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,
ac
cid
en
ts
a
n
d
n
at
u
r
al
c
au
s
e
s
.
So
t
h
e
r
i
s
k
m
an
a
g
e
m
e
n
t
i
s
t
h
e
id
en
t
if
icati
o
n
,
ass
e
s
s
m
e
n
t
a
n
d
p
r
io
r
itizatio
n
o
f
r
i
s
k
s
(
b
y
r
i
s
k
m
ea
s
u
r
e
m
en
ts
)
f
o
llo
w
ed
b
y
co
o
r
d
in
ated
an
d
ec
o
n
o
m
ical
a
p
p
licatio
n
o
f
r
eso
u
r
ce
s
to
m
i
n
i
m
ize,
m
o
n
ito
r
an
d
co
n
tr
o
l
th
e
p
r
o
b
ab
ilit
y
a
n
d
/o
r
i
m
p
ac
t
o
f
u
n
f
o
r
tu
n
ate
e
v
en
t
s
(
b
y
r
i
s
k
tr
ea
t
m
e
n
t
s
)
[
2
]
.
Ma
r
k
et
r
is
k
is
a
co
m
p
le
x
s
u
b
j
ec
t
w
ith
m
u
lt
ip
le
d
i
m
en
s
io
n
s
an
d
i
m
p
licatio
n
s
f
o
r
elec
tr
icit
y
b
u
s
i
n
es
s
ac
ti
v
i
t
y
.
A
n
a
l
y
t
ical
r
is
k
m
ea
s
u
r
es
s
u
c
h
as
tr
ad
itio
n
a
l
Gr
ee
k
m
ea
s
u
r
es
o
r
h
ig
h
-
o
r
d
er
an
d
cr
o
s
s
-
s
e
n
s
iti
v
ities
allo
w
u
s
to
co
n
tr
o
l
i
n
d
etail
m
ar
k
e
t
r
is
k
,
b
u
t
s
o
m
eti
m
es
th
ese
a
n
al
y
tical
r
is
k
m
ea
s
u
r
es
ar
e
to
o
tech
n
ical
to
b
e
u
n
d
er
s
to
o
d
b
y
n
o
n
-
tec
h
n
ical
s
ta
f
f
o
r
b
y
m
an
a
g
e
m
e
n
t
[
3
]
.
Hen
ce
,
it
i
s
n
ec
ess
ar
y
to
m
ak
e
a
s
y
n
t
h
esis
o
f
th
e
i
n
f
o
r
m
atio
n
co
n
t
ain
ed
in
an
al
y
tica
l
m
ea
s
u
r
es
in
to
a
m
o
r
e
in
te
lli
g
ib
le
f
o
r
m
.
T
h
e
n
at
u
r
al
w
a
y
o
f
cr
ea
ti
n
g
s
u
c
h
a
t
y
p
e
o
f
r
is
k
m
ea
s
u
r
e
is
th
at
o
f
ass
es
s
i
n
g
t
h
e
i
m
p
ac
t
o
f
r
is
k
y
ev
en
t
s
in
m
o
n
etar
y
ter
m
s
,
b
ec
au
s
e
n
o
n
-
tech
n
ical
p
eo
p
le
ar
e
also
ca
p
ab
le
o
f
u
n
d
er
s
ta
n
d
in
g
t
h
e
m
ea
n
i
n
g
o
f
a
p
o
ten
tial
m
o
n
etar
y
(
o
r
ec
o
n
o
m
ic)
lo
s
s
.
T
h
is
is
e
x
ac
tl
y
th
e
r
ea
s
o
n
w
h
y
s
y
n
t
h
etic
r
is
k
m
ea
s
u
r
es
h
a
v
e
b
ee
n
in
tr
o
d
u
ce
d
an
d
h
a
v
e
r
ea
ch
ed
a
v
er
y
h
ig
h
i
m
p
o
r
tan
ce
in
t
h
e
la
s
t
te
n
y
ea
r
s
[
1
]
.
A
t
y
p
ica
l
ex
a
m
p
le
is
r
ep
r
esen
ted
b
y
Val
u
e
at
R
i
s
k
,
b
u
t
t
h
is
in
d
ice
is
n
o
t
t
h
e
o
n
l
y
s
y
n
t
h
etic
r
is
k
m
ea
s
u
r
e
i
m
p
o
r
ta
n
t
to
co
n
s
id
e
r
in
t
h
e
elec
tr
icit
y
f
ield
,
esp
e
ciall
y
i
f
th
e
b
u
s
i
n
es
s
w
e
ar
e
in
te
r
ested
to
r
is
k
-
ass
es
s
is
n
o
t c
o
m
p
letel
y
b
ased
o
n
f
i
n
an
cia
l tr
ad
in
g
[
3
-
5
]
.
I
n
th
i
s
p
ap
er
r
is
k
m
ea
s
u
r
es
Va
lu
e
at
R
is
k
an
d
C
o
n
d
itio
n
a
l
V
alu
e
at
R
is
k
ar
e
u
s
ed
f
o
r
th
e
a
s
s
es
s
m
en
t
r
is
k
i
n
t
h
e
a
n
cil
lar
y
s
er
v
ice
s
p
r
o
v
is
io
n
.
At
t
h
e
b
eg
i
n
n
i
n
g
o
f
th
e
p
r
o
ce
d
u
r
e
to
s
p
e
ci
f
y
a
r
is
k
m
ea
s
u
r
e,
o
n
e
h
a
s
to
d
eter
m
i
n
e
a
p
r
o
b
ab
ilit
y
d
i
s
tr
ib
u
tio
n
f
u
n
ctio
n
o
f
th
e
v
a
r
iab
le
o
f
in
ter
est,
w
h
er
e
a
n
o
r
m
al
l
y
d
is
tr
ib
u
ted
f
u
n
ctio
n
ca
n
b
e
as
s
u
m
ed
.
T
h
i
s
as
s
u
m
p
tio
n
is
o
f
te
n
j
u
s
tifie
d
in
a
f
in
a
n
cial
co
n
te
x
t.
I
n
o
r
d
er
to
f
i
n
d
t
h
e
r
ig
h
t
p
ar
am
eter
s
o
f
th
e
d
is
tr
ib
u
tio
n
f
u
n
ct
io
n
,
f
o
r
in
s
ta
n
ce
s
m
e
an
v
al
u
e
an
d
s
tan
d
ar
d
d
ev
iatio
n
f
o
r
a
n
o
r
m
a
l
d
is
tr
ib
u
tio
n
,
o
n
e
h
a
s
to
h
av
e
s
o
m
e
d
ata
ab
o
u
t
th
e
v
ar
iab
le.
On
e
p
o
s
s
ib
ilit
y
i
s
to
tak
e
h
is
to
r
ic
v
alu
es
o
f
t
h
e
v
ar
iab
le
an
d
/o
r
it
s
f
r
eq
u
en
c
y
o
f
o
cc
u
r
r
en
ce
.
T
h
e
s
ec
o
n
d
p
o
s
s
ib
ilit
y
i
s
to
s
i
m
u
late
th
e
p
r
o
b
lem
(
i
n
Mo
n
te
-
C
ar
lo
s
i
m
u
lat
io
n
)
an
d
ex
tr
ac
t
o
u
t o
f
th
e
s
e
r
es
u
lts
t
h
e
n
ee
d
ed
d
ata
[
6
,
7
]
.
T
h
is
p
ap
er
is
o
r
g
an
ized
as
f
o
llo
w
s
.
Sec
tio
n
2
p
r
o
v
id
es
th
e
m
at
h
e
m
atica
l
f
r
a
m
e
w
o
r
k
f
o
r
r
is
k
m
ea
s
u
r
es
s
u
ch
as
Val
u
e
at
R
i
s
k
an
d
C
o
n
d
it
io
n
al
Val
u
e
at
R
is
k
.
I
n
Sectio
n
3
th
e
r
esu
lt
s
ar
e
p
r
esen
ted
an
d
d
is
cu
s
s
ed
.
T
h
e
co
n
clu
s
io
n
s
a
n
d
p
o
in
t to
f
u
t
u
r
e
r
esear
ch
ar
e
o
u
tli
n
ed
in
Sect
io
n
4
.
2.
VALU
E
-
AT
-
R
I
SK
AND
CO
NDIT
I
O
NA
L
VAL
UE
-
AT
-
RIS
K
AS A
R
I
SK
M
E
ASUR
E
S
I
n
th
i
s
c
h
ap
ter
th
e
m
at
h
e
m
at
ical
d
ef
i
n
iti
o
n
s
o
f
Valu
e
a
t
R
is
k
a
n
d
C
o
n
d
itio
n
al
Val
u
e
at
R
is
k
ar
e
g
iv
e
n
,
f
o
llo
w
ed
b
y
a
n
in
tu
i
tiv
e
d
escr
ip
tio
n
o
f
th
eir
p
r
o
p
er
ties
an
d
in
ter
ac
tio
n
s
.
2
.
1
.
V
a
lue a
t
Ris
k
Valu
e
at
R
is
k
(
Va
R
)
is
th
e
m
a
x
i
m
u
m
lo
s
s
c
t
h
at
w
i
ll
n
o
t
b
e
ex
ce
ed
ed
at
a
g
i
v
en
co
n
f
id
en
c
e
lev
el
.
Fo
r
a
g
i
v
e
n
p
o
r
tf
o
lio
,
p
r
o
b
a
b
ilit
y
d
i
s
tr
ib
u
tio
n
,
p
r
o
b
ab
ilit
y
le
v
el
a
n
d
ti
m
e
h
o
r
izo
n
,
V
aR
is
d
ef
i
n
ed
as
a
th
r
es
h
o
ld
v
al
u
e
s
u
ch
th
at
th
e
p
r
o
b
a
b
ilit
y
t
h
at
t
h
e
lo
s
s
o
n
t
h
e
p
o
r
tf
o
lio
,
b
ased
o
n
(
f
air
)
m
ar
k
et
p
r
ices
w
it
h
o
u
t
tr
ad
in
g
in
th
e
p
o
r
tf
o
lio
an
d
o
v
er
t
h
e
g
iv
e
n
ti
m
e
h
o
r
izo
n
,
e
x
ce
ed
s
t
h
i
s
v
alu
e
is
th
e
g
iv
e
n
p
r
o
b
ab
ilit
y
le
v
el.
T
h
is
g
iv
e
s
t
h
e
f
o
l
lo
w
i
n
g
m
at
h
e
m
atica
l d
ef
in
i
tio
n
o
f
Va
R
[
8
,
9
]
.
Let X
b
e
a
r
a
n
d
o
m
va
r
ia
b
le
r
e
p
r
esen
tin
g
lo
s
s
.
Give
n
p
a
r
a
me
ter 0
<
< 1
,
th
e
-
Va
R
o
f X
is
:
(
)
m
i
n
{
:
(
)
}
V
a
R
X
c
P
X
c
(
1
)
VaR c
an
h
av
e
s
ev
er
al
eq
u
iv
al
en
t in
ter
p
r
etatio
n
s
[
9
]:
VaR
(
X)
is
t
h
e
min
imu
m
lo
s
s
th
at
w
ill
n
o
t b
e
ex
ce
ed
ed
w
it
h
p
r
o
b
a
b
ilit
y
;
VaR
(
X)
is
t
h
e
-
q
u
a
n
tile o
f
t
h
e
d
is
tr
ib
u
tio
n
o
f
X
;
VaR
(
X
)
is
t
h
e
s
ma
lle
s
t
lo
s
s
i
n
th
e
(
1
–
)
1
0
0
%
w
o
r
s
t c
ases
;
VaR
(
X)
is
t
h
e
h
ig
h
est
lo
s
s
in
th
e
1
0
0
% b
est ca
s
es.
Su
p
p
o
s
e
th
e
c
u
r
r
en
t
p
r
ice
o
f
a
p
o
r
tf
o
lio
is
P
0
an
d
th
e
r
ate
o
f
r
etu
r
n
R
f
o
r
t
h
is
p
o
r
tf
o
lio
i
s
n
o
r
m
al
l
y
d
is
tr
ib
u
ted
w
it
h
m
ea
n
μ
an
d
s
tan
d
ar
d
d
ev
iatio
n
σ
.
T
h
en
th
e
p
o
r
tf
o
lio
v
alu
e
at
t
h
e
en
d
o
f
th
e
ti
m
e
h
o
r
izo
n
i
s
P
1
=
(1
+R)
×
P
0
w
it
h
m
ea
n
(1
+
μ
)
×
P
0
an
d
s
tan
d
ar
d
d
ev
iatio
n
σ
P
0
.
Firstl
y
,
w
e
d
en
o
te
t
h
e
lo
w
e
s
t
p
o
r
tf
o
lio
v
al
u
e
at
s
o
m
e
co
n
f
id
e
n
ce
lev
el
as
P
*
1
=
(
1
+
R
*
)
×
P
*
0
.
T
h
en
th
e
VaR
n
u
m
b
er
r
elativ
e
to
th
e
ex
p
ec
ted
r
etu
r
n
is
VaR
(
mea
n
)
=
E
[
P
1
]
–
P
*
1
=
(
1
+
μ)
×
P
0
–
(
1
+
R
*
)
×
P
0
=
(
μ
–
R
*
)
×
P
0
.
A
s
s
u
m
in
g
t
h
e
ex
p
e
cted
r
etu
r
n
is
ze
r
o
,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
R
is
k
a
s
s
es
s
men
t fo
r
a
n
cilla
r
y
s
ervices
(
Ome
r
Ha
d
z
ic
)
1563
VaR(
z
ero
)
=
−(
P
*
1
–
P
0
)
=
−
P
0
R
*
.
Nex
t
w
e
ca
n
ap
p
l
y
t
h
e
tr
a
n
s
f
o
r
m
atio
n
eq
u
atio
n
k
=
(
z
–
)/
,
w
it
h
z
=
–
Va
R
an
d
k
=
–
k
,
t
h
u
s
Va
R
=
k
–
(
k
–
n
u
m
b
er
co
r
r
esp
o
n
d
in
g
to
th
e
co
n
f
id
e
n
ce
le
v
el
)
.
T
h
en
it
is
R
*
=
–
(
k
–
)
.
B
y
s
u
b
s
t
itu
tin
g
R
*
in
to
t
h
e
ab
o
v
e
eq
u
atio
n
s
,
w
e
h
a
v
e
th
e
f
o
l
lo
w
i
n
g
t
w
o
f
o
r
m
u
las
[
8
]
:
VaR
(
z
ero
)
=
(
k
–
)
×
P
0
an
d
VaR(
mea
n
)
=
k
P
0
.
I
f
th
e
co
n
f
id
en
ce
le
v
el
i
s
g
i
v
e
n
an
d
th
e
p
o
r
tf
o
lio
v
al
u
e
is
k
n
o
wn
,
th
e
o
n
l
y
v
ar
iab
le
is
th
e
s
tan
d
ar
d
d
ev
iatio
n
o
f
th
e
r
ate
o
f
r
etu
r
n
,
.
T
h
e
to
o
l
th
at
is
ex
tr
e
m
el
y
u
s
e
f
u
l
to
m
an
a
g
e
r
i
s
k
is
t
h
e
co
m
p
o
n
en
t
VaR,
w
h
ic
h
is
a
p
ar
tit
io
n
o
f
t
h
e
p
o
r
tf
o
lio
VaR
t
h
at
in
d
icate
s
th
e
c
h
an
g
e
o
f
Va
R
i
f
a
g
iv
e
n
co
m
p
o
n
e
n
t
w
as
d
elete
d
.
T
h
e
co
m
p
o
n
e
n
t
Va
R
m
ea
s
u
r
es
th
e
to
tal
co
n
tr
ib
u
tio
n
o
f
co
m
p
o
n
en
t
i
to
th
e
o
v
er
al
l
p
o
r
tf
o
lio
VaR
a
n
d
ca
n
u
n
iq
u
el
y
b
e
a
ttrib
u
ted
to
ea
ch
o
f
th
e
in
d
i
v
id
u
al
co
m
p
o
n
en
ts
a
n
d
a
g
g
r
e
g
ate
li
n
ea
r
l
y
i
n
t
o
th
e
to
tal
d
i
v
er
s
i
f
ied
p
o
r
tf
o
li
o
VaR
.
W
e
ca
n
u
s
e
it
to
h
a
v
e
a
r
is
k
d
ec
o
m
p
o
s
itio
n
o
f
th
e
c
u
r
r
en
t
p
o
r
tf
o
lio
.
T
h
e
s
u
m
o
f
in
d
ivid
u
a
l
VaRs
i
s
n
o
t
s
o
u
s
ef
u
l
s
i
n
ce
it
d
is
ca
r
d
s
th
e
d
iv
er
s
i
f
icatio
n
e
f
f
ec
ts
.
T
h
u
s
,
w
e
d
e
f
in
e
th
e
co
m
p
o
n
en
t
Va
R
in
ter
m
o
f
ma
r
g
in
a
l
VaR
(
Δ
Va
R
)
as
f
o
llo
w
s
[
8
]
:
C
m
p
_
VaR
i
=
(
Δ
V
aR
i
)
×
w
i
P
=
(
VaR/P
)
×β
i
×w
i
P
=
VaR×(
β
i
w
i
)
.
No
te
th
at
th
e
s
u
m
o
f
all
co
m
p
o
n
en
t
VaRs
(
C
m
p
_
Va
R
)
is
th
e
VaR
f
o
r
t
h
e
en
tire
p
o
r
tf
o
lio
:
(i=
1;n
)
C
m
p
_
Va
R
i
=
VaR×(
(i=
1;n)
β
i
w
i
)
=
VaR
,
w
h
er
e
is
:
β
i
=
i,
P
/(
p
)
2
,
w
i
–
w
ei
g
h
t
d
ef
in
ed
b
y
w
i
=
P
i
/P
,
P
–
p
o
r
tf
o
lio
v
alu
e,
P
i
–
v
alu
e
o
f
ass
et
i
,
i,
P
–
c
o
v
ar
ian
ce
b
et
w
ee
n
i
th
as
s
et
(
i
th
t
y
p
e
o
f
a
n
cillar
y
s
er
v
ice
)
a
n
d
p
o
r
tf
o
lio
,
(
p
)
2
–
s
ta
n
d
ar
d
d
ev
iatio
n
f
o
r
r
etu
r
n
r
ate
o
f
t
h
e
p
o
r
tf
o
lio
.
I
n
tu
it
iv
el
y
th
e
m
ar
g
in
a
l
r
is
k
o
f
a
p
o
r
tf
o
lio
w
it
h
r
esp
ec
t
to
a
n
a
s
s
et,
i
s
t
h
e
in
cr
e
m
en
t
i
n
r
is
k
th
a
t
o
b
tain
ed
b
y
b
u
y
i
n
g
a
s
m
all
a
m
o
u
n
t
o
f
t
h
at
as
s
et.
T
h
e
m
ar
g
in
al
Va
R
is
d
ef
in
ed
as
t
h
e
p
ar
tial
d
er
iv
ati
v
e
w
it
h
r
esp
ec
t
to
th
e
co
m
p
o
n
en
t
w
e
ig
h
t.
A
cc
o
r
d
in
g
to
r
ef
er
en
ce
[
8
]
,
i
t
m
ea
s
u
r
e
is
d
e
fi
n
ed
to
b
e
th
e
ch
an
g
e
i
n
p
o
r
tf
o
lio
r
is
k
r
es
u
lti
n
g
f
r
o
m
tak
i
n
g
a
n
ad
d
itio
n
al
d
o
llar
to
a
co
m
p
o
n
e
n
t
:
Δ
Va
R
i
=
k
×
i
,P
/
p
=
k
×
p
×β
i
=
(
VaR
/P
)
×β
i
.
I
n
d
iv
id
u
a
l
V
A
R
r
ep
r
esen
ts
t
h
e
V
AR
as
s
o
ciate
d
to
in
d
iv
id
u
al
co
m
p
o
n
e
n
t
o
r
ass
et
co
n
s
tit
u
ti
n
g
p
o
r
tf
o
lio
n
o
t
ta
k
i
n
g
i
n
to
co
n
s
id
er
atio
n
d
i
v
er
s
i
f
icatio
n
b
en
ef
its
.
I
n
d
i
v
id
u
al
V
AR
i
s
ca
lc
u
lated
t
h
r
o
u
g
h
t
h
e
f
o
llo
w
in
g
eq
u
atio
n
:
VaR
i
=
k
i
P
i
,
w
h
er
e
i
r
ep
r
esen
ts
t
h
e
ass
et
(
t
y
p
e
o
f
an
ci
llar
y
s
er
v
ice
)
v
o
latilit
y
o
v
er
p
ast
p
er
io
d
.
2
.
2
.
C
o
nd
it
io
na
l V
a
lue a
t
R
is
k
T
h
e
g
en
er
al
d
e
fi
n
i
tio
n
o
f
C
o
n
d
itio
n
al
Valu
e
at
R
is
k
(
C
V
aR
)
is
a
s
f
o
llo
w
s
.
A
t
t
h
i
s
p
o
in
t,
o
n
l
y
th
e
C
VaR
d
e
fi
n
i
tio
n
f
o
r
co
n
tin
u
o
u
s
r
an
d
o
m
v
ar
iab
les
w
ill
b
e
g
iv
en
to
cr
ea
te
a
m
o
r
e
in
t
u
iti
v
e
in
tr
o
d
u
ctio
n
i
n
to
th
e
to
p
ic.
Fo
r
co
n
t
in
u
o
u
s
v
ar
iab
le
X
,
th
e
C
Va
R
i
s
t
h
e
e
x
p
ec
ted
lo
s
s
,
co
n
d
itio
n
a
l
o
n
th
e
f
ac
t
t
h
at
th
e
lo
s
s
ex
ce
ed
s
th
e
Va
R
at
t
h
e
g
i
v
e
n
co
n
fi
d
e
n
ce
lev
el
.
Let
X
b
e
a
co
n
tin
u
o
u
s
r
a
n
d
o
m
va
r
ia
b
le
r
ep
r
esen
tin
g
lo
s
s
.
Give
n
p
a
r
a
mete
r
0
<
<
1
,
t
h
e
-
C
Va
R
o
f X
is
:
(
)
[
|
(
)
]
C
V
a
R
X
X
X
V
a
R
X
E
(
2
)
Fu
n
d
a
m
e
n
tal
p
r
o
p
er
ties
o
f
C
VaR,
as
a
m
ea
s
u
r
e
o
f
r
is
k
w
i
th
s
ig
n
i
f
ica
n
t
ad
v
a
n
ta
g
es
o
v
e
r
VaR,
ar
e
d
er
iv
ed
f
o
r
lo
s
s
d
is
tr
ib
u
tio
n
s
in
f
i
n
an
ce
t
h
at
ca
n
i
n
v
o
l
v
e
d
is
cr
ee
tn
es
s
.
Su
c
h
d
is
tr
ib
u
tio
n
s
ar
e
o
f
p
ar
ticu
lar
i
m
p
o
r
tan
ce
i
n
ap
p
licatio
n
s
b
ec
au
s
e
o
f
t
h
e
p
r
ev
ale
n
ce
o
f
m
o
d
el
s
b
ased
o
n
s
ce
n
ar
io
s
a
n
d
f
in
ite
s
a
m
p
li
n
g
.
C
VaR
is
ab
le
to
q
u
a
n
ti
f
y
d
an
g
er
s
b
e
y
o
n
d
Va
R
,
an
d
m
o
r
eo
v
er
it
is
co
h
er
en
t.
I
t
p
r
o
v
id
es
o
p
ti
m
izat
io
n
s
h
o
r
tcu
t
s
w
h
ic
h
,
t
h
r
o
u
g
h
li
n
e
ar
p
r
o
g
r
am
m
i
n
g
tech
n
iq
u
es,
m
ak
e
p
r
ac
tical
m
a
n
y
lar
g
e
-
s
c
ale
ca
lcu
lat
io
n
s
th
at
co
u
ld
o
th
er
w
is
e
b
e
o
u
t o
f
r
ea
c
h
.
Fo
r
co
n
tin
u
o
u
s
lo
s
s
d
is
tr
ib
u
ti
o
n
s
,
t
h
e
C
VaR
at
a
g
i
v
e
n
co
n
f
id
en
ce
lev
el
is
t
h
e
e
x
p
ec
ted
lo
s
s
g
i
v
e
n
th
at
th
e
lo
s
s
is
g
r
ea
ter
t
h
a
n
t
h
e
VaR
at
t
h
at
lev
el,
o
r
f
o
r
t
h
a
t
m
a
tter
,
th
e
ex
p
ec
ted
lo
s
s
g
i
v
en
t
h
at
t
h
e
lo
s
s
is
g
r
ea
ter
t
h
an
o
r
eq
u
al
to
th
e
Va
R
.
Fo
r
d
is
tr
ib
u
tio
n
s
w
i
th
p
o
s
s
i
b
le
d
is
co
n
ti
n
u
ities
,
h
o
w
e
v
er
,
i
t h
a
s
a
m
o
r
e
s
u
b
tle
d
ef
in
i
tio
n
a
n
d
ca
n
d
if
er
f
r
o
m
eit
h
er
o
f
t
h
o
s
e
q
u
a
n
titi
e
s
,
w
h
ic
h
f
o
r
c
o
n
v
en
ien
ce
i
n
co
m
p
ar
is
io
n
ca
n
b
e
d
esig
n
ated
b
y
C
VaR
+
an
d
C
VaR
–
,
r
esp
ec
tiv
el
y
.
C
VaR
+
h
as
s
o
m
eti
m
es
b
ee
n
ca
lled
"
mea
n
s
h
o
r
tfa
ll
"
,
w
h
ile
"
ta
il V
a
R
"
is
a
ter
m
t
h
at
h
as b
ee
n
s
u
g
g
es
ted
f
o
r
C
VaR
–
.
Ge
n
er
all
y
[
9
]
:
C
VaR
+
(
u
p
p
er C
V
a
R
)
is
ex
p
ec
ted
va
lu
e
o
f X
s
tr
ictly
ex
ce
ed
in
g
V
a
R
:
(
)
[
|
(
)
]
CV
a
R
X
X
X
V
a
R
X
E
(
3
)
C
VaR
–
(
lo
w
e
r
C
V
a
R
)
is
ex
p
ec
ted
va
lu
e
o
f X
w
ea
kly
ex
ce
ed
i
n
g
V
a
R
:
(
)
[
|
(
)
]
CV
a
R
X
X
X
V
a
R
X
E
(
4
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
9
,
No
.
3
,
J
u
n
e
2
0
1
9
:
1
5
6
1
-
1
5
6
8
1564
C
VaR i
s
co
n
v
ex
,
b
u
t V
a
R
,
C
VaR
–
,
C
Va
R
+
m
a
y
b
e
n
o
n
-
co
n
v
ex
. T
h
e
f
o
llo
w
i
n
g
i
n
eq
u
a
liti
e
s
ar
e
v
alid
:
(
)
(
)
(
)
(
)
V
a
R
X
CV
a
R
X
CV
a
R
X
CV
a
R
X
(
5
)
Fig
u
r
e
1
s
h
o
w
s
t
h
e
Va
R
an
d
C
VaR
f
o
r
a
s
p
ec
if
i
c
co
n
tin
u
o
u
s
r
an
d
o
m
v
ar
iab
le
X.
T
h
e
cu
m
u
lat
iv
e
d
is
tr
ib
u
tio
n
f
u
n
ctio
n
o
f
X
ca
n
b
e
u
s
ed
to
f
i
n
d
VaR
(
X)
,
an
d
VaR
(
X)
ca
n
b
e
u
s
ed
i
n
tu
r
n
to
ca
lc
u
late
C
VaR
(
X)
.
Fig
u
r
e
1
.
VaR
(
X)
an
d
C
Va
R
(
X)
o
f
a
r
an
d
o
m
v
ar
iab
le
X
r
ep
r
esen
tin
g
lo
s
s
2
.
3
.
C
o
nd
it
io
na
l V
a
lue a
t
R
is
k
o
pti
m
iza
t
io
n t
heo
ry
P
o
r
tf
o
lio
co
n
s
is
ts
o
f
n
i
n
s
tr
u
m
en
ts
.
L
et
x
= (
x
1
,
x
2
,
…,
x
n
)
b
e
a
v
ec
to
r
o
f
p
o
s
itio
n
s
,
= (
1
,
2
,
…,
n
)
b
e
a
v
ec
to
r
o
f
in
itial
p
r
ices,
an
d
y
=
(
y
1
,
y
2
,
…,
y
n
)
b
e
a
v
ec
to
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Evaluation Warning : The document was created with Spire.PDF for Python.
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3.
NUM
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R
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ased
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n
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h
e
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lio
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n
t
h
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s
ca
s
e,
th
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ca
p
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f
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n
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f
o
r
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n
g
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ac
ce
p
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h
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(
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to
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f
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ch
in
s
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t)
as
illu
s
tr
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in
Fig
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r
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2.
Fo
r
th
e
an
al
y
ze
d
test
ca
s
e
an
d
h
is
to
r
ical
p
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ice
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f
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ch
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in
F
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,
p
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esen
ted
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ased
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to
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f
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ca
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n
t
h
is
m
et
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d
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d
a
co
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r
elatio
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an
d
v
o
latilit
y
m
atr
i
x
to
g
e
n
er
ate
th
e
r
an
d
o
m
s
ce
n
ar
io
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
9
,
No
.
3
,
J
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n
e
2
0
1
9
:
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5
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.
RE
F
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R
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NC
E
S
[1
]
S
.
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l,
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A
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lary
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.
[2
]
J.
Do
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g
e
,
M
.
F
e
h
r,
J.
Hin
z
,
H.
J.
L
u
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th
i,
M
.
W
il
h
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lm
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"
On
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2
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6
.
[3
]
S
.
F
i
o
re
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z
a
n
i,
“
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c
it
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trad
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lan
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0
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6
.
[4
]
M
.
L
iu
,
F
.
F
.
W
u
,
“
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su
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m
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IEE
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Po
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Soc
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M
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a
,
2
0
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[5
]
R.
Da
h
lg
re
n
,
C.
C.
L
iu
,
J.
L
a
wa
r
re
e
,
"
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s
s
m
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t
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"
IEE
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ra
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sa
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ms
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v
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1
8
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o
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2
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p
p
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1
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0
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3.
[6
]
C
.
Blan
c
o
,
"
V
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lu
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at
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Risk
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:
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R
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risk
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p
p
.
6
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,
1
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9
8
.
[7
]
P
.
Rich
tárik
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“
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me
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s in
fi
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”
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s n
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.
[8
]
D.
Bo
,
"
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Risk
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m
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M
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ti
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s,
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ti
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rsity
o
f
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p
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re
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2
0
0
1
.
[9
]
P
,
Jo
ri
o
n
,
Va
lu
e
a
t
Ri
sk
,
M
c
G
ra
w
-
Hill
,
2
n
d
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it
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n
,
2
0
0
1
.
[1
0
]
R.
T
.
Ro
c
k
a
fe
ll
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r,
S
.
P
.
Ury
a
s
e
v
,
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ti
m
iz
a
ti
o
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o
f
Co
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d
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ti
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l
V
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l
u
e
-
at
-
Risk
,
"
J
o
u
rn
a
l
o
f
Ri
sk
,
v
o
l.
2
,
p
p
.
2
1
-
4
2
,
2
0
0
0
.
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RAP
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AUTH
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RS
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s.
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