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Portfolio Value at Risk
M P Birla Institute of Management Page 1
A Research Report on
“Portfolio Value at Risk”
A Dissertation Submitted in partial fulfillment Of the requirements for the award of
M.B.A Degree of Bangalore University
Submitted By GUNJAN SHIKHA
Reg. No: 07XQCM6032
Under the Guidance of Prof. Praveen Bhagawan
(Internal Guide)
M.P.BIRLA INSTITUTE OF MANAGEMENT (Associate Bharathiya Vidya Bhavan)
#43, Race Course Road, BENGALURU-560001 2007-2009
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DECLARATION
I hereby declare that the research work embodied in this dissertation
entitle
“Portfolio Value at Risk”, An Analytical Study carried out by me under the
guidance and supervision of Prof. Praveen Bhagawan, M.P.Birla Institute of
Management, Bangalore, (Internal Guide) in partial fulfillment of degree of
Master of Business Administration program is my original work.
I also declare that this dissertation has not been submitted to any
University/Institution for the award of any Degree/Diploma, fellowship or other
similar title or prizes.
Place : Bengaluru Gunjan Shikha
Date : / / 2009 Reg.No.07XQCM6032
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GUIDE’S CERTIFICATE
I hereby declare that the research work embodied in this dissertation
entitled, “Portfolio Value at Risk” has been undertaken and completed by
GUNJAN SHIKHA bearing registration No.07XQCM6032 is a bonafide work
done carried under my guidance during the academic year 2007-09 in a
partial fulfillment of the requirement for the award of MBA degree by
Bangalore University.
I also certify that she has fulfilled all the requirements under the covenant
governing the submission of dissertation to the Bangalore University for the
award of MBA degree.
Place: Bengaluru Prof. Praveen Bhagawan
Date : / / 2009 Faculty , MPBIM
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PRINCIPAL’S CERTIFICATE
I hereby certify that this dissertation is an offshoot of the research work
undertaken and completed by Miss Gunjan Shikha under the guidance of
Prof. Praveen Bhagawan, MPBIM, Bangalore (Internal Guide).
I also declare that this dissertation has not been submitted to any
University/Institution for the award of any Degree/Diploma, fellowship or other
similar title or prizes. Place: Bangalore Dr. Nagesh S Malavalli
Date : / / 2009 Principal, MPBIM
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ACKNOWLEDGEMENT
It is my great pleasure to take this opportunity to thanks all those who
helped me directly or indirectly in the preparation of this research report. I am
happy to express my deep sense of gratitude to my internal guide Prof.
Praveen Bhagawan for his enormous guidance and assistance. He has been
my mentor and guide, his continuous encouragement and valuable suggestions
helped me at every stage of this project.
I would also like to express my thanks to Dr. Nagesh S Malavalli,
Principal, M.P Birla Institute of Management, Bangalore and I am also
thankful to the entire teaching faculty for having given me their valuable
guidance for preparing this research report successfully.
A special thanks to my friends and family for their encouragement and
help in completion of the study successfully.
Finally, I pray to the almighty to bestow upon me success and
progressing in my endeavor.
Gunjan Shikha
Reg. No. 07XQCM6032
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CONTENTS
S. No.
Chapters Page No.
1. Theoretical Background 12
1.1 Background 13-15
1.2 Value at Risk (VaR) 16-17
1.3 Mechanics of VaR Estimation 17
1.4 Steps in Constructing VaR 18
1.5 VaR and Confidence Levels 19-20
1.6 Identifying the Important Market Factors 21
1.7 VaR Methods 22
(i) Analytic Method 23-24
(ii) Historical Simulation Method 25-26
(iii) Monte Carlo Simulation Method 27-30
1.8 Review of Literature 31-37
2. Research Design 38
2.1 Statement of Problem 39
2.2 Research Objectives 39
2.3 Hypothesis 40
2.4 Scope of Study 40
2.5 Research Methodology 40-41
2.6 Limitations 41
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2.7 Chapter Scheme 41-43
3. Industry Profile 44
3.1 Equity 45-48
3.2 Bonds 49-50
3.3 Currencies 51-53
4. Analysis and Interpretation 54
4.1 ADF Tests for Equities, Bonds and Currencies 55-60
4.2 Historical Simulation 60-73
4.3 Analytic Approach 74-88
4.4 Single Asset Case 89-90
4.5 Two Asset Case 91-92
4.6 Monte Carlo Simulation 92-93
4.7 Value at Risk for Portfolio 94-96
5. Findings and Conclusion 97
5.1 Findings 98
5.2 Conclusion 99-100
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5.3 Recommendations 101
Selected Bibliography 102
Annexure 103-128
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List of Tables
S. No.
Chart & Tables Page No.
1. Chart 1: Historical Simulation 61
2. Table 1 : Equity ADF Test 47
3. Table 2 : Bonds ADF Test 49
4. Table 3 : Currency ADF Test 51
5. Table 4 : Data for calculation of VaR through Historical Simulation
54-59
6. Table 5 : Calculation of Daily Stock volatility 63-68
7. Table 6 : Simulated Index 77
8. Table 7 78
9. Table 8 79
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EXECUTIVE SUMMARY
What is the most I can lose on this investment? This is a question that
almost every investor who has invested or is considering investing in a risky
asset asks at some point in time. Financial institutions and corporate Treasuries
or individuals require a method to become aware of their risk and also require
the mechanism that can be scientifically rigorous. Optimal allocation of a given
capital between different available competing assets is a standard problem
which any fund manager faces. Hence given an IBM and a CISCO stock a fund
manager would have to decide how much money to allocate to each. The
decision would depend on the risk appetite of the person whose money is being
managed by the fund. We are considering Value at Risk, popularly known as
VaR, as a measure of risk. Value at Risk tries to provide an answer, at least
within a reasonable bound. In fact, it is misleading to consider Value at Risk, or
VaR as it is widely known, to be an alternative to risk adjusted value and
probabilistic approaches. After all, it borrows liberally from both. However, the
wide use of VaR as a tool for risk assessment, especially in financial service
firms, and the extensive literature that has developed around it, push us to
dedicate this report to its examination.
We begin with a general description of VaR and the view of risk that
underlies its measurement, and examine the history of its development and
applications. We then consider the various estimation issues and questions that
have come up in the context of measuring VaR and how analysts and
researchers have tried to deal with them, is discussed under Review Literature.
Next, we evaluate about the research design, where we elaborate about the
objectives, data and its source and chapter scheme of the research report. Then
in industry profile, a brief is about the equities, bonds and currencies which is
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selected for research report. Next, we calculate VaR using VaR models i.e.
Analytic Method, Historical Simulation and Monte Carlo Simulation and then
constructing portfolio and calculating VaR. In the final section, we focus on
Research findings, conclusion and recommendations on the research report.
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CHAPTER 1
THEORITICAL BACKGROUND
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1.1 Background
The concept and use of VaR is recent. Value-at-Risk was first used by
major financial firms in the late 1980s to measure the risks of their trading
portfolios. Since that time period, the use of Value-at-Risk has exploded. While
the term “Value at Risk” was not widely used prior to the mid 1990s, the origins
of the measure lie further back in time. The mathematics that underlies VaR
were largely developed in the context of portfolio theory by Harry Markowitz
and others, though their efforts were directed towards a different end – devising
optimal portfolios for equity investors. In particular, the focus on market risks
and the effects of the co movements in these risks are central to how VaR is
computed.
Value-at-Risk (VaR) has become one of the most popular risk measures
since it was recommended and adopted by the Bank of International Settlements
and USA regulatory agencies in 1988. The straightforward interpretation of
VaR makes this risk measure an intuitive criterion for asset management
decisions. The VaR concept has also been extended to the portfolio Value-at-
Risk (PVaR) measure used for managing risks and returns under a multiple-
asset portfolio. Although VaR and PVaR are widely used in practice, recent
criticisms have focused on the financial risks firms face if the VaR or PVaR
estimates are based on poor information. One potentially important source of
estimation error is in the assumptions regarding the probability model of asset
returns.
The impetus for the use of VaR measures, though, came from the crises
that beset financial service firms over time and the regulatory responses to these
crises. The first regulatory capital requirements for banks were enacted in the
aftermath of the Great Depression and the bank failures of the era, when the
Securities Exchange Act established the Securities Exchange Commission
(SEC) and required banks to keep their borrowings below 2000% of their equity
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capital. In the decades thereafter, banks devised risk measures and control
devices to ensure that they met these capital requirements. With the increased
risk created by the advent of derivative markets and floating exchange rates in
the early 1970s, capital requirements were refined and expanded in the SEC’s
Uniform Net Capital Rule (UNCR) that was promulgated in 1975, which
categorized the financial assets that banks held into twelve classes, based upon
risk, and required different capital requirements for each, ranging from 0% for
short term treasuries to 30% for equities. Banks were required to report on their
capital calculations in quarterly statements that were titled Financial and
Operating Combined Uniform Single (FOCUS) reports.
The first regulatory measures that evoke Value at Risk, though, were
initiated in 1980, when the SEC tied the capital requirements of financial
service firms to the losses that would be incurred, with 95% confidence over a
thirty-day interval, in different security classes; historical returns were used to
compute these potential losses. Although the measures were described as
haircuts and not as Value or Capital at Risk, it was clear the SEC was requiring
financial service firms to embark on the process of estimating one month 95%
VaRs and hold enough capital to cover the potential losses. At about the same
time, the trading portfolios of investment and commercial banks were becoming
larger and more volatile, creating a need for more sophisticated and timely risk
control measures. Ken Garbade at Banker’s Trust, in internal documents,
presented sophisticated measures of Value at Risk in 1986 for the firm’s fixed
income portfolios, based upon the covariance in yields on bonds of different
maturities. By the early 1990s,
many financial service firms had developed rudimentary measures of Value at
Risk, with wide variations on how it was measured. In the aftermath of
numerous disastrous losses associated with the use of derivatives and leverage
between 1993 and 1995, culminating with the failure of Barings, the British
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investment bank, as a result of unauthorized trading in Nikkei futures and
options by Nick Leeson, a young trader in Singapore, firms were ready for more
comprehensive risk measures. In 1995, J.P.Morgan provided public access to
data on the variances of and co-variances across various security and asset
classes, that it had used internally for almost a decade to manage risk, and
allowed software makers to develop software to measure risk. It titled the
service “Risk Metrics” and used the term Value at Risk to describe the risk
measure that emerged from the data. The measure found a ready audience with
commercial and investment banks, and the regulatory authorities overseeing
them, who warmed to its intuitive appeal. In the last decade, VaR has becomes
the established measure of risk exposure in financial service firms and has even
begun to find acceptance in non financial service firms.
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1.2 VaR
VaR is generally considered as a probability based measure of loss
potential. This definition is very general however and we need something more
specific. More formally, VaR is the loss that would be exceeded with a given
probability over a specified period of time. This definition has three important
elements. First, we see that VaR is a loss that could be exceeded. Hence, it is a
measure of a minimum loss. Second, we see that VaR is associated with a given
probability. Thus, we would state that there is a certain percent chance that a
particular loss would be exceeded with a given probability. Finally, VaR is
defined for a specific period of time. Therefore, the loss that would be exceeded
with a given probability is a loss that would be expected to occur over a
specified time period. Consider the following example of
VAR for an investment portfolio: The VaR for a portfolio is Rs. 15 million for
one day with a probability of 0.05. Consider what this statement says: There is a
5 percent chance that the portfolio will lose at least Rs. 15 million in a single
day. The emphasis here should be on the fact that the loss is a minimum, not a
maximum. Value at risk is a statistic that summarizes the exposures of
an asset or portfolio to market risks. VaR allows managers to quantify and
express risk. In other words, VaR is a measure of the maximum potential
change in the value of a portfolio of financial instruments with a given
probability over a pre-set horizon.
Thus, the value of VaR depends on: -
The Horizon over which the portfolio's change in value is measured.
The degree of confidence chosen for the measurement.
VaR is often considered a useful summary measure of market risk for
several reasons. One feature of VaR is its consistency as a measure of financial
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risk. VaR facilitates direct comparison of risk across different portfolios and
distinct financial products. Also it allows the managers or investors to examine
potential losses over a particular time horizon with which they are concerned.
Another relative advantage of is that it is largely tactical neutral. In other words,
VaR is calculated by examining the market risks of the individual instruments in
a portfolio, not using actual historical performance.
1.3 Mechanics of VaR Estimation
Establishing a VaR measure involves a number of decisions. Two
important ones are the choice of probability and the choice of the time period
over which the VaR will be measured. Once these parameters are chosen, one
can proceed to actually obtain the VaR estimate. The mechanics of VaR
estimation can be described as a 5-step process, which is explained with the
help of an example.
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1.4 Steps in Constructing VaR
Assume, for instance, that we need to measure the VaR of Rs.500 cr
equity portfolio over 10 days at the 99 percent confidence level. The following
steps are required to compute VaR: -
Mark-to-market of the current portfolio (e.g., Rs. 100 cr)
Measure the Variability of the risk factors(s) (e.g., 15 % annum)
Set the time horizon, or the holding period (e.g., adjust to 10 business
days)
assuming a normal distribution)
Report the worst loss by processing all the preceding information (e.g., a
Rs. 7 cr VaR)
This is a very simple method of calculating VaR for a given portfolio but
in reality the calculation of VaR for general, parametric and other complex
distribution is more complicated and different methods are used for calculating
VaR which are explained in detail in the subsequent part of the report.
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1.5 Value-at-Risk and Confidence levels
A more risk averse manager will want to determine VaR with greater
confidence -
Increasing the confidence level will increase VaR.
Decreasing the confidence level will decrease VaR.
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1.6 Identifying the important market factors
In order to compute VaR (or any other quantitative measure of market
risk), we need to identify the basic market rates and prices that affect the value
of the portfolio. These basic market rates and prices are the “market factors”. It
is necessary to identify a limited number of basic market factors simply because
otherwise the complexity of trying to come up with a portfolio level quantitative
measure of market risk explodes. Even if we restrict our attention to simple
instruments such as forward contracts, an almost countless number of different
contracts can exist, because virtually any forward price and delivery date are
possible. The market risk factors are inherent in most other instruments such as
swaps, loans, options, and exotic options of course are ever more complicated.
Thus, expressing the instrument’s values in terms of limited number of basic
market factors is an essential first step in the problem manageable. Typically,
market forces are identified by decomposing the instruments in the portfolio
into simpler instruments more directly related to basic market risk factors, and
then interpreting the actual instruments as portfolios of the simpler instruments.
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1.7 VaR Methods
There are three different methods for calculation of VaR namely: -
i. Analytic Method
ii. Historical Method
iii. Monte Carlo Simulation Method
i. Analytic Method
The analytic method follows the variance/ covariance approach, which
uses historic volatility and correlation data to predict the way markets are likely
to move in future. By assuming that underlying market factors follow normal
distribution, the VaR estimate can be calculated analytically for any confidence
interval.
There are essentially two types of analytic method: -
Delta-Normal Method : This method involves linear approximation of
the price changes. It is mainly suitable when the portfolio does not
contain non-linear products and when the movements in the risk factors
are small. This method can accommodate a large number of assets and is
simple to implement.
Delta-Gamma Method : This method improves upon the linear
approximation in the Delta-Normal Method by taking into account the
second order term also. However, inclusion of this term skews the
distribution of changes in portfolio values. Hence the simplicity of the
Delta-Normal approach is lost.
Risk metrics methodology is based on the analytic method. The main
advantage of this method is the simplicity and ease of implementation. This
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method is easy to communicate because of standardization. Delta Gamma
method performs well provided the Greeks are stable. Thus, it is not a good
measure of risk for At the money option, Near maturity money options, barrier
options where the price is close to the barrier etc.
Assessment
The strength of the Variance-Covariance approach is that the Value at
Risk is simple to compute, once you have made an assumption about the
distribution of returns and inputted the means, variances and co-variances of
returns. In the estimation process, though, lie the three key weaknesses of the
approach:
Wrong distributional assumption - If conditional returns are not
normally distributed, the computed VaR will understate the true VaR. In
other words, if there are far more outliers in the actual return distribution
than would be expected given the normality assumption, the actual Value
at Risk will be much higher than the computed Value at Risk.
Input error - Even if the standardized return distribution assumption
holds up, the VaR can still be wrong if the variances and co-variances
that are used to estimate it are incorrect. To the extent that these numbers
are estimated using historical data, there is a standard error associated
with each of the estimates. In other words, the variance - covariance
matrix that is input to the VaR measure is a collection of estimates, some
of which have very large error terms.
Non-stationary variables - A related problem occurs when the variances
and co-variances across assets change over time. This non-stationarity in
values is not uncommon because the fundamentals driving these numbers
do change over time.
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ii. Historical Simulation Method
The historical method identifies a portfolio's exposure to specific market
factors and calculates (say daily) observed changes in these market factors over
the time horizon (say 100 days) to be used in the VaR calculation. The portfolio
is then revalued as if each change occurred from today's levels, thus creating
100 possible changes to the portfolio's value. From these figures, a VaR number
corresponding to a given confidence level is determined. The method is
relatively simple to implement if historical data is easily available. By relying
on actual prices, the method allows non-linearity and non-normal distributions.
It does not rely on specific assumption about valuation models or the underlying
stochastic structure of the market. It accounts for "fat tails" and since it does not
rely on valuation models, it is not prone to model risk. However, the historical
simulation method uses only one path (i.e. the actual past). It also assumes that
the past represents the immediate future fairly. This method may miss situations
with temporarily elevated volatility. Further, the method puts the same weight
age on all observations in the window, including old data points. Thus, the
measure of risk change significantly after an old observation is dropped from
the window. Historical simulation becomes very cumbersome for large
portfolios with complicated structures.
Assessment
While historical simulations are popular and relatively easy to run, they
do come with baggage. In particular, the underlying assumptions of the model
generate give rise to its weaknesses:
(a) Past is not prologue – While all three approaches to estimating VaR
use historical data, historical simulations are much more reliant on them than
the other two approaches for the simple reason that the Value at Risk is
computed entirely from historical price changes. There is little room to overlay
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distributional assumptions (as we do with the Variance-covariance approach) or
to bring in subjective information (as we can with Monte Carlo simulations).
(b) Trends in the data - A related argument can be made about the way
in which we compute Value at Risk, using historical data, where all data points
are weighted equally. In other words, the price changes from trading days in
1992 affect the VaR in exactly the same proportion as price changes from
trading days in 1998. To the extent that there is a trend of increasing volatility
even within the historical time period, we will understate the Value at Risk.
(c) New assets or market risks - While this could be a critique of any of
the three approaches for estimating VaR, the historical simulation approach has
the most difficulty dealing with new risks and assets for an obvious reason:
there is no historic data available to compute the Value at Risk. Assessing the
Value at Risk to a firm from developments in online commerce in the late 1990s
would have been difficult to do, since the online business was in its nascent
stage.
Modifications
As with the other approaches to computing VaR, there have been
modifications suggested to the approach, largely directed at taking into account
some of the criticisms mentioned in the last section.
(a) Weighting the recent past more - A reasonable argument can be
made that returns in the recent past are better predictors of the immediate future
than are returns from the distant past. Boudoukh, Richardson and Whitelaw
present a variant on historical simulations, where recent data is weighted more,
using a decay factor as their time weighting mechanism. In simple terms, each
return, rather than being weighted equally, is assigned a probability weight
based on its recency. In other words, if the decay factor is 0.90, the most recent
observation has the probability weight p, the observation prior to it will be
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weighted 0.9p, the one before that is weighted 0.81p and so on. In fact, the
conventional historical simulation approach is a special case of this approach,
where the decay factor is set to 1.
(b) Combining historical simulation with time series models - Cabado
and Moya suggested that better estimates of VaR could be obtained by fitting at
time series model through the historical data and using the parameters of that
model to forecast the Value at Risk.
(c) Volatility Updating - Hull and White suggest a different way of
updating historical data for shifts in volatility. For assets where the recent
volatility is higher than historical volatility, they recommend that the historical
data be adjusted to reflect the change.
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iii. Monte – Carlo simulation
The Monte-Carlo simulation methodology has a number of similarities to
historical simulation. The main difference is that rather than carrying out the
simulation using the observed changes in the market factors over the last N
periods to generate N hypothetical portfolio profits and losses, one chooses a
statistical distribution that is believed to adequately capture or approximate the
possible changes in the market forces. Then, a pseudo-random number
generator is used to generate thousands or perhaps tens of thousands of
hypothetical changes in the market factors. These are then used to construct
thousands of hypothetical portfolio profits and losses on the current portfolio,
and the distribution of profits and losses. Finally, the value-at-risk is determined
from this distribution.
General Description
The first two steps in a Monte Carlo simulation mirror the first two steps
in the Variance-covariance method where we identify the markets risks that
affect the asset or assets in a portfolio and convert individual assets into
positions in standardized instruments. It is in the third step that the differences
emerge. Rather than compute the variances and co-variances across the market
risk factors, we take the simulation route, where we specify probability
distributions for each of the market risk factors and specify how these market
risk factors move together. Thus, in the example of the six-month Dollar/Euro
forward contract that we used earlier, the probability distributions for the 6-
month zero coupon $ bond, the 6-month zero coupon euro bond and the
dollar/euro spot rate will have to be specified, as will the correlation across
these instruments. While the estimation of parameters is easier if you assume
normal distributions for all variables, the power of Monte-Carlo simulations
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comes from the freedom you have to pick alternate distributions for the
variables. In addition, you can bring in subjective judgments to modify these
distributions. Once the distributions are specified, the simulation process starts.
In each run, the market risk variables take on different outcomes and the value
of the portfolio reflects the outcomes. After a repeated series of runs, numbering
usually in the thousands, you will have a distribution of portfolio values that can
be used to assess Value at Risk. For instance, assume that you run a series of
10,000 simulations and derive corresponding values for the portfolio. These
values can be ranked from highest to lowest, and the 95% percentile Value at
Risk will correspond to the 500th lowest value and the 99th percentile to the
100th lowest value.
Assessment
Much of what was said about the strengths and weaknesses of the
simulation approach in the last chapter apply to its use in computing Value at
Risk. Quickly reviewing the criticism, a simulation is only as good as the
probability distribution for the inputs that are fed into it. While Monte Carlo
simulations are often touted as more sophisticated than historical simulations,
many users directly draw on historical data to make their distributional
assumptions. In addition, as the number of market risk factors increases and
their co-movements become more complex, Monte Carlo simulations become
more difficult to run for two reasons. First, you now have to estimate the
probability distributions for hundreds of market risk variables rather than just
the handful that we talked about in the context of analyzing a single project or
asset. Second, the number of simulations that you need to run to obtain
reasonable estimate of Value at Risk will have to increase substantially (to the
tens of thousands from the thousands). The strengths of Monte Carlo
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simulations can be seen when compared to the other two approaches for
computing Value at Risk. Unlike the variance-covariance approach, we do not
have to make unrealistic assumptions about normality in returns. In contrast to
the historical simulation approach, we begin with historical data but are free to
bring in both subjective judgments and other information to improve forecasted
probability distributions. Finally, Monte Carlo simulations can be used to assess
the Value at Risk for any type of portfolio and are flexible enough to cover
options and option-like securities.
Modifications
As with the other approaches, the modifications to the Monte Carlo
simulation are directed at its biggest weakness, which is its computational
bulk. To provide a simple illustration, a yield curve model with 15 key rates and
four possible values for each will require 1,073,741,824 simulations (415) to be
complete. The modified versions narrow the focus, using different techniques,
and reduce the required number of simulations:-
(a) Scenario Simulation - One way to reduce the computation burden of
running Monte-Carlo simulations is to do the analysis over a number of discrete
scenarios. Frye suggests an approach that can be used to develop these scenarios
by applying a small set of pre-specified shocks to the system. Jamshidan and
Zhu (1997) suggest what they called scenario simulations where they use
principal component analysis as a first step to narrow the number of factors.
Rather than allow each risk variable to take on all of the potential values, they
look at likely combinations of these variables to arrive at scenarios. The values
are computed across these scenarios to arrive at the simulation results.
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(b) Monte Carlo Simulations with Variance-Covariance method
modification – The strength of the Variance-covariance method is its speed. If
you are willing to make the required distributional assumption about normality
in returns and have the variance-covariance matrix in hand, you can compute
the Value at Risk for any portfolio in minutes. The strength of the Monte Carlo
simulation approach is the flexibility it offers users to make different
distributional assumptions and deal with various types of risk, but it can be
painfully slow to run. Glasserman, Heidelberger and Shahabuddin use
approximations from the variance covariance approach to guide the sampling
process in Monte Carlo simulations and report a substantial savings in time and
resources, without any appreciable loss of precision. The trade off in each of
these modifications is simple. You give some of the power and precision of the
Monte Carlo approach but gain in terms of estimation requirements and
computational time.
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1.8 REVIEW LITERATURE
Literature was reviewed with an aim to gain an insight into two major
facets of our problem.
a) VaR Calculation
b) Finding the optimal allocation of VaR calculation methods
VaR is straightforward to estimate and interpret as a measure of risk
exposure, and these advantages often appeal to asset managers (Culp, Mensink,
and Neves 1998). However, most of the current research on Value-at-Risk
(VaR) estimation focuses on the one-dimension (univariate) case. One of the
first attempts to compute PVaR from a model of the joint returns distribution
was reported by Frauendorfer, Moix, and Schmid (1995), but applications of
this method are limited because the PVaR model cannot be stated in closed
form and can only be approximated with complex computational algorithms.
Alternatively, Wang and Wu (2001) use linear combinations of returns
models based on extreme value theory to approximate the tail areas of heavy-
tailed distributions, but this approach may be undesirable because it only
focuses on the lower-tail. The alternative approaches that are currently popular
include the variance-covariance (VC) method, Monte Carlo simulation, delta-
Normal simulation, and historical simulation (HS) (Dowd 1998).
In one of the studies of VaR on the Indian stock market, Varma (1999)
assumes a Generalized Error Distribution (GED) and uses GARCH(GED),
EWMA(GED) and EWMA(RM) models to estimate VaR. The study computes
the nominal coverage, i.e., the ratio of number of exceedences to the total
number of observations, and compares it with the true coverage. The study
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preferred the use of GARCH(GED) model over the other two models on the
basis of the results.
In another study of Nifty and S&P 500, Sarma et al. (2003) used four
models (GARCH, EWMA, Risk Metrics-RM, and Historical Simulation-HS)
and their different variations, on the basis of the number of data points used,
under the assumption of normally distributed errors, to find out that GARCH
and RM fare well, with the latter having a slight edge. They used Back Testing
methods for performance assessment of various models by testing for
conditional and unconditional coverage and independence that were perfected
by Christoffersen (1998) as well as loss functions developed by Lopez (1998).
In a study of the indices of the five of the developed countries, Angelidis
et al. (2004) used three variations of GARCH model (naïve, EGARCH,
TGARCH) and various orders of AR processes on normal GED and t-
distributions. They also tested for conditional and unconditional coverage using
Christoffersen's method, but could not point out any model as the `best' model.
They found student's-t distribution to be capturing the risk better than other
distributions.
Bao et al. (2004) checked the performance of VaR models in terms of
empirical coverage taking parametric and nonparametric models. They used
normal, historical simulation, Monte Carlo simulation non parametrically
estimated distribution and the extreme value distribution along with RM as
benchmark. They analyzed the model performance before, during, and after the
Asian financial crisis. In the pre-crisis period, RM was found to be quite a good
model, with normal not being far behind. Historical Simulation(HS), Non-
Parametric (NP) methods and Monte Carlo (MC) simulation were also seen to
be satisfactory. During the Aisan financial crisis, all the models understated the
VaR numbers, however, the EVT-based one did the best job. The post-crisis
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period results were found to be similar to the pre-crisis period results. From the
study, it is clear that the conventional models do a good job during the normal
periods.
Nath and Samantha (2003) studied the VaR for the Indian banking
system. They used one day return on the Government of India securities as the
variable. The models used were normal, historical simulation, risk metrics and
Hill's estimator. They found that VaR models under variance-covariance/normal
approach, particularly risk metrics, underestimated VaR numbers. The GARCH
(normal) model performed slightly better than the Risk Metrics model. While
HS provided quite reasonable estimates, Hill's estimator overestimated the VaR
numbers as the number of failures was less than theoretical expectation.
Value at Risk Models in the Indian Stock Market
by
Jayanth R. Varma
IIM A
This paper provides empirical tests of different risk management models
in the Value at Risk (VaR) framework in the Indian stock market. It is found
that the GARCH-GED (Generalised Auto-Regressive Conditional
Heteroscedasticity with Generalised Error Distribution residuals) performs
exceedingly well at all common risk levels (ranging from 0.25% to 10%). The
EWMA (Exponentially Weighted Moving Average) model used in J. P.
Morgan’s RiskMetrics® methodology does well at the 10% and 5% risk levels
but breaks down at the 1% and lower risk levels. The paper then suggests a way
of salvaging the EWMA model by using a larger number of standard deviations
to set the VaR limit. For example, the paper suggests using 3 standard
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deviations for a 1% VaR while the normal distribution indicates 2.58 standard
deviations and the GED indicates 2.85 standard deviations. With this
modification the EWMA model is shown to work quite well. Given its greater
simplicity and ease of interpretation, it may be more convenient in practice to
use this model than the more accurate GARCH-GED specification. The paper
also provides evidence suggesting that it may be possible to improve the
performance of the VaR models by taking into account the price movements in
foreign stock markets.
Decomposing Portfolio Value-at-Risk : A General Analysis
Winfried G. Hallerbach
Erasmus University Rotterdam and Tinbergen Institute Graduate School
of Economics
An intensive and still growing body of research focuses on estimating a
portfolio’s Value at Risk. Aside from the total portfolio’s VaR, there is a
growing need for information about (i) the marginal contribution of the
individual portfolio components to the diversified portfolio VaR, (ii) the
proportion of the diversified portfolio VaR that can be attributed to each of the
individual components consituting the portfolio, and (iii) theincremental effect
on VaR of adding a new instrument to the existing portfolio. For many
portfolios, however, the assumption of normally distributed returns is too
stringent. There exist to the best of our knowledge no procedures for estimating
marginal VaR, component VaR and incremental VaR in either a non-normal
analytical setting or a Monte Carlo / historical simulation context. This paper
tries to fill this gap by investigating these VaR concepts in a general
distribution-free setting. We derive a general expression for the marginal
contribution of an instrument to the diversified portfolio VaR whether this
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instrument is already included in the portfolio or not. We show how in a most
general way, the total portfolio VaR can be decomposed in partial VaRs that can
be attributed to the individual instruments comprised in the portfolio. These
component VaRs have the appealing property that they aggregate linearly into
the diversified portfolio VaR. We not only show how the standard results under
normality can be generalized to non-normal analytical VaR approaches but also
present an explicit procedure for estimating marginal VaRs in a simulation
framework. Given the marginal VaR estimate, component VaR and incremental
VaR readily follow. The proposed estimation approach pairs intuitive appeal
with computational efficiency. We evaluate various alternative estimation
methods in an application example and conclude that the proposed approach
displays an astounding accuracy and a promising outperformance.
Value-at-Risk for Fixed Income portfolios : A comparison of alternative
models
Gangadhar Darbha
National Stock Exchange, Mumbai, India
December 2001
The paper presents a case for a new method for computing the VaR for a
set of fixed income securities based on extreme value theory that models the tail
probabilities directly without making any assumption about the distribution of
entire return process. It compares the estimates of VaR for a portfolio of fixed
income securities across three methods: Variance-Covariance method,
Historical Simulation method and Extreme Value method and that extreme
value method provides the accurate VaR estimator in terms of correct failure
ratio and the size of VaR.
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Portfolio Value at Risk Based On Independent Components : Analysis
Ying Chen, Wolfgang H¨ardle1 and Vladimir Spokoiny
Risk management technology applied to high dimensional portfolios
needs simple and fast methods for calculation of Value-at-Risk (VaR). The
multivariate normal framework provides a simple off-the-shelf methodology but
lacks the heavy tailed distributional properties that are observed in data. A
principle component based method (tied closely to the elliptical structure of the
distribution) is therefore expected to be unsatisfactory. Here we propose and
analyze a technology that is based on Independent Component Analysis (ICA).
We study the proposed ICVaR methodology in an extensive simulation study
and apply it to a high dimensional portfolio situation. Our analysis yields very
accurate VaRs.
Evaluating Portfolio Value-at-Risk using Semi-Parametric GARCH
Models
Jeroen V.K. Rombouts and Marno Verbeek
December 2004
In this paper we examine the usefulness of multivariate semi-parametric
GARCH models for portfolio selection under a Value-at-Risk (VaR) constraint.
First, we specify and estimate several alternative multivariate GARCH models
for daily re- turns on the S&P 500 and Nasdaq indexes. Examining the within
sample VaRs of a set of given portfolios shows that the semi-parametric model
performs uniformly well, while parametric models in several cases have
unacceptable failure rates. Interestingly, distributional assumptions appear to
have a much larger impact on the performance of the VaR estimates than the
particular parametric specification chosen for the GARCH equations. Finally,
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we examine the economic value of the multivariate GARCH models by
determining optimal portfolios based on maximizing expected returns subject to
a VaR constraint, over a period of 500 consecutive days. Again, the superiority
and robustness of the semi-parametric model is confirmed.
Value-at-Risk Based Portfolio Optimization
Working Paper by
Amy v. Puelz
Edwin L. Cox School of Business, Southern Methodist University
The Value at Risk (VaR) metric, a widely reported and accepted measure
of financial risk across industry segments and market participants, is discrete by
nature measuring the probability of worst case portfolio performance. In this
paper I present four model frameworks that apply VaR to ex ante portfolio
decisions. The mean-variance model, Young's (1998) minimax model and Hiller
and Eckstein's (1993) stochastic programming model are extended to
incorporate VaR. A fourth model, that is new, implements stochastic
programming with a return aggregation technique. Performance tests are
conducted on the four models using empirical and simulated data. The new
model most closely matches the discrete nature of VaR exhibiting statistically
superior performance across the series of tests. Robustness tests of the four
model forms provides support for the argument that VaR-based investment
strategies lead to higher risk decision than those where the severity of worst
case performance is also considered.Conclusion
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CHAPTER 2
RESEARCH DESIGN
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2.1 Statement of Problem
In volatile financial markets, both market participants and market
regulators need models for measuring, managing and containing risks. Market
participants need risk management models to manage the risks involved in their
open positions. Market regulators on the other hand must ensure the financial
integrity of the stock exchanges and the clearing houses by appropriate
margining and risk containment systems.
How inaccurate VaR or PVaR estimates may lead to redundant amount of
risk capital maintained, which will reduce capital management efficiency as
well as increase the financial risk. Therefore an attempt is made in this study to
find out the right tool to measure Portfolio Value at Risk (PVaR).
2.2 Research Objectives
In this paper, my attempt is to propose an integrated method to compute
PVaR, and also,
to use VaR as a measure of risk of Portfolios
to know the application of various Value at-risk (VaR) Models
convenient to measure the risk of portfolios
to compare the results of various model
to give conclusion regarding the best method to be adopted to determine
Value at Risk
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2.3 Hypothesis
In order to see the stationarity of data collected, Augmented Dickey-Fuller Unit
Root Test has been incorporated where, Hypothesis is
H0 : Null Hypothesis is accepted as data is non-stationary, and
H1 : Alternate Hypothesis is rejected as data is stationary.
2.4 Scope of the Study
The scope of this study is
To become aware of Value at-risk (VaR) as a measure of risk of
portfolios
It gives simple information to layman investor to guide them in selection
of portfolio for their investment
2.5 Research Methodology
Actual collection of data : Data is the portfolio, consists of 10 elements. Four
Equities :
Reliance Industries Limited, DLF Limited, Bharti Airtel and Infosys
Technologies Limied.
Three AAA Rated Bonds :
Power Finance Corporation Limited, Indian Railway Finance Corporation and
Housing Development Finance Corporation Limited.
Three Currencies :
Dollar, Euro and Pound.
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Data Source : For equities closing price is taken for period ranging from July
2007 to March 2009, for bonds weighted average price is taken for period
ranging from December 2008 to March 2009 from nse-india.com and for
currencies exchange rate is taken from oanda.com for period ranging from July
2007 to March 2009.
Data analysis : The data generated would be subjected to rigorous statistical
treatment and inferences would be drawn accordingly. Appropriate statistical
tools would be applied. Excel sheets and appropriate graphs are used as
instruments for preparing the study.
2.6 Limitations
1. The time and resources were the major constrain in conducting the
research.
2. The bond instrument have been chosen on a random basis as AAA rated
trading bonds are very less. Also the ratings have changed over the period
of years.
2. The period of study is restricted only for one year.
2.7 Chapter Scheme
1. Introduction
1.1 Background
1.2 Value at Risk
1.3 Mechanics of VaR estimation
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1.4 Steps in Constructing VaR
1.5 VaR and Confidence Levels
1.6 Identifying the important Market Factors
1.7 VaR Methods
1.8 Review of Literature
2. Research Design
Statement of Problem
Research objective
Hypothesis
Scope of study
Research Methodology
Limitations
Chapter Scheme
3. Industry Profile
3.1 Equity
3.2 Bonds
3.3 Currency
4. Data Analysis and Interpretation
4.1 ADF Tests for Equities, Bonds and Currencies
4.2 Historical Simulation
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4.3 Analytic Approach
4.4 Single Asset Case
4.5 Two Asset Case
5. Findings, Conclusion and Suggestions
5.1 Comparing Approaches
5.2 Conclusion
5.3 Recommendations
Bibliography
Annexure
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CHAPTER 3
INDUSTRY PROFILE
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This report was not done in any industry, this research is all about risk
involved in portfolio. Here, I would like to comment on the various assets,
which I have included in the portfolio, and the brief description about the reason
for taking these assets in my portfolio. My portfolio consists of :
four equities,
three AAA rated bonds, and
three currencies.
3.1 EQUITIES
Four equities are from Reliance Industries Ltd., DLF Limited (Real
Estate), Bharti Airtel (Telecom) and Infosys Technologies Ltd.(IT Sector).
Reliance Industries Ltd. (Refineries)
The Reliance Group, founded by Dhirubhai H. Ambani (1932-2002), is
India's largest private sector enterprise, with businesses in the energy and
materials value chain. Group's annual revenues are in excess of US$ 34 billion.
The flagship company, Reliance Industries Limited, is a Fortune Global 500
company and is the largest private sector company in India. Reliance enjoys
global leadership in its businesses, being the largest polyester yarn and fiber
producer in the world and among the top five to ten producers in the world in
major petrochemical products.
Reliance Industries (RIL) is the country's largest private sector company.
The company has a 26% share of the total refining capacity in India and along
with its subsidiary, IPCL, controls over 70% of the country's domestic polymer
capacity. RIL is also a major player in the polyester fiber and yarn with a
combined capacity of 2 million tones. The company has also ventured into the
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upstream sector, whereby it has participating interests in existing oil and gas
fields. RIL has a large exploration acreage with 34 domestic exploration blocks
in addition to 1 exploration blocks each in Yemen and Oman. RIL also has
exploration and production rights to 5 coal bed methane (CBM) blocks. The
company also has a presence in the downstream segment and has commissioned
1,218 outlets out of permitted 5,849 outlets (FY06).
DLF Limited (Construction)
The DLF Group was founded in 1946. We developed some of the first
residential colonies in Delhi such as Krishna Nagar in East Delhi, which was
completed in 1949.
DLF is India's largest real estate company in terms of revenues, earnings,
market capitalization and developable area. In line with its current expansion
plans, DLF has over 751 million sq. ft. of development across its businesses,
including developed, on-going and planned projects. This land bank is spread
over 32 cities, mostly in metros and key urban areas across India. Already a
major player in locations across the country, DLF, with over six decades of
experience, is capitalizing on emerging market opportunities to deliver high-end
facilities and projects to its wide base of customers by constantly upgrading its
internal skills and resource capabilities.
All the intensified growth underlines DLF's commitment to quality, trust
and customer sensitivity and, delivering on its promise with agility and financial
prudence. This, in turn, has earned DLF the coveted 'Superbrand' ranking for
three years consecutively, including the current year.
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Bharti Airtel (Telecom)
Telecom giant Bharti Airtel is the flagship company of Bharti
Enterprises. The Bharti Group, has a diverse business portfolio and has created
global brands in the telecommunication sector.
Bharti Airtel is one of the topmost companies in the telecom sector in
India and is under the Bharti Enterprises Group. Airtel Bharti has been ranked
as the best performance company in the whole world by the Business Week
magazine in 2007. Airtel comes to you from Bharti Airtel Limited, India’s
largest integrated and the first private telecom services provider with a
footprint in all the 23 telecom circles. Bharti Airtel since its inception has been
at the forefront of technology and has steered the course of the telecom sector
in the country with its world class products and services. The businesses at
Bharti Airtel have been structured into three individual strategic business units
(SBU’s) - Mobile Services, Airtel Telemedia Services & Enterprise Services.
The mobile business provides mobile & fixed wireless services using GSM
technology across 23 telecom circles of India and is the largest mobile service
provider in the country, based on the number of customers, while the Airtel
Telemedia Services business offers broadband & telephone services in 94
cities. The Enterprise services focuses on delivering telecommunications
services as an integrated offering including mobile, broadband & telephone,
national and international long distance and data connectivity services to
corporate, small and medium scale enterprises.
The company has around 50 million customers in 2007 and its market
share of mobile subscribers in India is at 23.4%. The company Bharti Airtel
Limited's total revenue amounted to Rs.12,242 crore in 2006- 2007 and the
net profit stood at Rs.3,126 crore. Bharti's network spans over 364,000 non-
census towns and villages in India. During the period FY05 to FY08, the
company grew its sales and profits at compounded annual rates of 49% and
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74% respectively. Bharti Airtel has become a leading company in the telecom
sector in India due to the fact that it has provided the best quality of services to
its customers. And this has been possible for the company has a wide telecom
network that is of the latest technology.
Infosys Technologies Ltd. (IT Software)
Infosys Technologies Ltd, headquartered at Bangalore, India, is a leading
consulting & IT Service Solution Provider. Started in 1991, the company is
adept in technology driven and innovative business transformation initiatives.
The company works with global corporations and new generation technology
companies to deliver end- to-end solutions.
It offers services like software development, maintenance, consulting,
testing and packaging implementation. Infosys offers all these services through
its highly integrated and globally recognized delivery model. The company
achieved the US$ 3 billion revenue mark in FY07.
With a workforce of around 58,000 employees worldwide it has offices in
USA, Canada, Australia, China, UAE and European countries besides India.
Infosys is regarded as a pioneer in strategic offshore outsourcing of
software services. Its expertise is offered in Application Development and
Maintenance, Corporate Performance Management, Enterprise Quality Services,
Infrastructure Services, Packaged Application Services, Product Engineering
and Systems Integration.
Infosys has a global footprint with over 50 offices and development
centers in India, China, Australia, the Czech Republic, Poland, the UK, Canada
and Japan. Infosys has over 103,000 employees.
Infosys takes pride in building strategic long-term client relationships.
Over 97% of our revenues come from existing customers.
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3.2 AAA RATED BONDS
Three AAA rated Bonds are from Power Finance Corporation Limited,
Indian Railway Finance Corporation and Housing Development
Finance Corporation Limited.
Power Finance Corporation Limited
Power Finance Corporation Limited is an undertaking of the Government
of India. The Power Finance Corporation Limited is a Financial Institution (FI)
established in the year 1986. The main purpose of the Power Finance
Corporation Limited is to provide financial aid to the Power sector for the
integral development of power based infrastructure. In the year 1990 the Power
Finance Corporation Limited was notified as a Public Financial Institution
under the Companies Act of 1956. The organization is registered by the Reserve
Bank of India as a non banking financial company. The Power Finance
Corporation Limited also provides non-fund based consultancy services to
various clients. The ISIN CODE for bond of this government under taking is
INE134E08BH9 and its rating is AAA.
Indian Railway Finance Corporation
Indian Railway Finance Corporation is a dedicated financing arm of the
Ministry of Railways. Its sole objective is to raise money from the market to
part finance the plan outlay of Indian Railways. The money so made available
is used for acquisition of rolling stock assets and for meeting
other developmental needs of the Indian Railways. The ISIN CODE for bond
of this government under taking is INE053F09FM7 and its rating is AAA.
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Housing Development Finance Corporation Limited
Housing Development Finance Corporation
Limited or HDFC (BSE: 500010), founded 1977 by Ravi
Maurya and Hasmukhbhai Parekh, is an Indian NBFC, focusing on
home mortgages. HDFC's distribution network spans 243 outlets that include 49
offices of HDFC's distribution company, HDFC Sales Private Limited. In
addition, HDFC covers over 90 locations through its outreach programmes.
HDFC's marketing efforts continue to be concentrated on developing a stronger
distribution network. Home loans are also Sharcket through HDFC
Sales, HDFC Bank Limited and other third party Direct Selling Agents (DSA).
The Housing Development Finance Corporation Limited (HDFC) was amongst
the first to receive an 'in principle' approval from the Reserve Bank of India
(RBI) to set up a bank in the private sector, as part of the RBI's liberalization of
the Indian Banking Industry in 1994. The ISIN CODE for bond of this
government under taking is INE001A07DT1and its rating is AAA.
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3.3 CURRENCIES
The three currencies are Dollar, Euro and Pound.
Dollar (USA)
The United States dollar (sign: $; code: USD) is the unit of currency of
the United States and is defined by the Coinage Act of 1792 to be between 371
and 416 grains (27.0 g) of silver (depending on purity). The U.S. dollar is
normally abbreviated as the dollar sign, $, or as USD or US$ to distinguish it
from other dollar-denominated currencies and from others that use the $
symbol. It is divided into 100 cents (200 half-cents prior to 1857).
Taken over by the Congress of the Confederation of the United States on
July 6, 1785, the U.S. dollar is the currency most used in international
transactions. Although U.S. dollar is a fiat currency, several countries use it as
their official currency, and in many others it is the de facto currency.
The financial market turmoil that begun in August has put serious
pressure on the US dollar: by end-November the dollar had fallen by some 6%
since August against a trade-weighted currency basket tracked by the US
Federal Reserve. Dollar weakness is not a new issue: the currency has lost a
quarter of its value against a broader range of currencies over the past five
years. However, there are fears that, in the current environment, the dollar's
decline could turn into a rout.
Euro (European Union)
The Euro (€) is the official currency of 16 of the 27 member states of
the European Union (EU). The states, known collectively as the Euro zone,
are: Austria, Belgium, Cyprus, Finland, France, Germany, Greece, Ireland, Italy
, Luxembourg, Malta,the Netherlands,Portugal, Slovakia, Slovenia,and Spain.
The currency is also used in a further five European
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countries, with and without formal agreements and is consequently used daily
by some 327 million Europeans. Over 175 million people worldwide use
currencies which are pegged to the euro, including more than 150 million
people in Africa.
The euro is the second largest reserve currency and the second most
traded currency in the world after the U.S. dollar. As of November 2008, with
more than €751 billion in circulation, the euro is the currency with the highest
combined value of cash in circulation in the world, having surpassed the U.S.
dollar. Based on IMF estimates of 2008 GDP and purchasing power parity
among the various currencies, the Euro zone is the second largest economy in
the world.
The name euro was officially adopted on 16 December 1995. The euro
was introduced to world financial markets as an accounting currency on 1
January 1999, replacing the former European Currency Unit (ECU) at a ratio of
1:1. Euro coins and banknotes entered circulation on 1 January 2002.
With the dollar seemingly in terminal decline, there is little stopping the
euro from becoming the world's premier reserve currency. the fact that sterling
held the position of reserve currency until the Second World War, but lost it due
to imperial overreach. This, he warns, could happen to the dollar for a variety of
reasons, perhaps including multiple US interventions abroad. This weakness of
the US financial sector will play into the hands of the Europeans, whose
economies are better suited to overcoming the current credit crisis, the author
believes. However, despite gloomy predictions for the dollar, the euro is not in a
position to overtake it at the moment, he argues, pointing out that the euro holds
one-third of global reserves compared to the dollar's two-thirds.
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Pound (British)
The pound, a unit of currency, originated in England, as the value of
a pound mass of silver. For a long time, £1 worth of silver coins were a troy
pound in mass.
Today, the term may refer to a number of current (primarily British and
related) currencies, and a variety of now-obsolete currencies. Pound
sterling (GBP, represented by the pound sign: "£"), the currency of the United
Kingdom.
The global economic calendar fills out next week; but no other G10
currency can compete with the fundamental authority of the data that populates
the British pound’s docket. Over the past month, its seen that the sterling built
considerable strength against currencies that are considered far better positioned
than itself. This has developed along with the general recovery in risk
sentiment; because there has been no notable improvements in the UK’s
economic checkup recently. This leaves the pound in a precarious position. As
fundamentals continue to deteriorate from under the unit, its advance becomes
more and more dependent on a fragile and altogether volatile driver.
Alternatively, if the need for safety states the appetite for risk, all of the
Kingdom’s economic, interest rate and financial troubles will come rushing
back to the forefront. And, considering the level of event risk in the week ahead,
there is a lot at stake for fundamental traders.
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CHAPTER 4
DATA ANALYSIS & INTERPRETATION
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4.1 ADF Tests for Equities, Bonds and Currencies
Equity
The closing price of equity for this test is taken from April, 2008 to March, 2009. The data taken are raw rates. The unit root result is as under:
Table No 1: Equity ADF Test
Reliance Equity DLF Equity
Constraints ( log 0)
ADF values
Akaike Info Criterion
Constraints ( log 0)
ADF values Akaike Info Criterion
Intercept (1st
Difference)
-16.73502 1%(-3.4575521)5%(-2.8733935)10%(-2.573153)
Intercept (2nd
Difference)
-9.9722299 1%(-3.458384) 5%(-2.87375) 10%(-2.57334)
Trend & Intercept
(2nd Difference)
-9.0563009 1%( -3.998132) 5%( -3.429354) 10%( -3.13815)
Trend & Intercept
(1st Difference)
-2741.194 1%( -3.996618) 5%( -3.428622) 10%( -3.13772)
None (1st
difference)
-16.770080 1%( -2.574699) 5%( -1.942163) 10%( -1.61585)
None (2nd
difference)
-9.994969 1%( -2.574991) 5%( -1.942204) 10%( -1.61583)
Bharti Airtel Equity Infosys Equity
Constraints ( log 0)
ADF values
Akaike Info Criterion
Constraints ( log 0)
ADF values
Akaike Info Criterion
Intercept (1st
Difference)
-2530.684 1%(-3.45709) 5%(-2.87319) 10%(-2.5730)
Intercept (Level)
-2.820407 1%(-1.457552) 5%(-2.803393) 10%(-2.57315)
Trend & Intercept
(1st Difference)
-2509.315 1%(-3.99613) 5%(-3.42890) 10%(-3.1375)
Trend & Intercept(1st Difference)
-16.308672 1%( -3.99678) 5%( -3.42870) 10%( -3.1377)
None (2nd
difference)
-9.635644 1%(-2.57494) 5%(-1.94219) 10%(-1.6158)
None (1st
difference)
-11.45324 1%(-2.574739) 5%(-1.942169) 10%(-1.6158)
(** indicates acceptance of hypothesis) (* indicates rejection of hypothesis) Source:Excel(Add-Ins)
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Hypothesis:
H0 = ADF > critical values -- accept null hypothesis i.e., unit root exists.
H1 = ADF < critical values – reject null hypothesis i.e. unit root does not exist.
Interpretation
The above table tells that equities has a unit root problem i.e unit root
does not exists, so they are stationary in their 1st difference and 2nd difference at
various constraints i.e ADF is smaller than critical values at none, intercept and
trend and intercept.. The 1st
difference and 2nd difference level is nothing but the
log natural returns of the raw values. Log natural returns are to make series
mean and variance constant. Thus equity is stationary and null hypothesis is
rejected at 1%, 5% and 10% level of significance.
Portfolio Value at Risk
M P Birla Institute of Management Page 57
Bonds
The weighted average price of bonds for this test is taken from April, 2008 to March, 2009. The data taken are raw rates. The unit root result is as under:
Table No 2: Bonds ADF Test
Power Finance Corporation Ltd Indian Railway Finance Corporation
Constraints ( log 0)
ADF values
Akaike Info Criterion
Constraints ( log 0)
ADF values Akaike Info Criterion
Intercept(2nd Difference)
-6.3050974 1%( -3.520355) 5%( -2.90065) 10%( -2.58770)
Intercept (2nd Difference)
-7.725679 1%( -3.592494) 5%( -2.931407) 10%( -2.60396)
Trend & Intercept (Level)
-5.357081 1%( -4.080085) 5%( -3.468479) 10%( -3.16109)
Trend & Intercept (2nd Difference)
-1075.6289 1%( -4.170542) 5%( -3.510728) 10%( -3.18551)
None(Level) 6.3976235 1%( -2.594659) 5%( -1.944965) 10%( -1.61411)
None(2nd difference)
-7.7657838 1%( -2.619939) 5%( -1.948680) 10%( -1.61205)
Housing Development Finance Corporation
Constraints ( log 0)
ADF values Akaike Info Criterion
Intercept(2nd Difference) -7.725679 1%(-3.592494) 5%(-2.931407) 10%(-2.603966)
Trend & Intercept (2nd Difference)
-7.6579844 1%(-4.1864559) 5%(-3.5180744) 10%(-3.1897344)
None(1st difference) -1301.54327 1%(-2.616296) 5%(-1.948134) 10%(-1.612338)
(** indicates acceptance of hypothesis) (* indicates rejection of hypothesis) Source:Excel(Add-Ins)
Portfolio Value at Risk
M P Birla Institute of Management Page 58
Hypothesis:
H0 = ADF > critical values -- accept null hypothesis i.e., unit root exists.
H1 = ADF < critical values – reject null hypothesis i.e. unit root does not exist.
Interpretation
The above table tells that bonds has a unit root problem i.e unit root does
not exists, so they are stationary in their level, 1st difference and 2nd difference at
various constraints i.e ADF is smaller than critical values at none, intercept and
trend and intercept.. The 1st
difference and 2nd difference level is nothing but the
log natural returns of the raw values. Log natural returns are to make series
mean and variance constant. Thus equity is stationary and null hypothesis is
rejected at 1%, 5% and 10% level of significance.
Portfolio Value at Risk
M P Birla Institute of Management Page 59
Currencies
The Exchange rate of currencies for this test is taken from April, 2008 to March, 2009. The data taken are raw rates. The unit root result is as under:
Table No 3: Currencies ADF Test
Dollar Euro
Constraints ( log 0)
ADF values
Akaike Info Criterion
Constraints ( log 0)
ADF values Akaike Info Criterion
Intercept(1st Difference)
-11.12287 1%( -3.448307) 5%( -2.869336) 10%( -2.57097)
Intercept (1st
Difference)
-10.652492 1%( -3.448307) 5%( -2.869336) 10%( -2.57097)
Trend & Intercept
(1st Difference)
-11.175576 1%( -3.983691) 5%( -3.422363) 10%( -3.13402)
Trend & Intercept
(1st Difference)
-10.639176 1%( -3.983691) 5%( -3.422363) 10%( -3.13402)
None(2nd Difference)
-11.252048 1%( -2.571587) 5%( -1.941737) 10%( -1.61614)
None(2nd difference)
-11.099348 1%( -2.571587) 5%( -1.941737) 10%( -1.61614)
Pound
Constraints ( log 0)
ADF values Akaike Info Criterion
Intercept(2nd Difference) -12.546823 1%(-3.4487212) 5%( -2.869518) 10%( -2.571074)
Trend & Intercept (1st Difference)
-11.299146 1%( -3.9836916) 5%( -3.4223637) 10%( -3.1340235)
None(2nd difference) -12.563043 1%( -2.5715877) 5%( -1.9417379) 10%( -1.6161406)
(** indicates acceptance of hypothesis) (* indicates rejection of hypothesis) Source:Excel(Add-Ins)
Portfolio Value at Risk
M P Birla Institute of Management Page 60
Hypothesis:
H0 = ADF > critical values -- accept null hypothesis i.e., unit root exists.
H1 = ADF < critical values – reject null hypothesis i.e. unit root does not exist.
Interpretation
The above table tells that currencies has a unit root problem i.e unit root
does not exists, so they are stationary in their level, 1st difference and 2nd
difference at various constraints i.e ADF is smaller than critical values at none,
intercept and trend and intercept.. The 1st
difference and 2nd difference level is
nothing but the log natural returns of the raw values. Log natural returns are to
make series mean and variance constant. Thus equity is stationary and null
hypothesis is rejected at 1%, 5% and 10% level of significance.
Portfolio Value at Risk
M P Birla Institute of Management Page 61
4.2 Historical Simulation
The historical simulation methodology is illustrated below. Table 1 shows
observations on market variables affecting the portfolio of “Reliance Industries
Limited” from the period April 1, 2008 to March 31, 2009.
Historical Simulation can be described in the following steps:
(1) The first step is to identify the basic factors, i.e. equity prices in this
case.
(2) The next step is to obtain historical values of the market factors for
the last N periods. For our portfolio, this means collection of equity prices of the
stock for the last 243 trading days. Daily changes in these prices will be used to
construct hypothetical values of the market factors used in the calculation of
hypothetical profits and losses in Step 3 because the daily value at risk number
is a measure of the portfolio loss caused by changes over a one day holding
period, 1 April 2008 to March 31 2009.
(3) This is the key step. We subject the current portfolio to the changes in
market rates and prices experienced on each of the most recent 243 trading
days, calculating the daily profits and losses that would occur if comparable
daily changes in the market factors are experienced and the current portfolio is
marked-to-market.
To calculate the 100 daily profits and losses, we first calculate 243 sets of
hypothetical values of the market forces. The hypothetical market factors are
based upon, but not equal to, the historical values of the market factors over the
243 days. Rather, we calculate daily historical percentage changes in the market
factors, and then combine the historical percentage changes with the current
(March 31, 2009) market factors to compute 243 sets of hypothetical market
factors.
Portfolio Value at Risk
M P Birla Institute of Management Page 62
The observations are taken at some particular point in time during the day
(usually the close of trading). We denote the first day for which we have data as
Day 0; the second day as Day 1; and so on. Today is Day 100; tomorrow is Day
101. The values of the market variables tomorrow if their percentage changes
between today and tomorrow are the same as they were between Day (i – 1) and
Day i for 1 ≤ i ≤ 100. The first row shows the values of the market variables
tomorrow assuming their percentage changes between today and tomorrow are
the same as they were between Day 0 and Day 1; second row shows the values
of market variables tomorrow assuming their percentage changes between Day
1 and Day 2 occur; and so on. The 243 rows are the 243 scenario considered.
Reliance Industries Ltd
Table 4 : Data for Calculation of VaR through Historical Simulation
Date Equity price P&L VaR
1-Apr-08 2,345.25 0 0
2-Apr-08 2,343.55 1523.644755 9.450672643
3-Apr-08 2,396.05 1558.907315 -25.81188676
4-Apr-08 2,321.15 1477.086648 56.00878018
7-Apr-08 2,404.90 1579.76489 -46.66946225
8-Apr-08 2,381.75 1510.072482 23.02294578
9-Apr-08 2,418.25 1548.11659 -15.02116169
10-Apr-08 2,468.65 1556.528104 -23.432676
11-Apr-08 2,551.55 1575.952793 -42.85736502
15-Apr-08 2,611.80 1560.754071 -27.65864305
16-Apr-08 2,642.50 1542.672439 -9.577010548
17-Apr-08 2,640.05 1523.336325 9.759103497
Portfolio Value at Risk
M P Birla Institute of Management Page 63
21-Apr-08 2,643.60 1526.800288 6.295140127
23-Apr-08 2,577.60 1507.352523 25.74290532
25-Apr-08 2,624.50 1549.457486 -16.36205753
28-Apr-08 2,591.40 1505.519966 27.57546229
29-Apr-08 2,659.95 1565.084033 -31.98860476
30-Apr-08 2,614.50 1498.696921 34.39850701
2-May-08 2,674.75 1559.887192 -26.79176362
5-May-08 2,669.20 1521.586204 11.50922368
6-May-08 2,650.00 1513.782219 19.31320861
7-May-08 2,688.95 1547.160948 -14.06552011
8-May-08 2,667.25 1512.445169 20.65025888
9-May-08 2,528.40 1445.375537 87.7198914
12-May-08 2,553.85 1540.097606 -7.002178194
13-May-08 2,501.10 1493.256152 39.83927552
14-May-08 2,530.75 1542.825582 -9.730153744
15-May-08 2,622.95 1580.299521 -47.20409289
16-May-08 2,635.70 1532.161717 0.933711231
20-May-08 2,602.95 1505.804155 27.29127256
21-May-08 2,667.70 1562.679104 -29.58367648
22-May-08 2,626.05 1500.944536 32.15089245
23-May-08 2,556.20 1484.193351 48.90207677
26-May-08 2,515.60 1500.53247 32.56295793
27-May-08 2,495.10 1512.324585 20.77084341
28-May-08 2,522.50 1541.494078 -8.398650394
29-May-08 2,462.70 1488.6033 44.4921277
30-May-08 2,403.50 1488.097058 44.99836989
2-Jun-08 2,358.80 1496.392885 36.70254262
Portfolio Value at Risk
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3-Jun-08 2,406.65 1555.68068 -22.58525179
4-Jun-08 2,307.00 1461.616043 71.47938495
5-Jun-08 2,246.80 1484.962419 48.13300927
6-Jun-08 2,238.50 1519.117356 13.97807221
9-Jun-08 2,162.70 1473.118975 59.97645324
10-Jun-08 2,197.75 1549.461004 -16.36557561
11-Jun-08 2,261.40 1568.908952 -35.81352434
12-Jun-08 2,277.30 1535.470582 -2.375154383
13-Jun-08 2,270.40 1520.130154 12.96527387
16-Jun-08 2,282.35 1532.775353 0.320074538
17-Jun-08 2,332.90 1558.520505 -25.42507718
18-Jun-08 2,287.10 1494.815777 38.2796515
19-Jun-08 2,248.15 1498.783049 34.31237851
23-Jun-08 2,025.70 1471.363412 61.73201575
24-Jun-08 2,062.70 1552.600002 -19.50457447
26-Jun-08 2,239.55 1598.667539 -65.57211062
27-Jun-08 2,182.65 1486.010845 47.08458319
30-Jun-08 2,095.15 1463.624476 69.47095202
1-Jul-08 2,044.15 1487.634638 45.46078967
2-Jul-08 2,144.00 1599.229019 -66.1335914
3-Jul-08 2,070.10 1472.194485 60.90094339
4-Jul-08 2,097.90 1545.22633 -12.13090165
7-Jul-08 2,028.20 1474.092164 59.00326441
8-Jul-08 1,979.45 1488.10097 44.99445793
9-Jul-08 2,079.15 1601.547886 -68.45245778
10-Jul-08 2,046.65 1500.916041 32.17938659
11-Jul-08 2,016.10 1501.990313 31.10511456
Portfolio Value at Risk
M P Birla Institute of Management Page 65
14-Jul-08 2,043.45 1545.434446 -12.33901846
15-Jul-08 1,977.40 1475.46583 57.62959815
16-Jul-08 1,943.50 1498.610107 34.48532129
17-Jul-08 2,018.55 1583.629592 -50.53416423
18-Jul-08 2,113.20 1596.245671 -63.1502434
21-Jul-08 2,152.85 1553.358905 -20.26347674
22-Jul-08 2,152.15 1524.254227 8.841201045
23-Jul-08 2,267.30 1606.331192 -73.23576406
24-Jul-08 2,308.05 1552.154209 -19.05878075
25-Jul-08 2,147.10 1418.422792 114.672636
28-Jul-08 2,179.90 1548.04272 -14.94729241
29-Jul-08 2,083.10 1457.042399 76.05302927
30-Jul-08 2,165.50 1585.063667 -51.96823913
31-Jul-08 2,207.50 1554.322616 -21.22718802
1-Aug-08 2,297.60 1586.983284 -53.88785626
4-Aug-08 2,242.40 1488.117775 44.97765293
5-Aug-08 2,276.05 1547.630769 -14.53534149
6-Aug-08 2,298.60 1539.856484 -6.761055821
7-Aug-08 2,272.60 1507.503198 25.5922304
8-Aug-08 2,251.80 1510.794707 22.3007215
11-Aug-08 2,325.25 1574.484829 -41.3894008
12-Aug-08 2,347.25 1539.176191 -6.080762732
13-Aug-08 2,336.85 1517.994265 15.10116344
14-Aug-08 2,276.70 1485.503274 47.59215436
18-Aug-08 2,224.80 1489.991567 43.10386126
20-Aug-08 2,246.35 1543.125862 -10.03043364
21-Aug-08 2,212.65 1501.875526 31.21990215
Portfolio Value at Risk
M P Birla Institute of Management Page 66
22-Aug-08 2,244.80 1546.904752 -13.80932422
26-Aug-08 2,179.35 1488.783115 44.31231308
27-Aug-08 2,148.00 1502.816436 30.27899191
28-Aug-08 2,070.85 1469.985353 63.11007535
29-Aug-08 2,136.20 1572.866673 -39.77124511
1-Sep-08 2,141.65 1528.640033 4.455395466
2-Sep-08 2,212.75 1575.369721 -42.27429278
4-Sep-08 2,152.25 1483.060982 50.03444619
5-Sep-08 2,080.90 1474.202474 58.89295385
8-Sep-08 2,133.20 1563.072084 -29.97665619
9-Sep-08 2,142.55 1531.433111 1.662316946
10-Sep-08 2,082.65 1482.122045 50.97338301
11-Sep-08 1,997.40 1462.336758 70.75866954
12-Sep-08 1,932.65 1475.321962 57.7734657
15-Sep-08 1,886.95 1488.695321 44.40010681
16-Sep-08 1,928.05 1557.960856 -24.86542814
17-Sep-08 1,876.65 1484.101599 48.99382923
18-Sep-08 1,938.25 1574.799077 -41.70364881
19-Sep-08 2,055.10 1616.671598 -83.57617009
22-Sep-08 2,039.10 1512.879045 20.21638318
23-Sep-08 2,006.45 1500.335755 32.75967326
24-Sep-08 2,046.10 1554.880996 -21.78556829
25-Sep-08 2,025.70 1509.547957 23.54747091
26-Sep-08 1,963.20 1477.706077 55.38935109
29-Sep-08 1,932.85 1501.178198 31.91723041
30-Sep-08 1,949.35 1537.766207 -4.670778638
1-Oct-08 1,906.70 1491.389861 41.70556728
Portfolio Value at Risk
M P Birla Institute of Management Page 67
3-Oct-08 1,761.45 1408.596469 124.498959
6-Oct-08 1,641.60 1421.005195 112.0902334
7-Oct-08 1,674.65 1555.447483 -22.35205464
8-Oct-08 1,648.55 1500.986243 32.10918461
10-Oct-08 1,527.60 1412.882897 120.2125309
13-Oct-08 1,571.40 1568.468284 -35.37285558
14-Oct-08 1,621.05 1572.926045 -39.83061725
15-Oct-08 1,520.20 1429.891089 103.2043389
16-Oct-08 1,391.95 1396.116144 136.9792837
17-Oct-08 1,306.05 1430.654648 102.4407798
21-Oct-08 1,394.95 1610.227884 -77.1324564
22-Oct-08 1,316.80 1439.328148 93.76728004
23-Oct-08 1,217.65 1409.942161 123.1532671
24-Oct-08 1,019.50 1276.625159 256.4702689
28-Oct-08 1,153.00 1632.3461 -99.25067228
29-Oct-08 1,201.75 1589.217964 -56.12253601
31-Oct-08 1,375.45 1745.136166 -212.040738
3-Nov-08 1,441.70 1598.191192 -65.09576397
4-Nov-08 1,451.60 1535.220295 -2.124867484
5-Nov-08 1,269.05 1333.000818 200.0946099
6-Nov-08 1,170.55 1406.403304 126.6921243
7-Nov-08 1,220.75 1590.140158 -57.04473047
10-Nov-08 1,303.10 1627.607393 -94.511965
11-Nov-08 1,207.70 1413.122995 119.9724328
12-Nov-08 1,162.20 1467.305167 65.79026115
14-Nov-08 1,146.75 1504.48035 28.61507823
17-Nov-08 1,141.40 1517.636494 15.45893356
Portfolio Value at Risk
M P Birla Institute of Management Page 68
18-Nov-08 1,139.95 1522.813004 10.28242423
19-Nov-08 1,132.45 1514.71831 18.37711799
20-Nov-08 1,056.05 1421.883737 111.2116914
21-Nov-08 1,124.35 1623.363158 -90.26772952
24-Nov-08 1,144.80 1552.48259 -19.38716194
25-Nov-08 1,071.80 1427.521882 105.5735464
26-Nov-08 1,138.90 1620.206918 -87.11149027
28-Nov-08 1,134.45 1518.792376 14.30305158
1-Dec-08 1,109.40 1491.081714 42.01371439
2-Dec-08 1,073.95 1476.027819 57.06760891
3-Dec-08 1,069.10 1517.86417 15.23125835
4-Dec-08 1,159.10 1653.107965 -120.0125366
5-Dec-08 1,117.60 1470.158399 62.93702924
8-Dec-08 1,118.55 1526.046092 7.049335928
10-Dec-08 1,227.20 1672.856108 -139.7606804
11-Dec-08 1,259.00 1564.260308 -31.16488002
12-Dec-08 1,307.10 1583.002959 -49.9075307
15-Dec-08 1,340.55 1563.769882 -30.6744538
16-Dec-08 1,388.50 1579.288632 -46.19320353
17-Dec-08 1,351.40 1484.009471 49.08595735
18-Dec-08 1,361.00 1535.581434 -2.486006068
19-Dec-08 1,351.30 1513.882935 19.21249266
23-Dec-08 1,259.75 1494.14944 38.94598807
24-Dec-08 1,242.00 1503.266124 29.82930377
26-Dec-08 1,210.15 1485.649124 47.4463036
29-Dec-08 1,246.30 1570.297835 -37.20240698
30-Dec-08 1,250.50 1529.88837 3.207058426
Portfolio Value at Risk
M P Birla Institute of Management Page 69
1-Jan-09 1,254.65 1551.837426 -18.74199849
2-Jan-09 1,286.40 1563.335113 -30.23968538
5-Jan-09 1,365.85 1618.920855 -85.82542671
6-Jan-09 1,370.90 1530.387506 2.707922051
7-Jan-09 1,200.75 1335.504823 197.5906045
9-Jan-09 1,153.25 1464.433011 68.66241738
12-Jan-09 1,097.90 1451.569933 81.5254952
13-Jan-09 1,077.55 1496.488171 36.6072574
14-Jan-09 1,179.75 1669.364589 -136.2691607
15-Jan-09 1,142.35 1476.412937 56.68249094
16-Jan-09 1,217.35 1624.856141 -91.76071285
19-Jan-09 1,229.90 1540.469072 -7.373644165
20-Jan-09 1,183.65 1467.412259 65.68316887
21-Jan-09 1,119.85 1442.564345 90.53108254
22-Jan-09 1,136.30 1547.147765 -14.05233733
23-Jan-09 1,156.15 1551.385825 -18.29039661
27-Jan-09 1,225.95 1616.80341 -83.70798202
28-Jan-09 1,274.00 1584.511195 -51.4157674
29-Jan-09 1,270.10 1520.082398 13.01303004
30-Jan-09 1,323.60 1588.976537 -55.88110928
2-Feb-09 1,280.00 1474.524025 58.57140261
3-Feb-09 1,306.20 1555.959727 -22.86429856
4-Feb-09 1,307.50 1526.267513 6.827915368
5-Feb-09 1,288.80 1502.942868 30.15255993
6-Feb-09 1,344.85 1591.061482 -57.96605361
9-Feb-09 1,389.70 1575.599565 -42.50413701
10-Feb-09 1,401.95 1538.190446 -5.09501778
Portfolio Value at Risk
M P Birla Institute of Management Page 70
11-Feb-09 1,381.25 1502.23684 30.85858824
12-Feb-09 1,351.55 1491.964425 41.13100266
13-Feb-09 1,392.40 1570.834893 -37.73946527
16-Feb-09 1,320.20 1445.687267 87.40816141
17-Feb-09 1,267.30 1463.653746 69.44168236
18-Feb-09 1,295.15 1558.257684 -25.16225566
19-Feb-09 1,293.75 1523.101813 9.993615469
24-Feb-09 1,253.25 1524.567526 8.527901672
25-Feb-09 1,266.55 1540.931269 -7.835840701
26-Feb-09 1,290.80 1553.943626 -20.84819839
27-Feb-09 1,266.05 1495.514206 37.58122169
2-Mar-09 1,225.65 1476.094813 57.00061539
3-Mar-09 1,196.85 1488.921827 44.17360081
5-Mar-09 1,149.80 1447.57456 85.52086768
6-Mar-09 1,169.90 1551.404614 -18.30918585
9-Mar-09 1,153.35 1503.180112 29.91531645
12-Mar-09 1,202.00 1589.066198 -55.97077047
13-Mar-09 1,284.25 1629.085015 -95.98958656
16-Mar-09 1,327.60 1576.218104 -43.12267595
17-Mar-09 1,300.20 1493.281071 39.81435689
18-Mar-09 1,331.40 1561.338371 -28.24294302
19-Mar-09 1,345.70 1541.12669 -8.03126195
20-Mar-09 1,339.20 1517.385153 15.71027529
23-Mar-09 1,438.45 1637.751372 -104.6559441
24-Mar-09 1,452.45 1539.589932 -6.494503871
25-Mar-09 1,532.20 1608.469792 -75.37436442
26-Mar-09 1,565.50 1557.888086 -24.79265776
Portfolio Value at Risk
M P Birla Institute of Management Page 71
27-Mar-09 1,548.75 1508.436003 24.65942513
30-Mar-09 1,516.45 1492.950533 40.14489531
31-Mar-09 1,524.75 1533.095428 0
Source : www.nseindia.com
Calculations :
Reliance Industries Ltd(equity)
The 1 day loss (VaR) = 1334.0525
Similarly, loss of 10 day(VaR) = (√10*one day loss) = 4218.6444
Working Notes :
1. P&L = Equity price of nth day*
(Equity price of current day/Equity price of previous day
2. VaR = Value of nth day P&L – Value of current day P&L
Here, for Reliance equity price we have daily VaR, which shows that an
investor has risk of 56.00878 after taking the difference of nth day to current
day i.e. 4 April, 2008. So, for an investor it’s a point where he can calculate
daily risk with respect to nth day P&L to that of current day.
Portfolio Value at Risk
M P Birla Institute of Management Page 72
Chart 1. Showing Historical Simulation
Note:
In the above graph, x – axis denotes the scenario number and the y – axis
denotes the Value-at-Risk.
Likewise, worst daily VaR is calculated for all other equities, bonds and Currencies.
DLF Limited(equity)
The 1 day loss (VaR) = 141.9933
Bharti Airtel(equity)
The 1 day loss (VaR) = 576.5053
Portfolio Value at Risk
M P Birla Institute of Management Page 73
Infosys Technologies Ltd(equity)
The 1 day loss (VaR) = 1235.3038
Power Finance Corporation Limited(bond)
The 1 day loss (VaR) = 94.8263
Indian Railway Finance Corporation(bond)
The 1 day loss (VaR) = 49.9667
Housing Development Finance Corporation Limited(bond)
The 1 day loss (VaR) = 47.9506
Dollar(currency)
The 1 day loss (VaR) = 50.9104
Euro(currency)
The 1 day loss (VaR) = 67.2294
Pound(currency)
The 1 day loss (VaR) = 72.0662
Portfolio Value at Risk
M P Birla Institute of Management Page 74
4.3 Analytic Method (Variance-Covariance Approach)
Reliance Industries Ltd.
Table 5 : Calculation of Daily Stock Volatility
Date Equity price Price relative Daily Return
1-Apr-08 2,345.25 0 0
2-Apr-08 2,343.55 0.99927513 -0.00072513
3-Apr-08 2,396.05 1.02240191 0.02215467
4-Apr-08 2,321.15 0.96874022 -0.0317588
7-Apr-08 2,404.90 1.03608125 0.03544557
8-Apr-08 2,381.75 0.99037382 -0.00967281
9-Apr-08 2,418.25 1.01532487 0.01520863
10-Apr-08 2,468.65 1.02084152 0.0206273
11-Apr-08 2,551.55 1.03358111 0.03302958
15-Apr-08 2,611.80 1.0236131 0.02333862
16-Apr-08 2,642.50 1.01175435 0.0116858
17-Apr-08 2,640.05 0.99907285 -0.00092758
21-Apr-08 2,643.60 1.00134467 0.00134377
22-Apr-08 2,607.35 0.98628764 -0.01380724
23-Apr-08 2,577.60 0.98858995 -0.01147565
24-Apr-08 2,582.65 1.00195919 0.00195727
25-Apr-08 2,624.50 1.01620429 0.0160744
28-Apr-08 2,591.40 0.98738807 -0.01269213
29-Apr-08 2,659.95 1.02645288 0.02610906
Portfolio Value at Risk
M P Birla Institute of Management Page 75
30-Apr-08 2,614.50 0.98291321 -0.01723445
2-May-08 2,674.75 1.02304456 0.02278304
5-May-08 2,669.20 0.99792504 -0.00207712
6-May-08 2,650.00 0.99280683 -0.00721916
7-May-08 2,688.95 1.01469811 0.01459114
8-May-08 2,667.25 0.99192994 -0.0081028
9-May-08 2,528.40 0.94794264 -0.05346129
12-May-08 2,553.85 1.01006565 0.01001533
13-May-08 2,501.10 0.97934491 -0.02087139
14-May-08 2,530.75 1.01185478 0.01178507
15-May-08 2,622.95 1.03643189 0.03578394
16-May-08 2,635.70 1.00486094 0.00484916
20-May-08 2,602.95 0.98757446 -0.01250338
21-May-08 2,667.70 1.02487562 0.02457126
22-May-08 2,626.05 0.9843873 -0.01573586
23-May-08 2,556.20 0.97340112 -0.02695904
26-May-08 2,515.60 0.98411705 -0.01601044
27-May-08 2,495.10 0.99185085 -0.00818254
28-May-08 2,522.50 1.01098152 0.01092166
29-May-08 2,462.70 0.97629336 -0.02399216
30-May-08 2,403.50 0.97596134 -0.0243323
2-Jun-08 2,358.80 0.98140212 -0.01877299
3-Jun-08 2,406.65 1.02028574 0.02008272
4-Jun-08 2,307.00 0.9585939 -0.04228776
5-Jun-08 2,246.80 0.9739055 -0.026441
6-Jun-08 2,238.50 0.99630586 -0.00370098
9-Jun-08 2,162.70 0.96613804 -0.03444856
Portfolio Value at Risk
M P Birla Institute of Management Page 76
10-Jun-08 2,197.75 1.01620659 0.01607667
11-Jun-08 2,261.40 1.02896144 0.02854998
12-Jun-08 2,277.30 1.00703104 0.00700644
13-Jun-08 2,270.40 0.9969701 -0.0030345
16-Jun-08 2,282.35 1.00526339 0.00524959
17-Jun-08 2,332.90 1.02214822 0.02190651
18-Jun-08 2,287.10 0.98036778 -0.01982749
19-Jun-08 2,248.15 0.9829697 -0.01717698
20-Jun-08 2,099.20 0.93374552 -0.06855134
23-Jun-08 2,025.70 0.96498666 -0.035641
24-Jun-08 2,062.70 1.01826529 0.01810048
25-Jun-08 2,136.00 1.03553595 0.03491912
26-Jun-08 2,239.55 1.04847846 0.04734003
27-Jun-08 2,182.65 0.97459311 -0.02573522
30-Jun-08 2,095.15 0.95991112 -0.04091459
1-Jul-08 2,044.15 0.97565807 -0.02464309
2-Jul-08 2,144.00 1.04884671 0.04769119
3-Jul-08 2,070.10 0.96553172 -0.03507633
4-Jul-08 2,097.90 1.0134293 0.01333993
7-Jul-08 2,028.20 0.9667763 -0.03378814
8-Jul-08 1,979.45 0.97596391 -0.02432967
9-Jul-08 2,079.15 1.05036753 0.04914013
10-Jul-08 2,046.65 0.98436861 -0.01575485
11-Jul-08 2,016.10 0.98507317 -0.01503936
14-Jul-08 2,043.45 1.0135658 0.0134746
15-Jul-08 1,977.40 0.96767721 -0.03285671
16-Jul-08 1,943.50 0.98285628 -0.01729238
Portfolio Value at Risk
M P Birla Institute of Management Page 77
17-Jul-08 2,018.55 1.0386159 0.03788896
18-Jul-08 2,113.20 1.04689009 0.04582395
21-Jul-08 2,152.85 1.01876301 0.01858916
22-Jul-08 2,152.15 0.99967485 -0.0003252
23-Jul-08 2,267.30 1.05350463 0.05212235
24-Jul-08 2,308.05 1.01797292 0.01781332
25-Jul-08 2,147.10 0.93026581 -0.07228492
28-Jul-08 2,179.90 1.01527642 0.01516091
29-Jul-08 2,083.10 0.95559429 -0.04542184
30-Jul-08 2,165.50 1.03955643 0.03879411
31-Jul-08 2,207.50 1.01939506 0.01920937
1-Aug-08 2,297.60 1.0408154 0.04000445
4-Aug-08 2,242.40 0.97597493 -0.02431838
5-Aug-08 2,276.05 1.01500624 0.01489476
6-Aug-08 2,298.60 1.00990752 0.00985876
7-Aug-08 2,272.60 0.98868877 -0.01137569
8-Aug-08 2,251.80 0.99084749 -0.00919465
11-Aug-08 2,325.25 1.03261835 0.03209766
12-Aug-08 2,347.25 1.00946135 0.00941687
13-Aug-08 2,336.85 0.99556928 -0.00444056
14-Aug-08 2,276.70 0.97426022 -0.02607684
18-Aug-08 2,224.80 0.97720385 -0.02306
19-Aug-08 2,219.60 0.99766271 -0.00234002
20-Aug-08 2,246.35 1.01205172 0.01197968
21-Aug-08 2,212.65 0.98499789 -0.01511578
22-Aug-08 2,244.80 1.01453009 0.01442554
25-Aug-08 2,232.00 0.99429793 -0.00571839
Portfolio Value at Risk
M P Birla Institute of Management Page 78
26-Aug-08 2,179.35 0.97641129 -0.02387138
27-Aug-08 2,148.00 0.98561498 -0.01448949
28-Aug-08 2,070.85 0.96408287 -0.03657803
29-Aug-08 2,136.20 1.03155709 0.0310694
1-Sep-08 2,141.65 1.00255126 0.00254801
2-Sep-08 2,212.75 1.0331987 0.03265953
4-Sep-08 2,152.25 0.97265846 -0.02772228
5-Sep-08 2,080.90 0.96684865 -0.03371331
8-Sep-08 2,133.20 1.02513336 0.02482271
9-Sep-08 2,142.55 1.00438309 0.00437351
10-Sep-08 2,082.65 0.97204266 -0.02835559
11-Sep-08 1,997.40 0.95906657 -0.04179479
12-Sep-08 1,932.65 0.96758286 -0.03295422
15-Sep-08 1,886.95 0.97635371 -0.02393035
16-Sep-08 1,928.05 1.02178118 0.02154736
17-Sep-08 1,876.65 0.97334094 -0.02702086
18-Sep-08 1,938.25 1.03282445 0.03229723
19-Sep-08 2,055.10 1.06028634 0.058539
22-Sep-08 2,039.10 0.99221449 -0.00781597
23-Sep-08 2,006.45 0.98398803 -0.01614154
24-Sep-08 2,046.10 1.01976127 0.01956855
25-Sep-08 2,025.70 0.99002981 -0.01002022
26-Sep-08 1,963.20 0.96914647 -0.03133952
29-Sep-08 1,932.85 0.98454055 -0.0155802
30-Sep-08 1,949.35 1.00853662 0.00850039
1-Oct-08 1,906.70 0.97812091 -0.02212198
3-Oct-08 1,761.45 0.92382126 -0.07923667
Portfolio Value at Risk
M P Birla Institute of Management Page 79
6-Oct-08 1,641.60 0.93195947 -0.07046596
7-Oct-08 1,674.65 1.0201328 0.01993281
8-Oct-08 1,648.55 0.98441465 -0.01570807
10-Oct-08 1,527.60 0.9266325 -0.07619824
13-Oct-08 1,571.40 1.02867243 0.02826907
14-Oct-08 1,621.05 1.03159603 0.03110715
15-Oct-08 1,520.20 0.93778724 -0.06423218
16-Oct-08 1,391.95 0.9156361 -0.08813626
17-Oct-08 1,306.05 0.93828801 -0.06369833
20-Oct-08 1,320.90 1.01137016 0.01130601
21-Oct-08 1,394.95 1.05606026 0.05454525
22-Oct-08 1,316.80 0.94397649 -0.05765402
23-Oct-08 1,217.65 0.92470383 -0.07828178
24-Oct-08 1,019.50 0.83726851 -0.17761046
27-Oct-08 1,077.00 1.0564002 0.05486709
28-Oct-08 1,153.00 1.07056639 0.06818784
29-Oct-08 1,201.75 1.04228101 0.04141159
31-Oct-08 1,375.45 1.14453921 0.13500212
3-Nov-08 1,441.70 1.04816605 0.04704202
4-Nov-08 1,451.60 1.00686689 0.00684342
5-Nov-08 1,269.05 0.87424222 -0.13439781
6-Nov-08 1,170.55 0.92238288 -0.08079487
7-Nov-08 1,220.75 1.04288582 0.0419917
10-Nov-08 1,303.10 1.06745853 0.06528062
11-Nov-08 1,207.70 0.92678996 -0.07602832
12-Nov-08 1,162.20 0.96232508 -0.03840296
14-Nov-08 1,146.75 0.98670625 -0.01338291
Portfolio Value at Risk
M P Birla Institute of Management Page 80
17-Nov-08 1,141.40 0.99533464 -0.00467628
18-Nov-08 1,139.95 0.99872963 -0.00127118
19-Nov-08 1,132.45 0.99342076 -0.00660097
20-Nov-08 1,056.05 0.93253565 -0.06984789
21-Nov-08 1,124.35 1.06467497 0.06266956
24-Nov-08 1,144.80 1.01818829 0.01802486
25-Nov-08 1,071.80 0.9362334 -0.06589047
26-Nov-08 1,138.90 1.06260496 0.06072341
28-Nov-08 1,134.45 0.99609272 -0.00391493
1-Dec-08 1,109.40 0.97791882 -0.02232862
2-Dec-08 1,073.95 0.96804579 -0.03247589
3-Dec-08 1,069.10 0.99548396 -0.00452627
4-Dec-08 1,159.10 1.08418296 0.08082667
5-Dec-08 1,117.60 0.96419636 -0.03646031
8-Dec-08 1,118.55 1.00085004 0.00084967
10-Dec-08 1,227.20 1.09713468 0.09270195
11-Dec-08 1,259.00 1.02591265 0.0255826
12-Dec-08 1,307.10 1.03820492 0.03749319
15-Dec-08 1,340.55 1.025591 0.02526903
16-Dec-08 1,388.50 1.0357689 0.03514405
17-Dec-08 1,351.40 0.97328052 -0.02708294
18-Dec-08 1,361.00 1.00710374 0.00707863
19-Dec-08 1,351.30 0.99287289 -0.00715263
22-Dec-08 1,285.55 0.95134315 -0.04988045
23-Dec-08 1,259.75 0.97993077 -0.02027335
24-Dec-08 1,242.00 0.9859099 -0.01419031
26-Dec-08 1,210.15 0.97435588 -0.02597866
Portfolio Value at Risk
M P Birla Institute of Management Page 81
29-Dec-08 1,246.30 1.02987233 0.02943484
30-Dec-08 1,250.50 1.00336998 0.00336431
31-Dec-08 1,232.75 0.98580568 -0.01429603
1-Jan-09 1,254.65 1.01776516 0.0176092
2-Jan-09 1,286.40 1.02530586 0.02499097
5-Jan-09 1,365.85 1.0617615 0.05992933
6-Jan-09 1,370.90 1.00369733 0.00369051
7-Jan-09 1,200.75 0.87588446 -0.1325211
9-Jan-09 1,153.25 0.96044139 -0.04036232
12-Jan-09 1,097.90 0.9520052 -0.04918478
13-Jan-09 1,077.55 0.98146461 -0.01870932
14-Jan-09 1,179.75 1.09484479 0.09061261
15-Jan-09 1,142.35 0.96829837 -0.03221501
16-Jan-09 1,217.35 1.06565413 0.06358882
19-Jan-09 1,229.90 1.01030928 0.0102565
20-Jan-09 1,183.65 0.96239532 -0.03832998
21-Jan-09 1,119.85 0.94609893 -0.05540814
22-Jan-09 1,136.30 1.01468947 0.01458262
23-Jan-09 1,156.15 1.01746898 0.01731815
27-Jan-09 1,225.95 1.06037279 0.05862053
28-Jan-09 1,274.00 1.03919409 0.0384455
29-Jan-09 1,270.10 0.99693878 -0.00306592
30-Jan-09 1,323.60 1.04212267 0.04125966
2-Feb-09 1,280.00 0.96705953 -0.03349522
3-Feb-09 1,306.20 1.02046875 0.02026208
4-Feb-09 1,307.50 1.00099525 0.00099476
5-Feb-09 1,288.80 0.9856979 -0.01440536
Portfolio Value at Risk
M P Birla Institute of Management Page 82
6-Feb-09 1,344.85 1.04349007 0.04257093
9-Feb-09 1,389.70 1.03334944 0.03280541
10-Feb-09 1,401.95 1.00881485 0.00877623
11-Feb-09 1,381.25 0.98523485 -0.01487524
12-Feb-09 1,351.55 0.97849774 -0.0217368
13-Feb-09 1,392.40 1.03022456 0.02977679
16-Feb-09 1,320.20 0.94814708 -0.05324564
17-Feb-09 1,267.30 0.95993031 -0.04089459
18-Feb-09 1,295.15 1.02197585 0.02173787
19-Feb-09 1,293.75 0.99891904 -0.00108154
20-Feb-09 1,253.40 0.96881159 -0.03168512
24-Feb-09 1,253.25 0.99988033 -0.00011968
25-Feb-09 1,266.55 1.01061241 0.01055649
26-Feb-09 1,290.80 1.0191465 0.01896551
27-Feb-09 1,266.05 0.98082584 -0.01936036
2-Mar-09 1,225.65 0.96808973 -0.0324305
3-Mar-09 1,196.85 0.97650226 -0.02377821
4-Mar-09 1,211.10 1.01190625 0.01183593
5-Mar-09 1,149.80 0.94938486 -0.05194102
6-Mar-09 1,169.90 1.0174813 0.01733026
9-Mar-09 1,153.35 0.98585349 -0.01424752
12-Mar-09 1,202.00 1.04218147 0.04131608
13-Mar-09 1,284.25 1.06842762 0.06618805
16-Mar-09 1,327.60 1.03375511 0.03319791
17-Mar-09 1,300.20 0.97936125 -0.0208547
18-Mar-09 1,331.40 1.02399631 0.02371292
19-Mar-09 1,345.70 1.01074057 0.0106833
Portfolio Value at Risk
M P Birla Institute of Management Page 83
20-Mar-09 1,339.20 0.9951698 -0.0048419
23-Mar-09 1,438.45 1.07411141 0.07149372
24-Mar-09 1,452.45 1.0097327 0.00968564
25-Mar-09 1,532.20 1.05490723 0.05345283
26-Mar-09 1,565.50 1.02173346 0.02150065
27-Mar-09 1,548.75 0.98930054 -0.01075711
30-Mar-09 1,516.45 0.97914447 -0.02107608
31-Mar-09 1,524.75 1.00547331 0.00545839
Source : www.nseindia.com
Reliance Industries Limited (Equity)
Standard deviation of daily return = 0.038926896
Standard deviation of daily return in % = 3.892689561
Assuming that there is 252 trading days in = 252
Square root of 252 = 15.87450787
Volatility of stock per annum (S) = 0.617945311
Standard error of estimate =S/ Sqrt(2n)
SQRT (2n) =22.04540769
Standard error of estimate = 0.028030569
Likewise, the analytic approach is done for all equity, bonds and currencies.
Here are the results of all the equities, bonds and currencies.
Portfolio Value at Risk
M P Birla Institute of Management Page 84
DLF (Equity)
Standard deviation of daily return = 0.055192424
Standard deviation of daily return in % = 5.5192424
Assuming that there is 252 trading days in = 252
Square root of 252 = 15.87450787
Volatility of stock per annum (S) = 0.876152564
Standard error of estimate =S/ Sqrt(2n)
SQRT (2n) =22.04540769
Standard error of estimate = 0.039743087
Bharti Airtel (Equity)
Standard deviation of daily return = 0.032893102
Standard deviation of daily return in % = 3.2893102
Assuming that there is 252 trading days in = 252
Square root of 252 = 15.87450787
Volatility of stock per annum (S) = 0.522161808
Standard error of estimate =S/ Sqrt(2n)
SQRT (2n) =22.04540769
Standard error of estimate = 0.023685741
Portfolio Value at Risk
M P Birla Institute of Management Page 85
Infosys Technologies Ltd (Equity)
Standard deviation of daily return = 0.029106498
Standard deviation of daily return in % = 2.9106498
Assuming that there is 252 trading days in = 252
Square root of 252 = 15.87450787
Volatility of stock per annum (S) = 0.462051337
Standard error of estimate =S/ Sqrt(2n)
SQRT (2n) =22.04540769
Standard error of estimate = 0.020959074
Power Finance Corporation Limited (Bond)
Standard deviation of daily return = 0.010129182
Standard deviation of daily return in % = 1.0129182
Assuming that there is 252 trading days in = 252
Square root of 252 = 15.87450787
Volatility of stock per annum (S) = 0.160795783
Standard error of estimate =S/ Sqrt(2n)
SQRT (2n) =22.04540769
Standard error of estimate = 0.012405664
Portfolio Value at Risk
M P Birla Institute of Management Page 86
Indian railway Finance Corporation (Bond)
Standard deviation of daily return = 0.011765793
Standard deviation of daily return in % = 1.1765793
Assuming that there is 252 trading days in = 252
Square root of 252 = 15.87450787
Volatility of stock per annum (S) = 0.1867766179
Standard error of estimate =S/ Sqrt(2n)
SQRT (2n) =22.04540769
Standard error of estimate = 0.014410095
Housing Development Finance Corporation Limited (Bond)
Standard deviation of daily return = 0.004975903
Standard deviation of daily return in % = 0.497590312
Assuming that there is 252 trading days in = 252
Square root of 252 = 15.87450787
Volatility of stock per annum (S) = 0.078990013
Standard error of estimate =S/ Sqrt(2n)
SQRT (2n) =22.04540769
Standard error of estimate = 0.006094212
Portfolio Value at Risk
M P Birla Institute of Management Page 87
DOLLAR (Currency)
Standard deviation of daily return = 0.003128319
Standard deviation of daily return in % = 0. 3128319
Assuming that there is 252 trading days in = 252
Square root of 252 = 15.87450787
Volatility of stock per annum (S) = 0.049660518
Standard error of estimate =S/ Sqrt(2n)
SQRT (2n) =22.04540769
Standard error of estimate = 0.003831392
EURO (Currency)
Standard deviation of daily return = 0.004520986
Standard deviation of daily return in % = 0. 4520986
Assuming that there is 252 trading days in = 252
Square root of 252 = 15.87450787
Volatility of stock per annum (S) = 0.07176843
Standard error of estimate =S/ Sqrt(2n)
SQRT (2n) =22.04540769
Standard error of estimate = 0.005537055
Portfolio Value at Risk
M P Birla Institute of Management Page 88
POUND (Currency)
Standard deviation of daily return = 0.00378929
Standard deviation of daily return in % = 0.378929011
Assuming that there is 252 trading days in = 252
Square root of 252 = 15.87450787
Volatility of stock per annum (S) = 0.060153116
Standard error of estimate =S/ Sqrt(2n)
SQRT (2n) =22.04540769
Standard error of estimate = 0.004640194
Working Notes:
(1) Price relative =
Equity price of current period/Equity price of previous period
(2) Daily return = Natural logarithm of Price relative
(3) Standard deviation of daily return = √(Σ Daily return)
(4) Standard deviation in % = Standard deviation * 100
(5) Volatility of stock per annum = Standard deviation of daily return
* √252
(6) Standard error of estimate = Volatility of stock per annum / √2n
where, n = number of trading days = 252
Portfolio Value at Risk
M P Birla Institute of Management Page 89
4.4 Single Asset Case
We now consider how VaR is calculated using the analytic approach in a
very simple situation where the portfolio consists of a position in a single stock.
The portfolio we consider is one consisting of Rs.1,25,000 in Reliance
Industries Limited. We suppose that N = 10 and X = 99, so that we are
interested in the loss level over 10 days that we are 99% confident will not be
exceeded. Initially, we consider a one-day time horizon.
We assume that the volatility of 4% per day (corresponding to about 68%
per year). Because the size of the position is Rs.1,25,000, the standard deviation
of daily changes in the value of the position is 4% of
Rs.1,25,000, or Rs.5,000.
It is customary in the model-building approach to assume that the
expected change in a market variable over the time period considered is zero.
This is exactly not true, but it is a reasonable assumption. The expected change
in the price of a market variable over a short time period is generally small
when compared with the standard deviation of the change. Suppose, for
example, that Reliance Industries Limited has an expected return of 20% per
annum. Over a one-day period, the expected return is 0.20/252, or about 0.08%,
whereas the standard deviation of the return is 4%. Over a 10- day period, the
expected return is 0.08 * 10, or about 0.8%, whereas the standard deviation of
the return is 4√10, or about 12.65%. So far, we have established that the change
in the value of the portfolio of Cairn India Limited shares over a one-day period
has a standard deviation of Rs.5,000 and (at least approximately) a mean of
zero. We assume that the change is normally distributed. From the tables of
normal distribution, we find that N (- 2.33) = 0.01. This means that there is a
1% probability that a normally distributed variable will decrease in value by
more than 2.33 standard deviations. Equivalently, it means that we are 99%
Portfolio Value at Risk
M P Birla Institute of Management Page 90
certain that a normally distributed variable will not decrease in value by more
than 2.33 standard deviations. Therefore the one-day 99% VaR for our portfolio
consisting of a Rs.1,25,000 position of Reliance Industries Limited is,
= 2.33 * 5,000 = Rs. 11,650
As discussed earlier, the N-day VaR is calculated as √N times the one-day
VaR. The 10-day 99% VaR for Cairn India Limited is therefore,
= 11,650 * √10 = Rs. 36,841
Consider next a portfolio consisting of a Rs. 1,25,000 position in Infosys
Technologies Ltd, and suppose the daily volatility of Infosys is 3%
(approximately 48% per year). A similar calculation to that for Reliance
Industries Limited shows that the standard deviation of the change in the value
of the portfolio in one day is,
= 1,25,000 * 0.03 = Rs. 3,750
Assuming the change is normally distributed, the one-day 99% VaR is,
= 3,750 * 2.33 = Rs. 8,738
and the 10-day VaR is,
= 8,738 * √10 = Rs. 27,632
Portfolio Value at Risk
M P Birla Institute of Management Page 91
4.5 Two asset case:
Now consider a portfolio consisting of both Rs 1,25,000 of Reliance
Industries Limited and Rs.1,25,000 of Infosys technologies Ltd. We suppose
that the returns on the two shares have a bi-variate normal distribution with a
correlation of 0.3. A standard result in statistics tells us that, if two variables X
and Y have standard deviations σx and σy, with the coefficient of correlation
between them being equal to ρ, then the standard deviation of X + Y is given
by,
σx+y = √[(σx2 + σy
2 + 2 ρ σxσy]
To apply this result, we set X equal to the change in the value of the position in
Reliance Industries Limited over a one-day period and Y equal to the change in
the value of the position in Infosys Technologies Ltd over a one-day period.
The standard deviation of the change in the value of Reliance position in one
day,
σx = 1,25,000 * 4% = 5,000
The standard deviation of the change in the value of Infosys position in one day,
σy = 1,25,000 * 3% = 3,750
The standard deviation of the change in the portfolio value per day is therefore:
= √[(5,000 * 5,000) + (3,750 * 3,750) + (2 * 0.3 * 5,000 * 3,750)]
= Rs 6,339
The one-day 99% VaR is therefore = Rs 6,339 * 2.33 = Rs 14,770
The 10-day 99% VaR is = √10 * Rs 14,770 = Rs 46,706
Portfolio Value at Risk
M P Birla Institute of Management Page 92
The Benefits of Diversification by forming Portfolio
In the example we have just considered –
The 10-day 99% VaR for the portfolio of Reliance is Rs. 36,841.
The 10-day 99% VaR for the portfolio of Infosys is Rs. 27,632.
The 10-day 99% VaR for the portfolio of both Reliance and Infosys is
Rs. 46,706.
The amount i.e. = (36,841+27,632) – 46,706 = Rs. 17,767
represents the benefits of diversification. If Reliance and Infosys were perfectly
correlated, the VaR for the portfolio of both Reliance and Infosys would equal
the VaR for the Reliance the VaR for the Infosys. Less than perfect correlation
leads to some of the risk being “diversified away”.
4.6 Monte Carlo Simulation
The Monte Carlo approach entails simulations of possible portfolio
outcomes derived from random market moves taken from historical data. The
distribution of these simulated portfolio returns reveals the VAR. Like the
historical simulation approach, Monte Carlo analysis expresses the returns as a
histogram. This approach can provide a much greater range of outcomes than
historical simulation, and it is much more flexible than the other approaches.
Any distribution may be simulated, as long as the necessary parameters of the
assumed distribution can be estimated. However, all this makes Monte Carlo
analysis the most expensive and time-consuming method, and tends to make it
unsuitable for large, complex portfolios.
Portfolio Value at Risk
M P Birla Institute of Management Page 93
Here, I have taken Reliance equity prices for Monte Carlo simulation
Approach, where I got the result that equity above Rs 1700 is in better position,
its always in winning position.
Table 6. Simulated Index
Index probability
Cumulative
probability
RN
Internal
Random
Number
Below
1700
Above
1700
1100 0.1 0.1 0-999 8417
1300 0.16 0.26 1000-2599 1471 1+1+
1500 0.2 0.46 2600-4599 5660
2000 0.05 0.51 4600-5099 6066
2300 0.15 0.66 5100-6599 1545 1+1+1+
2600 0.34 1 6600-10000 5693 1+
Total 2 4
Above is the table where its shows the relevance of this interpretation.
The anticipated index is 1700.
Portfolio Value at Risk
M P Birla Institute of Management Page 94
4.7 Value at Risk for Portfolio
Now, considering the portfolio taken for measuring Value at risk.
Considering Value at Risk first seeing the Equity, Bonds and Currencies
separately.
Table 7
Equity Deviation(3%) 1 day 99% 10 day 99%
Reliance 15000 34950 110521.6042
DLF
Bharti Airtel
Infosys
Investment Rs 500000 for Equities
Bonds Deviation(4%) 1 day 99% 10 day 99%
Power Finance Corporation Ltd 20000 46600 147362.139
Indian Railway Finance Corporation
HDFC
Investment Rs 500000 for Bonds
Currency Deviation(5%) 1 day 99% 10 day 99%
Dollar 25000 58250 184202.6737
Euro
Pound
Investment Rs 500000 for Currency
The Table 7, shows the VaR for 1 day and 10 days separately for equities,
bonds and currencies because here we have invested Rs 5,00,000 individually in
equity, bonds and currencies, assuming deviation to be 3%, 4% and 5%
Portfolio Value at Risk
M P Birla Institute of Management Page 95
respectively for equities, bonds and currencies. The daily loss for equities,
bonds and currencies can be seen in the table separately.
Table 8
Constructing Portfolio Coefficient Correlation
of Equity & Bonds
Coefficient Correlation of Equity &
Currency
Coefficient Correlation of
Bonds & Currency
Equity, Bonds and Currencies 0.3 0.35 0.32
Investment Rs 1500000
Coefficient Correlation
(500000*0.03)15000 15000^2=225000000 2*correl• (equity &bond)*15000 180000000
(500000*0.04)20000 20000^2=400000000 2*correl• (equity
¤cy)*20000 262500000
(500000*0.05)25000 25000^2=625000000
2*correl• (bond
¤cy)*20000 320000000
The standard deviation of the change in the portfolio value per day is
44860.89611
one day 99% 10 day 99%
104525.8879 330539.88
•Correl: coefficient correlation
The Table 8, shows the VaR for 1 day and 10 days for portfolio, which consists
of equities, bonds and currencies where the investment is Rs 15,00,000,
assuming coeffient correlation to be 0.3, 0.35 and 0.32 respectively for equities,
bonds and currencies. The daily loss for equities, bonds and currencies can be
seen in the table separately.
Portfolio Value at Risk
M P Birla Institute of Management Page 96
The Benefits of Portfolio :
In this portfolio we have –
The 10-day 99% VaR for Equity is Rs. 1,10,521.
The 10-day 99% VaR for Bonds is Rs. 1,47,362.
The 10-day 99% VaR for Currency is Rs.1,84,202 .
The 10-day 99% VaR for the portfolio of Equity, Bonds and Currency is
Rs. 3,30,539.
The amount i.e. = (1,10,521 + 1,47,362 + 1,84,202 ) – 3,30,539
= Rs. 1,11,546
represents the benefits of portfolio construction. If Equity, Bonds and currency
were perfectly correlated, the VaR for the portfolio of equity, Bonds and
currency would equal the VaR for the portfolio of equity, Bonds and currency.
Less than perfect correlation leads to some of the risk being “diversified away”.
Portfolio Value at Risk
M P Birla Institute of Management Page 97
CHAPTER 5
FINDINGS, CONCLUSION
& Recommendations
Portfolio Value at Risk
M P Birla Institute of Management Page 98
5.1 Findings
Each of the three approaches for estimating Value at Risk has advantages
and comes with baggage.
In short, the question of which VaR approach is best answered by looking
at the task at hand. If you are assessing the Value at Risk for portfolios,
that do not include options, over very short time periods (a day or a
week), the variance-covariance approach does a reasonably good job.
If the Value at Risk is being computed for a risk source that is stable and
where there is substantial historical data (commodity prices, for instance),
historical simulations provide good estimates.
In the most general case of computing VaR for nonlinear portfolios
(which include options) over longer time periods, where the historical
data is volatile and non-stationary and the normality assumption is
questionable, Monte - Carlo simulations do best.
If Equity, Bonds and currency were taken together then probability of
loss is reduced than all are taken individually. Thus, portfolio leads to
some of the risk being “diversified away”.
Portfolio Value at Risk
M P Birla Institute of Management Page 99
5.2 CONCLUSION
Value at Risk has developed as a risk assessment tool at banks and other
financial service firms in the last decade. Its usage in these firms has been
driven by the failure of the risk tracking systems used until the early 1990s to
detect dangerous risk taking on the part of traders and it offered a key benefit: a
measure of capital at risk under extreme conditions in trading portfolios that
could be updated on a regular basis. While the notion of Value at Risk is
simple- the maximum amount that you can lose on an investment over a
particular period with a specified probability – there are three ways in which
Value at Risk can be measured. In the first approach, we run a portfolio through
historical data – a historical simulation – and estimate the probability that the
losses exceed specified values. In the second, we assume that the returns
generated by exposure to multiple market risks are normally distributed. We use
a variance-covariance matrix of all standardized instruments representing
various market risks to estimate the standard deviation in portfolio returns and
compute the Value at Risk from this standard deviation. In the third approach,
we assume return distributions for each of the individual market risks and run
Monte Carlo simulations to arrive at the Value at Risk. Each measure comes
with its own pluses and minuses: the Variance-covariance approach is simple to
implement but the normality assumption can be tough to sustain, historical
simulations assume that the past time periods used are representative of the
future and Monte Carlo simulations are time and computation intensive. All
three yield Value at Risk measures that are estimates and subject to judgment.
We understand why Value at Risk is a popular risk assessment tool in
financial service firms, where assets are primarily marketable securities; there is
limited capital at play and a regulatory overlay that emphasizes short term
exposure to extreme risks. We are hard pressed to see why Value at
Portfolio Value at Risk
M P Birla Institute of Management Page 100
Risk is of particular use to non-financial service firms, unless they are highly
levered and risk default if cash flows or value fall below a prespecified level.
Even in those cases, it would seem to us to be more prudent to use all of the
information in the probability distribution rather than a small slice of it.
Portfolio Value at Risk
M P Birla Institute of Management Page 101
5.3 Recommendations
Suggestions regarding this study are as follows :
Elements taken for portfolio is based on the personal choice for equities,
for bonds one should take on the basis of the day it is traded and for
currencies one should consider the strong currency compared to home
currency.
For calculating VaR, using the analytic approach it was done separately
for equities, bonds and currencies where its probability of loss for 99%
VaR was found and again, portfolio was constructed taking equities,
bonds and currencies together for 99% VaR, but this time loss was less in
respect of taking equities, bonds and currencies separately where,
investment was equal in each case. Thus, it shows that its always better to
invest in portfolio than individually and using VaR helps us to know the
probability of occurrence of loss.
In this study, measuring portfolio through VaR models i.e. Historical
Simulation , Analytic Approach and Monte Carlo Simulation, found that
if one is assessing the Value at Risk for portfolios, that do not include
options, over very short time periods (a day or a week), the variance-
covariance approach does a reasonably good job but for portfolios of
longer period with volatile and stationary data Monte Carlo Simulations
is best. Thus, for short periods data, go for variance-covariance approach
and for longer period data, go for Monte Carlo Simulation.
Portfolio Value at Risk
M P Birla Institute of Management Page 102
Selected Bibliography
Books & Journal
Pramod M Mantravadi, Value at Risk – Concepts and Applications, First
Edition: 2005
Websites
www.nse-india.com
www.jstor.com
www.icfai.org
Portfolio Value at Risk
M P Birla Institute of Management Page 103
Annexure
Portfolio Value at Risk
M P Birla Institute of Management Page 104
Date
Equity price of Reliance Date
Equity price of
DLF Date
Equity price of Airtel Date
Equity price of Infosys
1-Apr-08 2,345.25 1-Apr-08 626.95 1-Apr-08 803.1 1-Apr-08 1,423.05
2-Apr-08 2,343.55 2-Apr-08 619.55 2-Apr-08 823.1 2-Apr-08 1,482.80
3-Apr-08 2,396.05 3-Apr-08 623.7 3-Apr-08 819.7 3-Apr-08 1,522.50
4-Apr-08 2,321.15 4-Apr-08 606 4-Apr-08 783.9 4-Apr-08 1,485.45
7-Apr-08 2,404.90 7-Apr-08 617.65 7-Apr-08 818.8 7-Apr-08 1,492.05
8-Apr-08 2,381.75 8-Apr-08 616.25 8-Apr-08 828.45 8-Apr-08 1,466.95
9-Apr-08 2,418.25 9-Apr-08 610.25 9-Apr-08 821.9 9-Apr-08 1,479.95
10-Apr-08 2,468.65 10-Apr-08 600.6 10-Apr-08 798.65 10-Apr-08 1,452.60
11-Apr-08 2,551.55 11-Apr-08 598.05 11-Apr-08 805.1 11-Apr-08 1,421.90
15-Apr-08 2,611.80 15-Apr-08 616.65 15-Apr-08 816.7 15-Apr-08 1,510.40
16-Apr-08 2,642.50 16-Apr-08 624.45 16-Apr-08 807.55 16-Apr-08 1,600.20
17-Apr-08 2,640.05 17-Apr-08 649.85 17-Apr-08 823.4 17-Apr-08 1,659.10
21-Apr-08 2,643.60 21-Apr-08 653.75 21-Apr-08 855.9 21-Apr-08 1,645.80
22-Apr-08 2,607.35 22-Apr-08 676.25 22-Apr-08 856 22-Apr-08 1,598.60
23-Apr-08 2,577.60 23-Apr-08 684.45 23-Apr-08 845.05 23-Apr-08 1,647.95
24-Apr-08 2,582.65 24-Apr-08 676 24-Apr-08 843 24-Apr-08 1,696.05
25-Apr-08 2,624.50 25-Apr-08 667.85 25-Apr-08 922.4 25-Apr-08 1,684.30
28-Apr-08 2,591.40 28-Apr-08 668.5 28-Apr-08 928.05 28-Apr-08 1,663.65
29-Apr-08 2,659.95 29-Apr-08 725.55 29-Apr-08 901.2 29-Apr-08 1,749.55
30-Apr-08 2,614.50 30-Apr-08 705.15 30-Apr-08 898.25 30-Apr-08 1,753.10
2-May-08 2,674.75 2-May-08 721.35 2-May-08 900.25 2-May-08 1,787.45
5-May-08 2,669.20 5-May-08 705.15 5-May-08 893.95 5-May-08 1,785.95
6-May-08 2,650.00 6-May-08 668.2 6-May-08 846.35 6-May-08 1,807.45
7-May-08 2,688.95 7-May-08 650.75 7-May-08 816.15 7-May-08 1,844.15
8-May-08 2,667.25 8-May-08 644.15 8-May-08 829.2 8-May-08 1,784.65
9-May-08 2,528.40 9-May-08 630.65 9-May-08 841.8 9-May-08 1,749.35
12-May-08 2,553.85 12-May-08 621.9 12-May-08 837.95 12-May-08 1,773.40
Portfolio Value at Risk
M P Birla Institute of Management Page 105
13-May-08 2,501.10 13-May-08 615.1 13-May-08 820.05 13-May-08 1,748.75
14-May-08 2,530.75 14-May-08 623.35 14-May-08 848.85 14-May-08 1,824.90
15-May-08 2,622.95 15-May-08 644.9 15-May-08 856.2 15-May-08 1,893.10
16-May-08 2,635.70 16-May-08 649.75 16-May-08 852.45 16-May-08 1,871.40
20-May-08 2,602.95 20-May-08 636.75 20-May-08 829.6 20-May-08 1,887.70
21-May-08 2,667.70 21-May-08 631.55 21-May-08 822.45 21-May-08 1,870.65
22-May-08 2,626.05 22-May-08 620.75 22-May-08 817.7 22-May-08 1,863.65
23-May-08 2,556.20 23-May-08 609 23-May-08 837.65 23-May-08 1,828.75
26-May-08 2,515.60 26-May-08 600.8 26-May-08 863.45 26-May-08 1,885.05
27-May-08 2,495.10 27-May-08 596.6 27-May-08 862.75 27-May-08 1,883.10
28-May-08 2,522.50 28-May-08 601.7 28-May-08 884.25 28-May-08 1,911.60
29-May-08 2,462.70 29-May-08 587.85 29-May-08 858.45 29-May-08 1,885.45
30-May-08 2,403.50 30-May-08 586.85 30-May-08 875.95 30-May-08 1,962.80
2-Jun-08 2,358.80 2-Jun-08 567.7 2-Jun-08 877.45 2-Jun-08 1,950.80
3-Jun-08 2,406.65 3-Jun-08 582.65 3-Jun-08 841.4 3-Jun-08 1,922.25
4-Jun-08 2,307.00 4-Jun-08 554.1 4-Jun-08 810.35 4-Jun-08 1,870.70
5-Jun-08 2,246.80 5-Jun-08 538.95 5-Jun-08 822.55 5-Jun-08 1,979.55
6-Jun-08 2,238.50 6-Jun-08 518.4 6-Jun-08 802 6-Jun-08 1,993.55
9-Jun-08 2,162.70 9-Jun-08 480.9 9-Jun-08 781.3 9-Jun-08 1,901.50
10-Jun-08 2,197.75 10-Jun-08 480 10-Jun-08 776.3 10-Jun-08 1,854.05
11-Jun-08 2,261.40 11-Jun-08 512.15 11-Jun-08 806.9 11-Jun-08 1,892.85
12-Jun-08 2,277.30 12-Jun-08 497.45 12-Jun-08 820.05 12-Jun-08 1,874.90
13-Jun-08 2,270.40 13-Jun-08 480.25 13-Jun-08 816.4 13-Jun-08 1,866.65
16-Jun-08 2,282.35 16-Jun-08 492.3 16-Jun-08 840.35 16-Jun-08 1,907.80
17-Jun-08 2,332.90 17-Jun-08 507.65 17-Jun-08 832.55 17-Jun-08 1,911.55
18-Jun-08 2,287.10 18-Jun-08 492.6 18-Jun-08 812.05 18-Jun-08 1,862.45
19-Jun-08 2,248.15 19-Jun-08 477.2 19-Jun-08 805.35 19-Jun-08 1,862.40
20-Jun-08 2,099.20 20-Jun-08 456.8 20-Jun-08 765.7 20-Jun-08 1,827.00
23-Jun-08 2,025.70 23-Jun-08 445.8 23-Jun-08 758.75 23-Jun-08 1,845.80
Portfolio Value at Risk
M P Birla Institute of Management Page 106
24-Jun-08 2,062.70 24-Jun-08 439.95 24-Jun-08 750.7 24-Jun-08 1,794.45
25-Jun-08 2,136.00 25-Jun-08 458.85 25-Jun-08 779.9 25-Jun-08 1,746.75
26-Jun-08 2,239.55 26-Jun-08 451.8 26-Jun-08 769.45 26-Jun-08 1,785.10
27-Jun-08 2,182.65 27-Jun-08 425.1 27-Jun-08 747.95 27-Jun-08 1,705.45
30-Jun-08 2,095.15 30-Jun-08 396.4 30-Jun-08 721.25 30-Jun-08 1,736.80
1-Jul-08 2,044.15 1-Jul-08 369.1 1-Jul-08 706.9 1-Jul-08 1,717.00
2-Jul-08 2,144.00 2-Jul-08 423.45 2-Jul-08 742.3 2-Jul-08 1,820.60
3-Jul-08 2,070.10 3-Jul-08 382.85 3-Jul-08 707.75 3-Jul-08 1,743.05
4-Jul-08 2,097.90 4-Jul-08 415.45 4-Jul-08 717.15 4-Jul-08 1,755.80
7-Jul-08 2,028.20 7-Jul-08 425.75 7-Jul-08 727.65 7-Jul-08 1,799.80
8-Jul-08 1,979.45 8-Jul-08 429.6 8-Jul-08 711.05 8-Jul-08 1,736.60
9-Jul-08 2,079.15 9-Jul-08 450.3 9-Jul-08 746.4 9-Jul-08 1,821.90
10-Jul-08 2,046.65 10-Jul-08 459.95 10-Jul-08 741.75 10-Jul-08 1,805.25
11-Jul-08 2,016.10 11-Jul-08 452.7 11-Jul-08 744.9 11-Jul-08 1,676.85
14-Jul-08 2,043.45 14-Jul-08 457 14-Jul-08 735.85 14-Jul-08 1,555.55
15-Jul-08 1,977.40 15-Jul-08 427.35 15-Jul-08 710.05 15-Jul-08 1,544.65
16-Jul-08 1,943.50 16-Jul-08 393.4 16-Jul-08 730.95 16-Jul-08 1,547.60
17-Jul-08 2,018.55 17-Jul-08 430.15 16-Jul-08 710 17-Jul-08 1,582.30
18-Jul-08 2,113.20 18-Jul-08 460.95 17-Jul-08 749.45 18-Jul-08 1,547.40
21-Jul-08 2,152.85 21-Jul-08 463.95 18-Jul-08 803.1 21-Jul-08 1,561.30
22-Jul-08 2,152.15 22-Jul-08 453.95 21-Jul-08 798.55 22-Jul-08 1,578.30
23-Jul-08 2,267.30 23-Jul-08 495.5 22-Jul-08 778.2 23-Jul-08 1,602.25
24-Jul-08 2,308.05 24-Jul-08 507.75 23-Jul-08 816.2 24-Jul-08 1,567.30
25-Jul-08 2,147.10 25-Jul-08 489.5 24-Jul-08 799.65 25-Jul-08 1,550.65
28-Jul-08 2,179.90 28-Jul-08 499.45 25-Jul-08 796.85 28-Jul-08 1,539.45
29-Jul-08 2,083.10 29-Jul-08 472 28-Jul-08 795.15 29-Jul-08 1,540.70
30-Jul-08 2,165.50 30-Jul-08 489.65 29-Jul-08 777.9 30-Jul-08 1,602.05
31-Jul-08 2,207.50 31-Jul-08 511.5 30-Jul-08 809.9 31-Jul-08 1,583.45
1-Aug-08 2,297.60 1-Aug-08 520.15 31-Jul-08 798.7 1-Aug-08 1,639.45
Portfolio Value at Risk
M P Birla Institute of Management Page 107
4-Aug-08 2,242.40 4-Aug-08 514.7 1-Aug-08 820.15 4-Aug-08 1,658.85
5-Aug-08 2,276.05 5-Aug-08 552.1 4-Aug-08 816 5-Aug-08 1,678.55
6-Aug-08 2,298.60 6-Aug-08 545.1 5-Aug-08 840.1 6-Aug-08 1,699.35
7-Aug-08 2,272.60 7-Aug-08 557 6-Aug-08 869.4 7-Aug-08 1,726.15
8-Aug-08 2,251.80 8-Aug-08 549.75 7-Aug-08 849.35 8-Aug-08 1,679.60
11-Aug-08 2,325.25 11-Aug-08 568.75 8-Aug-08 840.85 11-Aug-08 1,670.40
12-Aug-08 2,347.25 12-Aug-08 567.1 11-Aug-08 845.55 12-Aug-08 1,603.90
13-Aug-08 2,336.85 13-Aug-08 548.25 12-Aug-08 823.25 13-Aug-08 1,625.95
14-Aug-08 2,276.70 14-Aug-08 500 13-Aug-08 820.85 14-Aug-08 1,693.50
18-Aug-08 2,224.80 18-Aug-08 500.15 14-Aug-08 818.6 18-Aug-08 1,704.35
19-Aug-08 2,219.60 19-Aug-08 500.85 18-Aug-08 809.1 19-Aug-08 1,692.75
20-Aug-08 2,246.35 20-Aug-08 510.5 19-Aug-08 791.9 20-Aug-08 1,699.30
21-Aug-08 2,212.65 21-Aug-08 481.75 20-Aug-08 816 21-Aug-08 1,661.45
22-Aug-08 2,244.80 22-Aug-08 484.9 21-Aug-08 799.25 22-Aug-08 1,696.50
25-Aug-08 2,232.00 25-Aug-08 495.2 22-Aug-08 809.7 25-Aug-08 1,706.45
26-Aug-08 2,179.35 26-Aug-08 498.05 25-Aug-08 816.5 26-Aug-08 1,698.00
27-Aug-08 2,148.00 27-Aug-08 478.85 26-Aug-08 808.8 27-Aug-08 1,708.20
28-Aug-08 2,070.85 28-Aug-08 469.5 27-Aug-08 804.85 28-Aug-08 1,699.40
29-Aug-08 2,136.20 29-Aug-08 493.1 28-Aug-08 804.5 29-Aug-08 1,749.10
1-Sep-08 2,141.65 1-Sep-08 494.05 29-Aug-08 837.5 1-Sep-08 1,723.30
2-Sep-08 2,212.75 2-Sep-08 530.2 1-Sep-08 816.2 2-Sep-08 1,775.25
4-Sep-08 2,152.25 4-Sep-08 523.05 2-Sep-08 834.65 4-Sep-08 1,789.00
5-Sep-08 2,080.90 5-Sep-08 494.15 4-Sep-08 825.8 5-Sep-08 1,712.30
8-Sep-08 2,133.20 8-Sep-08 512.35 5-Sep-08 803.4 8-Sep-08 1,750.05
9-Sep-08 2,142.55 9-Sep-08 502.6 8-Sep-08 819.8 9-Sep-08 1,749.45
10-Sep-08 2,082.65 10-Sep-08 501.75 9-Sep-08 836.9 10-Sep-08 1,759.75
11-Sep-08 1,997.40 11-Sep-08 484.95 10-Sep-08 812 11-Sep-08 1,750.65
12-Sep-08 1,932.65 12-Sep-08 466.45 11-Sep-08 776.95 12-Sep-08 1,644.00
15-Sep-08 1,886.95 15-Sep-08 433.15 12-Sep-08 778.85 15-Sep-08 1,578.15
Portfolio Value at Risk
M P Birla Institute of Management Page 108
16-Sep-08 1,928.05 16-Sep-08 422.9 15-Sep-08 766.15 16-Sep-08 1,566.40
17-Sep-08 1,876.65 17-Sep-08 409.2 16-Sep-08 774.1 17-Sep-08 1,579.00
18-Sep-08 1,938.25 18-Sep-08 395.9 17-Sep-08 770.15 18-Sep-08 1,525.40
19-Sep-08 2,055.10 19-Sep-08 426.5 18-Sep-08 761.25 19-Sep-08 1,624.95
22-Sep-08 2,039.10 22-Sep-08 421.25 19-Sep-08 805.85 22-Sep-08 1,629.90
23-Sep-08 2,006.45 23-Sep-08 394.7 22-Sep-08 808.8 23-Sep-08 1,543.95
24-Sep-08 2,046.10 24-Sep-08 400.35 23-Sep-08 792.65 24-Sep-08 1,524.30
25-Sep-08 2,025.70 25-Sep-08 389.25 24-Sep-08 810.55 25-Sep-08 1,506.85
26-Sep-08 1,963.20 26-Sep-08 369.65 25-Sep-08 790.9 26-Sep-08 1,446.90
29-Sep-08 1,932.85 29-Sep-08 350.35 26-Sep-08 776 29-Sep-08 1,393.30
30-Sep-08 1,949.35 30-Sep-08 352.65 29-Sep-08 750.25 30-Sep-08 1,398.05
1-Oct-08 1,906.70 1-Oct-08 345.3 30-Sep-08 784.85 1-Oct-08 1,449.30
3-Oct-08 1,761.45 3-Oct-08 336.35 1-Oct-08 790.7 3-Oct-08 1,391.90
6-Oct-08 1,641.60 6-Oct-08 301.45 3-Oct-08 756.3 6-Oct-08 1,318.65
7-Oct-08 1,674.65 7-Oct-08 303.15 6-Oct-08 727.75 7-Oct-08 1,301.80
8-Oct-08 1,648.55 8-Oct-08 308.85 7-Oct-08 748.25 8-Oct-08 1,254.25
10-Oct-08 1,527.60 10-Oct-08 281.9 8-Oct-08 733.35 10-Oct-08 1,225.20
13-Oct-08 1,571.40 13-Oct-08 302.3 10-Oct-08 692.3 13-Oct-08 1,318.55
14-Oct-08 1,621.05 14-Oct-08 311.15 13-Oct-08 739.85 14-Oct-08 1,397.95
15-Oct-08 1,520.20 15-Oct-08 300.8 14-Oct-08 766.6 15-Oct-08 1,335.40
16-Oct-08 1,391.95 16-Oct-08 324.95 15-Oct-08 714.85 16-Oct-08 1,263.00
17-Oct-08 1,306.05 17-Oct-08 291.9 16-Oct-08 730.65 17-Oct-08 1,202.75
20-Oct-08 1,320.90 20-Oct-08 272.65 17-Oct-08 677.15 20-Oct-08 1,294.55
21-Oct-08 1,394.95 21-Oct-08 286.35 20-Oct-08 708.1 21-Oct-08 1,347.60
22-Oct-08 1,316.80 22-Oct-08 271.9 21-Oct-08 724.35 22-Oct-08 1,299.55
23-Oct-08 1,217.65 23-Oct-08 268.05 22-Oct-08 666.25 23-Oct-08 1,283.25
24-Oct-08 1,019.50 24-Oct-08 204.6 23-Oct-08 615.05 24-Oct-08 1,246.10
27-Oct-08 1,077.00 27-Oct-08 198.1 24-Oct-08 537.1 27-Oct-08 1,251.80
28-Oct-08 1,153.00 28-Oct-08 217.85 27-Oct-08 564.6 28-Oct-08 1,279.35
Portfolio Value at Risk
M P Birla Institute of Management Page 109
29-Oct-08 1,201.75 29-Oct-08 202.6 28-Oct-08 610.7 29-Oct-08 1,304.35
31-Oct-08 1,375.45 31-Oct-08 220 29-Oct-08 616.45 31-Oct-08 1,388.95
3-Nov-08 1,441.70 3-Nov-08 253.25 31-Oct-08 653.75 3-Nov-08 1,378.60
4-Nov-08 1,451.60 4-Nov-08 289.75 3-Nov-08 691.5 4-Nov-08 1,331.90
5-Nov-08 1,269.05 5-Nov-08 265.65 4-Nov-08 716.95 5-Nov-08 1,322.00
6-Nov-08 1,170.55 6-Nov-08 272.2 5-Nov-08 685.35 6-Nov-08 1,245.70
7-Nov-08 1,220.75 7-Nov-08 280.6 6-Nov-08 639.4 7-Nov-08 1,262.80
10-Nov-08 1,303.10 10-Nov-08 298.9 7-Nov-08 648.05 10-Nov-08 1,338.50
11-Nov-08 1,207.70 11-Nov-08 268.6 10-Nov-08 711.65 11-Nov-08 1,256.70
12-Nov-08 1,162.20 12-Nov-08 244.9 11-Nov-08 659.45 12-Nov-08 1,259.45
14-Nov-08 1,146.75 14-Nov-08 241.45 12-Nov-08 631.95 14-Nov-08 1,213.20
17-Nov-08 1,141.40 17-Nov-08 232.5 14-Nov-08 647.5 17-Nov-08 1,232.40
18-Nov-08 1,139.95 18-Nov-08 224.3 17-Nov-08 664.4 18-Nov-08 1,180.50
19-Nov-08 1,132.45 19-Nov-08 225.45 18-Nov-08 622.9 19-Nov-08 1,172.70
20-Nov-08 1,056.05 20-Nov-08 205.25 19-Nov-08 612.35 20-Nov-08 1,121.85
21-Nov-08 1,124.35 21-Nov-08 198.25 20-Nov-08 592 21-Nov-08 1,184.65
24-Nov-08 1,144.80 24-Nov-08 190.95 21-Nov-08 618.85 24-Nov-08 1,196.20
25-Nov-08 1,071.80 25-Nov-08 188.2 24-Nov-08 637.8 25-Nov-08 1,181.95
26-Nov-08 1,138.90 26-Nov-08 198.5 25-Nov-08 627.8 26-Nov-08 1,187.10
28-Nov-08 1,134.45 28-Nov-08 198.4 26-Nov-08 653 28-Nov-08 1,243.85
1-Dec-08 1,109.40 1-Dec-08 178.7 28-Nov-08 671.05 1-Dec-08 1,231.20
2-Dec-08 1,073.95 2-Dec-08 181.95 1-Dec-08 650.7 2-Dec-08 1,208.35
3-Dec-08 1,069.10 3-Dec-08 191.95 2-Dec-08 670.85 3-Dec-08 1,156.55
4-Dec-08 1,159.10 4-Dec-08 213.45 3-Dec-08 664 4-Dec-08 1,192.40
5-Dec-08 1,117.60 5-Dec-08 203.45 4-Dec-08 685.7 5-Dec-08 1,134.50
8-Dec-08 1,118.55 8-Dec-08 221.55 5-Dec-08 664.8 8-Dec-08 1,157.75
10-Dec-08 1,227.20 10-Dec-08 262.9 8-Dec-08 701.25 10-Dec-08 1,173.75
11-Dec-08 1,259.00 11-Dec-08 255.35 10-Dec-08 735.8 11-Dec-08 1,134.90
12-Dec-08 1,307.10 12-Dec-08 277.15 11-Dec-08 742.4 12-Dec-08 1,104.85
Portfolio Value at Risk
M P Birla Institute of Management Page 110
15-Dec-08 1,340.55 15-Dec-08 280.85 12-Dec-08 723.5 15-Dec-08 1,102.30
16-Dec-08 1,388.50 16-Dec-08 276.2 15-Dec-08 737.4 16-Dec-08 1,124.80
17-Dec-08 1,351.40 17-Dec-08 253.5 16-Dec-08 744.8 17-Dec-08 1,141.55
18-Dec-08 1,361.00 18-Dec-08 277.4 17-Dec-08 709.25 18-Dec-08 1,177.05
19-Dec-08 1,351.30 19-Dec-08 307.9 18-Dec-08 710.15 19-Dec-08 1,191.30
22-Dec-08 1,285.55 22-Dec-08 316.55 19-Dec-08 721.7 22-Dec-08 1,186.80
23-Dec-08 1,259.75 23-Dec-08 301.8 22-Dec-08 722.1 23-Dec-08 1,176.05
24-Dec-08 1,242.00 24-Dec-08 294.9 23-Dec-08 710.85 24-Dec-08 1,174.00
26-Dec-08 1,210.15 26-Dec-08 275.45 24-Dec-08 689.4 26-Dec-08 1,109.25
29-Dec-08 1,246.30 29-Dec-08 276.35 26-Dec-08 686.4 29-Dec-08 1,113.75
30-Dec-08 1,250.50 30-Dec-08 285.3 29-Dec-08 712.05 30-Dec-08 1,127.90
31-Dec-08 1,232.75 31-Dec-08 282.15 30-Dec-08 722.6 31-Dec-08 1,115.45
1-Jan-09 1,254.65 1-Jan-09 291.75 31-Dec-08 715.5 1-Jan-09 1,148.15
2-Jan-09 1,286.40 2-Jan-09 300.55 1-Jan-09 719.95 2-Jan-09 1,132.10
5-Jan-09 1,365.85 5-Jan-09 295.95 2-Jan-09 704.9 5-Jan-09 1,172.80
6-Jan-09 1,370.90 6-Jan-09 278.5 5-Jan-09 685.65 6-Jan-09 1,168.55
7-Jan-09 1,200.75 7-Jan-09 233.85 6-Jan-09 657 7-Jan-09 1,187.55
9-Jan-09 1,153.25 9-Jan-09 216.35 7-Jan-09 650 9-Jan-09 1,203.40
12-Jan-09 1,097.90 12-Jan-09 204.75 9-Jan-09 638.9 12-Jan-09 1,159.70
13-Jan-09 1,077.55 13-Jan-09 205.45 12-Jan-09 623.85 13-Jan-09 1,228.15
14-Jan-09 1,179.75 14-Jan-09 212.4 13-Jan-09 607.05 14-Jan-09 1,303.60
15-Jan-09 1,142.35 15-Jan-09 202.2 14-Jan-09 622.75 15-Jan-09 1,251.70
16-Jan-09 1,217.35 16-Jan-09 195.55 15-Jan-09 603.65 16-Jan-09 1,267.40
19-Jan-09 1,229.90 19-Jan-09 195.15 15-Jan-09 600.5 19-Jan-09 1,262.05
20-Jan-09 1,183.65 20-Jan-09 188.7 16-Jan-09 634.75 20-Jan-09 1,251.75
21-Jan-09 1,119.85 21-Jan-09 181.35 19-Jan-09 646.85 21-Jan-09 1,223.85
22-Jan-09 1,136.30 22-Jan-09 164.15 20-Jan-09 616.35 22-Jan-09 1,230.90
23-Jan-09 1,156.15 23-Jan-09 161.3 21-Jan-09 583.05 23-Jan-09 1,204.65
27-Jan-09 1,225.95 27-Jan-09 166.5 22-Jan-09 619.2 27-Jan-09 1,252.10
Portfolio Value at Risk
M P Birla Institute of Management Page 111
28-Jan-09 1,274.00 28-Jan-09 177.45 23-Jan-09 613.15 28-Jan-09 1,286.70
29-Jan-09 1,270.10 29-Jan-09 164.05 27-Jan-09 647.4 29-Jan-09 1,310.15
30-Jan-09 1,323.60 30-Jan-09 177.65 28-Jan-09 652.15 30-Jan-09 1,306.65
2-Feb-09 1,280.00 2-Feb-09 153 29-Jan-09 627.4 2-Feb-09 1,279.65
3-Feb-09 1,306.20 3-Feb-09 132.85 30-Jan-09 633.95 3-Feb-09 1,282.50
4-Feb-09 1,307.50 4-Feb-09 139.45 2-Feb-09 615.8 4-Feb-09 1,281.65
5-Feb-09 1,288.80 5-Feb-09 140 3-Feb-09 625 5-Feb-09 1,255.85
6-Feb-09 1,344.85 6-Feb-09 137.95 4-Feb-09 634.3 6-Feb-09 1,288.90
9-Feb-09 1,389.70 9-Feb-09 139.95 5-Feb-09 628.65 9-Feb-09 1,312.10
10-Feb-09 1,401.95 10-Feb-09 152.6 6-Feb-09 647.05 10-Feb-09 1,308.40
12-Feb-09 1,351.55 12-Feb-09 156.6 10-Feb-09 663.55 12-Feb-09 1,254.55
13-Feb-09 1,392.40 13-Feb-09 160.55 11-Feb-09 673.45 13-Feb-09 1,251.65
16-Feb-09 1,320.20 16-Feb-09 156.65 12-Feb-09 650.85 16-Feb-09 1,221.45
17-Feb-09 1,267.30 17-Feb-09 148.25 13-Feb-09 651.75 17-Feb-09 1,173.95
18-Feb-09 1,295.15 18-Feb-09 158.75 16-Feb-09 637.75 18-Feb-09 1,178.80
19-Feb-09 1,293.75 19-Feb-09 156 17-Feb-09 632.65 19-Feb-09 1,208.85
20-Feb-09 1,253.40 20-Feb-09 154.85 18-Feb-09 640.85 20-Feb-09 1,177.15
24-Feb-09 1,253.25 24-Feb-09 158.15 19-Feb-09 648.45 24-Feb-09 1,185.55
25-Feb-09 1,266.55 25-Feb-09 154.8 20-Feb-09 642.7 25-Feb-09 1,217.45
26-Feb-09 1,290.80 26-Feb-09 156.2 24-Feb-09 636.65 26-Feb-09 1,236.00
27-Feb-09 1,266.05 27-Feb-09 152 25-Feb-09 640.75 27-Feb-09 1,231.25
3-Mar-09 1,196.85 3-Mar-09 148.35 27-Feb-09 638.5 3-Mar-09 1,197.60
4-Mar-09 1,211.10 4-Mar-09 147.65 2-Mar-09 616.8 4-Mar-09 1,199.20
5-Mar-09 1,149.80 5-Mar-09 146.85 3-Mar-09 601.15 5-Mar-09 1,181.95
6-Mar-09 1,169.90 6-Mar-09 145.8 4-Mar-09 600.65 6-Mar-09 1,219.35
9-Mar-09 1,153.35 9-Mar-09 138.75 5-Mar-09 589.75 9-Mar-09 1,202.05
12-Mar-09 1,202.00 12-Mar-09 136.65 5-Mar-09 601 12-Mar-09 1,227.80
13-Mar-09 1,284.25 13-Mar-09 152.65 6-Mar-09 602.25 13-Mar-09 1,297.05
16-Mar-09 1,327.60 16-Mar-09 161.95 9-Mar-09 587.75 16-Mar-09 1,287.80
Portfolio Value at Risk
M P Birla Institute of Management Page 112
17-Mar-09 1,300.20 17-Mar-09 159 12-Mar-09 548.45 17-Mar-09 1,265.50
18-Mar-09 1,331.40 18-Mar-09 172.3 13-Mar-09 557.65 18-Mar-09 1,278.05
19-Mar-09 1,345.70 19-Mar-09 174 16-Mar-09 571.3 19-Mar-09 1,297.20
20-Mar-09 1,339.20 20-Mar-09 171.5 17-Mar-09 571.55 20-Mar-09 1,296.20
23-Mar-09 1,438.45 23-Mar-09 166.65 18-Mar-09 569.75 23-Mar-09 1,331.30
24-Mar-09 1,452.45 24-Mar-09 166.3 19-Mar-09 570.7 24-Mar-09 1,320.75
25-Mar-09 1,532.20 25-Mar-09 176.65 20-Mar-09 569.4 25-Mar-09 1,338.95
26-Mar-09 1,565.50 26-Mar-09 176.35 23-Mar-09 593.2 26-Mar-09 1,379.15
27-Mar-09 1,548.75 27-Mar-09 182.8 24-Mar-09 603.7 27-Mar-09 1,344.90
30-Mar-09 1,516.45 30-Mar-09 165.5 25-Mar-09 591 30-Mar-09 1,301.30
31-Mar-09 1,524.75 31-Mar-09 167.3 26-Mar-09 621.85 31-Mar-09 1,323.90
27-Mar-09 621.85
30-Mar-09 609.65
31-Mar-09 625.75
Date
Weighted Average Price (Rs.) of PFCL Date
Weighted Average Price (Rs.) of IRFC Date
Weighted Average Price (Rs.) of HDFC
4-Dec-08 106.1302 16-Apr-08 97.9005 2-Apr-08 100.6671
5-Dec-08 105.7138 24-Apr-08 97.6224 3-Apr-08 100.6995
8-Dec-08 108.5793 25-Apr-08 97.6242 17-Apr-08 100.4636
10-Dec-08 109.8788 2-May-08 98.4558 28-Apr-08 100.4503
11-Dec-08 110.2972 8-May-08 98.2629 29-Apr-08 100.5818
12-Dec-08 113.5564 9-May-08 98.2994 30-Apr-08 100.6651
15-Dec-08 113.2589 12-May-08 98.1833 2-May-08 100.7385
16-Dec-08 113.3345 3-Jun-08 97.4791 5-May-08 100.7956
17-Dec-08 115.3004 4-Jun-08 97.3769 7-May-08 100.8008
Portfolio Value at Risk
M P Birla Institute of Management Page 113
18-Dec-08 116.324 5-Jun-08 97.3386 13-May-08 100.762
19-Dec-08 116.6233 6-Jun-08 97.3092 27-May-08 100.534
22-Dec-08 115.6992 20-Jun-08 95.4794 5-Jun-08 100.3139
23-Dec-08 115.1241 23-Jun-08 95.4856 9-Jun-08 100.2774
24-Dec-08 115.5771 25-Jun-08 94.5031 10-Jun-08 100.1821
26-Dec-08 115.5641 2-Jul-08 93.7075 13-Jun-08 99.9128
29-Dec-08 115.7616 3-Jul-08 93.8755 16-Jun-08 99.9195
30-Dec-08 116.4699 24-Jul-08 94.0921 18-Jun-08 99.9881
31-Dec-08 116.9221 25-Jul-08 94.2608 19-Jun-08 99.9102
1-Jan-09 116.8284 31-Jul-08 93.9185 20-Jun-08 99.9102
2-Jan-09 116.7426 8-Sep-08 96.4773 24-Jun-08 99.546
5-Jan-09 119.531 9-Sep-08 96.4595 9-Jul-08 98.71
6-Jan-09 119.15 10-Sep-08 96.4816 16-Jul-08 98.3655
7-Jan-09 115.3182 31-Oct-08 91.8076 18-Jul-08 98.36
9-Jan-09 114.1236 28-Nov-08 93.5805 28-Jul-08 98.4496
12-Jan-09 115.6353 5-Dec-08 96.7837 4-Aug-08 98.2534
13-Jan-09 116.2045 8-Dec-08 97.7193 6-Aug-08 98.2577
14-Jan-09 115.7981 10-Dec-08 97.8765 1-Sep-08 98.076
15-Jan-09 116.3177 11-Dec-08 99.7823 3-Oct-08 96.7154
16-Jan-09 116.4797 16-Dec-08 99.9854 7-Oct-08 96.7266
19-Jan-09 115.7576 18-Dec-08 100.6647 14-Nov-08 97.2949
20-Jan-09 115.6321 23-Dec-08 101.0713 3-Dec-08 98.0464
21-Jan-09 114.6265 24-Dec-08 101.0057 4-Dec-08 98.0464
22-Jan-09 114.1985 29-Dec-08 101.132 18-Dec-08 100.0802
23-Jan-09 114.8757 30-Dec-08 101.2493 19-Dec-08 100.2504
27-Jan-09 114.4195 1-Jan-09 101.4469 13-Jan-09 101.4442
28-Jan-09 113.9598 13-Jan-09 100.8325 14-Jan-09 101.5511
29-Jan-09 113.5184 28-Jan-09 100.4136 15-Jan-09 101.7126
30-Jan-09 113.198 30-Jan-09 100.4433 13-Feb-09 101.3687
Portfolio Value at Risk
M P Birla Institute of Management Page 114
3-Feb-09 113.1854 10-Feb-09 99.8498 27-Feb-09 101.9291
4-Feb-09 112.5258 27-Feb-09 101.3307 3-Mar-09 101.9135
5-Feb-09 112.6236 2-Mar-09 101.6278 6-Mar-09 101.7805
9-Feb-09 113.5627 4-Mar-09 101.689 18-Mar-09 101.7511
11-Feb-09 112.4814 5-Mar-09 101.8062 19-Mar-09 101.6574
12-Feb-09 112.82 20-Mar-09 101.2521 23-Mar-09 101.7252
13-Feb-09 112.9919 23-Mar-09 101.4268 31-Mar-09 102.1553
17-Feb-09 112.0221 2-Apr-09 102.0395 6-Apr-09 102.5982
18-Feb-09 111.9203 16-Apr-09 104.4128 24-Apr-09 103.5354
19-Feb-09 112.5648 17-Apr-09 105.049 27-Apr-09 103.458
24-Feb-09 112.3776 20-Apr-09 105.049
25-Feb-09 112.0372 24-Apr-09 104.9622
26-Feb-09 112.5568 28-Apr-09 105.0211
27-Feb-09 113.2551
2-Mar-09 113.5717
3-Mar-09 113.7237
6-Mar-09 112.0355
9-Mar-09 113.72
12-Mar-09 109.9913
13-Mar-09 110.0535
16-Mar-09 112.193
17-Mar-09 111.3056
18-Mar-09 110.9029
19-Mar-09 111.2793
20-Mar-09 111.905
23-Mar-09 112.2412
24-Mar-09 112.5665
25-Mar-09 112.3017
Portfolio Value at Risk
M P Birla Institute of Management Page 115
26-Mar-09 112.2243
31-Mar-09 112.6018
2-Apr-09 113.9368
6-Apr-09 113.8556
8-Apr-09 115.0941
9-Apr-09 115.0598
13-Apr-09 116.0479
15-Apr-09 116.0628
16-Apr-09 116.6922
17-Apr-09 117.1333
20-Apr-09 117.3871
21-Apr-09 117.9761
22-Apr-09 119.931
23-Apr-09 118.7522
24-Apr-09 118.6424
27-Apr-09 118.4951
28-Apr-09 118.1423
29-Apr-09 118.1216
Date Exchange
Rate of Dollar Date Exchange
Rate of Euro Date Exchange
Rate of Pound
4/1/2008 40.0611 4/1/2008 63.3022 4/1/2008 79.7261
4/2/2008 40.15 4/2/2008 63.0198 4/2/2008 79.5199
4/3/2008 39.9926 4/3/2008 62.4644 4/3/2008 79.2149
4/4/2008 40.0158 4/4/2008 62.5687 4/4/2008 79.5382
4/5/2008 39.9698 4/5/2008 62.8169 4/5/2008 79.7895
4/6/2008 39.9698 4/6/2008 62.9216 4/6/2008 79.6925
4/7/2008 39.955 4/7/2008 62.8988 4/7/2008 79.6639
Portfolio Value at Risk
M P Birla Institute of Management Page 116
4/8/2008 39.97 4/8/2008 62.7809 4/8/2008 79.4684
4/9/2008 40.0184 4/9/2008 62.9569 4/9/2008 79.2168
4/10/2008 40.0151 4/10/2008 62.9801 4/10/2008 78.8361
4/11/2008 39.9629 4/11/2008 63.2393 4/11/2008 78.9987
4/12/2008 39.9509 4/12/2008 63.0972 4/12/2008 78.8211
4/13/2008 39.9509 4/13/2008 63.1735 4/13/2008 78.6909
4/14/2008 39.9509 4/14/2008 63.1448 4/14/2008 78.6861
4/15/2008 39.96 4/15/2008 63.0209 4/15/2008 78.9102
4/16/2008 39.9704 4/16/2008 63.236 4/16/2008 78.7334
4/17/2008 39.9674 4/17/2008 63.425 4/17/2008 78.6922
4/18/2008 39.9306 4/18/2008 63.6027 4/18/2008 78.9796
4/19/2008 39.925 4/19/2008 63.2923 4/19/2008 79.5749
4/20/2008 39.925 4/20/2008 63.1685 4/20/2008 79.7881
4/21/2008 39.925 4/21/2008 63.1693 4/21/2008 79.7901
4/22/2008 39.93 4/22/2008 63.3226 4/22/2008 79.513
4/23/2008 39.9558 4/23/2008 63.6532 4/23/2008 79.3474
4/24/2008 40.0198 4/24/2008 63.8384 4/24/2008 79.6222
4/25/2008 40.185 4/25/2008 63.4389 4/25/2008 79.4144
4/26/2008 40.1626 4/26/2008 62.8379 4/26/2008 79.4472
4/27/2008 40.145 4/27/2008 62.7643 4/27/2008 79.7396
4/28/2008 40.145 4/28/2008 62.7527 4/28/2008 79.7404
4/29/2008 40.1699 4/29/2008 62.8193 4/29/2008 79.8036
4/30/2008 40.3823 4/30/2008 63.0308 4/30/2008 80.0374
5/1/2008 40.4968 5/1/2008 63.0547 5/1/2008 79.8535
5/2/2008 40.51 5/2/2008 63.017 5/2/2008 80.3475
5/3/2008 40.665 5/3/2008 62.8331 5/3/2008 80.4342
5/4/2008 40.665 5/4/2008 62.7424 5/4/2008 80.2024
5/5/2008 40.665 5/5/2008 62.742 5/5/2008 80.2011
5/6/2008 40.5968 5/6/2008 62.7919 5/6/2008 80.0834
Portfolio Value at Risk
M P Birla Institute of Management Page 117
5/7/2008 40.8724 5/7/2008 63.4343 5/7/2008 80.6318
5/8/2008 41.2755 5/8/2008 63.8275 5/8/2008 81.0055
5/9/2008 41.7216 5/9/2008 64.1043 5/9/2008 81.5427
5/10/2008 41.5908 5/10/2008 64.205 5/10/2008 81.2156
5/11/2008 41.5908 5/11/2008 64.415 5/11/2008 81.2838
5/12/2008 41.5908 5/12/2008 64.4125 5/12/2008 81.283
5/13/2008 41.9668 5/13/2008 64.8936 5/13/2008 82.0114
5/14/2008 42.0853 5/14/2008 65.2604 5/14/2008 82.1084
5/15/2008 42.427 5/15/2008 65.5751 5/15/2008 82.4814
5/16/2008 42.6065 5/16/2008 65.9523 5/16/2008 82.9016
5/17/2008 42.6413 5/17/2008 66.0897 5/17/2008 83.1517
5/18/2008 42.6413 5/18/2008 66.4449 5/18/2008 83.4681
5/19/2008 42.6413 5/19/2008 66.444 5/19/2008 83.4677
5/20/2008 42.6413 5/20/2008 66.3754 5/20/2008 83.3466
5/21/2008 42.6385 5/21/2008 66.479 5/21/2008 83.5267
5/22/2008 42.7858 5/22/2008 67.2101 5/22/2008 84.1648
5/23/2008 43.0374 5/23/2008 67.8209 5/23/2008 85.0474
5/24/2008 42.756 5/24/2008 67.3503 5/24/2008 84.6788
5/25/2008 42.756 5/25/2008 67.4147 5/25/2008 84.6646
5/26/2008 42.756 5/26/2008 67.4142 5/26/2008 84.6654
5/27/2008 42.6917 5/27/2008 67.3166 5/27/2008 84.5649
5/28/2008 42.9379 5/28/2008 67.6633 5/28/2008 84.9604
5/29/2008 42.7785 5/29/2008 67.063 5/29/2008 84.5902
5/30/2008 42.7958 5/30/2008 66.7036 5/30/2008 84.5965
5/31/2008 42.533 5/31/2008 66.0378 5/31/2008 84.0523
6/1/2008 42.533 6/1/2008 66.1766 6/1/2008 84.3339
6/2/2008 42.533 6/2/2008 66.1775 6/2/2008 84.3254
6/3/2008 42.3248 6/3/2008 65.7846 6/3/2008 83.3232
6/4/2008 42.612 6/4/2008 66.1714 6/4/2008 83.7463
Portfolio Value at Risk
M P Birla Institute of Management Page 118
6/5/2008 42.7228 6/5/2008 66.0097 6/5/2008 83.679
6/6/2008 42.9022 6/6/2008 66.3203 6/6/2008 83.7823
6/7/2008 42.7086 6/7/2008 66.8953 6/7/2008 83.8197
6/8/2008 42.7086 6/8/2008 67.2016 6/8/2008 84.1268
6/9/2008 42.7086 6/9/2008 67.407 6/9/2008 84.186
6/10/2008 42.8898 6/10/2008 67.5944 6/10/2008 84.6049
6/11/2008 42.944 6/11/2008 66.7894 6/11/2008 84.3204
6/12/2008 42.8786 6/12/2008 66.464 6/12/2008 83.9375
6/13/2008 42.844 6/13/2008 66.2548 6/13/2008 83.7052
6/14/2008 42.9217 6/14/2008 65.9867 6/14/2008 83.5443
6/15/2008 42.9217 6/15/2008 66.0376 6/15/2008 83.6051
6/16/2008 42.9217 6/16/2008 66.0492 6/16/2008 83.6081
6/17/2008 42.9393 6/17/2008 66.2626 6/17/2008 84.0141
6/18/2008 42.9104 6/18/2008 66.5253 6/18/2008 84.1018
6/19/2008 42.897 6/19/2008 66.5093 6/19/2008 83.8671
6/20/2008 42.962 6/20/2008 66.7016 6/20/2008 84.4445
6/21/2008 42.9537 6/21/2008 66.9833 6/21/2008 84.825
6/22/2008 42.9537 6/22/2008 67.0602 6/22/2008 84.8997
6/23/2008 42.9537 6/23/2008 67.0597 6/23/2008 84.8975
6/24/2008 42.9713 6/24/2008 66.8651 6/24/2008 84.5856
6/25/2008 42.9631 6/25/2008 66.8055 6/25/2008 84.5037
6/26/2008 42.7891 6/26/2008 66.6732 6/26/2008 84.3185
6/27/2008 42.7028 6/27/2008 67.0302 6/27/2008 84.5341
6/28/2008 42.8494 6/28/2008 67.4938 6/28/2008 85.2134
6/29/2008 42.8494 6/29/2008 67.6982 6/29/2008 85.503
6/30/2008 42.8494 6/30/2008 67.6969 6/30/2008 85.5026
7/1/2008 43.04 7/1/2008 67.9188 7/1/2008 85.8007
7/2/2008 43.3311 7/2/2008 68.3236 7/2/2008 86.4035
7/3/2008 43.2225 7/3/2008 68.3945 7/3/2008 86.1291
Portfolio Value at Risk
M P Birla Institute of Management Page 119
7/4/2008 43.2825 7/4/2008 68.5093 7/4/2008 86.053
7/5/2008 43.1674 7/5/2008 67.7737 7/5/2008 85.6053
7/6/2008 43.1674 7/6/2008 67.8341 7/6/2008 85.5928
7/7/2008 43.1674 7/7/2008 67.8345 7/7/2008 85.5928
7/8/2008 43.2476 7/8/2008 67.762 7/8/2008 85.4256
7/9/2008 43.3061 7/9/2008 67.9793 7/9/2008 85.4741
7/10/2008 43.1687 7/10/2008 67.7623 7/10/2008 85.1821
7/11/2008 43.2 7/11/2008 67.986 7/11/2008 85.4721
7/12/2008 43.2 7/12/2008 68.3567 7/12/2008 85.593
7/13/2008 43.2 7/13/2008 68.8699 7/13/2008 85.9369
7/14/2008 43.2 7/14/2008 68.8725 7/14/2008 85.9395
7/15/2008 42.8957 7/15/2008 68.2011 7/15/2008 85.2886
7/16/2008 42.8957 7/16/2008 68.3731 7/16/2008 85.8715
7/17/2008 42.8957 7/17/2008 68.1732 7/17/2008 85.9131
7/18/2008 42.8957 7/18/2008 67.9733 7/18/2008 85.8243
7/19/2008 42.8341 7/19/2008 67.8857 7/19/2008 85.5402
7/20/2008 42.69 7/20/2008 67.6747 7/20/2008 85.351
7/21/2008 42.69 7/21/2008 67.6739 7/21/2008 85.351
7/22/2008 42.7258 7/22/2008 67.787 7/22/2008 85.2674
7/23/2008 42.7382 7/23/2008 67.9084 7/23/2008 85.493
7/24/2008 42.3095 7/24/2008 66.6505 7/24/2008 84.4247
7/25/2008 42.0889 7/25/2008 65.9928 7/25/2008 83.8128
7/26/2008 42.2112 7/26/2008 66.3 7/26/2008 84.0208
7/27/2008 42.205 7/27/2008 66.3214 7/27/2008 84.0774
7/28/2008 42.205 7/28/2008 66.3226 7/28/2008 84.0774
7/29/2008 42.3752 7/29/2008 66.6383 7/29/2008 84.3325
7/30/2008 42.5858 7/30/2008 66.8568 7/30/2008 84.7287
7/31/2008 42.47 7/31/2008 66.1797 7/31/2008 84.1101
8/1/2008 42.5239 8/1/2008 66.3385 8/1/2008 84.2747
Portfolio Value at Risk
M P Birla Institute of Management Page 120
8/2/2008 42.28 8/2/2008 65.7116 8/2/2008 83.4438
8/3/2008 42.45 8/3/2008 66.0879 8/3/2008 83.8642
8/4/2008 42.45 8/4/2008 66.0896 8/4/2008 83.8651
8/5/2008 42.4088 8/5/2008 66.0831 8/5/2008 83.5559
8/6/2008 42.2735 8/6/2008 65.5873 8/6/2008 82.7588
8/7/2008 42.089 8/7/2008 65.0751 8/7/2008 82.2264
8/8/2008 42.0547 8/8/2008 64.8303 8/8/2008 81.9163
8/9/2008 42.2122 8/9/2008 64.0212 8/9/2008 81.4383
8/10/2008 42.18 8/10/2008 63.3156 8/10/2008 81.048
8/11/2008 42.1806 8/11/2008 63.3151 8/11/2008 81.0487
8/12/2008 42.1515 8/12/2008 63.1353 8/12/2008 80.8255
8/13/2008 42.3936 8/13/2008 63.161 8/13/2008 80.7323
8/14/2008 42.6474 8/14/2008 63.6068 8/14/2008 80.4632
8/15/2008 42.9413 8/15/2008 63.9255 8/15/2008 80.2852
8/16/2008 43.2526 8/16/2008 63.7596 8/16/2008 80.5922
8/17/2008 43.38 8/17/2008 63.73 8/17/2008 80.9631
8/18/2008 43.38 8/18/2008 63.7313 8/18/2008 80.9662
8/19/2008 43.3873 8/19/2008 63.8688 8/19/2008 81.0107
8/20/2008 43.684 8/20/2008 64.2037 8/20/2008 81.3663
8/21/2008 43.7931 8/21/2008 64.5878 8/21/2008 81.5795
8/22/2008 43.6431 8/22/2008 64.6023 8/22/2008 81.5084
8/23/2008 43.4617 8/23/2008 64.5358 8/23/2008 81.1091
8/24/2008 43.34 8/24/2008 64.1029 8/24/2008 80.353
8/25/2008 43.465 8/25/2008 64.3182 8/25/2008 80.5576
8/26/2008 43.6472 8/26/2008 64.4202 8/26/2008 80.7115
8/27/2008 43.9423 8/27/2008 64.4792 8/27/2008 80.9773
8/28/2008 43.8583 8/28/2008 64.4827 8/28/2008 80.7515
8/29/2008 43.8441 8/29/2008 64.6745 8/29/2008 80.4268
8/30/2008 43.9187 8/30/2008 64.6289 8/30/2008 80.2772
Portfolio Value at Risk
M P Birla Institute of Management Page 121
8/31/2008 44.095 8/31/2008 64.7248 8/31/2008 80.319
9/1/2008 44.095 9/1/2008 64.7297 9/1/2008 80.3111
9/2/2008 44.1851 9/2/2008 64.6874 9/2/2008 79.762
9/3/2008 44.3856 9/3/2008 64.5509 9/3/2008 79.3673
9/4/2008 44.5723 9/4/2008 64.4926 9/4/2008 79.2046
9/5/2008 44.5075 9/5/2008 64.3765 9/5/2008 79.0595
9/6/2008 44.655 9/6/2008 63.7173 9/6/2008 78.7009
9/7/2008 44.7 9/7/2008 63.7949 9/7/2008 78.963
9/8/2008 44.7003 9/8/2008 63.8074 9/8/2008 78.9805
9/9/2008 44.6106 9/9/2008 63.6973 9/9/2008 79.1241
9/10/2008 44.865 9/10/2008 63.4086 9/10/2008 78.966
9/11/2008 45.1367 9/11/2008 63.6915 9/11/2008 79.4496
9/12/2008 45.578 9/12/2008 63.5767 9/12/2008 79.8303
9/13/2008 45.7102 9/13/2008 64.3385 9/13/2008 80.8806
9/14/2008 45.7291 9/14/2008 65.0871 9/14/2008 82.0658
9/15/2008 45.73 9/15/2008 65.1012 9/15/2008 82.0716
9/16/2008 45.887 9/16/2008 65.5152 9/16/2008 82.465
9/17/2008 46.5038 9/17/2008 66.1084 9/17/2008 83.2622
9/18/2008 46.5564 9/18/2008 66.0896 9/18/2008 83.3531
9/19/2008 46.6639 9/19/2008 67.0938 9/19/2008 84.89
9/20/2008 46.6639 9/20/2008 66.7915 9/20/2008 84.7171
9/21/2008 45.37 9/21/2008 65.659 9/21/2008 83.1206
9/22/2008 45.3713 9/22/2008 65.6659 9/22/2008 83.1257
9/23/2008 45.3919 9/23/2008 66.1877 9/23/2008 83.5451
9/24/2008 45.8024 9/24/2008 67.5846 9/24/2008 84.9937
9/25/2008 46.2718 9/25/2008 67.9081 9/25/2008 85.8069
9/26/2008 46.6197 9/26/2008 68.4107 9/26/2008 86.3127
9/27/2008 46.54 9/27/2008 68.021 9/27/2008 85.6428
9/28/2008 47.095 9/28/2008 68.8392 9/28/2008 86.8874
Portfolio Value at Risk
M P Birla Institute of Management Page 122
9/29/2008 47.095 9/29/2008 68.8402 9/29/2008 86.8874
9/30/2008 47.3476 9/30/2008 68.4112 9/30/2008 86.0519
10/1/2008 47.9547 10/1/2008 68.5489 10/1/2008 86.2701
10/2/2008 47.3634 10/2/2008 66.7422 10/2/2008 84.2823
10/3/2008 47.3678 10/3/2008 65.908 10/3/2008 83.6846
10/4/2008 47.6787 10/4/2008 65.9349 10/4/2008 84.357
10/5/2008 46.9992 10/5/2008 64.7659 10/5/2008 83.2953
10/6/2008 47.0107 10/6/2008 64.7436 10/6/2008 83.2898
10/7/2008 47.9792 10/7/2008 65.2109 10/7/2008 84.2553
10/8/2008 48.4806 10/8/2008 65.8085 10/8/2008 84.7857
10/9/2008 48.9225 10/9/2008 66.6931 10/9/2008 85.381
10/10/2008 49.0448 10/10/2008 67.0413 10/10/2008 84.7038
10/11/2008 49.0448 10/11/2008 66.4082 10/11/2008 83.3659
10/12/2008 50.63 10/12/2008 67.8817 10/12/2008 86.349
10/13/2008 50.63 10/13/2008 67.9338 10/13/2008 86.3778
10/14/2008 49.8905 10/14/2008 67.7543 10/14/2008 85.8361
10/15/2008 49.156 10/15/2008 67.1712 10/15/2008 85.9832
10/16/2008 49.9261 10/16/2008 67.8346 10/16/2008 87.0936
10/17/2008 50.1418 10/17/2008 67.4362 10/17/2008 86.4946
10/18/2008 49.878 10/18/2008 67.1228 10/18/2008 86.3762
10/19/2008 50.3544 10/19/2008 67.5675 10/19/2008 87.102
10/20/2008 50.355 10/20/2008 67.5724 10/20/2008 87.1091
10/21/2008 50.303 10/21/2008 67.4966 10/21/2008 87.1058
10/22/2008 50.4755 10/22/2008 66.8694 10/22/2008 86.1444
10/23/2008 51.2318 10/23/2008 66.1818 10/23/2008 84.1237
10/24/2008 52.1082 10/24/2008 66.8047 10/24/2008 84.4908
10/25/2008 52.1082 10/25/2008 65.9489 10/25/2008 82.2297
10/26/2008 52.1082 10/26/2008 66.1926 10/26/2008 82.9299
10/27/2008 53.7639 10/27/2008 67.8963 10/27/2008 85.5701
Portfolio Value at Risk
M P Birla Institute of Management Page 123
10/28/2008 53.7322 10/28/2008 67.2614 10/28/2008 83.8814
10/29/2008 53.3624 10/29/2008 66.6534 10/29/2008 83.3084
10/30/2008 53.0901 10/30/2008 67.8778 10/30/2008 85.5318
10/31/2008 52.3572 10/31/2008 68.3528 10/31/2008 86.2835
11/1/2008 52.3572 11/1/2008 66.7114 11/1/2008 84.7189
11/2/2008 50.7156 11/2/2008 64.5721 11/2/2008 81.5907
11/3/2008 50.715 11/3/2008 64.5856 11/3/2008 81.602
11/4/2008 50.1156 11/4/2008 64.0973 11/4/2008 80.7833
11/5/2008 49.0971 11/5/2008 62.4922 11/5/2008 77.761
11/6/2008 48.0973 11/6/2008 62.223 11/6/2008 76.7089
11/7/2008 48.6559 11/7/2008 62.5554 11/7/2008 77.2115
11/8/2008 48.813 11/8/2008 62.2409 11/8/2008 76.6246
11/9/2008 48.9943 11/9/2008 62.336 11/9/2008 76.6894
11/10/2008 49.005 11/10/2008 62.3633 11/10/2008 76.7276
11/11/2008 48.4696 11/11/2008 62.2471 11/11/2008 76.4007
11/12/2008 48.5775 11/12/2008 61.6939 11/12/2008 75.6692
11/13/2008 49.7131 11/13/2008 62.3721 11/13/2008 76.0601
11/14/2008 50.4523 11/14/2008 63.0447 11/14/2008 75.0413
11/15/2008 50.2804 11/15/2008 63.9994 11/15/2008 74.5331
11/16/2008 49.876 11/16/2008 62.9271 11/16/2008 73.5916
11/17/2008 49.88 11/17/2008 62.8842 11/17/2008 73.5481
11/18/2008 49.9824 11/18/2008 63.0683 11/18/2008 74.1618
11/19/2008 49.9824 11/19/2008 63.1307 11/19/2008 74.9631
11/20/2008 50.7945 11/20/2008 64.1464 11/20/2008 76.2268
11/21/2008 51.4512 11/21/2008 64.4108 11/21/2008 76.6598
11/22/2008 51.2877 11/22/2008 64.2116 11/22/2008 76.1114
11/23/2008 51.2212 11/23/2008 64.4896 11/23/2008 76.4508
11/24/2008 51.23 11/24/2008 64.5119 11/24/2008 76.4874
11/25/2008 50.8706 11/25/2008 64.5573 11/25/2008 76.2026
Portfolio Value at Risk
M P Birla Institute of Management Page 124
11/26/2008 50.5224 11/26/2008 65.2351 11/26/2008 76.7047
11/27/2008 50.0581 11/27/2008 64.9318 11/27/2008 76.9187
11/28/2008 50.3776 11/28/2008 65.0148 11/28/2008 77.5417
11/29/2008 50.4807 11/29/2008 64.8051 11/29/2008 77.661
11/30/2008 51.0888 11/30/2008 64.8659 11/30/2008 78.5603
12/1/2008 51.105 12/1/2008 64.8916 12/1/2008 78.576
12/2/2008 50.8859 12/2/2008 64.4373 12/2/2008 77.167
12/3/2008 50.9114 12/3/2008 64.4126 12/3/2008 75.8854
12/4/2008 50.3924 12/4/2008 63.887 12/4/2008 74.7178
12/5/2008 50.1842 12/5/2008 63.7009 12/5/2008 73.801
12/6/2008 49.9567 12/6/2008 63.6104 12/6/2008 73.2436
12/7/2008 50.1627 12/7/2008 63.8215 12/7/2008 73.7632
12/8/2008 50.3633 12/8/2008 64.0833 12/8/2008 74.0512
12/9/2008 50.3355 12/9/2008 64.5845 12/9/2008 74.6033
12/10/2008 50.7878 12/10/2008 65.4746 12/10/2008 75.266
12/11/2008 50.3189 12/11/2008 65.2103 12/11/2008 74.4373
12/12/2008 49.7178 12/12/2008 65.3406 12/12/2008 74.1108
12/13/2008 48.4496 12/13/2008 64.6417 12/13/2008 72.4101
12/14/2008 48.2393 12/14/2008 64.5929 12/14/2008 71.7804
12/15/2008 48.4496 12/15/2008 64.8589 12/15/2008 72.2708
12/16/2008 49.5213 12/16/2008 66.8651 12/16/2008 74.6637
12/17/2008 49.3497 12/17/2008 67.7191 12/17/2008 75.5203
12/18/2008 48.9631 12/18/2008 69.2299 12/18/2008 76.0427
12/19/2008 48.4778 12/19/2008 69.9967 12/19/2008 74.6757
12/20/2008 48.5052 12/20/2008 68.4927 12/20/2008 72.8853
12/21/2008 48.7186 12/21/2008 67.8008 12/21/2008 72.7657
12/22/2008 48.725 12/22/2008 67.8145 12/22/2008 72.7986
12/23/2008 49.1347 12/23/2008 68.7045 12/23/2008 73.1252
12/24/2008 50.0344 12/24/2008 69.927 12/24/2008 74.0949
Portfolio Value at Risk
M P Birla Institute of Management Page 125
12/25/2008 50.3209 12/25/2008 70.3154 12/25/2008 74.2228
12/26/2008 50.4475 12/26/2008 70.7053 12/26/2008 74.3783
12/27/2008 49.4729 12/27/2008 69.5193 12/27/2008 72.9309
12/28/2008 49.205 12/28/2008 69.0686 12/28/2008 71.7281
12/29/2008 49.205 12/29/2008 69.0681 12/29/2008 71.7242
12/30/2008 49.6782 12/30/2008 70.4919 12/30/2008 72.8049
12/31/2008 49.7178 12/31/2008 70.0891 12/31/2008 71.9874
1/1/2009 49.9002 1/1/2009 70.0738 1/1/2009 72.3597
1/2/2009 50.095 1/2/2009 70.0238 1/2/2009 73.2975
1/3/2009 49.8918 1/3/2009 69.4753 1/3/2009 72.7512
1/4/2009 49.655 1/4/2009 69.1426 1/4/2009 72.2793
1/5/2009 49.655 1/5/2009 69.1337 1/5/2009 72.2714
1/6/2009 49.52 1/6/2009 68.1183 1/6/2009 72.0189
1/7/2009 49.6709 1/7/2009 67.0423 1/7/2009 72.9775
1/8/2009 49.7595 1/8/2009 67.5664 1/8/2009 74.5766
1/9/2009 50.255 1/9/2009 68.5569 1/9/2009 76.0299
1/10/2009 49.7903 1/10/2009 67.8279 1/10/2009 75.7381
1/11/2009 49.555 1/11/2009 66.7982 1/11/2009 75.168
1/12/2009 49.5555 1/12/2009 66.7999 1/12/2009 75.1574
1/13/2009 49.7973 1/13/2009 66.7338 1/13/2009 74.6775
1/14/2009 50.0963 1/14/2009 66.5143 1/14/2009 73.5013
1/15/2009 49.9717 1/15/2009 66.0296 1/15/2009 72.7938
1/16/2009 50.0063 1/16/2009 65.7747 1/16/2009 73.0401
1/17/2009 49.5689 1/17/2009 65.5852 1/17/2009 73.3467
1/18/2009 49.505 1/18/2009 65.707 1/18/2009 72.959
1/19/2009 49.505 1/19/2009 65.7436 1/19/2009 73.007
1/20/2009 49.4202 1/20/2009 65.5435 1/20/2009 72.6531
1/21/2009 49.8541 1/21/2009 64.7121 1/21/2009 70.4174
1/22/2009 49.9315 1/22/2009 64.5005 1/22/2009 69.1342
Portfolio Value at Risk
M P Birla Institute of Management Page 126
1/23/2009 49.6635 1/23/2009 64.5645 1/23/2009 68.8853
1/24/2009 49.7772 1/24/2009 64.2729 1/24/2009 68.3651
1/25/2009 49.685 1/25/2009 64.5552 1/25/2009 68.6249
1/26/2009 49.685 1/26/2009 64.5557 1/26/2009 68.6205
1/27/2009 49.7264 1/27/2009 64.6254 1/27/2009 68.4494
1/28/2009 49.5385 1/28/2009 65.4037 1/28/2009 69.7785
1/29/2009 49.3836 1/29/2009 65.3606 1/29/2009 70.3918
1/30/2009 49.4441 1/30/2009 64.7066 1/30/2009 70.3654
1/31/2009 49.432 1/31/2009 63.6358 1/31/2009 70.7234
2/1/2009 49.7295 2/1/2009 63.7383 2/1/2009 72.3276
2/2/2009 49.73 2/2/2009 63.728 2/2/2009 72.3084
2/3/2009 49.5849 2/3/2009 63.3214 2/3/2009 70.8905
2/4/2009 49.2849 2/4/2009 63.5115 2/4/2009 70.3429
2/5/2009 48.9808 2/5/2009 63.4017 2/5/2009 70.6945
2/6/2009 49.1162 2/6/2009 63.0588 2/6/2009 71.3187
2/7/2009 48.9725 2/7/2009 62.8082 2/7/2009 71.8975
2/8/2009 49.28 2/8/2009 63.793 2/8/2009 72.8999
2/9/2009 49.28 2/9/2009 63.7969 2/9/2009 72.895
2/10/2009 48.974 2/10/2009 63.5438 2/10/2009 72.6936
2/11/2009 49.04 2/11/2009 63.392 2/11/2009 72.4468
2/12/2009 49.0957 2/12/2009 63.3851 2/12/2009 70.8928
2/13/2009 49.0802 2/13/2009 63.122 2/13/2009 70.1537
2/14/2009 49.0117 2/14/2009 63.18 2/14/2009 70.5725
2/15/2009 49.255 2/15/2009 63.3794 2/15/2009 70.7376
2/16/2009 49.255 2/16/2009 63.3626 2/16/2009 70.6726
2/17/2009 49.0957 2/17/2009 62.7164 2/17/2009 69.9349
2/18/2009 49.7212 2/18/2009 62.9117 2/18/2009 70.7891
2/19/2009 50.2348 2/19/2009 63.2145 2/19/2009 71.4766
2/20/2009 50.1322 2/20/2009 63.328 2/20/2009 71.7407
Portfolio Value at Risk
M P Birla Institute of Management Page 127
2/21/2009 50.1789 2/21/2009 63.5014 2/21/2009 71.7067
2/22/2009 50.685 2/22/2009 65.038 2/22/2009 73.178
2/23/2009 50.685 2/23/2009 65.0136 2/23/2009 73.1547
2/24/2009 50.1541 2/24/2009 64.3603 2/24/2009 72.7355
2/25/2009 50.167 2/25/2009 63.9674 2/25/2009 72.7506
2/26/2009 50.1797 2/26/2009 64.2953 2/26/2009 72.4259
3/1/2009 51.975 3/1/2009 65.8752 3/1/2009 74.4246
3/2/2009 51.975 3/2/2009 65.8643 3/2/2009 74.4173
3/3/2009 52.1604 3/3/2009 65.6814 3/3/2009 73.8868
3/5/2009 51.93 3/5/2009 65.1415 3/5/2009 73.0935
3/6/2009 51.8575 3/6/2009 65.2669 3/6/2009 73.3437
3/7/2009 51.6796 3/7/2009 65.2873 3/7/2009 73.2698
3/8/2009 52.325 3/8/2009 66.2382 3/8/2009 73.7652
3/9/2009 52.325 3/9/2009 66.2403 3/9/2009 73.7662
3/10/2009 52.3981 3/10/2009 66.2679 3/10/2009 73.2971
3/11/2009 51.8655 3/11/2009 65.7996 3/11/2009 71.6625
3/12/2009 52.3691 3/12/2009 66.6088 3/12/2009 72.1054
3/13/2009 52.7869 3/13/2009 67.6384 3/13/2009 73.0512
3/14/2009 51.5464 3/14/2009 66.5402 3/14/2009 71.967
3/15/2009 52.9405 3/15/2009 68.4743 3/15/2009 74.1538
3/16/2009 52.94 3/16/2009 68.4705 3/16/2009 74.1578
3/17/2009 52.5265 3/17/2009 68.0401 3/17/2009 73.89
3/18/2009 52.473 3/18/2009 68.1457 3/18/2009 73.7734
3/19/2009 52.4997 3/19/2009 68.6748 3/19/2009 73.5961
3/20/2009 51.8076 3/20/2009 70.1532 3/20/2009 74.3621
3/21/2009 51.5785 3/21/2009 70.2886 3/21/2009 74.6831
3/22/2009 52.195 3/22/2009 70.9132 3/22/2009 75.5215
3/23/2009 52.195 3/23/2009 70.9147 3/23/2009 75.5199
3/24/2009 51.8337 3/24/2009 70.6613 3/24/2009 75.3118
Portfolio Value at Risk
M P Birla Institute of Management Page 128
3/25/2009 51.6914 3/25/2009 70.277 3/25/2009 75.8498
3/26/2009 51.8867 3/26/2009 70.0418 3/26/2009 75.932
3/27/2009 51.6635 3/27/2009 70.1275 3/27/2009 75.2008
3/28/2009 51.6458 3/28/2009 69.4847 3/28/2009 74.3642
3/29/2009 51.72 3/29/2009 68.7602 3/29/2009 74.0899
3/30/2009 51.72 3/30/2009 68.7509 3/30/2009 74.0558
3/31/2009 52.1743 3/31/2009 68.9097 3/31/2009 74.1579
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