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A non-Parametric Measure of Expected Shortfall (ES)

By Kostas Giannopoulos

UAE University

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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

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Criticisms of VaR

• violations of the assumptions with regard the distributional properties of the underlying risk factors

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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

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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))

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• 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 “volatility refers to the variance of a random variable, while extremes are a characteristic of variable, while extremes are a characteristic of the tails only”, Neftci (2000, p1). And VaR is a the tails only”, Neftci (2000, p1). And VaR is a scaled measure of risk based on the volatility of scaled measure of risk based on the volatility of the portfolio’s returns (the random variable).the portfolio’s returns (the random variable).

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• 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.  

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• 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

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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

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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).

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• The GEV can describe the density function, H, of the sample data located at the tails,

• H(x) =

0

0/1)1(1

if

if

e

exe

x

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• 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:

• G(yt,,) =

01

011

/

/1

ife

ify

ty

t

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• The parameters and can be estimated by maximising the following likelihood function:

• Ln(,) =

01

)ln(

01ln11

)ln(

1

1

ifyn

ifyn

n

ii

n

ii

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• Once the parameters and have been estimated the VaR for a given probability p is given by:

• VaRp = U +

1)1(

pn

N

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• Provided that <1 the ES for the given probability p can be computed as

• ESp =

1

UVaRp

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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

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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 Barone-Adesi 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

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• 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, (1997-2002), 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 U: 5th %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

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VaR and ES at 99% prob. for 1 Day horizon

MethodVaRESMaximum

GDP U:1.65*σ

5.2911.85NA

GDP U: 5th %3.87.95NA

FHS U: 1st %2.472.873.38

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VaR and ES at 99% prob. for 10 Day horizon

MethodVaRESMaximum

GDP U:1.65*σ

16.7337.22NA

GDP U: 5th% 12.0424.94NA

FHS U: 1st %2.484.555.28

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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