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A non-Parametric Measure of
Expected Shortfall (ES)
By Kostas Giannopoulos
UAE University
1
Early days
• When Value-at-Risk (VaR) was first
introduced it achieved much popularity
among regulators and financial institutions
• VaR is an estimator of the maximum
likelihood loss over a short period of time
for a predefined probability level
• VaR is the lower quantile on the tail of the
portfolio’s distribution of expected values
over the target horizon
2
Criticisms of VaR
• violations of the assumptions with regard
the distributional properties of the
underlying risk factors
3
VaR based on full Var-Cov Matrix;
Problems
• Stability of correlations
– Correlations measured from daily returns are
unstable. Even their sign is often ambiguous
• Dimensionality (correlation matrix)
• Correlations are necessary to optimise
portfolios; Not for monitoring their variance
4
Criticisms of VaR
• supporters of the EVT are arguing that the
VaR estimates do not take into account
the magnitude of extreme or rare losses
mapped outside the VaR quantile, (Artzner
et al (1997), Embrechts et al (1997),
Longin (1997), Embrechts et al (1998))
5
• In fact, the VaR considers mainly the frequency
of losses[1], but it is the severity of a loss that is
most important in risk management
• [1] “volatility refers to the variance of a random
variable, while extremes are a characteristic of
the tails only”, Neftci (2000, p1). And VaR is a
scaled measure of risk based on the volatility of
the portfolio’s returns (the random variable).
6
• Since the introduction of VaR, in early
1990's, a number of alternative
methodologies have been proposed
pointing at two main goals: (a) overcome
the limitations left behind by the other
methodologies, (b) match, much closely,
the distributional properties of the
underlying risk factors.
7
• Nevertheless, in each method the VaR is
always defined as the maximum possible
loss for a given probability
• Therefore, VaR does not consider any
rare, but possible, loss that is much larger
than VaR itself. And are these infrequent
and unusually large losses that can bring
the company to a collapse
8
five notions of risk that consider
losses beyond VaR
Acerbi and Tasche (2002)
• Conditional VaR (CVaR), Rockafellar and
Uryasev (2002).
• Expected shortfall (ES), Acerbi and Tasche
(2002).
• Tail conditional expectation, Artzner et al (1999).
• Worst conditional expectation, Artzner et al
(1999).
• Spectral risk measures
9
The EVT theory:
• enables us to estimate the quantiles and
the probabilities beyond the threshold
point of VaRp.
• The sample of the extreme losses can be
modelled as the generalised extreme
value distribution (GEV) (the one
parameter representation of three
probability distributions, the Frechet,
Weibul and Gumbel).
10
• The GEV can describe the density
function, H, of the sample data located at
the tails,
•
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 1(1x ) 1 / 
if
e

H(x) = 
x

e

if
e

 0
 0
• But the limited amount of data available for
the tail region renders the estimation of the
parameters describing the above
distributions extremely difficult.
• it is more appropriate to estimate the
function of a distribution of exceedances,
Fu, over a threshold, U, which, for a high
threshold U is well approximated by the
Generalised Pareto distribution (GPD),
see Pickands (1975).
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The generalised Pareto distribution
• Given the set of exceedances Y, i.e. losses
above the threshold U, the GPD -G is
described as:
•
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  yt 1/ 
 if   0
1  1 




G(yt,,) =

1  e  yt / 
if   0
• The parameters  and  can be estimated
by maximising the following likelihood
function:
• Ln(,) =
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
1  n   
 n ln(  )    1 ln 1  yi  if   0


 i 1   

1 n

if   0
 n ln(  )    yi
i 1

• Once the parameters  and  have been
estimated the VaR for a given probability p
is given by:
•
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
VaRp = U +

 N
  (1  p ) 
 n




 1


• Provided that <1 the ES for the given
probability p can be computed as
VaRp    U
• ESp =
1
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Overview of EVT
• Suitable for stress testing
– Distribution of extreme values is unknown
– Statistical theory of extreme values is that this distribution
converges in large samples to a known form of limiting
distribution
– Can infer extreme risks from this
• Handling of multivariate data not computationally
feasible
– Work underway with simulating portfolio distributions
• from this develop distribution of extremes
• Problems with this not solved e.g. constituent drivers of portfolio risk
can have different extreme distributions
17
EVT, secondary issues
• As Dowd (2002) points out, to apply the
GPD we need to take into account some
secondary issues like position size, left or
right tails, short or long etc.
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FHS, a nonparametric measure of
ES
• Barone-Adesi et al (1998) and BaroneAdesi et al (1999) introduced the FHS
algorithm, in order to generate correlated
pathways for a set of risk factors
• At each simulation trial, a value for each
risk factor is generated and all assets in
the portfolio are re-priced
19
• After running large number of simulation
trials a set of portfolio values is generated
that form the empirical distribution for the
predicted portfolio values at a certain
horizon.
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An Empirical Investigation
• DJIA - six years of daily prices, (19972002), a total of 1460 observations
• computed the GPD estimates of the ES for
different thresholds, U=
– 2.33*σ (i.e. 1% of normalised extremes)
– 1.65*σ (i.e. 5% of normalised extremes)
– 5th percentile (non-parametric threshold)
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Table 1 Parameter Estimates for
GPD
U : 1.65*std U : 2.33*std
22
th
U: 5 %ntile
No observations
exceeding U
65
26
83

0.549
1.018
0.515

0.043
0.076
0.041
ML
-197.14
-55.16
-226.23
VaR and ES at 99% prob. for 1 Day
horizon
Method
GDP
U:1.65*σ
th
GDP U: 5
st
FHS U: 1 %
23
%
VaR
ES
Maximum
5.29
11.85
NA
3.8
7.95
NA
2.47
2.87
3.38
VaR and ES at 99% prob. for 10
Day horizon
Method
GDP
U:1.65*σ
th
GDP U: 5
st
FHS U: 1 %
24
%
VaR
ES
Maximum
16.73
37.22
NA
12.04
24.94
NA
2.48
4.55
5.28
FHS-conclusions
• We get an accurate measure for the
volatility of the current portfolio without
using computationally intense multivariate
methodologies
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FHS-conclusions
• Simulation estimates out of sample
portfolio losses and is based on empirical
distribution of data
– takes account of “catastrophe risk”
– does not impose particular probability function
– no compression of tails or change to
skewness
• No use made of variance-covariance
matrix allows fast estimation of ES
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• Multiperiod risk measures are estimated
by re-pricing all contracts, thus :
– Expiring contracts are taken into account
– Implied volatility simulated pathways are
conditional to the underlying risk factor
(simulated) volatility pathways
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