Background component securities.

Banks and financial institutions experience Banks and financial institutions in

different types of risk, such as, business risk, developed as well as in underdeveloped

strategic risk, and financial risk. Financial countries are generally required by their

risk has been given more importance as it respective regulators to pass through

can be quantifiable while other risks have minimum Capital Adequacy. The 1988 Basel

some amount of subjectivity. It is caused by Capital Accord set capital requirements on

movements in financial markets. The the basis of risk adjusted assets, defined as

literature distinguishes four major the sum of asset positions multiplied by

categories of financial risk, viz., credit risk, asset-specific risk weights. These risk weights

operational risk, liquidity risk and market were intended to reflect primarily the credit

risk. Credit risk generally relates to the risk associated with a given asset. In 1996 the

potential loss due to the default on the part Accord was amended to include r market

of the counterparty to meet its obligations at risk, defined as the risk arising from

designated time. It has three basic movements in the market prices of trading

components: credit exposure, probability of positions (Basel Committee on Banking

default and loss in the event of default. S u p e r v i s i o n , 1 9 9 6 a ) . T h e 1 9 9 6

Operational risk takes into account the Amendment’s Internal Models Approach

errors that can be made in instructing (IMA) determines capital requirements on

payments or settling transactions, and the basis of the output of the financial

includes the risk of fraud and regulatory institutions’ internal risk measurement

risks. Liquidity risk is caused by an systems. Financial institutions are required

unexpected large and stressful negative cash to report daily their VaR’s at the 99%

flow over a short period. If a firm has highly confidence level over a one-day horizon and

illiquid assets and suddenly needs some over a two-week horizon (ten trading days).

liquidity, it may be compelled to sell some of Simply stated, VaR is the maximum loss

its assets at a discount. Finally, market risk of a trading portfolio over a given

estimates the loss of an investment portfolio horizon, at a given confidence level.

due to the changes in market price of Financial institutions are allowed to derive

A STEP BY STEP APPROACH TO VALUE AT RISK – RELEVANCE FOR INDIAN MARKET

Golaka.C.Nath*

*Shri Golaka C Nath is the Advisor, Economic Research & Surveillance

Department, The Clearing Corporation of India Limited.

their two-week VaR measure by scaling up suggested by the Basel Committee to address

the daily VaR by the square root of ten (see: this problem relies on “backtesting” (Basel

Basel Committee on Banking Supervision, Committee on Banking Supervision, 1996c):

1996b, p. 4). The new Capital Accord regulators should evaluate on a quarterly

(commonly known as BASEL II) likely to be basis the frequency of “exceptions” (that is,

implemented in 2006 would require the frequency of daily losses exceeding the

international banks to provide for reported VaR) for every financial institution

operations risk. in the most recent twelve-month period and

the multiplier used to determine the market The minimum capital requirement on a

risk charge should be increased (according to given day is then equal to the sum of a charge

a given scale varying between 3 and 4) if the to cover “general market risk” and a charge

frequency of exceptions is high. The reason to cover “credit risk” (or idiosyncratic risk),

to base backtesting on a daily VaR measure in where the market-risk charge is equal to a

spite of the fact that the market risk charge is multiple of the average reported two-week

based on a two-week VaR measure is due to VaR’s in the last 60 trading days and the

the fact that VaR measures are typically credit-risk charge is equal to 8% of risk-

computed ignoring portfolio revisions over adjusted assets. The US regulated banks and

the VaR horizon. The basic point to OTC derivatives dealers are subject to capital

remember here is that the bank is assumed to requirements determined on the basis of the

hold the same portfolio over a historical IMA. More precisely, the market-risk charge

time horizon from the date of VaR is equal to the larger of: (i) the average

estimation. According to the Basel reported two-week VaR’s in the last 60

Committee, “it is often argued that revalue-trading days times a multiplier and (ii) the

at-risk measures cannot be compared against last-reported two-week VaR. However, since

actual trading outcomes, since the actual the multiplier is not less than 3 (see below),

outcomes will inevitably be ‘contaminated’ the average of the reported VaR’s in the last

by changes in portfolio composition during 60 trading days times a multiplier typically

the holding period. This argument is exceeds the last-reported VaR.

persuasive with regard to the use of Value-at-

Of course, the reliance on the financial Risk measures based on price shocks

institution’s self-reported VaR to determine calibrated to longer holding periods. That is,

capital requirements creates an adverse comparing the ten-day, 99th percentile risk

selection and moral hazard problem, since measures from the internal models capital

the institution has an incentive to under- requirement with actual ten-day trading

report its true VaR in order to minimize outcomes would probably not be a

capital requirements The procedure meaningful exercise. In particular, in any

given ten day period, significant changes in through Internal Model Approach. Under

portfolio composition relative to the initial this approach, regulators do not provide any

positions are common at major trading specific VaR measurement technique to their

institutions. For this reason, the backtesting supervised banks – the banks are free to use

framework described here involves the use of their own model. But to eliminate the

risk measures calibrated to a one-day holding possible inertia of supervised banks to

period.” (Basel Committee on Banking underestimate VaR so as to reduce the capital

Supervision (1996c, p. 3). Additional requirements, BIS has prescribed minimum

corrective actions in response to a high standard of VaR estimates and also certain

number of exceptions are left to the tests, such as backtesting, of VaR models. If

discretion of regulators. VaR model of a bank fails in backtesting, a

penalty is imposed resulting to higher In recent years, VaR has become the standard

capital charge. measure that financial analysts use to

quantify the market risk. The great Thus, providing accurate estimates of VaR is

popularity that this instrument has achieved of crucial importance for all stakeholders. If

among financial practitioners is essentially the underlying risk is not properly

due to its conceptual simplicity: VaR reduces estimated, this may lead to a sub-optimal

the (market) risk associated with any capital allocation with consequences on the

portfolio to just one number that is the loss profitability or the financial stability of the

associated with a given probability and institutions. A bank would like to pick up a

horizon. model that would generate as low VaR as

possible but pass through the backtesting.VaR measures can have many applications. It

evaluates the performance of risk takers and There has been voluminous work done on

satisfies the regulatory requirements. VaR VaR in financial market all over the world,

has become an indispensable tool for the task of estimating/forecasting VaR still

monitoring risk and an integral part of remains challenging. The major difficulty

methodologies that allocate capital to lies in modeling the return series which has

various lines of business. Today regulators all normally heteroskedistic properties (being

over the globe have been forcing institutions skewed and/or having fatter tails than

to adopt internal models and calculate the normal distribution). Available VaR models

required capital charge based on VaR can be classified into four broad categories:

methodologies. The Basel Committee on the historical simulation method, the Monte

Banking Supervision of the BIS imposes Carlo simulation method, modelling return

requirements on banks to meet capital d i s t r i b u t i o n ( i n c l u d i n g t h e

requirements based on the VaR estimated variance/covariance method, which assumes

normality of the return distribution, and total exclusion of bank’s internal rating

methods under Extreme Value Theory system. However, RBI has issued detailed

(EVT). All these VaR estimation methods guidelines to Primary Dealers (PDs) for

adopt the classical approach: they deal with mandatory implementation of VaR methods

the statistical distribution of time series of while calculating the capital charge 2returns. required . The recent guidelines of RBI for

PDs states that the capital requirement will The main objective of this paper is to give a

be the higher of (i) the previous day’s value-theoretical orientation of the subject and to

at-risk number measured according to the d i s c u s s t h e s t e p s i n v o l v e d i n

above parameters specified in this section implementation of VaR models with

and (ii) the average of the daily value-at-risk backtesting as per BIS/RBI norms. Reserve

measures on each of the preceding sixty Bank of India (RBI) has issued detailed

b u s i n e s s d a y s , m u l t i p l i e d b y a guidelines on market risk for banks on the

multiplication factor prescribed by Reserve basis of the BIS framework though it has not

Bank of India (3.30 presently).become mandatory for banks to use VaR

1models . It is interesting to note that a We have restricted our analysis to only

significant number of banks in developed application of VaR methodologies using a

countries are opting for the Internal Ratings popular GOI bond index (IBEX – Principal

Based methods, while their counterparts in Return Index). The data period is from

developing countries (including India) August 1994 to June 2004, about 11 years of

follow the Standardized Approach. The data with 3622 data point. The data period is

main issue for moving towards the sufficiently large to cover various cyclical

Standardized Approach is the absence of and seasonal factors in the economy. For

rating culture in emerging markets like India the limited purpose of the study, we have

for which the unrated claims on Sovereigns, used Normal (Variance Covariance),

PSEs, banks, Security Firms and Corporates Historical Simulation and a weighted

would attract 100% risk weighting and hence Historical Simulation method for simplicity

higher capital charge requirement. The BIS and parsimony, though it is not difficult to

capital accord, for capital purposes, estimate VaR using EVT and Weighted

recognizes only external credit ratings, to the Normal.

1RBI circular BP./21.04.103/2001 dated March 26, 2002 and RBI/2004/263/DBDO.No.BP.BC.103/21.04.151/2003-04

dated June 24, 2004.

2 RBI circular IDMC.PDRS.PDC.3/03.64.00/2000-01 dated December 11, 2000 and RBI / 2004/7/IDMD 1/(PDRS)

03.64.00 /2003-04 dated January 7, 2004.

Literature Review Andersen and Bollerslev (1998) and

Andersen, et al. (1999), who use average There has been large volume of literature on

intraday squared returns to estimate daily VaR methodologies as well as on its

volatility.implementation. The concept received

tremendous response from banks all over the Several studies such as Danielsson and de

world. Banks management can apply the Vries (1997), Christoffersen (1998), and

VaR concept to set capital requirements Engle and Manganelli (1999) have found

because VaR models allow for an estimate of significant improvements possible when

capital loss due to market risk (see Duffie deviations from the relatively rigid

and Pan, 1997; Jackson, Maude and RiskMetrics framework are explored.

Perraudin, 1997; Jorion, 1997; Saunders, Choosing an appropriate VaR measure is an

1999; Friedmann and Sanddrof-Kohle, 2000; important and difficult task, and risk

Hartmann-Wendels, et al., 2000; Simons, managers have coined the term Model Risk

2000, among others). to cover the hazards from working with

potentially mis-specified models. Beder The computation of volatility is the most

(1995), for example, compares simulation-important aspect of any VaR estimation. The

based and parametric models on fixed volatility estimation should take care of the

income and stock option portfolios and most stylized facts of any financial asset class

finds apparently economically large – the important ones being fat tailed

differences in the VaRs from different property, volati l i ty clustering and

models applied to the same portfolio. asymmetry of return distribution. Once

Hendricks (1996) finds similar results these issues are identified in the distribution,

analyzing foreign exchange portfolios. In then calculating volatility is easy. Today

Indian context, Darbha (2001) made a GARCH family models have been

comparative study of three models – increasingly used by researchers to model

Normal, HS and Extreme Value Theory volatility. An important documentation in

while studying the portfolio of Gilts held by this regard has been the J P Morgan’s

PDs. Nath and Samanta (2003a and b) RiskMetrics that applied declining weights

looked at the issues in implemention as well to past returns to compute volatility with a

as selection of VaR methodologies for the decay factor 0.94 which is a variant of

portfolio of Government of India securities.IGARCH. Other measures of volatility,

which differs from the estimation of return Theoretical Issues

variance, include Garman and Klass (1980), As stated earlier, VaR is the maximum

and Gallant and Tauchen (1998), who amount of money that may be lost on a

incorporate daily high and low quotes, and portfolio over a given period of time, with a

given level of confidence and typically The PDs should calculate the capital

calculated for a one-day time horizon with requirement based on their internal Value at

95% or 99% confidence level. The capital Risk (VaR) model for market risk, as per the

charge would be different for different following minimum parameters:

holding period. (a) “Value-at-risk” must be computed on a

BIS requires that VaR be computed daily by daily basis.

Banks, using a 99th percentile, one-tailed (b) In calculating the value-at-risk, a 99th

confidence interval with a minimum price percentile, one-tailed confidence

shock equivalent to ten trading days interval is to be used.

(holding period) and the model incorporate

(c) An ins tantaneous pr ice shock a historical observation period of at least one

equivalent to a 15-day movement in year. The capital charge for a bank that uses a

prices is to be used, i.e. The minimum proprietary model will be higher of (i) The

“holding period” will be fifteen trading previous day’s VaR and (ii) an average of the

days. daily VaR of the preceding sixty business

days, multiplied by a multiplication factor. (d) Interest rate sensitivity of the entire

The multiplication factor may be 3 and this portfolio should be captured on an

may go up if the regulators feel that 3 is not integrated basis by including all f i x e d

sufficient to account for potential income securities like G o v e r n m e n t

weaknesses in the modeling process.securities, Corporate/PSU bonds, CPs

In the case of PDs, RBI prescribes all these and derivatives like IRS, FRAs,

above criteria except that (i) minimum Interest rate futures etc., based on the

holding period would be 15 trading days; (ii) mapping of the cash flows to work o u t the minimum length of the historical the portfolio VaR. Wherever da ta for observation period used for calculating VaR calculating volatilities is not available, should be one year or 250 trading days. For PDs may calculate the volatilities of PDs who use a weighting scheme or other such instruments using the G-Securities methods for the historical observation curve with appropriate spread. period, the “effective” observation period

However, the de ta i l s of such must be at least one year (that is, the

instruments and the spreads applied weighted average time lag of the individual

have to be reported and consistency of observations cannot be less than 6 months);

methodology should be ensured . and (iii) the multiplication factor is

Instruments which are part of trading presently fixed at 3.3. The RBI guidelines of

book, but found d i ff i cu l t to be January 2004 read as:

subjected to measurement of market

risk may be applied a flat market risk (h) The capital requirement will be the

measure of 15%. The instruments likely higher of (i) the previous day’s value-at-

to be applied the flat market risk risk number measured according to the

measure are units of MF, Unquoted above parameters specified in this

Equity, etc., and added arithmetically section and (ii) the average of the daily

to the measure obtained under VaR in value-at-risk measures on each of the

respect of other instruments. preceding s i x t y b u s i n e s s d a y s ,

multiplied by a multiplication factor (e) Underwr i t ing commitments a s

prescribed by Reserve Bank of India explained at the beginning of the

(3.30 presently). Appendix should also be mapped into

t h e VaR f r amewo rk f o r r i s k (i) No particular type of model is

measurement purposes. prescribed. So long as the model

used captures all the material risks(f) The unhedged foreign exchange

run by the PDs, they will be free to use position arising out of the foreign

models, based for example, on variance-c u r r e n c y b o r r o w i n g s u n d e r

covar iance matr ices , his tor ica l FCNR(B) loans scheme would

s imula t ions , o r Monte Car lo carry a market risk of 15% as

simulations or EVT etc.hitherto and the measure obtained will

be added0 arithmetically to the VaR The RBI has not restricted PDs for any

m e a s u r e o b t a i n e d f o r o t h e r particular model but the intention is that

instruments. PDs should use the model which gives them

the best comfort level with regard to capital (g) The choice of historical observation

requirement while not violating the period (sample period) for calculating

regulatory norms on backtesting. The value-at-risk will be constrained to a

weaknesses of the VaR models may be due to minimum length of one year and not

(a) market prices often display patterns less than 250 trading days. For PDs

(heteroskedastic) that differs from the who use a weighting scheme or other

statistical simplifications used in modeling, methods for the historical observation

(b) past not being always a good period, the “effective” observation

approximation of the future (October 1987 period must be at least one year (that is,

crash happened that did not have parallel in the weighted average time lag of the

historical data), (c) most of the models take individual observations cannot be less

ex-post volatility and not ex-ante, (d) VaR than 6 months).

estimations normally is based on end-of-day

positions and not take into account intra-

day risk, (e) models can not adequately

capture event risk arising from exceptional (3)

market circumstances. Since VaR heavily Where s is the covariance and r is the ij,t+1 ij,t+1relies on the availability of historical market

correlation between security i and j on day price data on the portfolio to understand its 2t+1 and for r = 1 and we write s = seffectiveness, it would be appropriate to use ij,t+1 i,t+1 ij,t+1

for all i.the long historical data to see if the stress

conditions can be replicated. The VaR of the portfolio is simply

The logic behind VaR is to consider today’s (4)

portfolio and find out what would have been

where Fp-1 is the p’th quantile of the rescaled its historical values (time series) and then

portfolio returns.construct the return series and then calculate

the VaR numbers. We have used a bond index When we use HS method, we write the VaR

that represents the most liquid basket of equation as

underlying Government securities and has a

long historical price sereis.

Select VaR MethodologiesBasic Statistics Related to VaR

There are few VaR methodologies that are The portfolio consists of many securities very simple and easy to implement, to name and in our case we are concerned with only a few are (a) Normal (parametric using Gilts. The basic price equation of the variance and covariance approach) and (b) portfolio can be written as follows:Historical simulation. Cleverly these simple

(1) methods have been extended with

application of weights – recent events are and the return on the portfolio is at time

given more weight and past is given less. defined as

However, different people have used (2)

different weighting methodologies .

Riskmetrics has used ‘exponentially moving Where the sum is taken over n securities in

average’ where the decay factor (l) has been the portfolio, wi denotes the proportionate

considered as 0.94 while Boudoukh, et al. value of the holding of security i at the end of

(1997) fixed it at 0.98. day t.

Variance-Covariance (Normal) MethodAnd the variance of the portfolio should

be written as The Variance-Covariance (Normal) method

}100,}{{ 11,pR m

rtFP =+1, percentileVaR tPFp

+ -=

1,1

1, * +=

+ å= ti

n

iitpf RwR

1p1tF,P1tPF,

p F*óVaR -

++ =

is the easiest of the VaR methodologies. Since and then revalues the same using the

we are considering a sovereign bond index historical price series. Once we calculate the

for our analysis, it is known that interest rate daily returns of the price series, then sorting

movement in sovereign bond market is the same in an ascending order and find out

unidirectional at any point of time. For our the required data point at desired percentiles.

purpose, the plain standard deviation would Linear interpolation can be used if the

be useful to calculate the require VaR. But required percentile falls in between 2 data

whether to take static variance of the entire points. The moot question is what length of

time series or conditional variance is a point price series should be used to compute VaR

for debate. It is argued that variance changes using HS method and what we should do if

over time horizons and hence we should not the price history is not available. It has to be

rely on unconditional variance for kept in mind that HS method does not allow

measuring VaR. for time-varying volatility.

Historical Simulation Method Another variant of HS method is a hybrid

approach put forward by Boudhoukh, et al. Historical simulation approach provides

(1997), that takes into account the some advantages over the normal method, as

exponential declining weights as well as HS it is not model based, although it is a

by extimating the percentiles of the return statistical measure of potential loss. The

directly, using declining weights on past main benefit is that it can cope with all

data. As described by Boudhoukh et al. portfolios that are either linear or non-

(1997, pp. 3), “the approach starts with linear. The method does not assume any

ordering the returns over the observation specific form of the distribution of price

period just like the HS approach. While the change/return. The method captures the

HS approach attributes equal weights to each characteristics of the price change

observation in building the conditional distribution of the portfolio, as VaR is

empirical distribution, the hybrid approach estimated on the basis of actual distribution.

attributes exponentially declining weights to This is very important, as the HS method

historical returns”. The process is simplified would be on the basis of available past data.

as follows:•Calculate the return series of If the past data does not contain highly

past price data of the security or the volatile periods, then HS method would not

portfolio from t-1 to t. be able to capture the same. Hence, HS

should be applied when we have very large • To each most recent K returns: R(t), R(t-

data points that are sufficiently large to take 1), ……R(t-K+1) assign a weight

into account all possible cyclical events. HS

method takes a portfolio at a point of time 1kk l)]l(1--k /l)[(1)],l/(1l)[(1 k /l)l,.....[(1)]l(1 -----

r e s p e c t i v e l y . T h e c o n s t a n t the risk of low-probability events that could

simply ensures that the lead to catastrophic losses. Yet traditional

weights sum to 1. VaR methods tend to ignore extreme events

and focus on risk measures that • Sor t the re turns in a scending

accommodate the whole empirical order.

distribution of returns. For example, it is

• In order to obtain p% VaR of the often assumed that returns are normally or

portfolio, start from the lowest lognormally distributed, and little attention

return and keep accumulating the is paid to the distribution of the extreme

we i gh t s un t i l p% i s re a ch ed . returns we are most concerned about. The

Linear interpolation may be used danger is then that our models are prone to

to achieve exac t ly p% of the fail just when they are needed most – in large

distribution. market moves, when we can suffer very large

losses. • In many studies lambda (l) has

been used as 0.98. One response to this problem is to use stress

tests and scenario analyses. These can Another Hybrid method is a weighting

simulate the changes in the value of our scheme suggested by Hull and White that

portfolio under hypothesized extreme allows us to transform the returns by

market conditions. These are certainly very multiplying the return series with a vector of

useful. However, they are inevitably limited – ratios of last day’s (the day for which VaR is

we cannot explore all possible scenarios – estimated) VaR and conditional volatilities

and by definition give us no indication of (using the Riskmetrics method with a decay

the likelihoods of the scenarios considered. factor) calculated for previous ‘n’ days.

Then take the appropriate percentile values This type of problem is not unique to risk

to represent the VaR numbers. The weighting management, but also occurs in other

scheme is justified on the ground that if the disciplines as well, particularly in hydrology

volatility on a previous day in the sample is and structural engineering, where the failure

lower than the current period volatility, VaR to take proper account of extreme values can

would be underestimated. The weighting have devastating consequences. Researchers

scheme makes them comparable during the and practitioners in these areas handle this

entire period. This is a method of problem by using Extreme Value Theory

normalizing the return series. (EVT) – a specialist branch of statistics that

attempts to make the best possible use of Extreme Value Theory (EVT)

what little information we have about the

Risk managers are primarily concerned with extremes of the distributions in which we are

)]l/(1l)[(1 k--

interested. To put it in simple, suppose we have 1000

data points of historical prices and we have The key to EVT is the extreme value theorem

some idea how to divide the period into – a cousin of the better-known central limit

smaller time buckets of, say,10-15 days. Once theorem – which tells us what the

we have a rationality of diving the entire distribution of extreme values should look

period into optimal time buckets (we may like in the limit, as our sample size increases.

use some statistical tests to identify the Suppose we have some return observations

optimal number), we take the extreme high but do not know the density function from

and low values and combine all these which they are drawn. Subject to certain

extreme high values – one from each time relatively innocuous conditions, this

bucket, to estimate the distribution theorem tell us that the distribution of

properties of the extreme series and estimate extreme returns converges asymptotically to:

the VaR using this extreme series as these are

If the tails in which we are interested rather

than taking the entire sample of 1000 data The parameters µ and s correspond to the points to estimate the tail.mean and standard deviation, and the third

EVT provides a natural approach to VaR parameter, x gives an indication of the

estimation, given that VaR is primarily heaviness of the tails: the bigger, x the

concerned with the tails of our return heavier the tail. This parameter is known as

distributions. To apply to VaR, we first the tail index, and the case of most interest in

estimate the parameters of the distribution, finance is where , x>0 which corresponds to and there are a number of standard the fat tails commonly founded in financial estimators available (in one of the earlier return data. In this case, our asymptotic work, the author has used Gauss codes to distribution takes the form of a Fréchet estimate EVT VaR). Once we have these, we distribution. can plug them into a number of alternative

formulas to obtain VaR estimates. To give a This theorem tells us that the limiting simple example, if we want to estimate a VaR distribution of extreme returns always has that is out of (i.e., more extreme than) our the same form – whatever the distribution of sample range, we can project the tail out the parent returns from which our extreme from an existing in-sample quantile Xk+1 – returns are drawn. It is important because it

where Xk+1 is the k+1-th most extreme allows us to estimate extreme probabilities

observation in our sample – and infer the and extreme quantiles, including VaRs,

(asymptotic) VaR from the projected tail without having to make strong assumptions

using the formula: about an unknown parent distribution.

H xx

e xxms

x

m s

x m s, ,

/

( )/( )exp( [ ( ) / ]

exp( )=

- + -

-

-

- -

1 1x

x

¹

=

0

0

ˆ-Xk+1[CL/k] x

VaR = Robust VaR Models - Backtesting

Where CL is the confidence level on which Any method used for VaR estimation need to

the VaR is predicated. EVT also gives us satisfy the criteria of back testing using the

expressions for the confidence intervals current data set. Suppose we calculate the

associated with our VaR estimates. VaR numbers with probability level 0.01. We

can check the accuracy of a VaR model by The EV approach to VaR has certain

counting the number of times VaR estimate advantages over traditional parametric and

fails (i.e. actual loss exceeds estimated VaR), non-parametric approaches to VaR.

say in 100 days. If we want to calculate VaR Parametric approaches estimate VaR by

of a one-day holding period with 99% fitting some distribution to a set of observed

confidence level, logically, we are allowing 1 returns. However, since most observations lie

failure in 100 days. But if the number is more close to the centre of any empirical

than 1, then the model is under predicting distribution, traditional parametric

VaR numbers and if we find less number of approaches tend to fit curves that

failures the model is over predicting. The accommodate the mass of central

Basle Committee provides guidelines for observations, rather than accommodate the

imposing penalty leading to higher tail observations that are more important for

multiplication factor, when the number of VaR purposes. Traditional parametric

failure is too high. However, no penalty is approaches also suffer from the drawback

imposed when the failure occurs with less that they impose distributions that make no

frequency than the expected number. Thus, sense for tail estimation and fly in the face of

selection of VaR model is a very difficult task. EV theory. By comparison, the EV approach

A model, which overestimates VaR, may is free of these problems and specifically

result in reduced number of failure but designed for tail estimation.

increase the required capital charge directly.

On the other hand if a model underestimates Non-parametric or historical simulation

VaR numbers, the number failures may be approaches estimate VaR by reading off the

too large which ultimately increases the VaR from an appropriate histogram of

multiplying factor and hence the required returns. However, they lead to less efficient

capital charge. Thus an ideal VaR model VaR estimates than EV approaches, because

would be the one, which produces VaR they make no use of the EV theory that gives

estimates, as minimum as possible and also us some indication of what the tails should

pass through the backtesting. Samanta and look like. More importantly, these

Nath (2003) have discussed the issue of approaches also have the very serious

selecting models on the basis of robust loss limitation that they can tell us nothing

functions.whatever about VaRs beyond our sample

range.

The BIS requires that models must For carrying out the Backtesting of a VaR

incorporate past 250 days data points (one model, realized day-to-day returns of the

year assuming Saturday/Sundays being non- portfolio are compared to the VaR of the

trading days). In Indian market, RBI has portfolio. The number of days when actual

issued guidelines for PDs to use one year and portfolio loss was higher that VaR provides

not less than 250 trading days for VaR an idea about the accuracy of the VaR model.

estimation. Since Saturday is a trading day in For a good VaR model, this number would

bond market in India, we have taken 290 approximately be equal to the 1 per cent (i.e.

days (a period of about one year) for our 100 times of VaR probability) of back-test

analysis. Accordingly the capital charge is trading days. If the number of violation (i.e.

the higher of (i) the previous day’s value-at- number of days when loss exceeds VaR) is too

risk number measured according to the high, a penalty is imposed by raising the

above parameters specified in this section multiplying factor (which is at least 3),

and (ii) the average of the daily value-at-risk resulting in an extra capital charge. The

measures on each of the preceding sixty penalty directives provided by the Basle

b u s i n e s s d a y s , m u l t i p l i e d b y a Committee for 250 back-testing trading days

multiplication factor prescribed by RBI is as follows; multiplying factor remains at

(3.30 presently for Pds). minimum (i.e. 3) for number of violation

upto 4, increases to 3.4 for 5 violations, 3.5 Basle Committee (1996b) provides following

for 6 violations, 3.65 for violations 8, 3.75 backtesting criteria for an internal VaR

for violations 8, 3.85 for violation 9, and model (see van den Goorbergh and Vlaar,

reaches at 4.00 for violations above 9 in 1999; Wong et al., 2003, among others)

which case the bank is likely to be obliged to

(1) One-day VaRs are compared with actual revise its internal model for risk

one-day trading outcomes. management (van den Goorbergh and Vlaar,

1999). (2) One-day VaRs are required to be correct

on 99% of backtesting days. There For the limited purpose of this paper, to do

should be at least 250 days (around one the back testing, we can think of an indicator

year) for backtesting.variable I(t) which is one if return of the day

(3) A VaR model fails in Backtesting when it is more than the VaR for the previous day provides 5% or more incorrect VaRs. and zero otherwise. Average of the indicator

(4) If a bank provides a VaR model that fails variable should be our VaR percent. in backtesting, it will have its capital

Datamultiplier adjusted upward, thus

increasing the amount of capital We have estimated VaRs as on June 30, 2004

charges. for two important asset classes relevant for

banks and institutions. We have used GOI the backtesting, we have used the full period

bond index (IBEX) developed by ICICI as well as various rolling periods of 500, 750,

Securities and the INR-USD Exchange rate. 1000, 1250, 1500 days. For example, rolling

The exchange rate has been taken from 500 days means, we will take 1 to 500

various RBI publications and website and observations and compute the VaR and

the data is from march 01, 1993. We have compare the same with next day’s losses and

chosen this data as our starting point then 2 to 501,……and so on. However, the use

because the unified exchange rate system was should see how the models work over various

introduced from this date. This IBEX index back periods with respect to estimation and

is widely used by market participants and has backtesting. All our calculations are for June

a long time series. The index as of June 2004 30, 2003.

has 19 underlying liquid bonds covering all The Chart-I gives the movement of IBEX PRI

the time buckets upto 15 years. The longest index used for the study while Chart-2 gives

maturity underlying is 2019 while the the daily return distribution during the

shortest underlying is 2005. For the details period. The Chart-3gives the movement of

on construction of IBEX, readers are exchange rate while Chart 4 gives the daily

requested to look at the website of ICICI returns of exchange rates.

Securities. While calculating VaR and doing

PR

I

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

900

1000

1100

1200

1300

1400

1500

Chart-1: Movement of IBEX (August 1994-June 2004)

The Table -1 gives the summary statistics of LCL Mean : 1.469139e-003

UCL Mean : 1.622274e-002The daily returns. The figures shows that the

Skewness : -1.355666e-001series can not be said to have properties that

Kurtosis : 6.392652e+001can fit a normal distribution.

Table-1 - Summary Statistics for Data: Table-2 - Summary Statistics for Data:

IBEXINR-USD Exchange Rate

PRINCIPAL RETURN Rtn

Min : -3.780000e+000Min : -3.297790e+000

1st Qu. : -2.000000e-0021st Qu. : -4.728692e-002

Mean : 8.845941e-003Mean : 1.242997e-002

Median : 0.000000e+000Median : 0.000000e+000

3rd Qu. : 5.000000e-0023rd Qu. : 5.766430e-002

Max : 3.830000e+000Max : 2.976490e+000

Total N : 3.622000e+003Total N : 2.891000e+003

NA’s : 0.000000e+000NA’s : 0.000000e+000

Variance : 5.127412e-002Variance : 8.281272e-002

Std Dev. : 2.264379e-001Std Dev. : 2.877720e-001

Sum : 3.204000e+001Sum : 3.593505e+001

SE Mean : 3.762486e-003SE Mean : 5.352103e-003

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

-5

-3

-1

1

3

PR

IRT

N

Chart-2: Distribution of Daily Returns

LCL Mean : 1.935647e-003 to estimate the VaR numbers.

UCL Mean : 2.292430e-002We first compute 1-day VaR numbers for all

Skewness : -5.563513e-002 methods and as well as the average of 1-day

Kurtosis: 2.495990e+001 VaRs in last 60 days in our sample. All VaR

estimates correspond to the probability level Estimates of VaRs and Capital Charge 0.01 (equivalently correspond to the

confidence level 0.99). For a given In this section we report our estimated VaR security/portfolio, maximum of these two figures and corresponding capital charges VaRs (i.e. 1-day VaR in last day and 60-day required All calculations are restricted to left-average of 1-day VaR) has been adjusted to tail (one tailed) of return distribution and arrive at VaR numbers corresponding to two probability level is fixed at 0.01 (equivalently alternative holding periods, viz., 10-days and confidence level of VaR estimates is set to

115-days . For calculating capital charge 0.99) strictly as per the RBI guidelines. Thus,

corresponding to a holding period h, h=10-the estimates we provide here actually refer

days or 15-days, the VaR with h-days holding to long-investment positions assuming that

period has been multiplied by the investment has been on the basis of the

multiplication factor 3.3 (as given in the RBI index. However, it is not difficult to take

circular for Pds). individual bonds or the portfolio of bonds

0 500 1000 1500 2000 2500 3000

30

35

40

45

50

Chart-3: Movement of in INR-USD Exchange Rate (March 1993-June 2004)

Pri

ce

The estimated results are given in Annexure- take only last one year data for backtesting,

1 for Normal Method and Annexure -2 for The Normal method requires less capital

Historical Simulation Method and charge vis-à-vis Historical simulation. But

Annexure-3 for Weighted Historical when we increase the backtesting horizon,

Simulation (Hull-White) Method for the Historical simulation gives better results and

GOI bond index IBEX and Annexure-4 for is more stable. Form a regulatory point of

Normal Method and Annexure -5 for view, Historical simulation would be

Historical Simulation Method and preferred.

Annexure-6 for Weighted Historical An important issue need to be mentioned

Simulation (Hull-White) Method for the here is that all VaR estimates provided in the

INR-USD exchange rate data.tables are in percentage form, and thus, may

We notice from the above tables that if we actually be termed as the relative VaR (Wong,

0 500 1000 1500 2000 2500 3000

-5

-3

-1

1

3

Rtn

Chart-4: Distribution of Daily Returns of Exchange Rate

3As per the Basle Committee guideline (1996), capital charge should be derived based on VaR numbers for

probability level 0.01 and holding periods 10-days. The VaR for 10-days holding period, however, are

calculated based on 1-day VaR numbers computed daily basis. In India, guidelines issued to PDs maintain

all attributes for capital charge computation except that VaR should have 15-days holding period (rather

than 10-days holding period prescribed in the Basle Committee).

et al., 2003), which refers to the percentage of tradeoff between the failures and higher

a portfolio value which may be lost after h- capital charge. From a regulatory point of

holding period with a specified probability view Historical Simulations and weighted

(i.e. the probability level of VaR). The historical simulations would be useful.

absolute VaR (i.e. the VaR expressed in Reference:

Rupees term) can easily be computed by

Altzner, P., F.Delbaen, J-M. Eber and D. multiplying the portfolio values with the

Heath, 1999, “Coherent Measures of Risk”, estimated relative VaR. Similarly, the capital

Mathematical Finance, 9, pp. 203-208.charge can also be represented in two

alternative forms, viz., relative (i.e. in Andersen, T. and T. Bollerslev (1998),

percentage) or absolute (i.e. in rupees terms). “Answering the Critics: Yes, ARCH Models

The additional information we require to do Provide Good Volatility Forecasts,”

convert a relative VaR/capital charge in a day International Economic Review, 39,

to a corresponding absolute term (i.e. rupees 885-905.

term) figures is the value of the portfolio.

Andersen, T., T. Bo llerslev, F. Diebold and P. The results show that normal method does

Labys (1999), “The Distribution of not provide better results and if fails in

Exchange Rate Volatility,” Journal of the backtesting when we apply the VaR methods

American Statistical Association, Website: at 99%. Historical Simulation and Weighted

http://citeseer.nj.nec.com/andersen99distriHistorical Simulation methods provide

bution.htmlbetter results.

Basel Committee on Banking Supervision, Conclusion:

1996a Amendment to the Capital Accord to

Incorporate Market Risk.This paper has experimented with two most

widely used VaR models, such as, variance-Basel Committee on Banking Supervision,

cova r i ance/normal and h i s to r i ca l 1996b Overview of the Amendment to the

simulation for estimating VaR using GOI Capital Accord to Incorporate Market Risk.

bond index IBEX as well as INR-USD

Basel Committee on Banking Supervision, exchange rate data. The results are given in

1996c Supervisory Framework for the Use of annexure I to 6. Historical simulation and

‘Backtesting’ in conjunction with the weighted historical simulations methods

Internal Models Approach to Market Risk provide better results in terms of back testing

Capital Requirements. in general and they require higher capital

charge while normal method requires less Basel Committee on Banking Supervision,

capital charge. It is upto the banks to decide 2001, The Standardized Approach to Credit

which method to choose depending on the

Risk. Duff, D. and J. Pan (1997), “An Overview of

Value at Risk,” Journal of Derivatives, Basle Committee on Banking Supervision

4, 7-49. (1995), An Internal Model-Based Approach

to Market Risk Capital Requirements, Basle, Embrechts, P. [Ed.] (2000), Extremes and

Bank for International Settlements. Integrated Risk Management, UBS Warburg.

Basle Committee on Banking Supervision Engle, R. and S. Manganelli (1999), “CAVaR:

(1996), Supplement to the Capital Accord Conditional Autoregressive Value at Risk by

to Incorporate Market Risks, Basle, Bank for Regression Quantiles,” Manuscript, UCSD.

International Settlements.Gallant, R. and G. Tauchen (1996), “Which

Boudoukh J., Matthew Richardson, and R. F. Moments to Match?,” Econometric Theory,

Whitelaw (1997), “The Best of both Worlds: 12,657-681.

A Hybrid Approach to Calculating Value at Gallant, R. and G. Tauchen (1998),

Risk”, Stern School of Business, NYU“Reprojecting Partially Observed Systems

Christoffersen, P. (1998), “Evaluating with Application to Interest Rate Diffusions,

Interval Forecasts,” International Economic Journal of the American Statistical

Review, 39, 841-862. Association, 93, 10-24.

Cruz, M (2002), Modeling, measuring and Garman, M. and M. Klass (1980), “On the

hedging operational Risk, John Wiley & Estimation of Security Price Volatilities

Sons, Ltd. ISBN no. 0471515604 from Historical Data,” Journal of Business,

53, 67-78.Danielsson, J. 2000, “The Emperor has no

clothes: limits to risk modelling”, Hull, John and Allan White, 1998, “Value at

Mimeog r aph , London S choo l o f Risk When daily Changes in market

Economics . (Internet site http:// Variables are not normally distributed”

www.riskresearch.org). Journal of Derivatives (Spring), 9-19.

Danielsson, J. and C.G. de Vries, 2000, Hull, John and Allan White, 1998,

“Value-at-Risk and Extreme Returns”, “Incorporating Volatility Updating into the

Mimeog r aph , London S choo l o f Historical Simulation Method for Value at

Economics . ( Internet s i t e ht tp :// Risk” Journal of Risk (Fall), 5-19.

www.riskresearch.org.Hendricks, D. (1996), “Evaluation of Value-

Darbha G, 2001, Value-at-Risk for Income at-Risk Models Using Historical Data,”

portfolios: A comparison of alternative Federal Reserve Bank of New York Economic

models, (www.nseindia.com) Policy Review, April, 39-70.

Hendricks, D., and B. Hirtle, 1997, “Bank Pagan, A., 1998, “The Econometrics of

Capital requiremnts for market risk: The Financial Markets”, Journal of Empirical

Internal models approach.”, Federal Finance, 1, 1-70.

Reserve Bank of New York Economic Policy Reserve Bank of India (RBI), Handbook of

Review, 4, 1-12.Statistics, 2002-03 and various other

Lopez, J.A, 1999, “Regulatory Evaluation of publications and circulations.

Value-at-Risk models”, Journal of Risk, 1, Samanta, G P & Nath G C, 2003, Selecting

201-242.Value-at-Risk Models for Government of

Longin, F., 1996, “The asymptotic India Fixed Income Securities, ICFAI

distribution of extreme stock market Journal of Applied Finance (forthcoming)

returns”, Journal of Business, 63, 383-406. (http://gloriamundi.org/detailpopup.asp?I

D=453056896)Longin, F., 2000, “From Value-at-Risk to

Stress testing: The Extreme Value Tsay, Ruey S. , 2002, Analysis of Financial

Approach”, in Embrechts, P. [Ed.], Extremes Time Series, Wiley Series in Probability and

and Integrated Risk Management, UBS Statistics, John Wiley & Sons, Inc.

Warburg.van den Goorbergh, R.W.J. and P.J.G. Vlaar

Nath, G C and Samanta, G P, 2003, Value at (1999), “Value-at-Risk Analysis of Stock

Risk: Concept and Its Implementation for Returns Historical Simulation, Variance

Indian Banking Sys t em, UTIICM Techniques or Tail Index Estimation?”, DNB

conference paper (http:// gloriamundi.org/ Staff Reports, No. 40, De Nederlandsche

detailpopup.asp?ID=453056842) Bank.

Nath G C and Reddy Y V, 2003, Value at Risk: Wong, Michael Chak Sham, Wai Yan Cheng

Issues and Implementation in Forex Market and Clement Yuk Pang Wong (2003),

in India, ICFAI Journal of Applied Finance, “Market Risk Management of Banks:

Nov 2003, http://gloriamundi.org/ Implications from the Accuracy of Value-at-

detailpopup.asp?ID=453056841 Risk Forecasts”, Journal of Forecasting, Vol.

22, pp. 23-33.Nelson, C.R. and A. F. Siegel, 1987,

“Parsimonious Modelling of Yield Curves”, Amexur 2: VaR estimation using Normal

Journal of Business, Vol. 60, pp. 473-89. Method as of June 30, 2004 (IBEX)

Ann

exur

e 1:

VaR

est

imat

ion

usin

g N

orm

al M

etho

d as

of

June

30, 2

004

(IB

EX

)

Vari

ance

-Cov

aria

nce

(Nor

mal

) M

etho

d

Des

crip

tion

of

Est

imat

es

Full

(9

9%)

Full

(95%

) R

ollin

g500

(9

9%)

Rol

ling

500

(95%

) R

ollin

g 75

0 (9

9%)

Rol

ling

750

(95%

)

Rol

ling

1000

(9

9%)

Rol

ling

1000

(9

5%)

Rol

ling

1500

(9

9%)

Rol

ling

1500

(9

5%)

Rol

ling

2000

(9

9%)

Rol

ling

2000

(9

5%)

DE

aR

0.51

87

0.42

48

0.56

47

0.47

38

0.55

04

0.46

09

0.60

06

0.50

25

0.53

81

0.45

02

0.48

74

0.40

76

60-d

ay A

vera

ge3.

3 1.

7075

1.

4001

2.

0270

1.

5602

2.

0270

1.

5602

1.

9884

1.

6616

1.

7543

1.

4663

1.

5870

1.

3262

Max

1.

7075

1.

4001

2.

0270

1.

5602

2.

0270

1.

5602

1.

9884

1.

6616

1.

7543

1.

4663

1.

5870

1.

3262

Cap

Cha

rge,

H=1

0-da

y (%

) 5.

3996

4.

4275

6.

4100

4.

9337

6.

4100

4.

9337

6.

2877

5.

2546

5.

5476

4.

6370

5.

0186

4.

1937

Cap

Cha

rge,

H=1

5-da

y (%

) 6.

6131

5.

4226

7.

8507

6.

0425

7.

8507

6.

0425

7.

7009

6.

4355

6.

7943

5.

6791

6.

1465

5.

1362

Bac

ktes

ting

-Fai

lure

s

Ove

r 1Y

ear

(4/1

9)

2 3

1 2

1 2

1 3

2 3

3 4

Ove

r 50

0 da

ys (

5/25

) 9

14

6 7

6 9

7 11

9

15

10

16

Ove

r 75

0 da

ys (

8/38

) 16

23

10

14

13

18

14

20

18

26

18

25

Ove

r 10

00 d

ays

(10/

50)

34

49

34

42

37

47

40

54

42

57

41

53

Ove

r 15

00 d

ays

(15/

75)

38

56

44

56

49

68

55

77

47

67

47

64

Ove

r 20

00 d

ays

(20/

100)

40

62

66

87

57

81

58

85

52

75

49

68

Full

(31/

156)

54

86

91

12

4 75

10

8 70

10

2 52

75

49

68

For

IBE

X, d

ata

is c

onti

nuou

s an

d he

nce

365

days

are

tak

en f

or 1

yea

rs

Ann

exur

e 2:

VaR

est

imat

ion

usin

g H

istor

ical

Sim

ulat

ion

as o

n Ju

ne 3

0, 2

004

(IBEX

)

Hist

oric

al S

imul

atio

n

Des

crip

tion

of E

stim

ates

Full

(99%

) Fu

ll (9

5%)

Rol

ling5

00 (9

9%)

Rol

ling5

00

(95%

) R

ollin

g750

(9

9%)

Rol

ling7

50 (9

5%)

Rol

ling1

000

(99%

)

Rol

ling

1000

(9

5%)

Rol

ling1

500

(99%

)

Rol

ling

1500

(9

5%)

Rol

ling2

000

(99%

) R

ollin

g200

0 (9

5%)

DEa

R

0.72

85

0.25

36

0.66

63

0.28

92

0.76

13

0.27

68

0.79

98

0.37

76

0.76

18

0.32

401

0.75

91

0.27

06

60-d

ay A

vera

ge3.

3 2.

4134

0.

8071

2.

6911

1.

0256

2.

5615

0.

9306

2.

5139

1.

2165

2.

5139

0.

9750

3 2.

5049

0.

8588

Max

2.

4134

0.

8071

2.

6911

1.

0256

2.

5615

0.

9306

2.

5139

1.

2165

2.

5139

0.

9750

3 2.

5049

0.

8588

Cap

Cha

rge,

H=1

0-da

y (%

) 7.

6319

2.

5523

8.

5100

3.

2432

8.

1002

2.

9429

7.

9497

3.

8470

7.

9497

3.

0833

3 7.

9212

2.

7157

Cap

Cha

rge,

H=1

5-da

y (%

) 9.

3471

3.

1259

10

.422

6 3.

9720

9.

9207

3.

6043

9.

7364

4.

7116

9.

7364

3.

7762

9 9.

7015

3.

326

Bac

ktes

ting

-Fai

lure

s

Ove

r 1Y

ear

(4/1

9)

1 11

1

8 1

4 1

4 1

8 1

10

Ove

r 50

0 da

ys (5

/25)

6

25

4 15

6

14

6 17

6

21

6 23

Ove

r 75

0 da

ys (8

/38)

8

49

4 24

7

26

7 31

9

45

8 44

Ove

r 10

00 d

ays (

10/5

0)

16

107

12

72

14

76

19

88

21

103

17

101

Ove

r 15

00 d

ays (

15/7

5)

18

137

14

103

19

121

25

139

23

132

19

133

Ove

r 20

00 d

ays (

20/1

00)

20

161

27

159

22

153

27

165

25

162

21

147

Full

(31/

156)

31

24

3 42

24

9 33

22

6 35

21

2 25

16

3 21

14

7

For

IBEX

, dat

a is

cont

inuo

us a

nd h

ence

365

day

s are

take

n fo

r 1

year

s

Ann

exur

e 3:

VaR

est

imat

ion

usin

g W

HS

Met

hod

as o

n Ju

ne 3

0, 2

004

(IBE

X)

Wei

ghte

d H

isto

rica

l Sim

ulat

ion

Met

hod(

Hul

l-Whi

te -

Lam

bda

= 0.

94)

Des

crip

tion

of

Estim

ates

Fu

ll (9

9%)

Full

(95%

) R

ollin

g500

(9

9%)

Rol

ling5

00

(95%

)

Rol

ling

750

(99%

)

Rol

ling7

50

(95%

) R

ollin

g100

0 (9

9%)

Rol

ling1

000

(95%

) R

ollin

g150

0 (9

9%)

Rol

ling1

500

(95%

) R

ollin

g200

0 (9

9%)

Rol

ling2

000

(95%

)

DEa

R

0.66

57

0.34

25

0.57

70

0.40

40

0.57

65

0.33

65

0.57

66

0.34

59

0.62

13

0.34

97

0.62

13

0.34

96

60-d

ay A

vera

ge3.

3 2.

1984

1.

1002

2.

0815

1.

2024

1.

8739

1.

0387

1.

9492

1.

0847

2.

0257

1.

0895

2.

0256

1.

0855

Max

2.

1984

1.

1002

2.

0815

1.

2024

1.

8739

1.

0387

1.

9492

1.

0847

2.

0257

1.

0895

2.

0256

1.

0855

Cap

Cha

rge,

H=1

0-da

y (%

) 6.

9519

3.

4790

6.

5823

3.

8025

5.

9257

3.

2847

6.

1639

3.

4301

6.

4058

3.

4454

6.

4054

3.

4328

Cap

Cha

rge,

H=1

5-da

y (%

) 8.

5143

4.

2609

8.

0616

4.

6571

7.

2575

4.

0229

7.

5492

4.

2010

7.

8455

4.

2197

7.

8450

4.

2043

Bac

ktes

ting

-Fai

lure

s

Ove

r 1Y

ear

(4/1

9)

1 8

2 8

2 9

2 9

1 8

1 8

Ove

r 50

0 da

ys (5

/25)

6

21

8 21

8

22

8 22

6

21

6 21

Ove

r 75

0 da

ys (8

/38)

9

33

13

33

12

34

12

34

10

33

9 33

Ove

r 10

00 d

ays

(10/

50)

17

66

24

71

21

67

22

66

19

68

18

66

Ove

r 15

00 d

ays

(15/

75)

19

77

26

82

23

78

24

79

21

79

20

78

Ove

r 20

00 d

ays

(20/

100)

21

86

28

94

25

90

26

89

23

88

22

84

Full

(31/

156)

28

13

2 39

14

2 35

13

7 34

12

1 23

89

22

84

For

IBEX

, dat

a is

con

tinuo

us a

nd h

ence

365

day

s ar

e ta

ken

for 1

yea

rs

Ann

exur

e-4: V

aR e

stim

atio

n us

ing

Nor

mal

Met

hod

as o

n Ju

ne 3

0, 2

004

(INR-U

SD E

xcha

nge

Rat

e)

Vari

ance

-Cov

aria

nce

(Nor

mal

) Met

hod

Des

crip

tion

of E

stim

ates

Full

(9

9%)

Full

(95%

) R

ollin

g500

(9

9%)

Rol

ling

500

(95%

)

Rol

ling

750

(99%

)

Rol

ling

750

(95%

)

Rol

ling

1000

(9

9%)

Rol

ling

1000

(9

5%)

Rol

ling

1500

(9

9%)

Rol

ling

1500

(9

5%)

Rol

ling

2000

(9

9%)

Rol

ling

2000

(9

5%)

DEa

R

0.76

31

0.62

87

1.06

70

0.89

96

0.42

13

0.35

48

0.39

50

0.33

18

0.36

52

0.30

63

0.48

51

0.40

61

60-d

ay A

vera

ge3.

3 2.

1694

1.

7832

1.

4873

1.

2600

1.

2606

1.

0633

1.

2606

1.

0232

1.

2239

1.

0268

1.

5721

1.

3161

Max

2.

1694

1.

7832

1.

4873

1.

2600

1.

2606

1.

0633

1.

2606

1.

0232

1.

2239

1.

0268

1.

5721

1.

3161

Cap

Cha

rge,

H=1

0-da

y (%

) 6.

8604

5.

6389

4.

7033

3.

9846

3.

9863

3.

3623

3.

9863

3.

2357

3.

8703

3.

2471

4.

9714

4.

1620

Cap

Cha

rge,

H=1

5-da

y (%

) 8.

4022

6.

9062

5.

7603

4.

8801

4.

8823

4.

1180

4.

8823

3.

9629

4.

7401

3.

9769

6.

0887

5.

0974

Bac

ktes

ting

-Fai

lure

s

Ove

r 1Y

ear

(3/1

5)

6 6

14

15

15

19

15

20

12

15

8 10

Ove

r 50

0 da

ys (5

/25)

6

6 18

19

17

23

19

24

12

15

8

10

Ove

r 75

0 da

ys (8

/38)

6

6 27

29

27

33

21

28

12

17

8

10

Ove

r 10

00 d

ays (

10/5

0)

6 7

44

50

29

37

23

31

13

18

8 10

Ove

r 15

00 d

ays (

15/7

5)

7 8

46

53

31

40

25

33

14

20

8 10

Ove

r 20

00 d

ays (

20/1

00)

15

23

56

71

39

55

37

50

14

20

8 10

Full

(28/

140)

29

43

79

10

0 44

61

37

50

14

20

8

10

Anne

xure

- 6:

VaR

esti

mat

ion

usin

g W

HS

Met

hod

as o

n Ju

ne 3

0, 2

004

(INR-U

SD E

xcha

nge

Rate

)

Wei

ghte

d H

istor

ical

Sim

ulat

ion

Met

hod(

Hul

l-Whi

te -

Lam

bda

= 0.

94)

Des

crip

tion

of

Estim

ates

Fu

ll (9

9%)

Full

(95%

) Ro

lling

500

(99%

) Ro

lling

500

(95%

)

Rolli

ng

750

(99%

)

Rolli

ng75

0 (9

5%)

Rolli

ng10

00

(99%

) Ro

lling

1000

(9

5%)

Rolli

ng15

00

(99%

) Ro

lling

1500

(9

5%)

Rolli

ng20

00

(99%

) Ro

lling

2000

(9

5%)

DEa

R 0.

8298

0.

5407

0.

9225

0.

6546

0.

8501

0.

6157

0.

8378

0.

5955

0.

8378

0.

5385

0.

8370

0.

5382

60-d

ay A

vera

ge3.

3 2.

7421

1.

7875

3.

0444

2.

1588

2.

8053

2.

0278

2.

7863

1.

9671

2.

7649

1.

7771

2.

7620

1.

7768

Max

2.

7421

1.

7875

3.

0444

2.

1588

2.

8053

2.

0278

2.

7863

1.

9671

2.

7649

1.

7771

2.

7620

1.

7768

Cap

Cha

rge,

H=1

0-day

(%

) 8.

6713

5.

6527

9.

6272

6.

8269

8.

8711

6.

4125

8.

8112

6.

2205

8.

7433

5.

6196

8.

7344

5.

6188

Cap

Cha

rge,

H=1

5-day

(%

) 10

.620

1 6.

9231

11

.790

9 8.

3612

10

.864

8 7.

8537

10

.791

4 7.

6185

10

.708

3 6.

8825

10

.697

4 6.

8816

Back

test

ing

-Fai

lure

s

Ove

r 1Ye

ar (3

/15)

4

6 2

6 4

6 4

6 4

7 4

6

Ove

r 500

day

s (5/

25)

4 6

2 6

4 6

4 6

4 7

4 6

Ove

r 750

day

s (8/

38)

4 7

2 7

4 7

4 7

4 8

4 7

Ove

r 100

0 da

ys (1

0/50

) 4

9 3

10

4 12

4

10

4 10

4

7

Ove

r 150

0 da

ys (1

5/75

) 5

13

4 14

5

16

5 14

5

13

4 7

Ove

r 200

0 da

ys

(20/

100)

15

34

14

37

15

39

15

38

5

13

4 7

Full

(28/

140)

30

60

29

64

21

46

15

38

5

13

4 7

Anne

xure

- 5:

VaR

esti

mat

ion

usin

g H

istor

ical

Sim

ulat

ion

as o

f Jun

e 20

04 (I

NR-U

SD E

xcha

nge

Rate

)

Hist

oric

al S

imul

atio

n

Des

crip

tion

of

Estim

ates

Fu

ll (9

9%)

Full

(95%

) Ro

lling

500

(99%

) Ro

lling

500

(95%

) Ro

lling

750

(99%

) Ro

lling

750

(95%

) Ro

lling

1000

(9

9%)

Rolli

ng10

00

(95%

) Ro

lling

1500

(9

9%)

Rolli

ng15

00

(95%

) Ro

lling

2000

(9

9%)

Rolli

ng20

00

(95%

)

DEa

R 0.

9587

0.

2950

0.

5054

0.

2418

0.

4857

0.

1922

0.

4598

0.

1898

0.

3981

0.

1708

0.

5038

0.

2069

60-d

ay A

vera

ge3.

3 3.

1622

0.

9681

1.

6179

0.

6433

1.

4260

0.

5824

1.

3267

0.

5778

1.

3203

0.

5624

1.

6407

0.

6639

Max

3.

1622

0.

9681

1.

6179

0.

6433

1.

4260

0.

5824

1.

3267

0.

5778

1.

3203

0.

5624

1.

6407

0.

6639

Cap

Cha

rge,

H=1

0-day

(%

) 9.

9997

3.

0615

5.

1161

2.

0343

4.

5093

1.

8418

4.

1955

1.

8272

4.

1753

1.

7783

5.

1884

2.

0994

Cap

Cha

rge,

H=1

5-day

(%

) 12

.247

1 3.

7495

6.

2659

2.

4915

5.

5227

2.

2558

5.

1384

2.

2378

5.

1136

2.

1780

6.

3544

2.

5712

Back

test

ing

-Fai

lure

s

Ove

r 1Ye

ar (3

/15)

2

17

11

32

11

37

12

37

10

35

8 26

Ove

r 500

day

s (5/

25)

2 18

15

44

15

50

15

49

10

39

8

30

Ove

r 750

day

s (8/

38)

2 24

23

68

24

75

18

65

10

49

8

40

Ove

r 100

0 da

ys (1

0/50

) 2

30

39

102

26

95

20

84

10

61

8 42

Ove

r 150

0 da

ys (1

5/75

) 2

34

41

111

29

102

22

90

10

64

8 42

Ove

r 200

0 da

ys

(20/

100)

7

57

50

145

34

135

28

132

10

64

8 42

Full

(28/

140)

16

81

69

19

1 39

14

2 28

13

2 10

64

8

42