Value at Risk and Expected Tail Loss
Conditional value at risk (CVaR) is also referred to as expected tail loss (ETL) [see
Acerbi and Tasche (2001)].
It has gained significant importance in the risk management
field and is our choice of downside risk measure.
Let
( )
p
Xt
be the value of an
investment at time
p
t
which assumed to be known and let
( ) ( )
1
pp
XX
t
X
t
+
=−
be
the future uncertain value of the investment profit or loss.
Let
( )
X
f
x
and
( )
X
Fx
be
the probability density and distribution functions of this future uncertain investment
value.
Given a confidence level
q
(
01
q
<
<
) and time period duration
1
tt
+
−
(i.e.
month or quarter), value at risk (VaR) is that quantity
( )
*
Xq
such that.
( )
( )
{ }
( )
**
1
pX
PXt
X q
F X q
q
+
⎡⎤
≤=
=
⎣⎦
(5.1)
Let
( )
( )
*

X
FxX Xq
<
be the conditional distribution for the future investment value
given that
( )
1
p
+
it is less than
( )
*
.
From basic probability theory, this
conditional distribution is related to the unconditional distribution
( )
X
in the
following way.
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 Spring '11
 Neveskyi
 Probability theory, coherent, Risk in finance, Value at risk, Expected shortfall, Risk measure

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