

Remark2: The sample goes from February 1977 through December 1993, yielding a sample size =202

Exchange Rates

The data below consist of Taiwan dollar daily exchange rates (interbank rates)

against the U.S. dollar. The historical sample covers the period June 1, 2000, to June 7,

2005 (1833 days) downloaded from: http://www.oanda.com/convert/fxhistory. The

variable of interest is the logarithm of the price ratio for two consecutive days. The data

were transformed accordingly, yielding n=1832 values for each currency. The summary

statistics for the transformed data are showed in Table 4-2, including the coefficient of

skewness and kurtosis.

Table 4-2. Summary statistics for the exchange rates, after taking logarithm conversion. Currency Mean S.D.

Min Max Q1 Q3 Skewness Kurtosis

TWD 0.00000 0.00197 -0.0222 0.02204 -0.0002 0.00015 0.0012129 50.365 Remark: The sample goes from June 1, 2000 through June 7, 2005, yielding a sample size =1833

The histogram and normal quantile plot of the data are presented in Figure 4-4.

From the histogram, it is quite obvious that the data have a high peak, which is close to 0.

Due to the high peak, we might expect the kurtosis to be larger than for a normal

distribution. In fact, the kurtosis is 50.63, which is extremely high. Besides, the normal

quantile plot confirms our findings that the data are not normally distributed. The data do

not fit a straight line, not even the 95% confidence interval.

32

Figure 4-4. Histogram and normal quantile plot of Taiwan Dollar daily exchange rates,

6/1/00 to 6/7/05, sample size=1833.

The remaining challenge is to confirm if AL distributions fit the data well, and we

will use the same strategy as that used for the interest rates data.

33

Figure 4-5. The Q-Q plot of Taiwan Dollar daily exchange rates, 6/1/00 to 6/7/05 vs.

fitted AL distribution

Remark: We compute 99 quantiles for the Q-Q plot

The quantile plots of the data set with theoretical AL distributions are presented in

Figure 4-5. We see only a slight departure from the straight line. Comparing to the

normal quantile plot in Figure 4-4, it is quite obvious that the AL distributions fit the data

much better than the normal distributions.

In Figure 4-6, the distance between the nonparametric estimators of VaR and CVaR

for each quantile (99 quantiles) of the exchange rates dataset is similar to that of AL

distributions.

34

0

0.001

0.002

0.003

0.004

0.005

0.006

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Quantiles

CV

aR-V

aR

0

0.001

0.002

0.003

0.004

0.005

0.006

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Quantiles

CV

aR-V

aR

Figure 4-6. The difference between CVaR and VaR for 99 quantiles of Taiwan Dollar

daily exchange rates (top), and that of simulated AL distributions (with different simulation seeds)(bottom).

Comparison

In the previous section, we have confirmed that both the data sets: interest rates on

30-year Treasury bonds and Taiwan Dollar daily exchange rates can be plausibly

modeled by AL distributions. Therefore, in the following section, a comparison of three

estimation approaches: parametric, nonparametric, and semiparametric will be made to

check if the results are consistent with those of Chapter 3: the parametric approach is the

best estimation method, and the nonparametric is the worst.

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.9 0.95 0.99

Alpha

MS

E (

VaR

)

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.9 0.95 0.99

Alpha

MS

E (

CV

aR)

Figure 4-7. The comparison of three estimators of interest rates on 30-year Treasury

bonds, from February 1977 to December 1993, at different confidence levels:α=0.9, 0.95 and 0.99. [n=200, reps=500]

35

0

0.000002

0.000004

0.000006

0.000008

0.00001

0.9 0.95 0.99

Alpha

MS

E (

VaR

)

0

0.000002

0.000004

0.000006

0.000008

0.00001

0.9 0.95 0.99

Alpha

MS

E (

CV

aR)

Figure 4-8.The comparison of three estimators of Taiwan Dollar daily exchange

rates,from 6/1/00 to 6/7/05, at different confidence levels: α =0.9, 0.95 and 0.99. [n=200, reps=500]

From Figure 4-7 and Figure 4-8, we can find that the results are quite consistent

with those of Chapter 3. The parametric estimation approach is the best method to

estimate the VaR and CVaR of the AL distribution because of the lower MSEs. Besides,

we can also notice that the MSEs are getting bigger while the confidence level is rising,

especially for the nonparametric approach. That is because the nonparametric approach

is more sensitive for extreme values, and it may overestimate the VaR and CVaR for

large α . The higher confidence level means a smaller tail, a cut-off point further to the

extreme and, therefore, a higher VaR and CVaR. As a result, the corresponding MSEs

would become higher.

CHAPTER 5 CONCLUSION

In this thesis, we compare three different estimates for the risk measures: VaR and

CVaR when sampling from an AL distribution: parametric, semiparametric ,and

nonparametric.

The standard AL case is investigated in chapter 3, and we found that in general, the

parametric approach is the best estimator since it has the smallest MSEs for both VaR

and CVaR. We then applied the AL distribution to interest rates and exchange rates data

and find it to be a plausible fit. This is because AL distributions can account for

leptokurtosis and skewness typically present in financial data sets. Finally, we compared

those three approaches again based on the empirical data sets and the results are

consistent with those obtained earlier.

36

APPENDIX

THE PROBABILITIES OF THE MLES OF VAR AND CVAR BEING DEGENERATE

α=0.5

κ n 5 10 15 20 25

0.25 1 1 1 1 1

0.50 1 0.99 0.99 0.94 0.92

0.75 1 0.93 0.89 0.73 0.71

1 0.99 0.96 0.84 0.69 0.58

α=0.625

κ n 5 10 15 20 25

0.25 1 1 1 1 1

0.50 1 0.99 0.98 0.93 0.9

0.75 1 0.94 0.87 0.72 0.59

1 0.99 0.98 0.87 0.71 0.51

α=0.75

κ n 5 10 15 20 25

0.25 1 1 1 1 1

0.50 1 0.97 0.96 0.97 0.92

0.75 1 0.93 0.94 0.72 0.65

1 1 0.93 0.79 0.66 0.58

α=0.625

κ n 5 10 15 20 25

0.25 1 1 1 1 1

0.50 1 0.99 0.94 0.96 0.95

0.75 1 0.94 0.89 0.8 0.66

1 1 0.95 0.85 0.73 0.63

37

LIST OF REFERENCES

1. Markowitz, H. 1952. Portfolio Selection, Journal of Finance, Vol. 7, page 78.

2. J.P. Morgan. 1997. RiskMetrics Technical Documents, 4th edition. New York.

3. Artzuer, P., Delbaen, F., Eber, J., and Heath, D. 1999. Coherent Measures of Risk. Mathematical Finance, Vol. 9, pages 203-228.

4. Rockefeller R. T. and Uryasev S. 2000. Optimization of Conditional Value-At-Risk. The Journal of Risk, Vol. 2, No. 3, pages 21-41.

5. Mandelbrot, B.1963. The Variation of Certain Speculative Prices. Journal of Business, Vol. 36, pages 394-419.

6. Kotz, S., Kozubowski, T., and Podgorski, K. 2001. The Laplace Distribution and Generalizations, A Revisit with Applications to Communications, Economics, Engineering, and Finance. Birkhauser. Boston.

7. Gaivoronski, A.A. Norwegian University of Science and Technology, Norway. Pflug, G. University of Vienna, Austria. 2000. Value at Risk in Portfolio Optimization: Properties and Computational Approach, working paper.

8. Ayebo, A. and Kozubowski, T. 2003. An Asymmetric Generalization of Gaussian and Laplace Laws. Journal of Probability and Statistical Science, Vol. 1(2), pages 187-210.

9. Bickel, D.R. 2002. Robust Estimators of the Mode and Skewness of Continuous data. Comput. Statist. Data Anal, Vol. 39, pages 153-163.

10. Vieu, P. 1996. A Note on Density Mode Estimation, Statist. Probab. Lett, Vol. 26, pages 297-307.

11. Ruppert, D. 2004. Statistics and Finance: An Introduction, pages 348-352. Spring Verlag. Berlin, Germany.

12. Kozubowski, T. and Podgorski, K. 1999. A Class of Asymmetric Distributions. Actuarial Research Clearing House, Vol. 1, pages 113-134.

13. Dowd, K. 2002. Measuring Market Risk. John Wiley & Sons Ltd. West Success, England.

38

39

14. Linden, M. 2001. A Model for Stock Return Distribution. International Journal of Finance and Economics, Vol. 6, pages 159–169.

BIOGRAPHICAL SKETCH

Hsiao-Hsiang Hsu was born in Taipei, Taiwan. She received her Bachelor of

Business Administration (B.B.A.) degree in international trade and finance from Fu-Jen

Catholic University, Taipei, Taiwan. In Fall 2003, she enrolled for graduate studies in

the Department of Statistics at the University of Florida and will receive her Master of

Science in Statistics degree in December 2005.

40