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


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.


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


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.


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


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:


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:


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:


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:


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


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


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.


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


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%


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.


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%


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


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 TAANX ,~ . Thus, if ,~ NX ,

then the matrix A must satisfy the matrix equation TAA . 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

18 182 181 18

18 22221

18 11211

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 TAA 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

1818 1822 1811 1818

2221212

1111

ZaZaZaX

ZaZaX

ZaX


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.


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:


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

,


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:


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.


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


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


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.


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


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 ppx

Txf

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.


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

Pro

ba

bili

ty

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

Pro

ba

bili

ty

No. of exceptions

Histogram of Inaccurate Model (150 observations, 97% CL)


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%


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%


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.


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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