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