Statistical Distributions
BYU
James B. McDonald
Statistical Distributions
James B. McDonald
Brigham Young University
May 2013
The research assistance of Brad Larsen, Patrick Turley, and Sean
Kerman is gratefully acknowledged as are comments from
Richard Michelfelder and Panayiotis Theodossiou.
Statistical Distributions
1.
2.
3.
4.
5.
6.
7.
8.
Introduction
Some families of statistical distributions
Regression applications
Censored regression
Qualitative response models
Option pricing
VaR (value at risk)
Conclusion
Statistical Distributions
Introduction
Some families of statistical distributions
1.
2.
a.
3.
4.
5.
6.
7.
8.
Families
Regression applications
Censored regression
Qualitative response models
Option pricing
VaR (value at risk)
Conclusion
Some families of statistical
distributions
Families
a.
i.
f(y;θ), θ = vector of parameters
GB: GB1, GB2, GG (0<Y)
GB distribution tree
Probability Density Functions
GB  y; a, b, c, p, q  
1  1  c  y / b 
B  p, q  1  c  y / b  
ay
b
ap
a q 1
ap 1
GB1 y; a, b, p, q   GB  y; a, b, c  0, p, q  
a
pq
ay
, 0  y a  ba / 1  c 
ap 1
1  y / b 
bap B  p, q 
a q 1
Probability Density Functions
GB2  y; a, b, p, q   GB  y; a, b, c  1, p, q  
GG  y; a,  , p  
a y
ap 1  y /  
e
 ap   p 
a controls peakedness
b is a scale parameter
c domain 0  y a  b a / 1  c 
p, q shape parameters
a
a y ap 1

b B  p, q  1   y / b 
ap
a

pq
Probability Density Functions
GB2 PDF evaluated at different parameter values:
Some families of statistical
distributions
Families
a.
i.
ii.
GB: GB1, GB2, GG
EGB: EGB1, EGB2, EGG (Y is real valued)
EGB distribution tree
Probability Density Functions
EGB  y; m,  , c, p, q  
for -<
y-m

e
1  1  c  e 
 B  p, q  1  ce  
p  y  m  /
 y  m  /
y  m /
 1 
 n

 1 c 
EGB1 y; m,  , p, q  
e
p y  m / 
1  e
 y m / 
 B  p, q 

q 1
q 1
pq
Probability Density Functions
EGB2  y; m,  , p, q  
EGG  y; m,  , p  
m controls location
 is a scale parameter
c defines the domain
p, q are shape parameters
e p y m  / 

 B  p, q  1  e
e
p  y  m  /   e y  m  / 
e
  p 
 y  m / 

pq
Probability Density Functions
EGB2 PDF evaluated at different parameter values:
Some families of statistical
distributions
Families
a.
i.
ii.
iii.
GB: GB1, GB2, GG
EGB: EGB1, EGB2, EGG
SGT (Skewed generalized t): SGED, GT, ST,
t, normal (Y is real valued)
SGT distribution tree
5 parameter
SGT
λ=0
q→∞
4 parameter
SGED
p=1
GT
SLaplace
λ=0
ST
λ=0
λ=0
q→∞
3 parameter
p=2
q→∞
p=2
GED
p=2
SNormal
p=1
p=2
λ=0
q=1/2
t
q→∞
SCauchy
q=1/2
λ=0
p→∞
2 parameter
Laplace
Uniform
Normal
Cauchy
Probability Density Functions
SGT  y ; m, ,  , p, q 

p


 2 q1/ p B 1 / p, q   1 




SGED  y; m,  ,  , p  
pe

1   sign  y  m  
 y  m / 1  sign y  m  

p
ym

p


2 1/ p 
m = mode (location parameter )
 = scale
1  

  skewness  area to left of m 
 , -1 <  < 1
2


p
p, q  shape parameters  tail thickness, moments of order  pq  df 
p
q p





q 1/ p




Probability Density Functions
SGT PDF evaluated at different parameter values:
Some families of statistical
distributions
Families
a.
i.
ii.
iii.
iv.
GB: GB1, GB2, GG
EGB: EGB1, EGB2, EGG
SGT (Skewed generalized t): SGED, GT, ST, t,
normal
IHS
Probability Density Functions
IHS
Y  a  b sinh    N  0,1 / k 
IHS  y;  ,  , k ,   
ke
 k2 
2

2
  ln   y     /    2   y     /  2       ln   

 2 



2    y      /  2  2
where
  1/  w ,   w /  w , w  .5  e  e


e
.5k 2
  mean
 2  variance
  skewness parameter
k  tail thickness
N  y; ,   limk  IHS  y; , , k,   0
2
2

, and  w  .5 e
2 k 2
e
2 k 2
2
 e 1
.5
k 2
.5
Probability Density Functions
IHS PDF evaluated at different parameter values:
Some families of statistical
distributions
Families
a.
i.
ii.
iii.
iv.
v.
GB: GB1, GB2, GG
EGB: EGB1, EGB2, EGG
SGT (Skewed generalized t): SGED, GT, ST, t,
normal
IHS
g-and-h distribution (Y is real valued)
g-and-h distribution
Definition:
 e  1  hZ 2 / 2
Yg ,h  Z   a  b 
e
 g 
gZ
where Z ~ N[0,1]
h>0
h<0
g-and-h distribution
Y0,0  Z   a  bZ ~ N  a,  2  b 2 
 e gZ  1 
Yg ,h0  Z   a  b 

 g 
Yg  0,h  Z   a  bZe gZ
2
/2
Is known as the g distribution
where the parameter g allows
for skewness.
Is known as the h distribution
• Symmetric
• Allows for thick tails
Probability Density Functions
g-and-h PDF evaluated at different parameter values with h>0:
Probability Density Functions
g-and-h PDF evaluated at different parameter values with h<0:
Some families of statistical
distributions
Families
a.
i.
ii.
iii.
iv.
v.
vi.
f(y;θ)
GB: GB1, GB2, GG
EGB: EGB1, EGB2, EGG
SGT (Skewed generalized t): SGED, GT, ST, t,
normal
IHS
g-and-h distribution
Other distributions: extreme value, Pearson
family, …
Some families of statistical
distributions
Families
a.
i.
ii.
iii.
iv.
v.
vi.
vii.
f(y;θ)
GB: GB1, GB2, GG
EGB: EGB1, EGB2, EGG
SGT (Skewed generalized t): SGED, GT, ST, t,
normal
IHS
g- and h-distribution
Other distributions: extreme value, Pearson
family, …
Extensions: 1.     x  , 2. Multivariate


Statistical Distributions
Introduction
Some families of statistical distributions
1.
2.
a.
b.
3.
4.
5.
6.
7.
8.
Families
Properties
Regression applications
Censored regression
Qualitative response models
Option pricing
VaR (value at risk)
Conclusion
Some families of statistical
distributions
Properties
b.
Moments
i.
1.
GB family
h
b
B  p  h / a, q 
 p  h / a, h / a; c 
h
EGB Y  
2 F1 

p

q

h
/
a
;
B  p, q 


for h < aq with c=1
Some families of statistical
distributions
Properties
b.
i.
Moments
1. GB family
a. GB1
b B  p  h / a, q 
EGB1 Y  
B  p, q 
h
h
Some families of statistical
distributions
Properties
b.
i.
Moments
1. GB family
a. GB1
b. GB2
b B  p  h / a, q  h / a 
EGB 2 Y  
- p  h/a  q
B  p, q 
h
h
Some families of statistical
distributions
Properties
b.
i.
Moments
1. GB family
a. GB1
b. GB2
c. GG
EGG Y h  
 h  p  h / a 
  p
for h / a  p
Some families of statistical
distributions
Properties
b.
i.
Moments
1. GB family
2. EGB family
t
e
B  p  t , q 
 p  t ,
ty
M EGB  t   E  e  
2 F1 
B  p, q 
 p+q+t
for t  q / σ with c  1
t ; c 


EGB moments
EGG
EGB1
EGB2
    p 
     p    p  q  
     p    q  
Variance
 2  '  p  
 2  '  p   '  p  q  
 2  '  p   '  q  
Skewness
 3  ''  p  
 3  ''  p   ''  p  q  
 3  ''  p   ''  q  
Excess kurtosis
 4  '''  p  
 4  '''  p   '''  p  q  
 4  '''  p   '''  q  
Mean
 s 
d n  s 
ds
EGB2 moment space
Some families of statistical
distributions
Properties
b.
i.
Moments
1. GB family
2. EGB family
3. SGT family
SGT family
ESGT  y  m 
h
 h/ p  h 1
h 
q
B
,
q


 p

 h 
p

  1   h 1  1 h 1   h 1
 
     


2


1
 
B ,q


p





for h < pq=d.f.
ESGED  y  m 
h
  h 1  


h 
     p  
h 1
h
h 1
 
1      1 1   

 2   1  
   
  p 



SGT moment space
SGT family moment space
Some families of statistical
distributions
Families
Properties
a.
b.
i.
Moments
1. GB family
2. EGB family
3. SGT family
4. IHS
IHS moment space
Some families of statistical
distributions
Families
Properties
a.
b.
i.
Moments
1. GB family
2. EGB family
3. SGT family
4. IHS
5. g-and-h family
g- and h-family
  i  j  g  2 
 

 21ih  


i
1  e


n
j
 n  n i i j  0

n
E  X g , h     a b
g i 1  ih
i  i 
i
j
Moments exist up to order 1/h (0<h)
g-and-h moment space (h>0)
(visually equivalent to the IHS)
Moment space for g-and-h (h>0)
and g-and-h (h real)
Moment space of SGT, EGB2,
IHS, and g-and-h
Some families of statistical
distributions
Properties
b.
Moments
Cumulative distribution functions (see
appendix)
i.
ii.
•
Involve the incomplete gamma and beta
functions
Some families of statistical
distributions
Properties
b.
Moments
Cumulative distribution functions (see appendix)
i.
ii.
•
iii.
Involve the incomplete gamma and beta functions
Gini coefficients (G)
Gini Coefficients (G)
Definition:
 1   
G 
 0 0 x  y f  x :   f  y :   dxdy
 2 
1  F  y   dy


G
1  
 1  F  y  dy

0

0
 G  
2
(Dorfman, 1979, RESTAT)
Gini Coefficients
Interpretation:
G = 2A
Gini Coefficients
Application:
Stochastic Dominance
Measures of income and wealth inequality
Some families of statistical
distributions
Properties
b.
i.
ii.
iii.
iv.
Moments
Cumulative distribution functions (see appendix)
Gini coefficients (G)
Incomplete moments
Incomplete moments
y

Definition:   y; h   
s h f  s  ds
E Y h 
Applications:
Option pricing formulas
Lorenz Curves
Incomplete moments
Convenient theoretical results:
Distribution
  y; h
LN
LN  y;   h 2 ,  2 
GG
GG  y; a,  , p  h / a 
GB2
GB 2  y; a, b, p  h / a, q  h / a 
Some families of statistical
distributions
Properties
b.
i.
ii.
iii.
iv.
v.
Moments
Cumulative distribution functions (see appendix)
Gini coefficients (G)
Incomplete moments
Mixture models
Mixture Models
Let f  y; ,  denote a structural or conditional
density of the random variable Y where 
and  denote vectors of distributional
parameters. Let the density of  be given by
the mixing distribution g  ;  . The observed
or mixed distribution can be written as
h  y;  ,     f  y;  ,  g  ;   d
Mixture Models
Observed model
Structural
model
Mixing
distribution
SGT  y; m, , , p, q 
SGED  y; m, , s, p 
IGG  s; p, q1/ p , q 
GT  y; , p, q 
GED  y; s, p 
IGG  s; p, q1/ p , q 
EGB2  y; , , p, q 
EGG  y; ,ln  s  , p 
 1

IGG  s; , e , q 
 

GB2  y; a, b, p, q 
GG  y; a, s, p 
IGG  s; a, b, q 
LT  y; , , q 
LN  y; , s 
IGG  s; a  1, q1/ 2 
N  y; , s 
IGA  s; q1/ 2 
t  y; , q 
Some families of statistical
distributions
Properties
b.
i.
ii.
iii.
iv.
v.
vi.
Moments
Cumulative distribution functions (see appendix)
Gini coefficients (G)
Incomplete moments
Mixture models
Hazard functions (Duration dependence)
Hazard functions
Definition:
Let f  s  denote the pdf of a spell (S) or duration of an
event.
1  F  s  is the probability that that S>s.
The corresponding hazard function is defined by
f s
h( s) 
1 F  s
which can be thought of as representing the rate or
likelihood that a spell will be completed after surviving
s periods.
Hazard functions
Applications:


Does the probability of ending a strike, unemployment spell,
expansion, or stock run depend on the length of the strike,
unemployment spell, or of the run?
With unemployment,



A job seeker might lower their reservation wage and become more likely to find a
job
Increasing hazard function
However, if being out of work is a signal of damaged goods, the longer they are
out of work might decrease employment opportunities
Decreasing hazard
function.
An alternative example might deal with attempts to model the
time between stock trades.


Engle and Russell (1998) Autoregressive conditional duration: a new model for
irregularly spaced transaction data. Econometrica 66: 1127-1162
Hazard function of time between trades is decreasing as t increases or the
longer the time between trades the less likely the next trade will occur.
Hazard functions
Applications:

Bubbles





McQueen and Thorley (1994) Bubbles, stock returns, and duration dependence.
Journal of Financial and Quantitative Analysis, 29:379-401
Efficient markets hypothesis, stock runs should not exhibit duration dependence
(constant hazard function)
McQueen and Thorley argue that asset prices may contain “bubbles” which grow
each period until they “burst” causing the stock market to crash. Hence, bubbles
cause runs of positive stock returns to exhibit duration dependence—the longer
the run the less likely it will end (decreasing hazard function), but runs of negative
stock returns exhibit no duration dependence
Grimshaw, McDonald, McQueen, and Thorley. 2005, Communications in
Statistics—Simulation and Computation, 34: 451-463.
What model should we use to characterize duration
dependence?
Exponential—constant

Gamma—the hazard function can increase, decrease, or be constant

Weibull—the hazard function can increase, decrease, or be constant

Generalized Gamma: the hazard function can be increasing, decreasing, constant,
 -shaped, or  -shaped

Hazard functions
Possible shapes for the GG hazard functions
Statistical Distributions
Introduction
Some families of statistical distributions
1.
2.
a.
b.
c.
3.
4.
5.
6.
7.
8.
Families
Properties
Model selection
Regression applications
Censored regression
Qualitative response models
Option pricing
VaR (value at risk)
Conclusion
Some families of statistical
distributions
Model selection
c.
i.
•
Goodness of fit statistics
Log-likelihood values
n
o
    n  f  yi :    for individual data
i 1
o
   n  n!  ni n  pi     n  ni 
g
for grouped data
i 1
Partition the data into g groups, Ii  Yi 1, Yi  , i  1,2,..., g
Empirical frequency: pi  ni / n, n 
Theoretical frequency: pi   
g
n
i 1
i
 f  y; dy
Ii
Model Selection
Goodness of fit statistics
i.
•
•
Log-likelihood values
Possible Measures
g
SAE   pi  pi  
i 1
g
SSE    pi  pi   
2
i 1
2
 ni

2
  n   pi    / pi   ~  2  g  # parameters 1

i 1  n
g
Model Selection
Goodness of fit statistics
i.
•
Log-likelihood values
Possible Measures
Akaike Information Criterion (AIC)
•
•
AIC  2     k 
•
•
A tool for model selection
Attaches a penalty to over-fitting a model
Model Selection
i.
ii.
Goodness of fit statistics
Testing nested models
HO : g    0
Examples:
1. HO : SGT  GT  HO :   0
2. HO : SGT  Normal  HO : p  2,   0, and q  
Testing nested models
Likelihood ratio tests
LR  2   * ~a  2  r  where r denotes the number
of independent restrictions
LR1  2 
LR2  2 
Wald test
SGT

SGT

a
2
*
~


1
GT
a
2
*
~

3
Normal 

W   g MLE   ' var  g MLE  

    
W1  ˆ  0 Var ˆ
1


1
g MLE  ~ a  2  r 
ˆ  0 ~a  2 1
Statistical Distributions
Introduction
Some families of statistical distributions
1.
2.
a.
b.
c.
d.
3.
4.
5.
6.
7.
Families
Properties
Model selection
An example: the distribution of stock returns
Regression applications
Qualitative response models
Option pricing
VaR (value at risk)
Conclusion
An example: the distribution of
stock returns
yt  n  Pt / Pt 1  ~
Pt 1  Pt Pt 1

1
Pt
Pt
Daily, weekly, and monthly excess returns (1/2/2002 –
12/29/2006) from CRSP database (NYSE, AMEX, and
NASDAQ)— 4547 companies


H0: skewness = 0
CI.95  2 6 / n , 2 6 / n
H0: excess kurtosis = 0
CI.95  2 24 / n , 2 24 / n
H0: returns ~ N(μ, σ2)
CI.95   0  JB  5.99


 skew2  excess kurtosis 2 
2

 ~ .05
 2   5.99
JB = n 
24
 6

An example: the distribution of
stock returns (continued)
% of stocks for which excess returns statistics are in 95% C.I.
HO: Skewness=0 HO:Excess kurtosis=0
HO: Normal
Daily
16.38%
0.04%
0.09%
Weekly
30.61%
4.88%
4.75%
Monthly
66.79%
56.65%
53.77%
An example: the distribution of
stock returns (continued)
Daily excess returns plotted with admissible moment space of flexible distributions
CRSP daily stocks--excess returns
60
CRSP stock
EGB2
SGT
IHS
bound
50
Kurtosis
40
30
20
10
0
-4
-3
-2
-1
0
Skewness
1
2
3
4
An example: the distribution of
stock returns (continued)
Weekly excess returns plotted with admissible moment space of flexible distributions
CRSP weekly stocks--excess returns
60
CRSP stock
EGB2
SGT
IHS
bound
50
Kurtosis
40
30
20
10
0
-4
-3
-2
-1
0
Skewness
1
2
3
4
An example: the distribution of
stock returns (continued)
Monthly excess returns plotted with admissible moment space of flexible distributions
CRSP monthly stocks--excess returns
60
CRSP stock
EGB2
SGT
IHS
bound
50
Kurtosis
40
30
20
10
0
-4
-3
-2
-1
0
Skewness
1
2
3
4
An example: the distribution of
stock returns (continued)
Fraction of stocks in the admissible skewness-kurtosis
space
daily
weekly
monthly
EGB2
15.48%
43.81%
50.80%
IHS
83.92%
84.39%
61.97%
SGT
87.62%
89.00%
95.10%
g-and-h
100.00%
99.98%
98.99%
An example: the distribution of
stock returns (continued)
Fitting a PDF to normal excess returns
Estimated PDFs for US Steel daily excess returns
Company Name Skew
US Steel
0.06
Kurtosis
Jb Stat
3.308
SSE
20
5.62
Estimated PDF
logL
SAE
Chi^2
Normal
2753.52 0.001
0.12
27.81
EGB2
2756.83 0.001
0.11
23.38
IHS
2756.76 0.001
0.11
23.46
SGT
2758.78 0.001
0.12
28.19
Returns
Normal
EGB2
IHS
SGT
18
16
14
12
10
8
6
4
2
0
-0.15
-0.1
-0.05
0
0.05
Excess returns
0.1
0.15
0.2
An example: the distribution of
stock returns (continued)
Fitting a PDF to leptokurtic excess returns
Estimated PDFs for iShares daily excess returns
Company Name Skew
Kurtosis
50
Jb Stat
iShares
-29.06
965.09 48733899.02
Estimated PDF
logL
Normal
2516.86 0.099
0.93 1433.33
EGB2
3713.99 0.002
0.13
43.47
IHS
3795.21 0.001
0.12
33.43
SGT
3810.07 0.003
0.21
79.35
SSE
SAE
Chi^2
Returns
Normal
EGB2
IHS
SGT
45
40
35
30
25
20
15
10
5
0
-0.06
-0.04
-0.02
0
0.02
Excess returns
0.04
0.06
0.08
Statistical Distributions
1.
2.
3.
4.
5.
6.
7.
8.
Introduction
Some families of statistical distributions
Regression applications
Censored regression
Qualitative response models
Option pricing
VaR (value at risk)
Conclusion
Statistical Distributions
1.
2.
3.
Introduction
Some families of statistical distributions
Regression applications
a.
4.
5.
6.
7.
8.
Background
Censored regression models
Qualitative response models
Option pricing
VaR (value at risk)
Conclusion
Regression applications-background
Model:
Yt  X t   t
X t 1xK vector of observations on the explanatory
variables
 Kx1 vector of unknown coefficients
 t independently and identically distributed random
disturbances with pdf f  ; 
Regression applications-background

If the errors are normally distributed


OLS will be unbiased and minimum variance
However, if the errors are not normally distributed


OLS will still be BLUE
There may be more efficient nonlinear estimators
Statistical Distributions
1.
2.
3.
Introduction
Some families of statistical distributions
Regression applications
a.
b.
4.
5.
6.
7.
8.
Background
Alternative estimators
Censored regression
Qualitative response models
Option pricing
VaR (value at risk)
Conclusion
Alternative Estimators
i.
Estimation
OLS
n
OLS  arg min    t  Y  X t  
t 1
LAD
n
 LAD  arg min    t  Y  X t 
t 1
Lp
n
 L  arg min    t  Y  X t 
p
t 1
p
2
Alternative Estimators
(continued)
i.
Estimation (continued)
M-estimators:
n
 M  arg min     t  Y  X t  
t 1


Includes OLS, LAD, and Lp as special cases
Includes MLE (QMLE or partially adaptive estimators) as a
special case where
  ;    n f  ; 
n
 MLE  arg min  ,    t  Y  X t  ; 
t 1




SGT
SGED
EGB2
IHS
Alternative Estimators
(continued)
i.
ii.
Estimation
Influence functions:
OLS
 ( )   '( )
LAD
Redescending
influence function
Alternative Estimators
(continued)
i.
ii.
iii.
Estimation
Influence functions
Asymptotic distribution of extremum
estimators min H  
ˆ ~ a N   ;  sandwich  A1 BA1 
where
 d 2 H   
 dH dH 
A  E
and
B

E



d

d

'
d

d

'




Alternative Estimators
(continued)
i.
ii.
iii.
iv.
Estimation
Influence functions
Asymptotic distribution of extremum estimators
Other estimators

Semiparametric (Kernel estimator, Adaptive MLE)
 n

 SP  arg min   n f K  t  Y  X t  
 t 1

where
 1  n    ei 
f K       K 

nh
h
  i 1 

ei  Yi  X i OLS
K

denotes a kernel, and h is the window width
Regression applications
(continued)
iv.
Other estimators (continued)

Generalized Method of Moments (GMM)
GMM  arg min g    ' Qg   
where
n
g      Zi h   i  Yi  X i  
i 1
Z denotes a vector of instruments (can be X)
Q is a positive definite matrix
Q  Var 1 ( g   )
Statistical Distributions
1.
2.
3.
Introduction
Some families of statistical distributions
Regression applications
a.
b.
c.
4.
5.
6.
7.
8.
Background
Alternative estimators
A Monte Carlo comparison of alternative
estimators
Censored regression models
Qualitative response models
Option pricing
VaR (value at risk)
Conclusion
A Monte Carlo comparison of
alternative estimators
c.
A Monte Carlo comparison of alternative estimators

Model: yt  1  X t   t

Error distributions: (zero mean and unitary variance)
Normal:
N 0;1
Mixture:
.9* N 0,1/ 9  .1* N 0,9
Skewness =0
Kurtosis =24.3
Skewed:
 LN  0,1  e  / e  e  1
Skewness=6.18
Kurtosis=113.9
.5
A Monte Carlo comparison of
alternative estimators
Kurtosis
Skewed
Mixture
Normal
Skewness
A Monte Carlo comparison of
alternative estimators
Sample size = 50, T=1000 replications
RMSE for slope estimators
Estimators
Normal
Mixture-thick tails
Skewed
OLS
.275
.287
.280
LAD
.332
.122
.159
SGED
.335
.128
.060
ST
.293
.112
.054
GT
.314
.133
.135
SGT
.335
.125
.073
EGB2
.287
.125
.049
IHS
.285
.119
.054
SP = AML
.285
.114
.128
GMM
.319
.115
.088
Statistical Distributions
1.
2.
3.
Introduction
Some families of statistical distributions
Regression applications
a.
b.
c.
d.
Background
Alternative estimators
A Monte Carlo comparison of alternative estimators
An application: CAPM
i. Error distribution effects
ii. ARCH effects
4.
5.
6.
7.
Censored regression
Qualitative response models
Option pricing
VaR (value at risk)
An application: CAPM
i.
CAPM and the error distribution
Daily, weekly, and monthly excess returns
(1/2/2002 – 12/29/2006) from CRSP database
(NYSE, AMEX, and NASDAQ)— 4547
companies
Percent of stocks for which OLS residual statistics are in 95% C.I.
HO: Skewness=0
HO:Excess kurtosis=0
HO: Normal (JB)
Daily
14.14%
0.02%
0%
Weekly
28.13%
3.91%
3.43%
Monthly
67.56%
57.14%
54.76%
An application: CAPM with and
without ARCH effects (ST)
i.
CAPM and the error distribution
Daily, weekly, and monthly excess returns
(1/2/2002 – 12/29/2006) from CRSP database
(NYSE, AMEX, and NASDAQ)— 4547
companies
Percent of stocks for which ST residual statistics are in 95% C.I.
HO: Skewness=0
HO:Excess kurtosis=0
HO: Normal (JB)
Daily
14.05%
0.02%
0%
Weekly
28.82%
3.83%
3.39%
Monthly
64.04%
54.72%
51.48%
An application: CAPM with and
without ARCH effects (IHS)
i.
CAPM and the error distribution
Daily, weekly, and monthly excess returns
(1/2/2002 – 12/29/2006) from CRSP database
(NYSE, AMEX, and NASDAQ)— 4547
companies
Percent of stocks for which IHS residual statistics are in 95% C.I.
HO: Skewness=0
HO:Excess kurtosis=0
HO: Normal (JB)
Daily
13.99%
0.02%
0%
Weekly
27.89%
3.83%
3.36%
Monthly
65.54%
55.71%
52.32%
An application: CAPM with
alternative error distributions
Statistics of OLS residuals
Company Name
Skewness
Kurtosis
JB stat
UNITED NATURAL FOODS INC -0.074
2.8004
0.1543
99 CENTS ONLY STORES
7.6594
85.0456
1.7541
Estimated Betas
Company Name
OLS
T
GT
SGED
EGB2
IHS
ST
SGT
UNITED NATURAL FOODS INC
0.313
0.313
0.335
0.334
0.303
0.302
0.314
0.335
99 CENTS ONLY STORES
0.184
0.125
0.125
0.110
0.109
0.106
0.110
0.110
An application: CAPM with and
without ARCH effects
CAPM and the error distribution
CAPM: how about ARCH effects?
i.
ii.
Review:





If errors are normal and no ARCH effects, OLS is MLE
If errors are not normal and no ARCH effects OLS is
BLUE, but not MLE nor efficient
If errors are normal and have ARCH effects OLS is
BLUE, but not efficient
If errors are not normal and have ARCH effects OLS
is BLUE,but not efficient
An application: CAPM with and
without ARCH effects
CAPM: ARCH effects (continued)
ii.

Model: Yt  X t   t
.5
2
1 t 1
 t  ut  0    
Percent of stocks exhibiting ARCH(1) effects (OLS)
(% rejecting HO : 1  0 )
0.10 level
0.05 level
Daily
63.2%
60.0%
Weekly
29.2%
24.1%
Monthly
18.7%
13.7%
An application: CAPM with and
without ARCH effects
Percent of stocks exhibiting ARCH(1) effects (ST)
(% rejecting
HO : 1  0 )
0.10 level
0.05 level
Daily
63.2%
59.9%
Weekly
29.1%
23.9%
Monthly
16.9%
12.3%
Percent of stocks exhibiting ARCH(1) effects (IHS)
(% rejecting H :   0 )
O
1
0.10 level
0.05 level
Daily
63.3%
60.0%
Weekly
29.3%
24.1%
Monthly
18.9%
13.9%
An application: CAPM with and
without ARCH effects
ii.

CAPM: ARCH effects (continued)
ARCH Simulations

yt  0  .9  X  excess market returnt   t , t= 1, …, 60

X monthly excess market returns, 1/2002 to 12/31/2006

Error distributions
 t ~ N  0,  2 

 u 
ARCH N 1 :  t  ut  0  1 t1
ARCHt 1 :  t
t
2
 1 t1
2
0


.5
.5
where ut ~ N 0,1
where ut ~ t (5)
An application: CAPM with and
without ARCH effects
ARCH Simulations (continued)
Root Mean Square Error (RMSE) for 10,000 replications
N ( 0 , σ^2 )
Er r o r s
N ( 0 ,1) , A r ch( 1)
t ( 5) , A r ch( 1)
Est imat io n
N o n- A R C H
ARCH
N o n- A R C H
ARCH
N o n- A R C H
ARCH
OLS/Normal
0.352
0.356
0.347
0.291
0.353
0.300
LAD
0.444
0.446
0.397
0.369
0.315
0.297
T
0.358
0.363
0.338
0.293
0.283
0.265
GED
0.381
0.389
0.357
0.318
0.306
0.285
GT
0.387
0.396
0.362
0.322
0.306
0.286
SGED
0.406
0.417
0.374
0.341
0.318
0.297
EGB2
0.371
0.376
0.352
0.312
0.300
0.281
IHS
0.368
0.377
0.348
0.319
0.291
0.275
ST
0.375
0.382
0.350
0.310
0.293
0.277
SGT
0.409
0.420
0.376
0.344
0.316
0.297
Statistical Distributions
Introduction
2. Some families of statistical distributions
3. Regression applications
Y  X  
4. Censored regression models
a. Basic framework
b. Simulation study
5. Qualitative response models
6. Option pricing
7, VaR (value at risk)
8. Conclusion
1.
*
i
i
i
Censored Regression
a. Basic Framework

Model:
yi*  X i   i
yi  yi* if y*i  0
*
i
= 0 if y < 0

Log-likelihood function:



  ,     n  f  yi  X i  ;     n  F   X i  ;  
yi  0

yi*  0 


b. Censored regression: nonnormality and heteroskedasticity
Qualitative Response Models
Statistical Distributions
Introduction
Some families of statistical distributions
Regression applications
Censored Regression models
Qualitative response models
1.
2.
3.
4.
5.
a.
6.
7.
8.
Basic framework
Option pricing
VaR (value at risk)
Conclusion
Qualitative Response—
Basic Framework

yi*  X i   i
Model:
yi  1
if
y  0 and 0 otherwise
*
i
Pr  yi  1 X i   Pr  yi*  X i    i   X i 
 Pr  i  X i   

Log-likelihood function:
Xi 
 f  s; ds  F  X  ; 
i

  ,    yi n  F  X i  ;    1  yi  n 1  F  X i  ;  
n
i 1
Qualitative Response—
Basic Framework (continued)

MLE of
as
will be consistent and asymptotically distributed
ˆ

1
2






d
ˆ ~ a N   ;     E 
 

  d d '   
if the model is correctly specified.

Probit and logit estimators will be inconsistent if
 The error distribution is incorrectly specified
 heteroskedasticity exists, e.g. unmeasured heterogeneity is
present
 relevant variables have been omitted
 The index appears in a nonlinear form

Similar results are associated with Censored &
Truncated regression models
Statistical Distributions
Introduction
Some families of statistical distributions
Regression applications
Censored regression
Qualitative response models
1.
2.
3.
4.
5.
a.
b.
6.
7.
8.
Basic framework
An application: fraud detection
Option pricing
VaR (value at risk)
Conclusion
Qualitative response—
An application: fraud detection


Prediction of corporate fraud (Y=1 fraud)
 Compare financial ratios of companies with averages
of five largest companies (“virtual” firm)
 228 companies (114 fraud and 114 non-fraud)
 Variables: accruals to assets, asset quality, asset
turnover, days sales in receivables, deferred charges
to assets, depreciation, gross margin, increase in
intangibles, inventory growth, leverage, operating
performance margin, percent uncollectables,
receivables growth, sales growth, working capital
turnover.
SGT, EGB2, & IHS formulations improve predictions
Statistical Distributions
Introduction
Some families of statistical distributions
Regression applications
Censored regression
Qualitative response models
1.
2.
3.
4.
5.
a.
b.
c.
6.
7.
Basic framework
An application
Some related issues
Option pricing
VaR (value at risk)
Qualitative response—
Some related issues

Cost of misclassification
Choice-based sampling

Heterogeneity

Semi-parametric estimation procedures

Statistical Distributions
1.
2.
3.
4.
5.
6.
7.
8.
Introduction
Some families of statistical distributions
Regression applications
Censored regression
Qualitative response models
Option pricing: European call option
VaR (value at risk)
Conclusion
Statistical Distributions
Introduction
Some families of statistical distributions
Regression applications
Qualitative response models
Option pricing: European call option
1.
2.
3.
4.
5.
a.
6.
7.
The Black-Scholes option pricing formula
VaR (value at risk)
Conclusion
Option pricing—
Black-Scholes
a.
The Black Scholes option pricing formula
The equilibrium price of a European call option is equal to the
present value of its expected return at expiration:
C f  ST , T , X   e
 rT
E  C  S0 , 0    e
 rt

 S  X  f S S
T
, T  dS
X
X 
 ST   ;1  e rT X 
 ST . 
X 
 ;0 
 St 

y
 s f  s ds  s f  s ds
h
h
where   y; h   1  
Ey
h

incomplete” moments

y
E  yh 
involve “normalized
  y ; h   1    y ; h  
Statistical Distributions
Introduction
Some families of statistical distributions
Regression applications
Qualitative response models
Option pricing: European call option
1.
2.
3.
4.
5.
a.
b.
6.
7.
The Black-Scholes option pricing formula
Some background and alternative
formulations
VaR (value at risk)
Conclusion
Option pricing– Some background
and alternative formulations
The Black Scholes (1973) option pricing formula corresponds to f  s  being the
lognormal


 LN
The Black Scholes formula (Bookstaber and McDonald, 1991) corresponding to the
Generalized Gamma is obtained from




h
GG  y; h   GG  y; a,  , p   , the cdf for the GG
a

The Black Scholes formula ( Bookstaber and McDonald, 1991) corresponding to the
GB2 is obtained from



 y; h   LN  y;   h 2 ,  2,the cdf for the lognormal


h
h
GB 2  y; h   GB 2  y; a, b, p  , q  ,
a
a

the cdf for the GB2
Rebonato (1999) applied CGB2  ST , T , X  to the Deutschemark
Option pricing– Some background
and alternative formulations

Sherrick, Garcia, and Tirupattur (1996) used CBurr 3  ST , T , X  to price soybean futures.

Theodosiou (2000) developed the CSGED  ST , T , X 

Savickas (2001) explored the use of CWeibull  ST , T , X 

Dutta and Babbel (2005) explore the g- and h- family (4-parameter) of option pricing
formulas, Cg &h  ST , T , X , based on Tukey’s nonlinear transformation of a standard
normal.

Applied the g-and-h to pricing 1-month and 3-month London Inter Bank Offer Rates
(LIBOR)

g- and- h distribution and GB2 perform much better (errors fairly highly correlated)
than the Lognormal, Burr 3, and Weibull distributions
Statistical Distributions
Introduction
Some families of statistical distributions
Regression applications
Qualitative response models
Option pricing: European call option
1.
2.
3.
4.
5.
a.
b.
c.
6.
7.
The Black-Scholes option pricing formula
Some background and alternative formulations
A comparison of pricing behavior
VaR (value at risk)
Conclusion
A comparison of pricing
behavior
A comparison of pricing behavior (Dutta and Babbel, Journal of
Business, 2005)
c.

Calculates the difference between the market price and predicted price
for the g-and-h, GB2, lognormal, Burr3, and Weibull distributions
Option Pricing
Statistical Distributions
1.
2.
3.
4.
5.
6.
7.
Introduction
Some families of statistical distributions
Regression applications
Qualitative response models
Option pricing: European call option
VaR (value at risk)
Conclusion
Statistical Distributions
Introduction
Some families of statistical distributions
Regression applications
Censored regression
Qualitative response models
Option pricing: European call option
VaR (value at risk)
1.
2.
3.
4.
5.
6.
7.
a.
8.
Background and definitions
Conclusion
VaR—Background and
definitions
i.
Value at risk (VaR) is the maximum expected
loss on a portfolio of assets over a certain time
period for a given probability level.
R  
 f  R; dR  

R is the return on the asset

θ denotes the distributional parameters

α is the predetermined confidence level or coverage
probability

R  is the corresponding maximum expected loss or
conditional threshold
R   FR1  :  
VaR—Background and
definitions
ii. Standardized returns
R    z
R    



Z 



R
 

f Z  z; dz   , Z    FZ1  :  
    FZ1  :  
VaR—Background and
definitions
iii. Unconditional VaR formulation
Estimate f(R;θ)
VaR—Background and
definitions
iv. Conditional VaR formulation (AR(1) ABSGARCH(1,1))
Rt  0  1Rt 1  Zt t  t  zt t
 t  0  1 zt 1  t 1  2 t 1
 
Rt  t   t F
1
Z
 :  
Statistical Distributions
Introduction
Some families of statistical distributions
Regression applications
Censored regression
Qualitative response models
Option pricing: European call option
VaR (value at risk)
1.
2.
3.
4.
5.
6.
7.
a.
b.
8.
Background and definitions
Models and applications
Conclusion
VaR—
Models and applications
Unconditional VaR formulation
i.








Exponential: (Hogg, R. V. and S. A. Klugman (1983))
Gamma: (Cummins, et al. 1990)
Log-gamma: (Ramlau-Hansen (1988)), (Hogg, R. V. and
S. A. Klugman (1983))
Lognormal: (Ramlau-Hansen (1988))
Stable: (Paulson and Faris (1985)
Pareto: (Hogg, R. V. and S. A. Klugman (1983))
Log-t: (Hogg, R. V. and S. A. Klugman (1983))
Weibull: (Cummins et al. (1990))
VaR—
Models and applications
Unconditional VaR formulation (continued)
i.






Burr: (Hogg, R. V. and S. A. Klugman (1983))
Generalized Pareto: (Hogg, R. V. and S. A.
Klugman (1983))
GB2: (Cummins (1990, 1999, 2007)
Pearson family: Aiuppa (1988)
Extreme value distribution: Bali (2003), Bali and
Theodossiou (2008)
IHS: Bali and Theodossiou (2008)
VaR—
Models and applications
ii.
Conditional VaR formulations
(Bali and Theodossiou, JRI, 2008)
Data:






S&P500 composite index, 1/4/50 – 12/29/2000 (n=12,832)
Daily percentage log-returns: (Sample mean = .0341,
maximum=8.71, minimum=-22.90
standard deviation = .874
skewness =1.622
kurtosis=45.52
VaR—
Models and applications
ii.
Conditional distributions (Bali and Theodossiou, JRI, 2008)
(continued)

Models





Generalized extreme value
EGB2
SGT
IHS
Findings



Out of sample VaR estimates are rejected for most unconditional
specifications
Thresholds exhibit time varying behavior
Out of sample VaR estimates for the conditional specifications
corresponding to the SGT, IHS, and EGB2 perform better than the
extreme value distributions
Selected references for option pricing and VaR



















Aiuppa, T. A. 1988. “Evaluation of Pearson curves as an approximation of the maximum probable annual aggregate loss.”
Journal of Risk and Insurance 55, 425-441
Bali, T. G., 2003. “An Extreme Value Approach to Estimating Volatility and Value at Risk,” Journal of Business, 76:83-108
Bali, T. G. and P. Theodossiou, 2007. “A Conditional-SGT-VaR Approach with Alternative GARCH Models,” Annals of
Operations Research, 151: 241-267.
Bali, T. G. and P. Theodossiou, 2008. “Risk Measurement Performance of Alternaitve Distribution Functions,” Journal of Risk
and Insurance, 75: 411-437.
Black, F (1976). The Pricing of Commodity Contracts. Journal of Financial Economics 3:169-179.
Cummins, J. D., G. Dionne, J. B. McDonald, and B. M. Pritchett 1990. “Applications of the GB2 family of distributions in
modeling insurance loss processes.” Insurance: Mathematics and Economics 9, 257-272.
Cummins, J. D., C. Merrill, and J. B. McDonald, 2007. “Risky Loss Distributions and Modeling the Loss Reserve Pay-out Tail,”
Review of Applied Economics 3.
Cummins, J. D., R. D. Phillips, and S. D. Smith 2001. “Pricing Excess of Loss Reinsurance Contracts against catastrophic
loss.” In Kenneth Froot, ed., The Financing of Catastrophe Risk (Chicago: University of Chicago Press)
Dutta, K. K. and D. F. Babbel 2005. “Extracting Probabilistic Information from the Prices of Interest Rate Options: Tests of
Distributional Assumptions.” Journal of Business 78:841-870
Hogg, R. V. and S. A. Klugman, 1983. “On the Estimation of Long Tailed Skewed Distributions with Actuarial Applications.”
Journal of Econometrics 23, 91-102.
McDonald, J. B. and R. M. Bookstaber (1991). “Option Pricing for Generalized Distributions.” Communications in Statistics:
Theory and Methods, 20(12), 4053-4068.
Rebonato, R. (1999). Volatility and correlations in the pricing of equity. FX and interest-rate options. New York: John Wiley.
Paulson, A. S. and N. J. Faris (1985). “A Practical Approach to Measuring the Distribuiton of Total Annual Claims.” In J. D.
Cummins, ed., Strategic Planning and Modeling in Property-Liability Insurance. Norwell, MA: Kluwer Academic Publishers.
Ramlau-Hansen, H. (1988). “A Solvency Study in Non-life Insurance. Part 1. Analysis of Fire, Windstorm, and Glass Claims.”
Scandinavian Actuarial Journal, pp. 3-34.
Rebonato, R. 1999. Volatility and correlations in the pricing of equity, FX and interest-rate options. New York: John Wiley.
Reid, D. H. (1978). “Claim Reserves in General Insurance,” Journal of the Institute of Actuaries 105: 211-296
Savickas, R. (2001). A Simple option-pricing formula. Working paper, Department of Finance, George Washington University,
Washington, DC.
Sherrick, B. J., P. Garcia, and V. Tirupattur (1996). Recovering probabilistic information for options markets: Tests of
distributional assumptions. Journal of Futures Markets 16:545-560.
Theodossiou, Panayiotis, “Skewed Generalized Error Distribution of Financial Assets and Option Pricing,”
Statistical Distributions
1.
2.
3.
4.
5.
6.
7.
8.
Introduction
Some families of statistical distributions
Regression applications
Censored regression
Qualitative response models
Option pricing: European call option
VaR (value at risk)
Conclusion
Conclusion
END OF PRESENTATION
Appendices
Cumulative distribution functions

1.
2.
3.
4.
5.
6.


GB, GB1, GB2, GG
EGB2
SGT
SGED
IHS
g-and-h distribution
Option pricing basics
VaR—Models and applications discussion
Appendices—
Cumulative distribution functions
1.
GB, GB1, GB2, and GG
GB 1  y; a, b, p, q  
z p 2 F1  p,1  q; p  1; z 
pB  p, q 
 Bz  p, q 
where z   y / b 
z
Bz  p, q  
a
and
p 1
s
 1  s  ds
0
q 1
B  p, q 
denotes the incomplete beta function
Appendices—
Cumulative distribution functions
1.
GB, GB1, GB2, and GG (continued)
GB 2  y; a, b, p, q  
z p 2 F1  p,1  q; p  1; z 
pB  p, q 
 Bz  p, q 
y / b

z
a
1  y / b
a
where
Appendices—
Cumulative distribution functions
1.
GB, GB1, GB2, and GG (continued)
GG  y; a, b, p  
e
y/
  p  1
 y /  
a
ap
a


F
1;
p

1;
y
/



1 1

 z  p 
where
z y/
a
z
and
z  p 
p 1  s
s
 e ds
0
  p
denotes the incomplete gamma function
Abramowitz and Stegun (1970, p. 932), McDonald (1984), and Rainville (1960,p. 60 and 125)
Appendices—
Cumulative distribution functions
2.
EGB2
EGB 2  y; m,, p, q   Bz  p, q 
e y m /
where z 
y  m /
1  e 
3.
SGT
1   1   sign  y  m  
SGT  y; m,  ,  , p, q  

sign  y  m  Bz 1/ p, q 
2
2
where
z
ym
y  m  q
p
p
p
1   sign  y  m  
p
Appendix—
Cumulative distribution functions
4.
SGED
1    1   sign  y  m  
SGED  y; m,  ,  , p  

 sign  y  m   z 1/ p 
2
2


where z 
ym
p
 1   sign  y  m  
p
p
Appendices—
Cumulative distribution functions
5.
IHS
IHS  y; , , k,    Pr Y  y   Pr  Z  z 
where
N  z;   0,  2  1  Pr  Z  z 
and
 1 3  z 2  1  sign  z  
1
 
 z2 / 2  
1 F1  ; ;

 2
2
2 2 2  2
2
 y  a 

y

a


z  k n
 
 1   k  with


 b 

 b 

2
.5
.5
2


2 k
2   k
k 2

/
.5
e

e

2
e

1
b   /w 




2
a    bw    b .5  e  e  e.5k
1
z
 
2
2





Appendices—
Cumulative distribution functions
6.


g- and h-distribution
Numeric procedures, based on the use of order statistics as
outlined in Exploring Data Tables, Trends, and Shapes by
Hoaglin,, Mosteller, and Tukey (1985), Wiley.
For h > 0, the transformation
 e gZ  1  hZ 2 / 2
Yg ,h  Z   a  b 
e
 g 
is one-to-one, (Martinez, J. and B. Iglewicz . 1984. “Some
Properties of Tukey g and h family of distributions,”
Communications in Statistics—Theory and Methods 13,
353-369). Even without an explicit functional form for the
inverse, numerical “MLE” estimates” can be obtained.
Appendices
Cumulative distribution functions
Option pricing basics


1.
2.
3.
4.
5.
6.

European call option
Put option
Definitions of terms
Assumptions
Volatility
The Greeks
VaR—Models and applications discussion
Appendices—
Option pricing basics
1.
European call option

C f  ST , T , X , r   e  rT E  C  S0 , 0, X , r    e  rt   S  X  f  S ST , T  dS
X
 X   rT
 ST   ;1  e X 
 ST 
2.
X 
 ;0 
 St 
 BS: S   d   e
T
1
Put option
BS Put formula : erT X   d2   ST   -d1 
 rT
X   d2 
Appendices—
Option pricing basics
3.
Definitions of terms:




T = time to expiration
ST = Current market price
r = interest rate (risk free rate)
X = strike price (or exercise price)

call options: price at which the instrument can be purchased
up to expiration
 ST  X  profit per share gained upon exercising or selling the
option
 ST  X  >0 in the money
 ST  X  <0 out of the money

put options: price at which the instrument can be sold up to
expiration
Appendices—
Option pricing basics
4.
Assumptions:







5.
Can short sell the underlying instrument
No arbitrage opportunities
Continuous trading in the instrument
No taxes or transaction costs
Securities are perfectly divisible
Can borrow or lend at a constant risk free rate
The instrument does not pay a dividend
Volatility (in the BS option pricing formula—
based on the LN)
Appendices—
Option pricing basics
6.

The Greeks:
(delta) measures the change in value of the instrument to a change in
the current market price



  C f  ST , T , X , r  
ST
(kappa or vega) measures the responsiveness of the value of the
instrument in response to a change in volatility
  C f  ST , T , X , r  

 (volatility )
(theta) responsiveness of the value of the instrument to T (time to
expiration)
 

X 
   ;1
 ST 
  C f  ST , T , X , r  
T
(rho) responsiveness to changes in the risk free rate

  C f  ST , T , X , r  
r
Appendices



Cumulative distribution functions
Option pricing basics
VaR—Models and applications
discussion
Appendices—VaR: Models and
applications discussion
Paulson and Faris (1985) used the stable family and Aiuppa (1988) used
the Pearson family to model insurance losses
Ramlau-Hansen (1988) modeled fire, windstorm, and glass claims using
the log-gamma and lognormal
Cummins, et al. (1990) modeled fire losses using the GB2
Cummins, Lewis, and Phillips (1999) used the LN, Burr 12, and GB2 to
model hurricane and earthquake losses.
Hogg, R. V. and S. A. Klugman, 1983. “On the Estimation of Long Tailed
Skewed Distributions with Actuarial Applications.” Journal of Econometrics
23, 91-102









Models loss distributions (a. Hurricaines (1949-1980), b. malpractice claims
paid for insured hospitals in 1975)
Considers exponential, pareto (mixture of an exponential and inverse
gamma), generalized pareto (mixture of gamma and inverse gamma), Burr
distribution (mixture of a Weibull and inverse gamma), log-t (mixture of a
lognormal and inverse gamma) and a log-gamma.
Consider alternative estimation procedures: maximum likelihood and
minimum distance estimators
Many loss distributions are characterized by skewness and long tails such
as associated with the flexible distributions coming from mixtures.
Appendices—VaR: Models and
applications discussion

Cummins, J. D., G. Dionne, J. B. McDonald, and B. M. Pritchett,
1990. “Applications of the GB2 family of distributions in
modeling insurance loss processes.” Insurance: Mathematics
and Economics 9, 257-272.




Models fire losses
Considers the GB2 and special cases GG, BR3, BR12, LN, W, and
GA to model the fire loss data. MLE estimates of distributional
parameters and Maximum Probably Yearly Aggregate Loss (MPY)
were obtained at the .01 level.
Important to use distributions which permit thick tails
Bali, T. G., 2003. “An Extreme Value Approach to Estimating
Volatility and Value at Risk,” Journal of Business, 76:83-108
Appendices—VaR: Models and
applications discussion

Cummins, J. D., C. Merrill, and J. B. McDonald, 2007. “Risky
Loss Distributions and Modeling the Loss Reserve Pay-out Tail,”
Review of Applied Economics 3.
Estimate aggregate loss distribution associated with claims incurred
in a given year, but settled in different years





Data: U.S. products liability insurance paid claims (Insurance Services
Office (ISO))
Mixture model:

Consider different GB2 distributions for each cell (year)

Multinomial distribution for fraction of claims settled at different lags
Single aggregate GB2 distribution for each year GB2 provides a
significantly better fit to severity data than the LN, gamma, Weibull,
Burr12, or generalized gamma
The Aggregate GB2 distribution has a thicker tail than does the mixture
distribution
Appendices—VaR:
Models and

applications discussion

Bali, T. G. and P. Theodossiou, 2008. “Risk Measurement
Performance of Alternative Distribution Functions,” Journal of
Risk and Insurance, 75: 411-437.
Models: Unconditional formulations









Generalized Pareto
Generalized extreme value
Box-Cox extreme value
SGED
SGT
EGB2
IHS
Models: Conditional formulations (model time-varying
VaR thresholds)
Rt  0  1Rt 1  zt t  t  zt t
t  0  1 zt 1 t 1  2t 1
Lt
Appendices—VaR: Models and
applications discussion

Bali, T. G. and P. Theodossiou, 2008. “Risk Measurement
Performance of Alternative Distribution Functions,” Journal
of Risk and Insurance, 75: 411-437. (continued)
Data


S&P500 composite index (1/4/1950 to 12/29/2000)

Daily percentage log-returns: (n=12,832

maximum=8.71

minimum=-22.90

skewness =1.622

kurtosis=45.52
Findings




Out of sample VaR estimates are rejected for most unconditional
specifications
Thresholds exhibit time varying behavior
Out of sample VaR estimates for the conditional specifications
corresponding to the SGT, IHS, and EGB2 perform better than the
extreme value distributions
END OF APPENDICES