Transcript Stress testing

Stress testing and Extreme Value Theory
By A V Vedpuriswar
September 12, 2009
Introduction
If a bank uses a 99% confidence level to calculate its value at
risk, it generally expects to suffer a loss exceeding the value
at risk on one day out of every 100.
 What happens, however, on the one day when the value at
risk is exceeded?
How large is the loss on this day?
 Could this be the one bad day required to break the bank?
Such possibilities are considered under stress testing.
Stress testing refers to techniques used by financial
institutions to analyze the effects of exceptional but plausible
events in the market on a portfolio's value.
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Usefulness of stress tests
Stress tests help financial institutions to:
– overcome the shortfall of VAR models (as they deal with tail events
neglected by many such models)
– communicate extreme scenarios throughout the institution, thereby
enabling management to take the necessary precautions (limit systems,
additional capital, and so on)
– manage risk better in more volatile and less liquid markets
– bear in mind, during less volatile periods, that the probability of
disastrous events occurring should not be neglected
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Single-Factor Stress Testing
Sometimes referred to as sensitivity testing
Single-factor stress testing involves applying a shift to a
specific risk factor affecting a portfolio .
Risk factors commonly used in sensitivity testing include
changes in
–
interest rates,
– equity prices and
– exchange rates.
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Standardised single factor shocks
Standardised single-factor shocks have been issued by
several organizations, the most prominent of which is
probably the Derivatives Policy Group (DPG).
The standardized movements in the risk factors suggested by
the DPG include:
– A parallel shift in the yield curve of 100 basis points up and down
– Yield curve steepening/flattening by 25 basis points
– Stock index changes of 10% up and down
– Movements of 6% up and down in major currencies (20% for other
currencies) relative to the US dollar
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Subjective Shocks
Rather then use standardized changes in the risk factors,
many banks choose to run sensitivity tests based on their own
subjective opinion of a relevant risk factor shock and its
magnitude.
As this is an entirely subjective approach, it depends critically
on the ability of individual risk managers to choose risk factor
shocks and magnitudes that are both plausible and relevant to
their portfolio.
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Multiple factor stress testing
Many financial institutions run stress test scenarios in
addition to sensitivity tests.
Scenario analysis involves applying simultaneous moves in
multiple risk factors such as interest rates, exchange rates
and stock prices, to a portfolio.
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Historical and hypothetical scenarios
Historical scenario testing involves revaluing a portfolio using
values for the risk factors that existed during historical stress
events.
Hypothetical scenarios can be used when:
– no historical stress event is suitable for the portfolio in question
– risk managers want to stress test new or different combinations of risk
factors
– hypothetical scenarios can be created by imagining extreme, but
plausible, events that have not yet happened.
– Such scenarios may build upon, or expand, historical scenarios
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One of the most popular methods of creating hypothetical
scenarios is to combine worst-case movements in the risk
factors.
Unfortunately, the worst-case scenario method can also
create implausible scenarios as it ignores any correlation
between the different risk factors.
It is more useful to construct scenarios that reflect the
combined effects of multiple risk factors and therefore
incorporate possible correlation among the risk factors in
times of stress.
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Conducting stress tests
Once a set of scenarios has been developed, the next step is
to analyze the effect of each scenario on the value of the
portfolio.
This can sometimes be done in the same way as a simulation
to calculate VAR.
Stress tests can be run by inputting the stressed values of the
risk factors into VAR models and recalculating the portfolio
value using the new data.
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Extreme Value Theory
Extreme value theory (EVT) is a branch of statistics dealing
with extreme deviations from the mean of statistical
distributions.
In other words, it is the study of the tails of distributions.
Focus on extreme tail behavior is important because it has
been shown that actual return distributions display a higher
probability for extreme events (fat tails).
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Extreme Value Theorem
The key aspect of EVT is the extreme value theorem.
Given certain conditions, the distribution of extreme returns in
large samples converges to a particular known form, whatever
the initial or parent distribution of the returns.
The theorem tells us what the distribution of extreme values
should look like in the limit, as the sample size increases.
This distribution is characterized by three parameters:
– location (corresponds to the mean)
– scale (corresponds to the standard deviation)
– shape (or tail)
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The tail parameter (or tail index) defines the specific
distribution to be used and is the most important as it gives an
indication of the heaviness or fatness of the tails of the
distribution.
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Extreme Value Distributions
Suppose we have a sample of return observations from some
unknown distribution.
Then, using extreme value theory, we can say that for a large
class of underlying distributions, the distribution of excess
returns x converges to a Generalized Pareto distribution
(GPD) as the threshold u is progressively raised.
A GPD is a distribution that models the excess losses above a
threshold:
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How Useful is EVT?
Parametric VAR models work by fitting a certain distribution
(usually normal) to observed return data.
However, because most observations lie close to the center of
any empirical distribution, these approaches tend to fit curves
that accommodate these central observations.
 For the purposes of VAR, however, it is the observations in
the tail of a distribution that are the most important.
The EV approach, on the other hand, is specifically designed
for tail behavior and is therefore free of these problems.
The EV approach to value at risk calculation is very useful
because it does not make very strong assumptions about the
shape of this unknown distribution.
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