Risk reports are based on end-of-day positions. This means that the risk figures give
the loss at the chosen confidence interval over the holding period for the portfolio that
is held at the end of that business day. With a 1-day holding period, the risk figure
should be compared with the P&L from the following business day. The P&L, if
unwanted components are removed, gives the change in value from market movements
of the portfolio the risk was measured for. Therefore, the risk figures and P&L figures
used for comparison must be skewed by 1 business day for meaningful backtesting.
27 Şubat 2011 Pazar
Comparing risk measurements and P&L
Holding period
For regulatory purposes, the maximum loss over a 10-business-day period at the
99% confidence level must be calculated. This measurement assumes a static
portfolio over the holding period. In a realistic trading environment, however, portfolios
usually change significantly over 10 days, so a comparison of 10-day P&L with
market risk would be of questionable value. A confidence level of 99% and a holding
period of 10 days means that one exception would be expected in 1000 business
days (about 4 years). If exceptions are so infrequent, a very long run of data has to be
observed to obtain a statistically significant conclusion about the risk measurement
model. Because of this, regulators require a holding period of one day to be used for
backtesting. This gives an expected 2.5 events per year where actual loss exceeds
the market risk figure. Figure 9.2 shows simulated backtesting results. Even with
this number of expected events, the simple number of exceptions in one year has
only limited power to distinguish between an accurate risk measurement model and
an inaccurate one.
As noted above, risk figures are often calculated for a holding period of 10 days.
For backtesting, risks should ideally be recalculated using a 1-day holding period.
For the most accurate possible calculation, this would use extreme moves of risk
factors and correlations based on 1-day historical moves rather than 10-day moves.
Then the risk figures would be recalculated. The simplest possible approach is simply
to scale risk figures by the square root of 10. The effectiveness of a simple scaling
approach depends on whether the values of the portfolios in question depend almost
linearly on the underlying risk factors. For instance, portfolios of bonds or equities
depend almost linearly on interest rates or equity prices respectively. If the portfolio
has a significant non-linear component (significant gamma risk), the scaling would
be inaccurate. For example, the value of a portfolio of equity index options would
typically not depend linearly on the value of the underlying equity index. Also, if the
underlying risk factors are strongly mean reverting (e.g. spreads between prices of
two grades of crude oil, or natural gas prices), 10-day moves and 1-day moves would
not be related by the square root of time. In practice, the simple scaling approach is
often used. At the whole bank level, this is likely to be reasonably accurate, as
typically the majority of the risk of a whole bank is not in options portfolios. Clearly,
this would not be so for specialist businesses such as derivative product subsidiaries,
or banks with extensive derivative portfolios.
For regulatory purposes, the maximum loss over a 10-business-day period at the
99% confidence level must be calculated. This measurement assumes a static
portfolio over the holding period. In a realistic trading environment, however, portfolios
usually change significantly over 10 days, so a comparison of 10-day P&L with
market risk would be of questionable value. A confidence level of 99% and a holding
period of 10 days means that one exception would be expected in 1000 business
days (about 4 years). If exceptions are so infrequent, a very long run of data has to be
observed to obtain a statistically significant conclusion about the risk measurement
model. Because of this, regulators require a holding period of one day to be used for
backtesting. This gives an expected 2.5 events per year where actual loss exceeds
the market risk figure. Figure 9.2 shows simulated backtesting results. Even with
this number of expected events, the simple number of exceptions in one year has
only limited power to distinguish between an accurate risk measurement model and
an inaccurate one.
As noted above, risk figures are often calculated for a holding period of 10 days.
For backtesting, risks should ideally be recalculated using a 1-day holding period.
For the most accurate possible calculation, this would use extreme moves of risk
factors and correlations based on 1-day historical moves rather than 10-day moves.
Then the risk figures would be recalculated. The simplest possible approach is simply
to scale risk figures by the square root of 10. The effectiveness of a simple scaling
approach depends on whether the values of the portfolios in question depend almost
linearly on the underlying risk factors. For instance, portfolios of bonds or equities
depend almost linearly on interest rates or equity prices respectively. If the portfolio
has a significant non-linear component (significant gamma risk), the scaling would
be inaccurate. For example, the value of a portfolio of equity index options would
typically not depend linearly on the value of the underlying equity index. Also, if the
underlying risk factors are strongly mean reverting (e.g. spreads between prices of
two grades of crude oil, or natural gas prices), 10-day moves and 1-day moves would
not be related by the square root of time. In practice, the simple scaling approach is
often used. At the whole bank level, this is likely to be reasonably accurate, as
typically the majority of the risk of a whole bank is not in options portfolios. Clearly,
this would not be so for specialist businesses such as derivative product subsidiaries,
or banks with extensive derivative portfolios.
Backtesting
MARK DEANS
The aim of backtesting is to test the effectiveness of market risk measurement by
comparing market risk figures with the volatility of actual trading results. Banks
must carry out backtesting if they are to meet the requirements laid down by the
Basel Committee on Banking Supervision in the Amendment to the Capital Accord to
incorporate market risks (1996a). If the results of the backtesting exercise are
unsatisfactory, the local regulator may impose higher capital requirements on a
bank. Further, when performed at a business line or trading desk level, backtesting
is a useful tool to evaluate risk measurement methods.
Backtesting is a requirement for banks that want to use internal models to calculate
their regulatory capital requirements for market risk. The process consists of comparing
daily profit and loss (P&L) figures with corresponding market risk figures over a
period of time. Depending on the confidence interval used for the market risk
measurement, a certain proportion of the P&L figures are expected to show a loss
greater than the market risk amount. The result of the backtest is the number of
losses greater than their corresponding market risk figures: the ‘number of exceptions’.
According to this number, the regulators will decide on the multiplier used for
determining the regulatory capital requirement.
Regulations require that backtesting is done at the whole bank level. Regulators
may also require testing to be broken down by trading desk (Figure 9.1). When there
is an exception, this breakdown allows the source of the loss to be analysed in more
detail. For instance, the loss might come from one trading desk, or from the sum of
losses across a number of different business areas.
In addition to the regulatory requirements, backtesting is a useful tool for evaluating
market risk measurement and aggregation methods within a bank. At the whole
bank level, the comparison between risk and P&L gives only a broad overall picture of
the effectiveness of the chosen risk measurement methods. Satisfactory backtesting
results at the aggregate level could hide poor risk measurement methods at a lower
level. For instance, risks may be overestimated for equity trading, but underestimated
for fixed income trading. Coincidentally, the total risk measured could be approximately
correct. Alternatively, risks could be underestimated for each broad risk
category (interest rate, equity, FX, and commodity risk), but this fact could be hidden
by a very conservative simple sum aggregation method.
Backtesting at the portfolio level, rather than just for the whole bank, allows individual market risk measurement models to be tested in practice. The lower the
level at which backtesting is applied, the more information becomes available about
the risk measurement methods used. This allows areas to be identified where market
risk is not measured accurately enough, or where risks are being taken that are not
detected by the risk measurement system.
Backtesting is usually carried out within the risk management department of a
bank where risk data is relatively easily obtained. However, P&L figures, often
calculated by a business unit control or accounting department, are equally important
for backtesting. The requirements of these departments when calculating P&L
are different from those of the risk management department. The accounting principle
of prudence means that it is important not to overstate the value of the portfolio, so
where there is uncertainty about the value of positions, a conservative valuation will
be taken. When backtesting, the volatility of the P&L is most important, so capturing
daily changes in value of the portfolio is more important than having a conservative
or prudent valuation. This difference in aims means that P&L as usually calculated
for accounting purposes is often not ideal for backtesting. It may include unwanted
contributions from provisions or intraday trading. Also, the bank’s breakdown of P&L
by business line may not be the same as the breakdown used for risk management.
To achieve effective backtesting, the risk and P&L data must be brought together
in a single system. This system should be able to identify exceptions, and produce
suitable reports. The data must be processed in a timely manner, as some regulators
(e.g. the FSA) require an exception to be reported to them not more than one business
day after it occurs.
In the last few years, investment banks have been providing an increasing amount
of information about their risk management activities in their annual reports. The
final part of this chapter reviews the backtesting information given in the annual
reports of some major banks.
The aim of backtesting is to test the effectiveness of market risk measurement by
comparing market risk figures with the volatility of actual trading results. Banks
must carry out backtesting if they are to meet the requirements laid down by the
Basel Committee on Banking Supervision in the Amendment to the Capital Accord to
incorporate market risks (1996a). If the results of the backtesting exercise are
unsatisfactory, the local regulator may impose higher capital requirements on a
bank. Further, when performed at a business line or trading desk level, backtesting
is a useful tool to evaluate risk measurement methods.
Backtesting is a requirement for banks that want to use internal models to calculate
their regulatory capital requirements for market risk. The process consists of comparing
daily profit and loss (P&L) figures with corresponding market risk figures over a
period of time. Depending on the confidence interval used for the market risk
measurement, a certain proportion of the P&L figures are expected to show a loss
greater than the market risk amount. The result of the backtest is the number of
losses greater than their corresponding market risk figures: the ‘number of exceptions’.
According to this number, the regulators will decide on the multiplier used for
determining the regulatory capital requirement.
Regulations require that backtesting is done at the whole bank level. Regulators
may also require testing to be broken down by trading desk (Figure 9.1). When there
is an exception, this breakdown allows the source of the loss to be analysed in more
detail. For instance, the loss might come from one trading desk, or from the sum of
losses across a number of different business areas.
In addition to the regulatory requirements, backtesting is a useful tool for evaluating
market risk measurement and aggregation methods within a bank. At the whole
bank level, the comparison between risk and P&L gives only a broad overall picture of
the effectiveness of the chosen risk measurement methods. Satisfactory backtesting
results at the aggregate level could hide poor risk measurement methods at a lower
level. For instance, risks may be overestimated for equity trading, but underestimated
for fixed income trading. Coincidentally, the total risk measured could be approximately
correct. Alternatively, risks could be underestimated for each broad risk
category (interest rate, equity, FX, and commodity risk), but this fact could be hidden
by a very conservative simple sum aggregation method.
Backtesting at the portfolio level, rather than just for the whole bank, allows individual market risk measurement models to be tested in practice. The lower the
level at which backtesting is applied, the more information becomes available about
the risk measurement methods used. This allows areas to be identified where market
risk is not measured accurately enough, or where risks are being taken that are not
detected by the risk measurement system.
Backtesting is usually carried out within the risk management department of a
bank where risk data is relatively easily obtained. However, P&L figures, often
calculated by a business unit control or accounting department, are equally important
for backtesting. The requirements of these departments when calculating P&L
are different from those of the risk management department. The accounting principle
of prudence means that it is important not to overstate the value of the portfolio, so
where there is uncertainty about the value of positions, a conservative valuation will
be taken. When backtesting, the volatility of the P&L is most important, so capturing
daily changes in value of the portfolio is more important than having a conservative
or prudent valuation. This difference in aims means that P&L as usually calculated
for accounting purposes is often not ideal for backtesting. It may include unwanted
contributions from provisions or intraday trading. Also, the bank’s breakdown of P&L
by business line may not be the same as the breakdown used for risk management.
To achieve effective backtesting, the risk and P&L data must be brought together
in a single system. This system should be able to identify exceptions, and produce
suitable reports. The data must be processed in a timely manner, as some regulators
(e.g. the FSA) require an exception to be reported to them not more than one business
day after it occurs.
In the last few years, investment banks have been providing an increasing amount
of information about their risk management activities in their annual reports. The
final part of this chapter reviews the backtesting information given in the annual
reports of some major banks.
26 Şubat 2011 Cumartesi
Acknowledgements
Certain sections of this chapter were drawn from Implementing Value at Risk, by
Philip Best, John Wiley, 1998. John Wiley’s permission to reproduce these sections
is kindly acknowledged.
The author would also like to thank Con Keating for his invaluable assistance in
reviewing this chapter and for writing the appendix on Extreme Value Theory. This
chapter also benefited from the comments of Gurpreet Dehal and Patricia Ladkin.
Notes
1 Note that observing other market parameters, such as the volatility of short-term interest
rates, might have warned the risk manager that a currency devaluation was possible.
Observed by a central risk management function in a different country, however, the chances
of spotting the danger are much reduced.
2 That is, twenty times the return volatility prior to the crisis.
3 Z score of binomial distribution of exceptions: 1.072, i.e. the VaR model would not be rejected
by a Type I error test.
4 Extreme price changes that have an almost infinitesimally small probability in a normal
distribution but which we know occur with far greater regularity in financial markets.
5 For a more comprehensive coverage of EVT see Embrechs et al. (1997).
6 This is the number of ways of selecting n assets from a set of 10, all multiplied by the
number of scenarios – 69 for this example.
7 Counterparty A’s liquidators would expect the bank to perform on the contracts, thus their
value at the time of default would have to be written off. Once written off, of course, there is
no potential for future beneficial market moves to improve the situation.
8 Bond price curvature – the slight non-linearity of bond prices for a given change in yield.
9 Note that this is not the same as the group of countries who have chosen to ‘peg’ their
currencies to the US dollar.
10 For a more formal and complete introduction to EVT see Embrechs et al. (1997), Reiss and
Thomas (1997) and Beirlant et al. (1996). Readers interested in either the rapidly developing
multivariate theory or available software should contact the author. 261
Philip Best, John Wiley, 1998. John Wiley’s permission to reproduce these sections
is kindly acknowledged.
The author would also like to thank Con Keating for his invaluable assistance in
reviewing this chapter and for writing the appendix on Extreme Value Theory. This
chapter also benefited from the comments of Gurpreet Dehal and Patricia Ladkin.
Notes
1 Note that observing other market parameters, such as the volatility of short-term interest
rates, might have warned the risk manager that a currency devaluation was possible.
Observed by a central risk management function in a different country, however, the chances
of spotting the danger are much reduced.
2 That is, twenty times the return volatility prior to the crisis.
3 Z score of binomial distribution of exceptions: 1.072, i.e. the VaR model would not be rejected
by a Type I error test.
4 Extreme price changes that have an almost infinitesimally small probability in a normal
distribution but which we know occur with far greater regularity in financial markets.
5 For a more comprehensive coverage of EVT see Embrechs et al. (1997).
6 This is the number of ways of selecting n assets from a set of 10, all multiplied by the
number of scenarios – 69 for this example.
7 Counterparty A’s liquidators would expect the bank to perform on the contracts, thus their
value at the time of default would have to be written off. Once written off, of course, there is
no potential for future beneficial market moves to improve the situation.
8 Bond price curvature – the slight non-linearity of bond prices for a given change in yield.
9 Note that this is not the same as the group of countries who have chosen to ‘peg’ their
currencies to the US dollar.
10 For a more formal and complete introduction to EVT see Embrechs et al. (1997), Reiss and
Thomas (1997) and Beirlant et al. (1996). Readers interested in either the rapidly developing
multivariate theory or available software should contact the author. 261
Appendix: The theory of extreme value theory – an introduction © Con Keating
Appendix: The theory of extreme value theory –
an introduction © Con Keating
In 1900 Bachelier introduced the normal distribution to financial analysis (see also
Cootner, 1964); today most students of the subject would be able to offer a critique
of the shortcomings of this most basic (but useful) model. Most would point immediately
to the ‘fat tails’ evident in the distributions of many financial time series.
Benoit Mandelbrot (1997), now better known for his work on fractals, and his
doctoral student Eugene Fama published extensive studies of the empirical properties
of the distributions of a wide range of financial series in the 1960s and 1970s which
convincingly demonstrate this non-normality. Over the past twenty years, both
academia and the finance profession have developed a variety of new techniques,
such as the ARCH family, to simulate the observed oddities of actual series. The
majority fall short of delivering an entirely satisfactory result.
At first sight the presence of skewness or kurtosis in the distributions suggests
that of a central limit theorem failing but, of course, the central limit theorem should
only be expected to apply strongly to the central region, the kernel of the distribution.
Now this presents problems for the risk manager who naturally is concerned with the
more unusual (or extreme) behavior of markets, i.e. the probability and magnitudes of
the events forming the tails of the distributions.
There is also a common misunderstanding that the central limit theorem implies
that any mixture of distributions or samplings from a distribution will result in a
normal distribution, but a further condition exists, which often passes ignored, that
these samplings should be independent.
Extreme value theory (EVT) is precisely concerned with the analysis of tail behaviour.
It has its roots in the work of Fisher and Tippett first published in 1928 and a
long tradition of application in the fields of hydrology and insurance. EVT considers
the asymptotic (limiting) behavior of series and, subject to the assumptions listed
below, states which is read as: F is a realization in the maximum domain of attraction of H.
To illustrate the concept of a maximum domain of attraction, consider a fairground
game: tossing ping-pong balls into a collection of funnels. Once inside a funnel, the
ball would descend to its tip – a point attractor. The domain of attraction is the
region within a particular funnel and the maximum domain of attraction is any
trajectory for a ping-pong ball which results in its coming to rest in a particular
funnel. This concept of a stable, limiting, equilibrium organization to which dynamic
systems are attracted is actually widespread in economics and financial analysis.
The assumptions are that bn and an[0 (location and scaling parameters) exist
such that the financial time series, X, demonstrates regular limiting behavior and that
the distribution is not degenerate. These are mathematical technicalities necessary to
ensure that we do not descend inadvertently into paradox and logical nonsenses. m is
referred to as a shape parameter. There is (in the derivation of equation (A1)) an
inherent assumption that the realisations of X, x0, x1, . . . , xn are independently
and identically distributed. If this assumption were relaxed, the result, for serially
dependent data, would be slower convergence to the asymptotic limit.
The distribution Hm (x) is defined as the generalized extreme value distribution
(GEV) and has the functional form:
The distributions where the value of the tail index, m, is greater than zero, equal to
zero or less than zero are known, correspondingly, as Fre´chet, Gumbel and Weibull
distributions. The Fre´chet class includes Student’s T, Pareto and many other distributions
occasionally used in financial analysis; all these distributions have heavy
tails. The normal distribution is a particular instance of the Gumbel class where m
is zero.
This constitutes the theory underlying the application of EVT techniques but it
should be noted that this exposition was limited to univariate data.10 Extensions of
EVT to multivariate data are considerably more intricate involving measure theory,
the theory of regular variations and more advanced probability theory. Though much
of the multivariate theory does not yet exist, some methods based upon the use of
copulas (bivariate distributions whose marginal distributions are uniform on the
unit interval) seem promising.
Before addressing questions of practical implementation, a major question needs
to be considered. At what point (which quantile) should it be considered that the
asymptotic arguments or the maximum domain of attraction applies? Many studies
have used the 95th percentile as the point beyond which the tail is estimated. It is
far from clear that the arguments do apply in this still broad range and as yet there
are no simulation studies of the significance of the implicit approximation of this
choice.
The first decision when attempting an implementation is whether to use simply
the ordered extreme values of the entire sample set, or to use maxima or minima in
defined time periods (blocks) of, say, one month or one year. The decision trade-off
is the number of data points available for the estimation and fitting of the curve parameters versus the nearness to the i.i.d. assumption. Block maxima or minima
should be expected to approximate an i.i.d. series more closely than the peaks over
a threshold of the whole series but at the cost of losing many data-points and
enlarging parameter estimation uncertainty. This point is evident from examination
of the following block maxima and peaks over threshold diagrams.
Implementation based upon the whole data series, usually known as peaks over
threshold (POT), uses the value of the realization (returns, in most financial applications)
beyond some (arbitrarily chosen) level or threshold. Anyone involved in the
insurance industry will recognize this as the liability profile of an unlimited excess
of loss policy. Figures 8A.1 and 8A.2 illustrate these two approaches:
Figure 8A.1 shows the minimum values in each 25-day period and may be
compared with the whole series data-set below. It should be noted that both series
are highly autocorrelated and therefore convergence to the asymptotic limit should
be expected to be slow.
Figure 8A.2 shows the entire data-set and the peaks under an arbitrary value (1.5).
In this instance, this value has clearly been chosen too close to the mean of the
distribution – approximately one standard deviation. In a recent paper, Danielsson
and De Vries (1997) develop a bootstrap method for the automatic choice of this cutoff
point but as yet, there is inadequate knowledge of the performance of small
sample estimators. Descriptive statistics of the two (EVT) series are given in Table
8A.1.
Notice that the data-set for estimation in the case of block minima has declined to
just 108 observations and further that neither series possesses ‘fat tails’ (positive
kurtosis). Figure 8A.3 presents these series.
The process of implementing EVT is first to decide which approach, then the level
of the tail boundary and only then to fit a parametric model of the GEV class to the
processed data. This parametric model is used to generate values for particular VaR
quantiles. It is standard practice to fit generalized Pareto distributions (GPD) to POT
data:
omitting the u and x subscripts. The numerical MLE solution should not prove
problematic provided m[ñ1
2 which should prove the case for most financial data. A
quotation from R. L. Smith is appropriate: ‘The big advantage of maximum likelihood
procedures is that they can be generalized, with very little change in the basic
methodology, to much more complicated models in which trends or other effects may
be present.’ Estimation of the parameters may also be achieved by either linear (see,
for example, Kearns and Pagan, 1997) or non-linear regression after suitable
algebraic manipulation of the distribution function.
It should be immediately obvious that there is one potential significant danger for
the risk manager in using EVT; that the estimates of the parameters introduce
error non-linearly into the estimate of the VaR quantile. However, by using profile
likelihood, it should be possible to produce confidence intervals for these estimates,
even if the confidence interval is often unbounded.
Perhaps the final point to make is that it becomes trivial to estimate the mean
expected loss beyond VaR in this framework; that is, we can estimate the expected
loss given a violation of the VaR limit – an event which can cause changes in
management behavior and cost jobs.
This brief appendix has attempted to give a broad overview of the subject. Of
necessity, it has omitted some of the classical approaches such as Pickand’s and
Hill’s estimators. An interested reader would find the introductory texts10 listed far
more comprehensive.
There has been much hyperbole surrounding extreme value theory and its application
to financial time series. The reality is that more structure (ARCH, for example)
needs to be introduced into the data-generating processes before it can be said that
the method offers significant advantages over conventional methods. Applications,
however, do seem most likely in the context of stress tests of portfolios.
an introduction © Con Keating
In 1900 Bachelier introduced the normal distribution to financial analysis (see also
Cootner, 1964); today most students of the subject would be able to offer a critique
of the shortcomings of this most basic (but useful) model. Most would point immediately
to the ‘fat tails’ evident in the distributions of many financial time series.
Benoit Mandelbrot (1997), now better known for his work on fractals, and his
doctoral student Eugene Fama published extensive studies of the empirical properties
of the distributions of a wide range of financial series in the 1960s and 1970s which
convincingly demonstrate this non-normality. Over the past twenty years, both
academia and the finance profession have developed a variety of new techniques,
such as the ARCH family, to simulate the observed oddities of actual series. The
majority fall short of delivering an entirely satisfactory result.
At first sight the presence of skewness or kurtosis in the distributions suggests
that of a central limit theorem failing but, of course, the central limit theorem should
only be expected to apply strongly to the central region, the kernel of the distribution.
Now this presents problems for the risk manager who naturally is concerned with the
more unusual (or extreme) behavior of markets, i.e. the probability and magnitudes of
the events forming the tails of the distributions.
There is also a common misunderstanding that the central limit theorem implies
that any mixture of distributions or samplings from a distribution will result in a
normal distribution, but a further condition exists, which often passes ignored, that
these samplings should be independent.
Extreme value theory (EVT) is precisely concerned with the analysis of tail behaviour.
It has its roots in the work of Fisher and Tippett first published in 1928 and a
long tradition of application in the fields of hydrology and insurance. EVT considers
the asymptotic (limiting) behavior of series and, subject to the assumptions listed
below, states which is read as: F is a realization in the maximum domain of attraction of H.
To illustrate the concept of a maximum domain of attraction, consider a fairground
game: tossing ping-pong balls into a collection of funnels. Once inside a funnel, the
ball would descend to its tip – a point attractor. The domain of attraction is the
region within a particular funnel and the maximum domain of attraction is any
trajectory for a ping-pong ball which results in its coming to rest in a particular
funnel. This concept of a stable, limiting, equilibrium organization to which dynamic
systems are attracted is actually widespread in economics and financial analysis.
The assumptions are that bn and an[0 (location and scaling parameters) exist
such that the financial time series, X, demonstrates regular limiting behavior and that
the distribution is not degenerate. These are mathematical technicalities necessary to
ensure that we do not descend inadvertently into paradox and logical nonsenses. m is
referred to as a shape parameter. There is (in the derivation of equation (A1)) an
inherent assumption that the realisations of X, x0, x1, . . . , xn are independently
and identically distributed. If this assumption were relaxed, the result, for serially
dependent data, would be slower convergence to the asymptotic limit.
The distribution Hm (x) is defined as the generalized extreme value distribution
(GEV) and has the functional form:
The distributions where the value of the tail index, m, is greater than zero, equal to
zero or less than zero are known, correspondingly, as Fre´chet, Gumbel and Weibull
distributions. The Fre´chet class includes Student’s T, Pareto and many other distributions
occasionally used in financial analysis; all these distributions have heavy
tails. The normal distribution is a particular instance of the Gumbel class where m
is zero.
This constitutes the theory underlying the application of EVT techniques but it
should be noted that this exposition was limited to univariate data.10 Extensions of
EVT to multivariate data are considerably more intricate involving measure theory,
the theory of regular variations and more advanced probability theory. Though much
of the multivariate theory does not yet exist, some methods based upon the use of
copulas (bivariate distributions whose marginal distributions are uniform on the
unit interval) seem promising.
Before addressing questions of practical implementation, a major question needs
to be considered. At what point (which quantile) should it be considered that the
asymptotic arguments or the maximum domain of attraction applies? Many studies
have used the 95th percentile as the point beyond which the tail is estimated. It is
far from clear that the arguments do apply in this still broad range and as yet there
are no simulation studies of the significance of the implicit approximation of this
choice.
The first decision when attempting an implementation is whether to use simply
the ordered extreme values of the entire sample set, or to use maxima or minima in
defined time periods (blocks) of, say, one month or one year. The decision trade-off
is the number of data points available for the estimation and fitting of the curve parameters versus the nearness to the i.i.d. assumption. Block maxima or minima
should be expected to approximate an i.i.d. series more closely than the peaks over
a threshold of the whole series but at the cost of losing many data-points and
enlarging parameter estimation uncertainty. This point is evident from examination
of the following block maxima and peaks over threshold diagrams.
Implementation based upon the whole data series, usually known as peaks over
threshold (POT), uses the value of the realization (returns, in most financial applications)
beyond some (arbitrarily chosen) level or threshold. Anyone involved in the
insurance industry will recognize this as the liability profile of an unlimited excess
of loss policy. Figures 8A.1 and 8A.2 illustrate these two approaches:
Figure 8A.1 shows the minimum values in each 25-day period and may be
compared with the whole series data-set below. It should be noted that both series
are highly autocorrelated and therefore convergence to the asymptotic limit should
be expected to be slow.
Figure 8A.2 shows the entire data-set and the peaks under an arbitrary value (1.5).
In this instance, this value has clearly been chosen too close to the mean of the
distribution – approximately one standard deviation. In a recent paper, Danielsson
and De Vries (1997) develop a bootstrap method for the automatic choice of this cutoff
point but as yet, there is inadequate knowledge of the performance of small
sample estimators. Descriptive statistics of the two (EVT) series are given in Table
8A.1.
Notice that the data-set for estimation in the case of block minima has declined to
just 108 observations and further that neither series possesses ‘fat tails’ (positive
kurtosis). Figure 8A.3 presents these series.
The process of implementing EVT is first to decide which approach, then the level
of the tail boundary and only then to fit a parametric model of the GEV class to the
processed data. This parametric model is used to generate values for particular VaR
quantiles. It is standard practice to fit generalized Pareto distributions (GPD) to POT
data:
omitting the u and x subscripts. The numerical MLE solution should not prove
problematic provided m[ñ1
2 which should prove the case for most financial data. A
quotation from R. L. Smith is appropriate: ‘The big advantage of maximum likelihood
procedures is that they can be generalized, with very little change in the basic
methodology, to much more complicated models in which trends or other effects may
be present.’ Estimation of the parameters may also be achieved by either linear (see,
for example, Kearns and Pagan, 1997) or non-linear regression after suitable
algebraic manipulation of the distribution function.
It should be immediately obvious that there is one potential significant danger for
the risk manager in using EVT; that the estimates of the parameters introduce
error non-linearly into the estimate of the VaR quantile. However, by using profile
likelihood, it should be possible to produce confidence intervals for these estimates,
even if the confidence interval is often unbounded.
Perhaps the final point to make is that it becomes trivial to estimate the mean
expected loss beyond VaR in this framework; that is, we can estimate the expected
loss given a violation of the VaR limit – an event which can cause changes in
management behavior and cost jobs.
This brief appendix has attempted to give a broad overview of the subject. Of
necessity, it has omitted some of the classical approaches such as Pickand’s and
Hill’s estimators. An interested reader would find the introductory texts10 listed far
more comprehensive.
There has been much hyperbole surrounding extreme value theory and its application
to financial time series. The reality is that more structure (ARCH, for example)
needs to be introduced into the data-generating processes before it can be said that
the method offers significant advantages over conventional methods. Applications,
however, do seem most likely in the context of stress tests of portfolios.
Conclusionzz
Due to the extreme price shocks experienced regularly in the world’s financial
markets VaR is not an adequate measure of risk. Stress testing must be used to
complement VaR. The primary objective of stress testing is to identify the scenarios
that would cause a significant loss and to put a limit on risk exposures that would
cause such losses.
Stress testing must be undertaken in a systematic way. Ad-hoc scenario tests may
produce interesting results but are unlikely to identify the worst-case loss a bank
could suffer. Care must be taken to identify the stress tests required by examining
the types of risk in the bank’s portfolio. Stress tests should be run daily as a bank’s
portfolio can change significantly over a 24-hour period.
A bank’s risk appetite should be set with reference to VaR and to the worst-case
loss a bank is prepared to countenance under extreme market conditions. This is
best done with reference to the frequency with which a certain loss can be tolerated.
Stress test limits can then be established to ensure that the bank does not create
positions that could give rise, in severe market circumstances, to a loss greater than
the bank’s absolute tolerance of loss. Stress testing should be an integral part of a
bank’s risk management framework and stress test limits should be used along side
other risk limits, such as VaR limits.
markets VaR is not an adequate measure of risk. Stress testing must be used to
complement VaR. The primary objective of stress testing is to identify the scenarios
that would cause a significant loss and to put a limit on risk exposures that would
cause such losses.
Stress testing must be undertaken in a systematic way. Ad-hoc scenario tests may
produce interesting results but are unlikely to identify the worst-case loss a bank
could suffer. Care must be taken to identify the stress tests required by examining
the types of risk in the bank’s portfolio. Stress tests should be run daily as a bank’s
portfolio can change significantly over a 24-hour period.
A bank’s risk appetite should be set with reference to VaR and to the worst-case
loss a bank is prepared to countenance under extreme market conditions. This is
best done with reference to the frequency with which a certain loss can be tolerated.
Stress test limits can then be established to ensure that the bank does not create
positions that could give rise, in severe market circumstances, to a loss greater than
the bank’s absolute tolerance of loss. Stress testing should be an integral part of a
bank’s risk management framework and stress test limits should be used along side
other risk limits, such as VaR limits.
Stress test limits
A bank must limit the amount it is prepared to lose due to extreme market moves,
this is best achieved by stress test limits. As VaR only controls day-to-day risk, stress
test limits are required in addition to VaR limits. Stress test limits are entirely
separate from VaR limits and can be used in a variety of ways (see below). However
they are used, it is essential to ensure that stress test limits are consistent with the
bank’s VaR risk management limits, i.e. the stress test limits should not be out of
proportion with the VaR limits. Stress test limits should be set at a magnitude that
is consistent with the ‘occasional loss’ figure from Figure 8.10, above. However,
significantly larger price shocks should also be tested to ensure that the ‘extreme
tolerance number is not breached’.
Stress test limits are especially useful for certain classes of products, particularly
options. Traditional limits for options were based around the greeks; delta, gamma,
vega, rho and theta. A single matrix of stress tests can replace the first three greeks.
The advantage of stress tests over the greeks is that stress tests quantify the loss on
a portfolio in a given market scenario. The greeks, particularly, gamma, can provide
misleading figures. When options are at-the-money and close to expiry gamma can
become almost infinitely large. This has nothing to do with potential losses and
everything to do with the option pricing function (Black–Scholes). Figure 8.11 gives
an example of stress limits for an interest rate option portfolio. This example of stress
test limits could be used for the matrix of stress tests given above in Table 8.3.
Figure 8.11 shows three stress test limits, which increase in magnitude with the
size of the shift in interest rates and the change in volatility. Stress tests limits like
this make it easy for trading management to see that losses due to specified ranges
of market shifts are limited to a given figure. Using the greeks, the loss caused by
specific market shifts is not specified (except for the tiny shifts used by the greeks).
Stress test limits can be used as are standard VaR, or other, risk limits, i.e. when
a stress test identifies that a portfolio could give rise to a loss greater than specified
by the stress test limit, then exposure cannot be increased and must be decreased.
This approach establishes stress test limits as ‘hard’ limits and therefore, along with
standard risk limits, as absolute constraints on positions and exposures that can be
created.
Another approach is to set stress test limits but use them as ‘trigger points’ for
discussion. Such limits need to be well within the bank’s absolute tolerance of loss.
When a stress test indicates that the bank’s portfolio could give rise to a specified
loss, the circumstances that would cause such a loss are distributed to senior
management, along with details of the position or portfolio. An informed discussion
can then take place as to whether the bank is happy to run with such a risk.
Although EVT and statistics can help, the judgement will be largely subjective and
will be based on the experience of the management making the decision.
this is best achieved by stress test limits. As VaR only controls day-to-day risk, stress
test limits are required in addition to VaR limits. Stress test limits are entirely
separate from VaR limits and can be used in a variety of ways (see below). However
they are used, it is essential to ensure that stress test limits are consistent with the
bank’s VaR risk management limits, i.e. the stress test limits should not be out of
proportion with the VaR limits. Stress test limits should be set at a magnitude that
is consistent with the ‘occasional loss’ figure from Figure 8.10, above. However,
significantly larger price shocks should also be tested to ensure that the ‘extreme
tolerance number is not breached’.
Stress test limits are especially useful for certain classes of products, particularly
options. Traditional limits for options were based around the greeks; delta, gamma,
vega, rho and theta. A single matrix of stress tests can replace the first three greeks.
The advantage of stress tests over the greeks is that stress tests quantify the loss on
a portfolio in a given market scenario. The greeks, particularly, gamma, can provide
misleading figures. When options are at-the-money and close to expiry gamma can
become almost infinitely large. This has nothing to do with potential losses and
everything to do with the option pricing function (Black–Scholes). Figure 8.11 gives
an example of stress limits for an interest rate option portfolio. This example of stress
test limits could be used for the matrix of stress tests given above in Table 8.3.
Figure 8.11 shows three stress test limits, which increase in magnitude with the
size of the shift in interest rates and the change in volatility. Stress tests limits like
this make it easy for trading management to see that losses due to specified ranges
of market shifts are limited to a given figure. Using the greeks, the loss caused by
specific market shifts is not specified (except for the tiny shifts used by the greeks).
Stress test limits can be used as are standard VaR, or other, risk limits, i.e. when
a stress test identifies that a portfolio could give rise to a loss greater than specified
by the stress test limit, then exposure cannot be increased and must be decreased.
This approach establishes stress test limits as ‘hard’ limits and therefore, along with
standard risk limits, as absolute constraints on positions and exposures that can be
created.
Another approach is to set stress test limits but use them as ‘trigger points’ for
discussion. Such limits need to be well within the bank’s absolute tolerance of loss.
When a stress test indicates that the bank’s portfolio could give rise to a specified
loss, the circumstances that would cause such a loss are distributed to senior
management, along with details of the position or portfolio. An informed discussion
can then take place as to whether the bank is happy to run with such a risk.
Although EVT and statistics can help, the judgement will be largely subjective and
will be based on the experience of the management making the decision.
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