Market Risk Paper

Diversified Portfolio
Value-at-Risk
Calculations for Equities
and Fixed Income
Instruments
Keith Rivera Buenaventura
Ismael Jaime Cruz
Martin Manuel Infante
A paper submitted to the Ateneo de Manila
Department of Mathematics in partial
fulfillment of the degree Master of Applied
Mathematics Major in Mathematical
Finance

1
Buenaventura, Cruz, Infante
No part of this paper may be reproduced without permission from the
authors and/ or the Ateneo de Manila Department of Mathematics

2
Abstract
This paper discusses three methodologies in computing the Value at Risk of a given
portfolio; Historical Simulations, Linear Modeling, and Monte Carlo Simulation. Both
Equities and Fixed Income portfolios are taken into consideration. Importance of
Diversification is highlighted. After which, their application to a local financial
institution’s portfolios are presented. Each method is then compared to the other
models and their respective advantages and disadvantages are discussed. Finally,
back-testing of the models were done to validate them.

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Table of Contents
1. Introduction………………………………………………………………….……………..4
2. VaR Definition…………………………………………………………….………………..4
3. Equities – Linear Model……………………………………………..…………...…….6
4. Linear Model – Advantages & Disadvantages….…………….……..………11
5. Equities – Historical Simulations………………………………………..……….12
6. Historical Simulations – Advantages & Disadvantages….................16
7. Equities – Monte Carlo Simulation…………………………………..……..…...17
8. Monte Carlo Simulation – Advantages & Disadvantages…………...…17
9. Fixed Income – Linear Model………………………….……………………………20
10.Back Testing……………………………………………………..………….……………...24
11.Conclusion………………………………………………………..………….………………32
12.Bibliography………………………………………………..……………….……...………33

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Introduction
The risk management department of a financial institution is required to
efficiently monitor and manage the financial risk its organization is exposed to.
Risk can be measured through the use of various complex mathematical models.
One of the most commonly used models across the financial industry is the
Value-at-Risk (VaR) measure. This provides a way of quantifying the total risk a
financial institution is exposed to and calculates the worst expected loss over a
given time horizon (N days) at a given confidence level (X%) under normal
market conditions. VaR can be used to measure different types of risk such as
market, credit, and operational. In this paper, we will discuss market risk.
VaR Definition
Mismatched positions in a portfolio that is marked-to-market periodically
(daily, weekly, monthly, etc.) based on movements in market parameters (prices,
interest rates, volatilities, etc.) are what constitute market risk. The VaR measure
(hereon referred to as V) is used to summarize the likelihood of an unfavorable
outcome and is used to complete the statement:
I am X percent certain there will not be a loss of more than V pesos in the
next N days.
It measures the loss level over N days that has a probability of (100-X)%
of being exceeded. It therefore involves two parameters, the time horizon and
the confidence level. Given a portfolio over the time horizon N days, it is the loss
corresponding to the (100-X)th percentile of the distribution of the change in the
value of the portfolio. For example, suppose a P100 million portfolio has a 10 day
VaR of P3 million with a 99% confidence level. VaR means that,
I am 99 percent certain there will not be a loss of more than 3 million pesos
in the next 10 days.
It must be noted however, certain constraints involving the VaR measure.
First, VaR is an estimate and not a uniquely defined value. Also, it assumes that
the trading positions in the portfolio are fixed during the given time horizon.

5
Moreover, the measure does not estimate the distribution of the potential losses
in the event the VaR measure is exceeded.
For the parameters, holding periods are usually 1 day or is set N = 1 and
the confidence level 99%. The holding period may also depend on the investment
horizon or reporting horizon and the regulatory requirement. The confidence
level expresses the accuracy or reliability of the result and one can expect the
VaR to approach its true value with a higher confidence level.
There are different methodologies to compute VaR, depending also on the
kind of portfolio or asset class in question. First consider a portfolio of equities.
Here, the market parameters are the closing prices of stocks. The three main
methodologies for this kind of portfolio are Parametric or Model Building,
Historical Simulations, and Monte Carlo Simulations.
To begin, assume a lognormal property of stocks. This means that or
the stock price at a future time T has a lognormal distribution. Let be the stock
price at time 0, μ the expected return on stock per year, and σ the volatility of the
stock per year. So that
From this, it follows that
and
The standard deviation σ in the equation above can be estimated using
historical data.
Define the following:
n + 1: Number of observations (stock prices)
: Stock price at the end of the ith interval, with i = 0, 1, … , n

6
τ: Length of time interval in years
for i = 1, 2, … , n
The estimate, s, of the standard deviation of is given by the equation
where is the mean of .
Equities – Linear Model
In the Model Building approach, daily volatility day is approximately
equal to the standard deviation of the percentage change in the asset price in one
day. Daily volatility is used in this case, in order to represent the uncertainty of
the asset price tomorrow. For a 10-day VaR for example, the 10-day volatility of
the asset price would be used and so on. This can be computed using the
following equation:
year = day 2 0 day =
year
2 0
Given a position , the change in value over one day will not exceed more
than about 2.326 standard deviations from the mean of the market variable ,
where is a random variable that follows the following distribution:
day
This is because for a 1-day 99% VaR, the probability that the change in the
asset price, denoted as , will only exceed a certain limit is .01 or one percent.
The limit mentioned here is a level of loss corresponding to exposure to that
asset. Let be the portfolio such that it is the position multiplied by the asset
price:

7
The change in portfolio value should exceed a certain only 1% of the
time
r = 01
r
day day
= 01
1 r
day day
= 1 01
1 r
day day
=
Because it has been transformed to follow a Standard Normal
Distribution, take the cdf values such that =
day
. Therefore:
day
=
day
= 1
= day
1
= day
1
where V is the 1-day 99% VaR.
Suppose the Portfolio now is made up of two market variables X and Y. To
simply add their daily volatilities to get the portfolio’s daily volatility would not
take into consideration how the securities are correlated with each other. It
would be tantamount to assuming they are independent with each other and
therefore, uncorrelated. This is not always the case. So denote as the
correlation measure between the two market variables such that:

8
where is defined as . Define and as the
percentage changes in and :
= , =
In this case the volatility will be given by:
= 2 2 2
So the new portfolio VaR, with portfolio mean and portfolio volatility
, will be = 1
.
Extending this to n-assets with positions , market variables
, daily volatilities 1 … , n, the Linear Model is given by:
with
or in matrix form:
; where for i=j.
There are several assumptions for this model. First, that the stock prices
follow a lognormal distribution. Secondly, the mean change in daily returns is 0.
In practice, the daily mean is so small that it can be taken as negligible. Thirdly,
the stocks pay no dividends. And lastly, there are 260 trading days in a year.
The Linear Model discussed above is implemented on a portfolio
consisting of stocks traded on the Philippine Stock Exchange (PSE). The portfolio
in question is the equities portfolio of Philippine National Bank (PNB) as of
September 30, 2011. The portfolio consists of positions in 18 stocks as follows:

9
Stock Code Position ( )
Metropoilitan Bank & Trust Co. MBT 111,000.00
Energy Development Corporation EDC 3,400,000.00
Petron Corporation PCOR 200,000.00
First Philippine Holdings FPH 170,000.00
First Gen Corporation FGEN 960,000.00
Alliance Global Group, Inc. AGI 1,020,000.00
Ayala Corporation AC 28,000.00
JG Summit Holdings, Inc. JGS 90,000.00
DMCI Holdings, Inc. DMC 380,000.00
Jollibee Foods Corporation JFC 180,000.00
San Miguel Corporation SMC 223,500.00
Universal Robina Corporation URC 100,000.00
International Container Terminal Services,
Inc. ICT 240,000.00
Philex Mining Corporation PX 370,000.00
Semirara Mining Corporation SCC 37,500.00
Ayala Land, Inc. ALI 545,100.00
SM Prime Holdings, Inc. SMPH 350,000.00
Megaworld Corporation MEG 8,800,000.00
The next step is getting the daily volatilities for each stock. One way of
getting the day of each stock is by acquiring them from a Bloomberg terminal.
Another would be to estimate them by using historical stock prices.
Assuming 260 trading days in a year, the stock prices for each position
dating back to 260 days prior to September 30, 2011 are listed down. Out of the
261 stock prices for each (including the current stock price), 260 daily geometric
returns are computed as shown:

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Next s is found using:
Then set each result as the day of each stock:
Code Position ( ) Daily Volatility
MBT 111,000.00 2.00%
EDC 3,400,000.00 1.71%
PCOR 200,000.00 3.24%
FPH 170,000.00 1.63%
FGEN 960,000.00 1.83%
AGI 1,020,000.00 2.50%
AC 28,000.00 1.72%
JGS 90,000.00 2.43%
DMC 380,000.00 2.42%
JFC 180,000.00 2.20%
SMC 223,500.00 3.42%
URC 100,000.00 2.12%
ICT 240,000.00 2.45%
PX 370,000.00 2.63%
SCC 37,500.00 2.29%
ALI 545,100.00 2.36%
SMPH 350,000.00 2.01%
MEG 8,800,000.00 2.50%
From here, the VaR of each position may be computed and, ultimately, the
portfolio VaR may be attained by simply adding each individual VaR. Doing so

11
would tend to overstate the computed portfolio VaR because it does not take into
consideration, the benefits of diversification. By simple getting the sum of the
individual VaR measures, it assumes that the stocks are completely uncorrelated
with each other, which is not always the case. Hence, given the same historical
stock prices, a correlation matrix is formed to measure how much each stock is
correlated with one another
Finally, the daily portfolio volatility for the 18 stock portfolio is
computed:
which was found to be = 2,205,680.88 PHP. Thus, the 1-day 99% VaR is given
by = 1
= 2,20 , 0 1
= 5,131,181.02 PHP. This is
considerably less than the computed undiversified VaR whose comparison is
shown in the following table:
Portfolio VaR w/o Diversification 10,168,737.39
Portfolio VaR w/ Diversification 5,131,181.02
The benefit of diversification can be quantified to be the difference between the
two results. The VaR with diversification is lower by about 50% and so
diversification provides a huge benefit of 5,037556.37 PHP.
Linear Model – Advantages and Disadvantages
An advantage of the Model Building approach is its simplicity of
implementation. It is rather easy to compute for VaR using the parametric

12
approach and it requires few parameters. Its main factor is the expected
volatility, so since we used historical data to compute for volatility then we can
consider the correlation of today’s volatility with yesterday’s In this case, a
common volatility model used to compute for VaR is Generalized Autoregressive
Conditional Heteroscedasticity (GARCH). Also, it is the most time-efficient since
there are no simulations involved. It will only be computer-intensive in the part
of calculating the n(n-1)/2 terms of the Variance-Covariance matrix. A
disadvantage of this approach arises from its assumption of normal distribution.
It is not always the case that historical returns or the change in the price of an
asset follows a normal distribution. Another is that this approach will not
produce an accurate estimate of VaR for securities with non-linear payoff
distributions such as options. Lastly, in the case where historical data show
heavy tails, then VaR with the assumption of normal distribution will be
underestimated at high confidence levels and be overestimated at low
confidence levels.
Equities - Historical Simulation
In Historical Simulations VaR, the assumption in the Linear Model of
normal distribution is addressed. This is such since the Historical Simulations
approach works with the empirical distribution of the returns of the asset. In this
case, it is more logical to use the empirical distribution that captures the
historical behavior of assets and thus the portfolio and additionally, reflects the
correlations between the assets.
Since this model uses historical data, the simple assumption is that the
past performance is an indicator of what future performance will be. That is, the
past will reproduce itself in the near-future.
The first step of the model is to calculate the returns of the assets in the
portfolio between the set time intervals. The time interval may vary from daily,
monthly, quarterly, etc. In this case, since the 1-day VaR is measured, it is daily
returns.

13
Next, the computed price changes are applied to the current value of each asset.
If is the stock price today then,
This is done until the beginning of the chosen period.
, , … ,
for scenarios 1, 2, … , n, m From here, the total value of the portfolio is computed
This is based on the assumption that previous price changes are possible
scenarios that may occur tomorrow with the same likelihood. In other words, it
is assumed that each possible price change has the same probability. Thus it
gives a number of possible values of what the portfolio may be tomorrow. So,
260 historical prices will yield 259 simulations or scenarios for the value of the
portfolio tomorrow. The simulated portfolio values are then compared to the
mark-to-market portfolio in order to give the corresponding profit and loss
values (P&L). The P&Ls are then sorted in order to obtain the worst expected
loss over N days with probability of (100-X)%. In other words, the P&Ls are
sorted in ascending order. Now, based on the chosen confidence interval, the
(100-X)% smallest value in the sorted simulations will be the VaR of the
portfolio. So given a 99% confidence interval over a period of 260 simulations,
the VaR would be the 1% lowest value or the 2.6th smallest value. From here, the
VaR may be found using interpolation between the two successive time intervals
that surround the (100-X)% VaR.
To implement the model for a single-asset portfolio, refer to the table
below. First, the historical prices are listed down. Here, 261 prices are listed in
order to get 260 scenarios, so = 2 0. Next, the returns between each time
interval are computed. Then 260 simulated prices are obtained by applying the
returns to the current price of the asset which is P66.00. If the confidence
interval is 99%, then the VaR will refer to the 1% smallest value among the 260
scenarios. So the 2.6th smallest value will give the desired VaR. This is found by
interpolating the 2nd and 3rd smallest value, 3.68 and 3.47 respectively. This gives

14
3.55 which is then multiplied to the position in the stock to give a 1-day 99% VaR
as 394,146.69 PHP.
For the portfolio consisting of 18 assets, the computation of VaR is similar.
Remember that there is no need to compute for a correlation matrix for this
model since correlations are already embedded in the price changes of the
stocks. This provides a little ease in that it avoids the computation of a
correlation matrix. Similar steps are done for the other assets and the simulated
P&Ls of each are then summed together in order to obtain the corresponding
portfolio P&Ls.
The historical stock prices for each position are listed and here .
0
10
20
30
40
50
60
-5.7
%
-4.9
%
-4.2
%
-3.4
%
-2.6
%
-1.9
%
-1.1
%
-0.3
%
0.5% 1.2% 2.0% 2.8% 3.6% 4.3% 5.1% 5.9%
Histogram of Returns
99% VaR
394,147
5.4%

15
Similarly, 260 returns give simulations for the stock price of each position.
These simulated stock prices are then used to compute for the portfolio value.
Thus there are 260 scenarios for the portfolio value tomorrow.

16
When these values are sorted and then interpolated to find the 2.6th smallest
value, 6,590,478.62 PHP is obtained.
In comparison, the VaR of the portfolio without diversification being taken into
account gives 10,223,039.76 PHP. Again, this is the sum of the individual VaRs of
the stocks. The VaR with diversification is lower by 36% or shows a benefit of
3,632,561.14 PHP.
Historical Simulations – Advantages and Disadvantages
As mentioned earlier, the advantage of using Historical Simulations is
there is no need to make any assumption on the return distribution of the assets
in the portfolio. Another is in the number of computations since estimates for
individual volatilities and correlations are not needed. This is because they are
already implicitly captured in the historical data of the assets. Next, in contrast to
the Model Building approach, fat tails and extreme events in the distribution are
0
10
20
30
40
50
60
-5.7%-5.1%-4.5%-4.0%-3.4%-2.9%-2.3%-1.7%-1.2%-0.6%-0.1%0.5% 1.0% 1.6% 2.2% 2.7%
Histogram of Returns
99% VaR
6,590,479
3.6%

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captured in the historical data, provided the data have covered such events.
However, from its first advantage there also arises a shortfall. Because it is based
on historical data, the computed VaR will be biased to the characteristics of the
data used. For example, with historical data over a bull market, the VaR measure
may be underestimated. Another variable that may lead to inaccuracy is the
range of the data. Using data over a short period of time may distort the VaR
measure while using data over too long a period may include market trends or
cycles that are irrelevant. Another drawback is that for large portfolios, the
model may not be computationally efficient.
Equities - Monte Carlo Simulation
Another method to compute the market VaR is by using the Monte Carlo
simulation approach. Monte Carlo simulation corresponds to an algorithm of
generating random numbers that are used to compute for a formula that does
not have a closed analytical form. Monte Carlo simulation method is similar to
historical simulation approach in computing for the VaR. The only substantial
difference is that Monte Carlo simulation uses random numbers for the behavior
of stock prices while the historical simulation uses empirical or historical past
prices. Random numbers are used to estimate the arithmetic return of the price
of the stock over a given time horizon.
The following procedures, as discussed in one of the classes in Risk
Management course, are followed in generating the random numbers for the
Monte Carlo simulation approach. For the eighteen different stocks considered in
computing for the September 30, 2011 VaR of PNB, generate random numbers
18321 ,,,, UUUU  from the uniform distribution  1,0 using the Excel function
RAND( ). From these, independent standard normal random variables
18321 ,,,, ZZZZ  are obtained using the Excel function NORMSINV(RAND( ) ).
This means that )(NORMSINV ii UZ  .
Assume that Z, which contains component vectors 18321 ,,,, ZZZZ  , is
normally distributed with the zero matrix as its mean and the identity matrix as

18
its standard deviation, that is,  INZ ,~  . It must be noted that even if all the
component vectors 18321 ,,,, ZZZZ  are normally distributed, then Z is not
necessarily normal. In this procedure, the normality of Z is just an assumption of
the joint distribution of the component vectors. From the linear transformation
property, if  INZ ,~  and AZX  , then  T
AANX ,~  . Thus, if  ,~ NX ,
then the matrix A must satisfy the matrix equation T
AA . Such matrix A is not
unique, however, the next step is to generate a possible matrix A. The method of
finding such matrix is called the Cholesky decomposition or factorization. The
covariance matrix is given by





































18
2
1
1818218118
1822221
1811211
18
2
1
00
00
00
00
00
00





















Among all such matrices, a lower triangular one is particularly convenient
since it reduces the calculation of AZ to the following:
A representation of  as T
AA with a lower triangular matrix A is called
the Cholesky decomposition of . For n = 18 stocks and using the matrix
equation AZX  , each iX are computed as described by the systems of
equations above.
Now, in computing for the VaR, the portfolio today is valued in the same way as
historical simulation approach using the current values of market variables.
Then, a sample from multivariate normal probability distribution of the
percentage change of the stock price is generated using the Cholesky
1818182218111818
2221212
1111
ZaZaZaX
ZaZaX
ZaX






19
factorization method described above. These percentage changes of the stock
price are used to value each of the prices of stock for tomorrow. Then, the
portfolio tomorrow is revalued in the same way as in the historical simulation
approach. A sample for the change in portfolio is obtained by subtracting the
generated value of the portfolio tomorrow by the value of the portfolio today.
These steps are to be repeated thousands of times to generate a probability
distribution for P , the change in portfolio’s value over a one day horizon Note
that for a 15,000 values of P that are generated, the 1-day, 99% VaR is the
150th worst outcome.
Monte Carlo Simulation – Advantages and Disadvantages
Monte Carlo simulation has some advantages over the other two methods
in computing for the VaR. The main advantage of using Monte Carlo simulation
is that it can model instruments with non-linear and path-dependent payoff
functions, especially complex derivatives. Monte Carlo simulation is also not
covered by extreme shocks in contrast to the Historical simulation approach.
Moreover, the statistical distribution used is normal for this study but actually,
any statistical distribution can be used as an underlying assumption for the
simulation model.
The main disadvantage of using Monte Carlo simulation in computing for
the aR lies obviously on the power of the bank’s computer that will be required
to perform all the simulations. The longer time it takes to run the simulation is
also a disadvantage. If, for example, a portfolio is composed of a thousand
different stocks, a thousand simulations on each stock would mean a million
simulation runs all in all. Moreover, the cost associated with developing a
computer program that uses Monte Carlo simulation in computing the VaR is
another drawback. Meanwhile, using Monte Carlo simulation has been the
industry standard for estimating the VaR of a portfolio of assets. The advantages
outweigh the disadvantages by far.

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Fixed Income – Linear Model
For a portfolio of Fixed Income instruments, a similar method may be
used to compute for its 1-day 99% VaR. Historical changes in the bond prices
may be used to generate possible scenarios for the bond price tomorrow. This,
however, may be tedious for most treasury departments which usually have
positions in more than a hundred fixed income instruments. Furthermore, the
market values for such instruments may not be as readily available as the prices
of stocks, which are publicly declared. Thus, another market variable is used,
namely, the Yield to Maturity (YTM). Defined as the rate of return given by a
bond if it is held until maturity, it may be used to emulate small changes in bond
price. To do so, one must first recall the concept of Modified Duration.
Given that a bond’s current price is the present value of all future cash
flows, the (Macaulay) duration of a bond is defined to be the weighted average of
those future cash flows. Modified duration (MD) is therefore duration which
accounts for changing interest rates as shown below. Because as interest rates go
up (down), bond prices tend to go down (up), it can be established that the
market value of bond prices (MV) and YTM are inversely related. This essential
feature of the modified duration is what is then used to capture a bond’s
volatility with respect to yield changes.
Getting the derivative of the market value with respect to the YTM, a
formula that relates this inverse proportionality of the market value to YTM
changes is captured. For coupons paid times a year, cash flows as , and
letting YTM simply be , the above equations may be written as such:

21
Since ,
Therefore, for small changes in the YTM, , and the following holds:
Getting the 1-day 99% VaR for a single bond with market value MV is the
same as finding V such that:
Letting , we get
Assuming that the daily arithmetic return ,

22
Substituting back,
The standard deviation of the continuously compounded returns is taken
and is used to approximate the standard deviation of the daily arithmetic
returns. So the in the above equation is taken from the historical yields of the
bond.
Doing the same process for all the fixed income instruments in a portfolio,
simply summing up these values will attain an undiversified portfolio VaR. For a
portfolio of instruments:
To take in consideration how each bond is correlated with each other, a
correlation matrix is formed from the arithmetic returns of the historical yields.
The diversified portfolio VaR is then given by the following:
To illustrate this method, a portfolio consisting of three corporate bonds
is used. From the historical yields of each bond, daily arithmetic and continuous
compounded returns are taken and the mean and standard deviation of the latter
is computed:

23
The VaR of each bond is calculated using the current yield, the computed
modified duration, and its current market value.
Next, a correlation matrix is produced from the historical yields.
This is used to get the diversified Portfolio VaR.

24
Back-testing
The accuracy and reliability of the VaR measure has little value without
back-testing the models used. It is essential that the VaR measures obtained from
the selected models are validated against realized and actual profit and losses.
Back-testing is a model validation procedure designated by the Basel
Committee on Banking Supervision referred to as “the Basel Committee” as a
tool to check the quality and accuracy of Value-at-Risk models of banks. The
rationale of back-testing procedure is to compare the actual trading outcomes
with the model-generated risk measures such as the Market VaR. If comparison
of the two values is close enough, no issues regarding the quality of risk
measurement model can be raised. In some cases, however, when comparison
gives such a big difference, it can be said that the problem lies on either the
methodologies of the model or the choice of assumptions of the back-testing
procedure. In between these two cases where comparison cannot be easily
determined lies a grey area in which the test on the accuracy of the model is
inconclusive.
The procedure on back-testing is sometimes referred to as the reality
check of the bank’s aR models This procedure will provide ideas for
improvement on faulty assumptions, wrong parameters, inaccurate modeling, or
erroneous risk measurement methodology.
Essentially, back-testing procedure consists of a statistical test on the
periodic comparison of the bank’s daily market aR with the subsequent daily
trading outcomes, which are given by the actual profit and loss (referred to as
“ &L” of the bank’s portfolio Comparing the model’s risk measures with the
corresponding daily P&L means that the bank has to count the number of times
that the actual trading outcome exceeds the risk measure. Since VaR measure
provides an estimate of the amount that a bank can lose on a particular portfolio
of stock positions due to general movements on several market variables over a
given holding period under a specified statistical level of confidence, the back-
testing procedure should be a comparison on whether the observed percentage

25
of trading outcomes covered by the VaR measure is actually consistent with the
specified level of confidence. In this case, specifically, a 99% level of confidence is
used. This means that the risk measurement model is perfectly calibrated when
the number of P&L exceedences against the daily VaR is statistically in line with
the specified confidence level. With too many exceedences, the model is said to
be underestimating the risk. This yields a major problem to banks since too little
amount of capital may be allocated to risk-taking stocks and penalties may also
be imposed by regulators such as the Bangko Sentral ng Pilipinas.
Before addressing the statistical part of the back-testing procedure, Jorion
imposes a serious data problem that needs to be initially recognized. One of the
assumptions on measuring value-at-risk is that the current portfolio is “frozen”
over the time horizon, that is, only the end-of-day trading positions are used in
measuring the aR, which asses the possible change in the value of the “frozen”
portfolio due to price and rate movements over the time horizon. In practice,
however, the trading portfolio changes dynamically during the day. Thus,
according to Jorion, the actual trading portfolio is said to be “contaminated” by
changes in its composition during the day. Simply put, this argument means that
using a longer holding period on VaR measure (i.e. a ten-day, 99% VaR) will
surely have a “contaminated” trading portfolio since too many significant
changes in the composition of the portfolio will have been happened during a
longer time horizon. For this reason, the Basel Committee suggests a benchmark
in the back-testing framework of comparing risk measures with actual trading
outcomes over only a one-day holding period.
For a meaningful verification, Jorion states that, “the risk manager should
track both the actual portfolio return tR and the hypothetical return *
tR that
most closely matches the VaR forecast. The hypothetical return *
tR represents a
frozen portfolio, obtained from fixed positions applied to the actual returns on all
securities, measured from close to close ” Using the actual portfolio return tR
employs a dirty back-testing approach while using the hypothetical return *
tR
employs a clean back-testing approaching. The Basel Committee urges banks to

26
develop different back-testing frameworks which make use of both dirty and
clean approach. The two back-testing approaches have different values, both of
which “provide a strong understanding of the relation between calculated risk
measures and trading outcomes ” In this project, since the only available
information is the close-to-close historical prices and frozen positions on stocks,
only the hypothetical returns can be utilized in the back-testing procedure. Thus,
only the clean back-testing can be presented.
Back-testing procedure is done to both the historical simulation and
linear model approaches. For the back-testing procedure of PNB, the past 150
trading days from March 28, 2011 up to October 28, 2011 are used. The choice of
number of trading days is based on the constraint on the historical data of stock
prices and daily returns. Since the VaR measure for historical simulation
approach uses the past 260 historical prices but the earliest data covered in this
study is February 16, 2010, only the trading days from March to October 2011
have the desired number of historical prices data. Thus, the VaR for historical
simulation approach as well as the stock portfolio’s daily value are computed for
each trading day. In doing the back-testing framework, the VaR on day i is
compared with the trading outcome on day i + 1. For example, the VaR amount
on September 30, 2011 is compared with the actual P&L of the next trading day
which is October 3, 2011. In this case, the September 30 VaR amounted to PhP
3,108,958.38 is compared with the actual P&L on October 3 which is a loss of
PhP 1,856,303.50. This means that the September 30 VaR has properly estimated
the October 3 P&L. For historical simulation approach, the reported number of
exceedences is three. One example is during September 23, 2011 when the VaR
amounted to PhP 2,638,279.95 while the September 26 trading reported a loss of
PhP 4,509,877.85. The other two exceedences happened during August 9 and
September 26.
The following graph below shows the Profit and Loss in blue data points
and the VaR estimates (negative limits) in red data points. The three
exceedences, in blue data points, are shown clearly below the red data points.

27
These are the only three exceedences that occurred in using historical simulation
approach for the back-testing procedure.
Similarly, the back-testing procedure is also done on the linear model
approach Current trading day’s aR value is compared with the next trading
day’s actual &L for the past 1 0-trading period of March to October 2011. The
following parameters are computed for getting the VaR on linear model
approach: the volatility of each stock in the portfolio and the correlation matrix
which is used to diversify the risk for the whole portfolio. The back-testing
simulation done in Excel requires a longer time to compute each daily VaR since
the simulation computes for many parameters simultaneously before computing
finally for the VaR (i.e. the historical geometric returns, daily volatility, and daily
correlation matrix). Back-testing is done to the VaR for both the diversified and
non-diversified portfolio as well as to each of the eighteen stocks considered in
this study. The VaR for diversified portfolio makes use of the matrix of
correlations between each of the stocks in the portfolio while the VaR for non-
diversified portfolio just add the aR for each of the portfolio’s stocks Generally,
the reason for diversification is to lower the VaR estimates compared to just
summing the VaR of each stock of the portfolio.
A very serious problem arises in this back-testing procedure because a
significant number of exceedences is reported. For the back-testing of VaR for
diversified portfolio of stocks, the reported number of exceedences totaled to 18
(5,000,000.00)
(4,000,000.00)
(3,000,000.00)
(2,000,000.00)
(1,000,000.00)
-
1,000,000.00
2,000,000.00
3,000,000.00
4,000,000.00
0 20 40 60 80 100 120 140 160

28
while the reported number of excedences for the back-testing of VaR for non-
diversified portfolio of stocks totaled to 14. Obviously, this means that there may
be something erroneous in the model. In contrast to the result obtained from
back-testing the historical model, the number of exceedances shows that the
linear model must be reviewed and possibly re-calibrated. Also, some
assumptions should be revised because one or more may be causing an
underestimation of the VaR.
The following model verification based on failure rates is discussed by
Jorion “The simplest method to verify the accuracy of the model is to record
failure rate, which gives the proportion of times VaR is exceeded in a given
sample ” Suppose B provides a VaR estimate at the left-tail level (p = 1 – c) for
a total of T days. Then define N to be the number of times the total portfolio loss
exceeds the previous day’s aR estimate Moreover, let N/T, the ratio between
the exceedences and the number of trading days, be the failure rate. Ideally, the
failure rate should give an unbiased estimate for p, that is, should converge to p
as sample size increases.
The setup for this test makes use of the Bernoulli probability distribution.
The number of exceedences x follows the following distribution:
  xTx
pp
x
T
xf







 1)(
This binomial distribution can be used to test whether the number of
exceedences is acceptably small. The following figure below describes the
distribution when the model is correctly calibrated, that is, when p = 0.01, and
when T = 150. The graph shows that under null, more than three exceedences
will be observed 6.5% of the time. The 6.5% number describes the probability of
committing type I error, that is, of rejecting a correct model.

29
Next, the following figure below describes the distribution of exceedences
when the model is incorrectly calibrated, that is, when p = 0.03 instead of 0.01.
The graph shows that the incorrect model will not be rejected more than 33.8%
of the time. This describes the probability of committing a type 2 error, that is, of
not rejecting an incorrect model.
When designing a verification test, the tradeoff between these two types
of error is faced. The table below summarizes the two states of the world, correct
versus incorrect model, and the decision. For back-testing purposes, VaR models
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
40.0%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Probability
No. of exceptions
Histogram of Accurate Model
(150 observations, 99% CL)
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Probability
No. of exceptions
Histogram of Inaccurate Model
(150 observations, 97% CL)

30
are needed to balance type 1 errors against type 2 errors. Ideally, according to
Jorion, one would want to set a low type 1 error and then have a test that creates
a very low type 2 error, in which case the test is said to be powerful.
Decision: Correct Model Incorrect Model
Accept OK Type 2 error
Reject Type 1 error OK
Interpreting the back-testing results, the Basel Committee introduced the
three-zone approach. For a sample size of 150 trading days, the risk model falls
into any of the three zones: green zone, yellow zone, and red zone. In defining the
three zones, The Basel Committee agreed to set the boundaries based on the
cumulative binomial probabilities of certain range of exceptions. The boundary
for yellow zone lies within the points where the cumulative probability equaling
or exceeding 95% and the boundary for red zone lies within the points where the
cumulative probability equaling or exceeding 99.99%. The following table below
shows the number of exceptions or exceedences within the following three
zones.
Zone
Number of
Exceptions
Cumulative
Probability
Green
0 22.10%
1 55.70%
2 80.90%
3 93.50%
Yellow
4 98.20%
5 99.60%

31
The green zone suggests that no problems with the quality or accuracy of
the model occur. In other words, it is improbable that the bank will conclude that
the model is inaccurate. For the VaR measure using historical simulation
approach, the number of exceedences falls into this zone. This might be due to
the use of historical path of prices over the past 260 trading days which certainly
capture the possible change of stock price the following day.
If the number of exceedences is four to six, the model falls into the yellow
zone and will incur corresponding penalty. Within the yellow zone, there are
several possible explanations for an exceedence, some of which go to the
following categories as presented by the Basel Committee. As stated in the
committee’s “Supervisory Framework for the Use of Back-testing in Conjunction
with the Internal Models Approach to Market Risk Capital Requirements” on
January 1996, classifying the reasons for each exceedence is of useful exercise to
the supervisor assessing the bank’s risk measurement model The following
categories are summarized by Jorion as follows:
 Basic integrity of the model: The bank’s program code might not capture
the risk of the positions or an incorrect calculation for model volatilities
and correlations.
 Model’s accuracy could be improved: The model does not measure risk of
some stocks with enough precision.
 Inta-day trading: Positions changed during the day.
 Bad luck: Markets were particularly volatile or correlations changed.
In contrast to the yellow zone, outcomes in the red zone generally lead to an
automatic presumption that a problem exists with the bank’s model This is
because it is extremely unlikely that an accurate model would independently
generate seven or more exceedences from a sample of 150 trading outcomes.
6 99.90%
Red 7 or more 99.99%

32
Falling into the red zone, that is, exceeding the VaR amounts for more than 7
times, automatically generates a penalty to the bank “It should be stressed,
however, that the Basel Committee believes that these exceedences should be
allowed only under the most extraordinary circumstances, and that it is
committed to an automatic and non-discretionary increase in a bank’s capital
requirement for back-testing results that fall into the red zone ”
Conclusion
Value-at-Risk (VaR) is an important tool for measuring financial risk and
uncertainty a financial institution is exposed to. It may be useful to management,
traders, and even investors as it is a single number that can be easily understood
from the phrase, “I am X percent certain there will not be a loss of more than V
pesos in the next N days.” However, it must not be pigeon-holed into just a single
parametric value. It has numerous implications to its user that must not be
neglected. The different methodologies in computing for the VaR measure each
yield its own set of advantages and disadvantages. It is of great value that they
are understood so that its application gives the most accurate result. The VaR of
a portfolio of one asset class may be captured more accurately using a
methodology different from another asset class. In addition, back-testing the
methodologies validates and fine tunes the models to fit the portfolio more
accurately. It provides a way of assessing the assumptions, which may lead to
updates and corrections.

33
Bibliography
Jorion, Philippe, Value at Risk, The New Benchmark for Managing Financial Risk,
2nd ed. McGraw Hill, 2002. pp. 129-142.
Prepared by Jose Oliver Q. Suaiso. Backtesting – Theory and Application.
Philippine National Bank Risk Management Division. Oct. 21, 2005.
Basle Committee on Banking Supervision. Supervisory Framework for the Use of
“Backtesting” in Conjunction with the Internal Models Approach to Market Risk
Capital Requirements. January 1996.
Prof. Elvira de Lara-Tuprio. Lecture Notes.
Hull, John C. Options, Futures and Other Derivatives, 7th ed. Prentice Hall, 2009. Pp.
443-462.
Smith, Donald J. A Primer on Bond Portfolio Value at Risk. 2008.
Retrieved January 2012.
http://www.abe.sju.edu/proc2008/smith.pdf
Wimpro Technologies. Generalized VaR Framework. 2008.
Retrieved January 2012.
http://www.wipro.com/documents/insights/whitepaper/generalized_value_at_
risk_framework.pdf
Barry, Romain. Value-at-Risk: An Overview of Analytical VaR. March 2009.
Retrieved December 2010.
http://www.slideshare.net/sharegiant/var-methodologies-jp-morgan-5283452

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