Figure 13.3: Histogram of the Transform Probability from the 10% Largest Losses
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Backtesting Only the Left Tail of the
Distribution
60
• We have simply zoomed in on the leftmost 10% of the
histogram from Figure 13.2
• The systematic deviation from a flat histogram is again
obvious
• To do formal statistical testing, we can again construct an
alternative hypothesis as in
• for t+1 such that RPF,t+1 < -VaRpt+1
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Backtesting Only the Left Tail of the
Distribution
61
• We can then calculate a likelihood ratio test
• where nb again is the number of elements in the parameter
vector b1
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
Stress Testing
62
Due to the practical constraints from managing large
portfolios, risk managers often work with relatively short
data samples
• This can be a serious issue if the historical data available
do not adequately reflect the potential risks going forward
• The available data may, for example, lack extreme events
such as an equity market crash, which occurs very
infrequently
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Stress Testing
63
• To make up for the inadequacies of the available data, it can
be useful to artificially generate extreme scenarios of main
factors driving portfolio returns and then assess the resulting
output from the risk model
• This is referred to as stress testing, since we are stressing
the model by exposing it to data different from the data used
when specifying and estimating the model
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Stress Testing
64
• At first pass, the idea of stress testing may seem vague and
ad hoc
• Two key issues appear to be
– how should we interpret the output of the risk model
from the stress scenarios, and
– how should we create the scenarios in the first place?
• We deal with each of these issues in turn
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Combining Distributions for
Coherent Stress Testing
65
• VaR and ES are proper probabilistic statements:
– What is the loss such that I will loose more only 1% of
the time (VaR)?
– What is the expected loss when I violate my VaR (ES)?
• Standard stress testing does not tell the portfolio manager
anything about the probability of the scenario happening,
and it is therefore not at all clear what the portfolio
rebalancing decision should be
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Combining Distributions for
Coherent Stress Testing
66
• Once scenario probabilities are assigned, then stress testing
can be very useful
• To be explicit, consider a simple example of one stress
scenario, which we define as a probability distribution
fstress() of the vector of factor returns
• We simulate a vector of risk factor returns from the risk
model, calling it f (), and we also simulate from the
scenario distribution, fstress()
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
Combining Distributions for
Coherent Stress Testing
67
If we assign a probability of a draw from the scenario
distribution occurring, then we can combine the two
distributions as in
• Data from the combined distribution is generated by
drawing a random variable Ui from a Uniform(0,1)
distribution
• If Ui is smaller than , then we draw a return from fstress();
otherwise we draw it from f()
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Combining Distributions for
Coherent Stress Testing
68
• Once we have simulated data from the combined data set,
we can calculate the VaR or ES risk measure on the
combined data.
• If the risk measure is viewed to be inappropriately high
then the portfolio can be rebalanced.
• Assigning the probability, , also allows the risk manager
to backtest the VaR system using the combined probability
distribution fcomb()
• Any of these tests can be used to test the risk model using
the data drawn from fcomb()
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Choosing Scenarios
69
• Having decided to do stress testing, a key challenge to the
risk manager is to create relevant scenarios
• The risk manager ought to do the following:
• Simulate shocks which are more likely to occur than the
historical data base suggests
• Simulate shocks that have never occurred but could
• Simulate shocks reflecting the possibility that current
statistical patterns could break down
• Simulate shocks which reflect structural breaks which
could occur
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
Choosing Scenarios
70
While largely portfolio specific, the long and colorful
history of financial crises may give inspiration for scenario
generation.
• Scenarios could include crises set off by political events or
natural disasters.
• Scenarios could be the culmination of pressures such as a
continuing real appreciation building over time resulting in
a loss of international competitiveness.
• The effects of market crises can also be very different
• They can result in relatively brief market corrections or
they can have longer lasting effects
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen