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1.
NSE-VaR system
Value-at-Risk (VaR) has been widely promoted by regulatory
authorities as a way of monitoring and managing market risk and
as a basis for setting regulatory minimum capital standards.
The revised Basle Accord, implemented in January 1998, makes
it
mandatory for banks to use VaR as a basis for determining the
amount of regulatory capital adequate for covering market risk
beyond that required for credit risk. Within the realm of the
fixed income portfolios of financial sector players, market
related risk has become more relevant and important on account
of their trading activities and market positions. For players
in the Indian financial sector, the need to develop risk
measurement models would prove critical as regulation
progressively moves from uniform prudential standards to
entity-specific risk coverage requirements. Specifically, the
guidelines call for linking of each entity’s market risk
capital charge to the riskiness of its assets as measured by the
chosen VaR model. Accuracy of measurement would prove critical
as regulation would not specify ‘a’ single model
for measurement of risk; - the choice of model would be left
to
market participants who would also be required to furnish
details of back-testing for the chosen VaR model. While a
conservative estimate of risk would lead to very large capital
holdings, a liberal estimate would result in inadequate coverage
of loss and excessive number of model failures historically,
which would in turn attract penalties from the regulator. It
would therefore be in the interest of market participants to
develop models that accurately measure the riskiness of their
portfolios and furnish estimates of capital charge that would
provide adequate cover. An important consideration in this
context is that setting up of risk measurement systems by each
individual participant for estimating portfolio risk under
alternative models and scenarios would involve significant
costs.
In line with its endeavour to develop market infrastructure,
NSE has taken initiative in developing a VaR system for measuring
the market risk inherent in Government of India (GoI)
securities. The NSE-VaR system builds on the NSE database of
daily yield curves - the NSE-ZCYC is now well accepted in terms
of its conceptual soundness and empirical performance, and is
increasingly being used by market participants as a basis for
valuation of fixed income instruments. The NSE-VaR system
provides measures of VaR using 5 alternative methods -
variance-covariance (normal) and historical simulation methods,
together with weighted normal, weighted historical simulation
and the recently developed extreme value method [a technical
paper explaining these methods is available here]. While the
first set of methods are easier to implement and therefore more
popular, they may not provide
accurate assessment of risk in volatile market conditions. To
this end, we provide estimates based on the latter set of
methods that are specifically suited for this purpose. Together,
the 5 methods would provide a range of options for market
participants to choose from.
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2.
Standard approaches to VaR estimation
VaR is the measure of the maximum (worst case) trading loss for a given
portfolio over a certain holding period and for a given confidence
interval. The VaR number is thus essentially determined by two
parameters, - (holding) time period and a confidence level, and is a
measure of the loss (expressed in say Rupees crore) on the portfolio
that will not be exceeded by the end of the time period with the
specified confidence level.
The most difficult part in VaR estimations is the derivation of the
portfolio returns distribution. Two popular approaches in the literature
for the calculation of VaR are variance-covariance analysis and
historical simulation. Variance-covariance analysis relies on the
assumption that financial returns are normally distributed. This method
is easy to implement because the VaR can be computed from a simple
linear formula with variances and co-variances of the returns as the
only inputs. Its major drawback is the assumption that financial market
returns are normally distributed, an assumption that has been shown to
be invalid in thousands of empirical studies on asset returns. Financial
returns are typically characterized by fat-tails and volatility
clustering. Fat-tail property of asset returns implies that losses are
much more frequent than predicted by the variance-covariance analysis.
The variance-covariance analysis is particularly weak where the demands
from a VaR model for regulatory purposes and risk control are strong,
i.e. in the prediction of extreme quantiles or large losses.
Another variant of variance-covariance analysis is the Exponential
weighting approach. This approach applies exponentially declining
weights to past returns to calculate conditional volatilities. This
technique is justified by the presence of conditional heteroskedasticity
or volatility clustering in the data, meaning that a volatile day is
typically followed by volatile days. The exponential approach also has
the drawback that a conditional normality assumption needs to be made
to calculate the VaR of a portfolio from its conditional volatility,
an
assumption that, more often than not, is not satisfied by financial
data. Although the exponential smoothing approach addresses the issue
of non-normality in the (unconditional) distribution of returns, it may
not
be applicable for regulatory VaR purposes on three grounds. First, while
daily returns reflect strong volatility clustering, they can hardly be
detected in bi-weekly returns such as the regulatory 10-day holding
period. Second, the volatility clustering observed in the data largely
emanates from medium and small range volatility periods. Extreme events,
such as losses at or beyond 99% confidence interval scatter rather
independently over time. Finally, as has been established by the
theoretical and empirical literature, interest rate and bond return
processes typically display a complicated dependence structure in both
mean and variance, thereby making simple exponential weighting schemes
- so often used in computing the equity VaR - inapplicable.
The historical simulation (HS) method, by using empirical percentiles
from the historical return distribution, gets around the problem of
making distributional assumptions. By applying the full empirical market
return distribution to all the items in the current trading portfolio,
the outcome exactly reflects the historical frequency of the large
losses over specific data window. A second advantage of this approach
compared to variance-covariance analysis is that it can incorporate
non-linear positions, such as derivative positions, in a natural way,
a property that is also useful in the context of fixed income portfolios.
The problem with the HS method is that it is very sensitive to the
particular data window, which the Basle committee has chosen to be at
least one year of past returns. In other words, whether returns from
a
highly volatile or a crash period is included or not makes a huge
difference for the value-at-risk predicted. Hence VaR predictions based
on HS exhibit high variability.
A hybrid approach proposed by Boudokh, Richardson and Whitelaw (Risk,
1998) draws on the strengths of the exponential and the HS approaches
to estimate the percentiles of returns directly using declining weights
on
past observations. Ordering returns over the historical simulation
period, the hybrid approach attributes exponentially diminishing weights
to each observation in building the conditional empirical distribution.
Although this method combines the strengths of above mentioned methods,
it still suffers from the tail-discreteness problem that we discussed
earlier and it may not be very effective particularly in predicting the
extremely large losses.
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3. Extreme Value theory and Value-at-Risk
As pointed out above, financial risk management, either for regulatory
purposes or for internal control, is intimately concerned with tail
quantiles. Traditional parametric models, such as variance-covariance
method and exponential weighting method, implicitly strive to produce
a good fit in regions where most of the data fall, potentially
at the
expense of a good fit in the tails, where, by definition, few
observations fall. It is common, moreover, to require estimates of
quantiles and probabilities not only near the boundary of the range of
observed data, but also beyond the boundary. That is, one would like
to allow for the possibility that the expected loss on any future
date to
be greater than that observed in the past.
A key idea in Extreme Value theory is that one can estimate extreme
quantiles and probabilities by fitting a “model” to the empirical
survival function (or one minus the cumulative distribution function (CDF))
of a set of data using only the extreme event data rather than all the
data, thereby fitting the tail and only the tail. EVT uses statistical
techniques that focus only on that part of a sample of returns data that
carry information about extreme behavior. Typically, the sample is
divided into N blocks of non-overlapping returns with say ‘n’
returns in each block. From each block the largest rise and biggest fall
in returns are extracted to create a series of maxima and minima
respectively. An extreme value model (more specifically a Generalised
Extreme Value (GEV) or Generalised Pareto (GP) distributions) is fitted
to either of these series, via Maximum likelihood or method of moments,
to estimate the ‘tail index’ parameter that characterizes the way
the extreme events in the data can occur. Once an estimate of the
‘tail index’ is available, one can compute the probability
of occurrence of a large event from the CDF, or VaR value for a given
probability .
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4. Issues in Fixed Income VaR
The computation of VaR for a fixed income portfolio
differs in important ways from that of an equity
portfolio. First, unlike in the case of an
equity portfolio where observed prices can be directly used for the
computation of VaR, the price of each fixed income instrument in a
portfolio is an outcome of many security-specific attributes, in
addition to the fundamental factor, the underlying term structure. This
rules out the use of prices directly for the computation of VaR if the
objective is to measure the interest rate risk that the portfolio is
subject to . Use of a zero coupon yield curve (ZCYC) is central to the
exercise, as yield-to-maturity (YTM) based approaches are also subject
to the same problem as with use of observed prices. Movements of the
ZCYC, inasmuch as they depict the changes in the interest rate
structure, are reflective of changes in the value of the portfolio
occurring on account of interest rate changes alone.
Estimation of VaR for a portfolio of fixed income securities is
complicated by two reasons: one, the changes in market values of the
securities are non-linearly related to changes in spot interest rates
leading to difficulties in making simple assumptions about the
distribution of the portfolio returns. A related point is that since
one needs to know the entire term structure of interest rates to value
a
fixed income security (up to the relevant maturity), to study the VaR
of the security we need to model the distribution of a great number of
interest rates. The popular practice of cash-flow mapping considers a
selected set of interest rates and maps the cash flow timings to that
of
the tenor of the selected interest rates through linear interpolations.
Underling in this strategy is interest rates are distributed as normal
or conditional normal, an assumption not typically supported by the
data. In addition, the cash-flow interpolations may also lead to
significant approximation errors. A better strategy would be to generate ‘returns’ on
(a portfolio of) fixed income instruments at the first stage by valuing
the said portfolio on observed yield curves, and
estimate the VaR directly from the returns on the bond portfolio. A
major advantage of this approach is that it does not require an
assumption about the interest rates. Since the VaR is estimated based
on bond portfolio returns, this approach has the disadvantage of being
portfolio specific thereby necessitating the model parametrization, and
estimation to be done for each portfolio separately. The NSE-VaR system
follows the latter approach of generating returns via historical
simulation and fitting a model of VaR to the return series.
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5. VaR over multi-day horizon
A first important observation is that VaR applies to the extreme
lower tail of the return distribution, i.e. large losses far
way from the
mean. Bank regulators have recognized this and typically chosen a (the
confidence level) equal to 99%. This number obviously reflects
regulators’ natural tendency for conservativeness in their prudential
supervision of banks. The same tendency also comes out in Basle
regulators’ choice of holding period, the second important model
parameter. A 10-day horizon is prescribed on the assumption that
positions cannot be liquidated quicker than within 10 business days. To
compute the 10-day VaR, the application of a simple square-root-of-time
rule is permitted, ie. 10-day VaR is derived as the product of Ö10 and
the 1-day VaR. For the purpose of setting adequate capital, the Basle
accord provides, in addition, a multiplicative factor of 3 by which the
computed VaR for the 10-day period should be multiplied. It is important
to mention here that the square-root-of-time rule is the appropriate
scaling factor for deriving multi-day VaR figures from 1-day figures
only under the assumption of normality. Once we take into consideration
the fat-tail property of underlying risk factors, the scale factor would
be a-root-of-time, where a is the tail index . A point of interest in
this context is that while simple models, such as Normal, under-estimate
the 1-day VaR when the return series have ‘fat’ tails, they
over estimate the multi-day VaR because the square-root rule turns out
to be
too conservative for fat tailed series. For more detailed discussion
on this and an empirical illustration see the technical
document available.
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Resource Persons:
Dr.Gangadhar Darbha
Dr.(Ms.)Vardhana Pawaskar
Consultants,
National Stock Exchange,
Bandra-Kurla Complex,
Bandra (E), Mumbai.
Phone: 659-8291,
Fax: 659-8288
E-mail: gdarbha@nse.co.in
vpawaskar@nse.co.in
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