Friends or Foes: A Story of Value at Risk and Expected Tail Loss

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Friends or Foes A Story of Value at Risk and
Expected Tail Loss

• 14th Dubrovnik Economic Conference
• June 25 - 29, 2008, Dubrovnik, Croatia
• organized by the Croatian National Bank
• Dr.sc. Saa ikovic
• Faculty of Economics Rijeka

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Motivation

• With the latest market turmoil stemming from US
  sub-prime mortgage crises it is clear that there
  is a need for an approach that comes to terms
  with problems posed by extreme event estimation.
• VaR is not a coherent risk measure because it
  does not necessarily satisfy the sub-additivity
  condition.
• (Sub-additivity a portfolio will risk an
  amount, which is at most the sum of the separate
  amounts risked by its subportfolios)

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Motivation

• VaR provides no handle on the extent of the
  losses that might be suffered beyond a certain
  threshold. VaR is incapable of distinguishing
  between situations where losses in the tail are
  only a bit worse and those where they are
  overwhelming.
• An alternative measure that is coherent and
  quantifies the losses that might be encountered
  in the tail is the expected tail loss (ETL).
• Since the introduction by Artzner (1999) of
  coherent risk measures and the new dawn of
  measuring extreme losses it seems as if the
  academic community is willing to sacrifice all
  the advances made in the field of measuring VaR.

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Motivation

• The field of ETL estimation and model comparison
  is just beginning to develop and there is an
  obvious lack of empirical research.
• VaR and ETL are inherently connected in the sense
  that from the VaR surface of the tail ETL figures
  can be easily calculated.
• Advances that have been made in VaR estimation
  should not be lost with the adoption of coherent
  risk measures into regulatory framework.
• Superior VaR techniques can be employed to yield
  superior ETL forecasts.
• VaR and ETL should be regarded as partners not
  rivals.

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Contribution of the paper

• Add to currently scarce literature on ETL
  empirical testing and model comparison.
• Validating risk measurement models based on both
  their VaR and ETL performance.
• Connecting VaR and ETL models.
• Developing a new hybrid ETL model based on
  advanced VaR modelling techniques.
• Developing a new loss function for evaluating ETL
  forecasts.

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Literature review

• Artzner et al. (1999) introduced the Expected
  Shortfall risk measure, which equals the expected
  value of the loss, given that a VaR violation
  occurred.
• Yamai and Yoshiba (2002) compared the two
  measures and argued that VaR is not reliable
  during market turmoil, whereas ETL can be a
  better choice overall.
• Angelidis, Degiannakis (2007) test the
  performance of various parametric VaR and ETL
  models. They find that different volatility
  models are optimal for different assets.
• Although ETL is a superior risk measure to VaR,
  it lacks the depth of the theoretical and
  empirical research that VaR has. Investigation
  into the theoretical properties of ETL is still
  in its early stages.

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Value at Risk (VaR)

• VaR is usually defined as
• VaR is the maximum potential loss that a
  portfolio can suffer within a fixed confidence
  level (cl) during a holding period.
• This definition can be misleading because VaR
  does not represent the maximum loss - a
  portfolio can lose much more than suggested by
  VaR depending on the shape of the tail of the
  distribution.

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Value at Risk (VaR)

• A better definition of VaR
• VaR is the minimum potential loss that a
  portfolio can suffer in the 100(1-cl) worst
  cases during a holding period. OR
• VaR is the maximum potential loss that a
  portfolio can suffer in the 100(1-cl) best cases
  during a holding period.
• Is VaR the most appropriate measure to describe
  the risks associated with holding a certain
  position?

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Value at Risk (VaR)

• Advantages of VaR over traditional measures of
  risk
• - VaR applies to any financial instrument and can
  be expressed in any voluntary unit of measure.
  More traditional measures, such as the greeks,
  are measures created ad hoc for specific
  instruments or risk variables and are expressed
  in different units.
• - VaR includes an estimate of future events and
  allows the risk of the portfolio to be expressed
  in a single number. Unlike VaR, greeks amount
  to a what if risk measure that does not make
  any connection between the probability and
  severity of future events.

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Coherent risk measures

• A coherent risk measure ? assigns to each loss X
  a risk measure ?(X) such that the following
  conditions are satisfied
• ?(tX) t?(X) (homogeneity)
• ?(X) ?(Y), if X Y (monotonicity)
• ?(X n) ?(X) - n (risk-free condition)
• ?(X) ?(Y) ?(X Y) (sub-additivity)
• These conditions guarantee that the risk function
  is convex, which in turn corresponds to risk
  aversion
• ?(tX (1 - t)Y) t?(X) (1 - t)?(Y)

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Coherent risk measures

• VaR is not a coherent risk measure because it
  does not necessarily satisfy the sub-additivity
  condition. VaR can only be made sub-additive if a
  usually implausible assumption is imposed on
  returns being normally distributed.
• For a sub-additive measure, which ETL is,
  portfolio diversification always leads to risk
  reduction, while for VaR, diversification may
  produce an increase in its value even when
  partial risks are triggered by mutually exclusive
  events.

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Coherent risk measures

• Sub-additivity matters because
• adding risks together would give an overestimate
  of combined risk - a sum of risks can be used as
  a conservative estimate of combined risk.
• if regulators use non-sub-additive risk measures
  to set capital requirements, a bank might be
  tempted to break itself up to reduce its
  regulatory capital requirements.
• non-sub-additive risk measures can inspire
  traders to break up their accounts, with separate
  accounts for separate risks, in order to reduce
  their margin requirements.

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Coherent risk measures

• VaR provides no handle on the extent of the
  losses that might be suffered beyond the
  threshold amount.
• VaR is incapable of distinguishing between
  situations where losses in the tail are only a
  bit worse, and those where they are overwhelming.
  
• ETL quantifies the losses that might be
  encountered in the tail.
• ETL is the expected value of the loss of the
  portfolio in the 100(1-cl) worst cases during a
  holding period.
• ETLcl(X) EX X VaRcl(X)

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VaR and ETL
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Advantages of ETL over VaR

• Example 1
• We are faced with two different portfolios with
  one being clearly riskier than the other. Despite
  this, VaR tells us that we face exactly the same
  risk when investing in these two portfolios. This
  is because VaR ignores extreme losses if they are
  rare enough.
• Because ETL weights extreme losses it can differ
  between such portfolios and correctly identify
  the riskier one. ETL is far more sensitive to
  extreme events no matter how rare they are.

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Advantages of ETL over VaR

• Example 2
• VaR paradox VaR negative values -gt are the
  securities with low enough probabilities of
  losses trully risk free?
• If a bond has a default probability of 2 and we
  are using a 95 VaR as our risk management tool
  we are convinced that this bond can bring us only
  profit without any loss, because of this our VaR
  lt 0!
• By only looking at VaR we would be convinced that
  this bond is completely risk free.

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VaR/ETL models using Extreme value theory (EVT)

• Most VaR models use the Central limit theorem
  (assumption of normality), which is completely
  wrong for our purpose - we are interested in the
  distribution of the tails not the central mass.
• The key to estimating the distribution of such
  events is the EVT, which governs the distribution
  of extreme values.
• EVT provides a framework in which an estimate of
  anticipated forces could be made using historical
  data.
• Extreme events are rare, meaning that their
  estimates are often required for levels of a
  process that are greater than those in the
  available data set.
• Potential problems
• EV models are developed using asymptotic
  arguments.
• EV models are derived under idealized
  circumstances - need not be true for the process
  being modeled.

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• Fisher-Tippett theorem - as n gets large the
  distribution of tail of X converges to
  Generalized extreme value distribution (GEV)

• If ? gt 0, GEV distribution becomes a Fréchet
  distribution, meaning that F(x) is leptokurtotic.
  
• If ? 0, GEV distribution becomes a Gumbel
  distribution, meaning that F(x) has normal
  kurtosis
• EV VaR

(Fréchet EV VaR, ? gt 0)
(Gumbel EV VaR, ? 0 )
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• There are no closed form ETL formulas for Fréchet
  and Gumbel distributions but EV ETL can be
  derived from EV VaR estimates using average-tail
  VaR algorithm.
• It is easily shown that ETL is indeed estimable
  in a consistent way as the average of 100cl
  worst cases

Hybrid historical simulation (HHS) ETL developed
in this paper can be expressed as
are order statistics from volatility scaled
bootstrapped series
Where
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Model comparison and backtesting

• Blanco-Ihle loss function compares VaR with tail
  losses, which makes no practical sense because
  VaR forecasts only the best scenario for the
  tail losses. Blanco-Ihle loss function actually
  measures the discrepancy between the lowest
  possible and actual tail losses - not especially
  useful.
• Blanco-Ihle loss function can be modified to
  compare ETL with the actual value of the tail
  loss - exactly what the loss function should be
  measuring.
• Suggested modification

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• Backtesting results for VaR forecasts (DOW JONES
  index,
• cl 0.99, period 22.3.2004 - 12.3.2008). Symbols
  and denote significance at 5 and 10 levels

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• Backtesting results for VaR forecasts (NASDAQ
  index,
• cl 0.99, period 22.3.2004 - 12.3.2008 )

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• Backtesting results for VaR forecasts (SP500
  index,
• cl 0.99, period 22.3.2004 - 12.3.2008 )

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• Backtesting results for ETL forecasts (DOW JONES
  index,
• ? 0.2, cl 0.95, 0.99, period 22.3.2004 -
  12.3.2008)

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• Backtesting results for ETL forecasts (NASDAQ
  index,
• ? 0.31, cl 0.95, 0.99, period 22.3.2004 -
  12.3.2008)

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• Backtesting results for ETL forecasts (SP500
  index,
• ? 0.24, cl 0.95, 0.99, period 22.3.2004 -
  12.3.2008)

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VaR backtesting results

• Data daily returns from DOW JONES, NASDAQ,
  SP500, CAC, DAX and FTSE
• Period 01.01.2000 - 12.3.2008
• Backtesting out-of-the-sample latest 1.000
  observations, 1 day holding period, confidence
  level 95 and 99
• 5/6 tested VaR models continually failed the
  Basel criteria.
• HHS was the only model, out of the tested models,
  that passed all of the tests, for both the Basel
  criteria and independence test of VaR failures.
• Worst performers VCV, HS250 and RiskMetrics
  models
• Results are consistent with the results for
  transitional markets reported in ikovic (2007).

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ETL backtesting results

• Bootstrapped HHS ETL approach was the best
  performing ETL measure across all of the tested
  indexes with the exception of NASDAQ index at 99
  cut-off level - Bootstrapped HS500 ETL.
• Worst performers VCV approach based on Fréchet
  distribution and GARCH RM approach with Fréchet
  distribution.
• These models greatly overestimated the expected
  averages of tail loss. Models that used Gumbel
  distribution performed far better compared to
  those with Fréchet distribution - two possible
  reasons
• 1) Tail indexes have been incorrectly calculated
  (they are too high)
• 2) The use of GEV distributions in ETL
  estimation provides overly conservative estimates
  of average tail losses.

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Tail losses and ETL for NASDAQ index (cl 0.95,
? 0.31)
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Conclusion

• Advances that have been made in VaR should not be
  lost with the (probable) adoption of coherent
  risk measures into regulatory framework. Superior
  quality of VaR techniques should yield superior
  ETL forecasts showing that VaR and ETL should be
  regarded as partners not rivals.
• The weak points of risk measurement models cannot
  be ignored and they will continually come back to
  haunt us even when we switch from one risk
  measure to another. The problems remain the same
  regardless whether we are estimating VaR or ETL.
• Overall the results of VaR model comparison
  obtained for the tested US and selected European
  stock indexes are in line with the results
  reported by ikovic (2007) for stock indexes from
  transitional markets.

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Conclusion

• For both developing and developed stock markets
  simpler VaR models consistently fail their task -
  provide risk managers with falsely optimistic
  data about the levels of risk that the financial
  institutions are exposed to. GARCH based
  volatility models, even at lower confidence
  level, continually outperform VaR models based on
  the assumption of simpler models of volatility
  such as SMA and EWMA.
• Bootstrapped HHS ETL approach was the best
  performing ETL measure across all of the tested
  indexes with the exception of NASDAQ index at 99
  cut-off level.
• For the tested stock indexes the use of GEV
  distributions in ETL estimation provides overly
  conservative estimates of ETL.

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Conclusion

• The strong points and weaknesses of every model
  remain with them and that is why knowledge
  obtained in developing VaR models must not be
  wasted. VaR techniques can easily be adopted to
  serve a new superior risk measure ETL.
• Research in VaR estimation should by no means be
  discouraged, because now it can serve a dual
  purpose improving VaR estimates but also
  improving ETL estimates.
• The focus of future research should be on 1)
  improving both VaR and ETL techniques, 2) finding
  optimal combinations of VaR-ETL models.
• Only complete information can serve as a solid
  basis for decision making in financial
  institutions and reveal actual risk exposure both
  to investors and regulators.

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• Thank you for your attention!
• Questions?