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Figure 13.3: Histogram of the Transform Probability from the 10% Largest Losses

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



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Figure 13.3: Histogram of the Transform Probability from the 10% Largest Losses

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