Transcript CHAPTER 8

CHAPTER 8
Testing VaR Results to Ensure
Proper Risk Measurement
INTRODUCTION
• Up to this point, we have discussed how
positions change value, and have used
relatively complicated math to calculate
the statistics of those changes
• This has enabled us to construct estimates
of the probability distributions of the future
losses and therefore estimate VaR
• Now testing is required to tie the results
back to reality and give confidence that
VaR is a true measure of the risks
INTRODUCTION
• This is especially important now that the
Basel Committee allows banks to use their
own VaR models to assess the amount of
regulatory capital that they hold for market
risks
• The goal of this chapter is to detail the
tests that should be carried out on VaR
calculators to ensure their validity
VAR-TESTING METHODOLOGIES
• There are three different types of tests
– Software-installation test
– Profit-and-Loss (P&L) reconciliation test
– Modeled-probability-distribution back-test
Back-Testing the Modeled
Probability Distribution
• Back-testing requires many days of data
• The purpose of this test is to make sure
that the probability distribution (e.g., the
VaR) is consistent with actual losses
• Back-testing compares the loss on any
given day with the VaR predicted for that
day.
• Figure 8-1 illustrates VaR and the
experienced losses over 100 days.
Back-Testing the Modeled
Probability Distribution
Back-Testing the Modeled
Probability Distribution
• The VaR changes slowly from day to day
as positions change and as the market
volatility changes
• In 100 trading days, we would expect one
exception (as on day 73 in the figure)
• In a year of 250 trading days, we would
expect 2 to 3 exceptions.
Back-Testing the Modeled
Probability Distribution
• If it was the case that we always got a
representative sample, then we could say
that our VaR was a good representation of
the actual distribution if we only
experience exceptions 1% of the time
• If we experience exceptions more or less
often, we would conclude that the VaR
was not an accurate representation of the
distribution of losses
Back-Testing the Modeled
Probability Distribution
• Unfortunately, there is additional complication
because the number of exceptions is in itself a
random number
• Sometimes the bank will be lucky and the
random market movements will cause fewer
losses than usual; sometimes they will be
unlucky and suffer many losses
• This uncertainty in sampling means that it is
difficult to tell whether the experienced number
of exceptions is due to a poor model or to bad
luck.
Back-Testing the Modeled
Probability Distribution
• Fortunately, there is a framework to
calculate the probability of having a given
number of exceptions
• The exceptions are a binomial variable
• Binomial variables are those that can have
a value of zero or one
• Exceptions are binomial because on any
given day there either is or is not an
exception
Back-Testing the Modeled
Probability Distribution
• If the VaR calculator is correct, then on
each day there is a 1% chance of an
exception and a 99% chance of there
being no exception
Back-Testing the Modeled
Probability Distribution
• The number of exceptions over 250 days
has a Bernoulli distribution
• The Bernoulli distribution describes the
probability of having a given number of
outcomes that are equal to one if a
binomial variable is sampled multiple times
• From the Bernoulli distribution, we can
calculate the probability of a given number
of exceptions occurring, as shown in Table
8-1
Back-Testing the Modeled
Probability Distribution
• From this table, we can see that if the VaR
calculator is correct, there is a 13%
chance of having 4 exceptions in 250
trading days and an 89% chance that
there will be 0 to 4 exceptions
• We can also see that there is only a 0.01%
chance of there being 10 or more
exceptions.
Back-Testing the Modeled
Probability Distribution
• We can interpret this by saying that it is
very unlikely to get 10 or more exceptions
if the VaR model is correct i.e., if 10 or
more exceptions do occur, it is likely that
the model is incorrect.
Back-Testing the Modeled
Probability Distribution
Back-Testing the Modeled
Probability Distribution
• This principle is used by the Basel
Committee to check that a bank's VaR
calculator is performing well
• If more than 4 exceptions have occurred in
the last 250 trading days, the Capital
Accords for market risk require that the
bank should hold additional capital to
compensate for the possible unreliability of
the bank's calculator
Back-Testing the Modeled
Probability Distribution
• Table 8-2 shows that each number of
exceptions puts the calculator into a green,
yellow, or red "zone.“
• Corresponding to each number of
exceptions, there is a multiplier by which
the amount of market-risk capital must be
increased
Back-Testing the Modeled
Probability Distribution
• We investigate capital further in the next
chapter
• Back-testing should not only be carried out
for the whole portfolio, but also for
subportfolios
Assessment
• A stock portfolio=stock A+ stock B
• Using parametric VaR method
• One-year learning window: using one-year
(250 trading days) historical data to
estimate the parameters, such as
variances and correlation
• Five back-testing periods with 250 trading
days for each period
Assessment
Table 1 Variation of Portfolio Returns in Various Testing Periods
Periods
I
II
IIII
IV
V
Variance of
Portfolio Returns
6.0993
5.4289
6.8715
7.6510
4.5296
Number of Exceptions
21
14
17
15
4
Corresponding Zone by
linear-based VaR
Red
Red
Red
Red
Green
Assessment
Table 1 Variation of Portfolio Returns in Various Testing Periods
Periods
I
II
IIII
IV
V
Variance of
Portfolio Returns
6.0993
5.4289
6.8715
7.6510
4.5296
Number of Exceptions
21
14
17
15
4
Corresponding Zone by
linear-based VaR
Red
Red
Red
Red
Green