<s>
Expected	B-Algorithm
shortfall	I-Algorithm
(	O
ES	O
)	O
is	O
a	O
risk	O
measure	O
—	O
a	O
concept	O
used	O
in	O
the	O
field	O
of	O
financial	O
risk	O
measurement	O
to	O
evaluate	O
the	O
market	O
risk	O
or	O
credit	O
risk	O
of	O
a	O
portfolio	O
.	O
</s>
<s>
The	O
"	O
expected	B-Algorithm
shortfall	I-Algorithm
at	O
q%	O
level	O
"	O
is	O
the	O
expected	O
return	O
on	O
the	O
portfolio	O
in	O
the	O
worst	O
of	O
cases	O
.	O
</s>
<s>
Expected	B-Algorithm
shortfall	I-Algorithm
is	O
also	O
called	O
conditional	B-Algorithm
value	I-Algorithm
at	I-Algorithm
risk	I-Algorithm
(	O
CVaR	B-Algorithm
)	O
,	O
average	B-Algorithm
value	I-Algorithm
at	I-Algorithm
risk	I-Algorithm
(	O
AVaR	O
)	O
,	O
expected	B-Algorithm
tail	I-Algorithm
loss	I-Algorithm
(	O
ETL	O
)	O
,	O
and	O
superquantile	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
unlike	O
the	O
discounted	O
maximum	O
loss	O
,	O
even	O
for	O
lower	O
values	O
of	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
does	O
not	O
consider	O
only	O
the	O
single	O
most	O
catastrophic	O
outcome	O
.	O
</s>
<s>
Expected	B-Algorithm
shortfall	I-Algorithm
is	O
considered	O
a	O
more	O
useful	O
risk	O
measure	O
than	O
VaR	O
because	O
it	O
is	O
a	O
coherent	O
spectral	O
measure	O
of	O
financial	O
portfolio	O
risk	O
.	O
</s>
<s>
Expected	B-Algorithm
shortfall	I-Algorithm
can	O
be	O
generalized	O
to	O
a	O
general	O
class	O
of	O
coherent	O
risk	O
measures	O
on	O
spaces	O
(	O
Lp	O
space	O
)	O
with	O
a	O
corresponding	O
dual	O
characterization	O
in	O
the	O
corresponding	O
dual	O
space	O
.	O
</s>
<s>
If	O
the	O
underlying	O
distribution	O
for	O
is	O
a	O
continuous	O
distribution	O
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equivalent	O
to	O
the	O
tail	O
conditional	O
expectation	O
defined	O
by	O
.	O
</s>
<s>
If	O
we	O
believe	O
our	O
average	O
loss	O
on	O
the	O
worst	O
5%	O
of	O
the	O
possible	O
outcomes	O
for	O
our	O
portfolio	O
is	O
EUR	O
1000	O
,	O
then	O
we	O
could	O
say	O
our	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
EUR	O
1000	O
for	O
the	O
5%	O
tail	O
.	O
</s>
<s>
From	O
this	O
table	O
let	O
us	O
calculate	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
for	O
a	O
few	O
values	O
of	O
:	O
</s>
<s>
The	O
expected	B-Algorithm
shortfall	I-Algorithm
increases	O
as	O
decreases	O
.	O
</s>
<s>
The	O
100%	O
-quantile	O
expected	B-Algorithm
shortfall	I-Algorithm
equals	O
negative	O
of	O
the	O
expected	O
value	O
of	O
the	O
portfolio	O
.	O
</s>
<s>
For	O
a	O
given	O
portfolio	O
,	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
greater	O
than	O
or	O
equal	O
to	O
the	O
Value	O
at	O
Risk	O
at	O
the	O
same	O
level	O
.	O
</s>
<s>
Expected	B-Algorithm
shortfall	I-Algorithm
,	O
in	O
its	O
standard	O
form	O
,	O
is	O
known	O
to	O
lead	O
to	O
a	O
generally	O
non-convex	O
optimization	O
problem	O
.	O
</s>
<s>
However	O
,	O
it	O
is	O
possible	O
to	O
transform	O
the	O
problem	O
into	O
a	O
linear	B-Algorithm
program	I-Algorithm
and	O
find	O
the	O
global	O
solution	O
.	O
</s>
<s>
This	O
property	O
makes	O
expected	B-Algorithm
shortfall	I-Algorithm
a	O
cornerstone	O
of	O
alternatives	O
to	O
mean-variance	O
portfolio	O
optimization	O
,	O
which	O
account	O
for	O
the	O
higher	O
moments	O
(	O
e.g.	O
,	O
skewness	O
and	O
kurtosis	O
)	O
of	O
a	O
return	O
distribution	O
.	O
</s>
<s>
Suppose	O
that	O
we	O
want	O
to	O
minimize	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
of	O
a	O
portfolio	O
.	O
</s>
<s>
Rockafellar/Uryasev	O
proved	O
that	O
is	O
convex	O
with	O
respect	O
to	O
and	O
is	O
equivalent	O
to	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
at	O
the	O
minimum	O
point	O
.	O
</s>
<s>
To	O
numerically	O
compute	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
for	O
a	O
set	O
of	O
portfolio	O
returns	O
,	O
it	O
is	O
necessary	O
to	O
generate	O
simulations	O
of	O
the	O
portfolio	O
constituents	O
;	O
this	O
is	O
often	O
done	O
using	O
copulas	O
.	O
</s>
<s>
With	O
these	O
simulations	O
in	O
hand	O
,	O
the	O
auxiliary	O
function	O
may	O
be	O
approximated	O
by:This	O
is	O
equivalent	O
to	O
the	O
formulation	O
:	O
Finally	O
,	O
choosing	O
a	O
linear	O
loss	O
function	O
turns	O
the	O
optimization	O
problem	O
into	O
a	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Using	O
standard	O
methods	O
,	O
it	O
is	O
then	O
easy	O
to	O
find	O
the	O
portfolio	O
that	O
minimizes	O
expected	B-Algorithm
shortfall	I-Algorithm
.	O
</s>
<s>
Closed-form	O
formulas	O
exist	O
for	O
calculating	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
when	O
the	O
payoff	O
of	O
a	O
portfolio	O
or	O
a	O
corresponding	O
loss	O
follows	O
a	O
specific	O
continuous	O
distribution	O
.	O
</s>
<s>
In	O
the	O
former	O
case	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
corresponds	O
to	O
the	O
opposite	O
number	O
of	O
the	O
left-tail	O
conditional	O
expectation	O
below	O
:	O
</s>
<s>
For	O
engineering	O
or	O
actuarial	O
applications	O
it	O
is	O
more	O
common	O
to	O
consider	O
the	O
distribution	O
of	O
losses	O
,	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
in	O
this	O
case	O
corresponds	O
to	O
the	O
right-tail	O
conditional	O
expectation	O
above	O
and	O
the	O
typical	O
values	O
of	O
are	O
95%	O
and	O
99%	O
:	O
</s>
<s>
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
,	O
where	O
is	O
the	O
standard	O
normal	O
p.d.f.	O
,	O
is	O
the	O
standard	O
normal	O
c.d.f.	O
,	O
so	O
is	O
the	O
standard	O
normal	O
quantile	O
.	O
</s>
<s>
If	O
the	O
loss	O
of	O
a	O
portfolio	O
follows	O
the	O
normal	O
distribution	O
,	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
.	O
</s>
<s>
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
,	O
where	O
is	O
the	O
standard	O
t-distribution	O
p.d.f.	O
,	O
is	O
the	O
standard	O
t-distribution	O
c.d.f.	O
,	O
so	O
is	O
the	O
standard	O
t-distribution	O
quantile	O
.	O
</s>
<s>
If	O
the	O
loss	O
of	O
a	O
portfolio	O
follows	O
generalized	O
Student	O
's	O
t-distribution	O
,	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
.	O
</s>
<s>
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
for	O
.	O
</s>
<s>
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
.	O
</s>
<s>
If	O
the	O
loss	O
of	O
a	O
portfolio	O
follows	O
the	O
logistic	O
distribution	O
,	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
.	O
</s>
<s>
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
.	O
</s>
<s>
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
.	O
</s>
<s>
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
,	O
where	O
is	O
the	O
upper	O
incomplete	O
gamma	O
function	O
.	O
</s>
<s>
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
and	O
the	O
VaR	O
is	O
equal	O
to	O
,	O
where	O
is	O
the	O
upper	O
incomplete	O
gamma	O
function	O
,	O
is	O
the	O
logarithmic	O
integral	O
function	O
.	O
</s>
<s>
If	O
the	O
loss	O
of	O
a	O
portfolio	O
follows	O
the	O
GEV	O
,	O
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
,	O
where	O
is	O
the	O
lower	O
incomplete	O
gamma	O
function	O
,	O
is	O
the	O
Euler-Mascheroni	O
constant	O
.	O
</s>
<s>
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
,	O
where	O
is	O
the	O
Spence	O
's	O
function	O
,	O
is	O
the	O
imaginary	O
unit	O
.	O
</s>
<s>
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
,	O
where	O
is	O
the	O
c.d.f.	O
</s>
<s>
,	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
,	O
where	O
is	O
the	O
hypergeometric	O
function	O
.	O
</s>
<s>
,	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
,	O
where	O
is	O
the	O
hypergeometric	O
function	O
.	O
</s>
<s>
,	O
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
,	O
where	O
is	O
the	O
standard	O
normal	O
c.d.f.	O
,	O
so	O
is	O
the	O
standard	O
normal	O
quantile	O
.	O
</s>
<s>
,	O
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
,	O
where	O
is	O
the	O
regularized	O
incomplete	O
beta	O
function	O
,	O
.	O
</s>
<s>
As	O
the	O
incomplete	O
beta	O
function	O
is	O
defined	O
only	O
for	O
positive	O
arguments	O
,	O
for	O
a	O
more	O
generic	O
case	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
can	O
be	O
expressed	O
with	O
the	O
hypergeometric	O
function	O
:	O
.	O
</s>
<s>
,	O
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
,	O
where	O
is	O
the	O
incomplete	O
beta	O
function	O
.	O
</s>
<s>
,	O
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
.	O
</s>
<s>
,	O
then	O
the	O
expected	B-Algorithm
shortfall	I-Algorithm
is	O
equal	O
to	O
,	O
where	O
is	O
the	O
hypergeometric	O
function	O
.	O
</s>
