1. RISK MANAGEMENT
SPOTLIGHT
VALUE-AT-RISK: AN OVERVIEW OF ANALYTICAL VAR
by Romain Berry
J.P. Morgan Investment Analytics and Consulting
romain.p.berry@jpmorgan.com
In the last issue, we discussed the principles of a sound risk management function to efficiently manage and monitor the
financial risks within an organization. To many risk managers, the heart of a robust risk management department lies in
risk measurement through various complex mathematical models. But even one who is a strong believer in quantitative risk
management would have to admit that a risk management function that heavily relies on these sophisticated models cannot
add value beyond the limits of understanding and expertise that the managers themselves have towards these very models.
Risk managers relying exclusively on models are exposing their organization to events similar to that of the sub-prime crisis,
whereby some extremely complex models failed to accurately estimate the probability of default of the most senior tranches of
CDOs1. Irrespective of how you put it, there is some sort of human or operational risk in every team within any given organ-
ization. Models are valuable tools but merely represent a means to manage the financial risks of an organization.
This article aims at giving an overview of one of the most wide- that should always be kept in mind when handling VaR.
spread models in use in most of risk management departments VaR involves two arbitrarily chosen parameters: the holding
across the financial industry: Value-at-Risk (or VaR)2. VaR calcu- period and the confidence level. The holding period corre-
lates the worst expected loss over a given horizon at a given sponds to the horizon of the risk analysis. In other words, when
confidence level under normal market conditions. VaR esti- computing a daily VaR, we are interested in estimating the
mates can be calculated for various types of risk: market, credit, worst expected loss that may occur by the end of the next
operational, etc. We will only focus on market risk in this article. trading day at a certain confidence level under normal market
Market risk arises from mismatched positions in a portfolio that conditions. The usual holding periods are one day or one
is marked-to-market periodically (generally daily) based on month. The holding period can depend on the fund’s invest-
uncertain movements in prices, rates, volatilities and other rele- ment and/or reporting horizons, and/or on the local regulatory
vant market parameters. In such a context, VaR provides a requirements. The confidence level is intuitively a reliability
single number summarizing the organization’s exposure to measure that expresses the accuracy of the result. The higher
market risk and the likelihood of an unfavorable move. There the confidence level, the more likely we expect VaR to approach
are mainly three designated methodologies to compute VaR: its true value or to be within a pre-specified interval. It is there-
Analytical (also called Parametric), Historical Simulations, and fore no surprise that most regulators require a 95% or 99%
Monte Carlo Simulations. For now, we will focus only on the confidence interval to compute VaR.
Analytical form of VaR. The two other methodologies will be
treated separately in the upcoming issues of this newsletter. PART 2: FORMALIZATION AND APPLICATIONS
Part 1 of this article defines what VaR is and what it is not, and
describes the main parameters. Then, in Part 2, we mathemati- Analytical VaR is also called Parametric VaR because one of its
cally express VaR, work through a few examples and play with fundamental assumptions is that the return distribution
varying the parameters. Part 3 and 4 briefly touch upon two crit- belongs to a family of parametric distributions such as the
ical but complex steps to computing VaR: mapping positions to normal or the lognormal distributions. Analytical VaR can
risk factors and selecting the volatility model of a portfolio. simply be expressed as:
Finally, in Part 5, we discuss the pros and cons of Analytical
VaR. (1)
PART 1: DEFINITION OF ANALYTICAL VAR where
α
• VaRα is the estimated VaR at the confidence level
VaR is a predictive (ex-ante) tool used to prevent portfolio
managers from exceeding risk tolerances that have been devel- 100 × (1 – α)%.
oped in the portfolio policies. It can be measured at the port- • xα is the left-tail α percentile of a normal distribution
folio, sector, asset class, and security level. Multiple VaR is described in the expression
methodologies are available and each has its own benefits and where R is the expected return. In order for VaR to be mean-
drawbacks. To illustrate, suppose a $100 million portfolio has a ingful, we generally choose a confidence level of 95% or
monthly VaR of $8.3 million with a 99% confidence level. VaR 99%. xα is generally negative.
simply means that there is a 1% chance for losses greater than
$8.3 million in any given month of a defined holding period • P is the marked-to-market value of the portfolio.
under normal market conditions. The Central Limit Theorem states that the sum of a large number
It is worth noting that VaR is an estimate, not a uniquely defined of independent and identically distributed random variables
value. Moreover, the trading positions under review are fixed will be approximately normally distributed (i.e., following a
for the period in question. Finally, VaR does not address the Gaussian distribution, or bell-shaped curve) if the random vari-
distribution of potential losses on those rare occasions when ables have a finite variance. But even if we have a large enough
the VaR estimate is exceeded. We should also bear in mind sample of historical returns, is it realistic to assume that the
these constraints when using VaR. The ease of using VaR is also returns of any given fund follow a normal distribution? Thus, we
its pitfall. VaR summarizes within one number the risk exposure need to associate the return distribution to a standard normal
of a portfolio. But it is valid only under a set of assumptions distribution which has a zero mean and a standard deviation of
1
CDO stands for Collaterized Debt Obligation. These instruments repackage a portfolio of
average- or poor-quality debt into high-quality debt (generally rated AAA) by splitting a
portfolio of corporate bonds or bank loans into four classes of securities, called tranches.
2
Pronounced V’ah’R.
SEPTEMBER 2008 EDITION — 7

2. RISK MANAGEMENT
SPOTLIGHT
one. Using a standard normal distribution enables us to replace replace in (4) the mean of the asset by the weighted mean of
xα by zα through the following permutation: the portfolio, μp and the standard deviation (or volatility) of the
asset by the volatility of the portfolio, σ p. The volatility of a
(2) portfolio composed of two assets is given by:
which yields:
(6)
(3) where
zα is the left-tail α percentile of a standard normal distribution. • w1 is the weighting of the first asset
Consequently, we can re-write (1) as: • w2 is the weighting of the second asset
• σ1 is the standard deviation or volatility of the first asset
(4)
• σ2 is the standard deviation or volatility of the second asset
EXAMPLE 1 – ANALYTICAL VAR OF A SINGLE ASSET • ρ1,2 is the correlation coefficient between the two assets
Suppose we want to calculate the Analytical VaR at a 95% confi- And (4) can be re-written as:
dence level and over a holding period of 1 day for an asset in
which we have invested $1 million. We have estimated3 μ (7)
(mean) and σ (standard deviation) to be 0.3% and 3% respec- Let us assume that we want to calculate Analytical VaR at a 95%
tively. The Analytical VaR of that asset would be: confidence level over a one-day horizon on a portfolio
composed of two assets with the following assumptions:
• P = $100 million
This means that there is a 5% chance that this asset may lose at • w1 = w2 = 50%6
least $46,347 at the end of the next trading day under normal • μ1 = 0.3%
market conditions.
• σ1 = 3%
EXAMPLE 2 – CONVERSION OF THE CONFIDENCE • μ2 = 0.5%
LEVEL4 • σ2 = 5%
Assume now that we are interested in a 99% Analytical VaR of • ρ1,2 = 30%
the same asset over the same one-day holding period. The
corresponding VaR would simply be:
(8)
There is a 1% chance that this asset may experience a loss of at
least $66,789 at the end of the next trading day. As you can
see, the higher the confidence level, the higher the VaR as we EXAMPLE 5 – ANALYTICAL VAR OF A PORTFOLIO
travel downwards along the tail of the distribution (further left COMPOSED OF N ASSETS
on the x-axis).
From the previous example, we can generalize these calcula-
EXAMPLE 3 – CONVERSION OF THE HOLDING tions to a portfolio composed of n assets. In order to keep the
mathematical formulation handy, we use matrix notation and
PERIOD can re-write the volatility of the portfolio as:
If we want to calculate a one-month (21 trading days on
average) VaR of that asset using the same inputs, we can simply
apply the square root of the time5:
(5) (9)
Applying this rule to our examples above yields the following where
VaR for the two confidence levels:
• w is the vector of the weights of the n assets
• w’ is the transpose vector of w
• Σ is the covariance matrix of the n assets
Practically, we could design a spreadsheet in Excel (Exhibit 1) to
EXAMPLE 4 – ANALYTICAL VAR OF A PORTFOLIO calculate Analytical VaR on the portfolio in Example 4.
OF TWO ASSETS
Let us assume now that we have a portfolio worth $100 million
that is equally invested in two distinct assets. One of the main Note that these parameters have to be estimated. They are not the historical parame-
3
ters derived from the series.
reasons to invest in two different assets would be to diversify 4
Note that zα is to be read in the statistical table of a standard normal distribution.
the risk of the portfolio. Therefore, the main underlying ques- 5
This rule stems from the fact that the sum of n consecutive one-day log returns is the n-
tion here is how one asset would behave if the other asset were day log return and the standard deviation of n-day returns is √n × standard deviation of
to move against us. In other words, how will the correlation one-day returns.
6
These weights correspond to the weights of the two assets at the end of the holding
between these two assets affect the VaR of the portfolio? As we period. Because of market movements, there is little likelihood that they will be the
aggregate one level up the calculation of Analytical VaR, we same as the weights at the beginning of the holding period.
SEPTEMBER 2008 EDITION — 8

3. RISK MANAGEMENT
SPOTLIGHT
Exhibit 1 – Excel Spreadsheet to calculate Analytical VaR for PART 4: VOLATILITY MODELS
a portfolio of two assets We can guess from the various expressions of Analytical VaR we
Analytical VaR have used that its main driver is the expected volatility (of the
asset or the portfolio) since we multiply it by a constant factor
Expected
greater than 1 (1.6449 for a 95% VaR, for instance) – as
parameters
opposed to the expected mean, which is simply added to the
p 100,000,000 Asset 1 Asset 2 expected volatility. Hence, if we have used historical data to
Standard derive the expected volatility, we could consider how today’s
w1 50% Deviation 0.03 0.05 volatility is positively correlated with yesterday’s volatility. In
w2 50% that case, we may try to estimate the conditional volatility of the
Correlation asset or the portfolio. The two most common volatility models
μ1 0.3% Matrix 1 0.3 used to compute VaR are the Exponential Weighted Moving
σ1
Average (EWMA) and the Generalized Autoregressive
3% 0.3 1
Conditional Heteroscedasticity (GARCH). Again, in order to be
μ2 0.5% exhaustive on this very important part in computing VaR, we
σ2 5% will discuss these models in a future article.
Covariance
p1,2 30% Matrix PART 5: ADVANTAGES AND DISADVANTAGES OF
Σ 0.00090 0.00045 ANALYTICAL VAR
μp 0.40% 0.00045 0.00250 Analytical VaR is the simplest methodology to compute VaR and
σp 3.28% is rather easy to implement for a fund. The input data is rather
Exposures limited, and since there are no simulations involved, the
Confidence computation time is minimal. Its simplicity is also its main
level w1 0.5 0.5 drawback. First, Analytical VaR assumes not only that the histor-
95% -1.6449 ical returns follow a normal distribution, but also that the
changes in price of the assets included in the portfolio follow a
Σw 0.00068
normal distribution. And this very rarely survives the test of
0.00148
reality. Second, Analytical VaR does not cope very well with
securities that have a non-linear payoff distribution like options
σ 2=w’Σw 0.00108 or mortgage-backed securities. Finally, if our historical series
exhibits heavy tails, then computing Analytical VaR using a
σ 0.03279 normal distribution will underestimate VaR at high confidence
levels and overestimate VaR at low confidence levels.
VaR 4,993,012 .77
Grey = input cells
CONCLUSION
Source: J.P. Morgan Investment Analytics & Consulting. As we have demonstrated, Analytical VaR is easy to implement
as long as we follow these steps. First, we need to collect
It is easy from there to expand the calculation to a portfolio of n
historical data on each security in the portfolio (we advise using
assets. But be aware that you will soon reach the limits of Excel
at least one year of historical data – except if one security has
as we will have to calculate n(n-1)/2 terms for your covariance
experienced high volatility, which would suggest a shorter
matrix.
period of time). Second, if the portfolio has a large number of
underlying positions, then we would need to map them against
PART 3: RISK MAPPING a more manageable set of risk factors. Third, we need to calcu-
In order to cope with an increasing covariance matrix each time late the historical parameters (mean, standard deviation, etc.)
you diversify your portfolio further, we can map each security of and need to estimate the expected prices, volatilities and corre-
the portfolio to common fundamental risk factors and base our lations. Finally we apply (7) to find the Analytical VaR estimate
calculations of Analytical VaR on these risk factors. This process of the portfolio.
is called reverse engineering and aims at reducing the size of
As always when building a model, it is important to make sure
the covariance matrix and speeding up the computational time
that it has been reviewed, fully tested and approved, that a User
of transposing and multiplying matrices. We generally consider
Guide (including any potential code) has been documented and
four main risk factors: Spot FX, Equity, Zero-Coupon Bonds and
will be updated if necessary, that a training has been designed
Futures/Forward. The complexity of this process goes beyond
and delivered to the members of the risk management team and
the scope of this overview of Analytical VaR and will need to be
to the recipients of the outputs of the risk management function,
treated separately in a future article.
and finally that a capable person has been allocated the over-
sight of the model, its current use, and regular refinement.
Opinions and estimates offered in this Investment Analytics and Consulting newsletter constitute our judgment and are subject to change without notice, as are statements
of financial market trends, which are based on current market conditions. We believe the information provided here is reliable, but do not warrant its accuracy or complete-
ness. References to specific asset classes, financial markets, and investment strategies are for information purposes only and are not intended to be, and should not be
interpreted as, recommendations or a substitute for obtaining your own investment advice.
This document contains information that is the property of JPMorgan Chase & Co. It may not be copied, published, or used in whole or in part for any purposes other than
expressly authorized by JPMorgan Chase & Co.
www.jpmorgan.com/visit/iac
SEPTEMBER 2008 EDITION — 9
Digitally signed by Sreehari Menon
Signature Not Verified
Sreehari Menon DN: cn=Sreehari Menon, c=IN
Date: 2010.09.25 07:21:45 Z

4. RISK MANAGEMENT
AN OVERVIEW OF VALUE-AT-RISK:
PART II – HISTORICAL SIMULATIONS VAR
by Romain Berry
J.P. Morgan Investment Analytics and Consulting
romain.p.berry@jpmorgan.com
This article is the third in a series of articles exploring risk management for institutional investors.
In the previous issue, we looked at Analytical Value-at-Risk, whose cornerstone is the Variance-Covariance matrix. In this
article, we continue to explore VaR as an indicator to measure the market risk of a portfolio of financial instruments, but
we touch on a very different methodology.
We indicated in the previous article that the main benefits of from a Local Valuation method in which we only use the
Analytical VaR were that it requires very few parameters, is information about the initial price and the exposure at the
easy to implement and is quick to run computations (with an origin to deduce VaR.
appropriate mapping of the risk factors). Its main drawbacks Step 1 – Calculate the returns (or price changes) of all the
lie in the significant (and inconsistent across asset classes assets in the portfolio between each time interval.
and markets) assumption that price changes in the financial
The first step lies in setting the time interval and then calcu-
markets follow a normal distribution, and that this method-
lating the returns of each asset between two successive
ology may be computer-intensive since we need to calculate
periods of time. Generally, we use a daily horizon to calcu-
the n(n-1)/2 terms of the Variance-Covariance matrix (in the
late the returns, but we could use monthly returns if we were
case where we do not proceed to a risk mapping of the
to compute the VaR of a portfolio invested in alternative
various instruments that composed the portfolio). With the
investments (Hedge Funds, Private Equity, Venture Capital
increasing power of our computers, the second limitation
and Real Estate) where the reporting period is either
will barely force you to move away from spreadsheets to
monthly or quarterly. Historical Simulations VaR requires a
programming. But the first assumption in the case of a port-
long history of returns in order to get a meaningful VaR.
folio containing a non-negligible portion of derivatives
Indeed, computing a VaR on a portfolio of Hedge Funds with
(minimum 10%-15% depending on the complexity and
only a year of return history will not provide a good VaR esti-
exposure or leverage) may result in the Analytical VaR being
mate.
seriously underestimated because these derivatives have
non-linear payoffs. Step 2 – Apply the price changes calculated to the current
mark-to-market value of the assets and re-value your port-
One solution to circumvent that theoretical constraint is
folio.
merely to work only with the empirical distribution of the
returns to arrive at Historical Simulations VaR. Indeed, is it Once we have calculated the returns of all the assets from
not more logical to work with the empirical distribution that today back to the first day of the period of time that is being
captures the actual behavior of the portfolio and encom- considered – let us assume one year comprised of 265 days
passes all the correlations between the assets composing – we now consider that these returns may occur tomorrow
the portfolio? The answer to this question is not so clear-cut. with the same likelihood. For instance, we start by looking
Computing VaR using Historical Simulations seems more at the returns of every asset yesterday and apply these
intuitive initially but has its own pitfalls as we will see. But returns to the value of these assets today. That gives us new
first, how do we compute VaR using Historical Simulations? values for all these assets and consequently a new value of
the portfolio. Then, we go back in time by one more time
HISTORICAL SIMULATIONS VAR METHODOLOGY interval to two days ago. We take the returns that have been
The fundamental assumption of the Historical Simulations calculated for every asset on that day and assume that
methodology is that you look back at the past performance those returns may occur tomorrow with the same likelihood
of your portfolio and make the assumption – there is no as the returns that occurred yesterday. We re-value every
escape from making assumptions with VaR modeling – that asset with these new price changes and then the portfolio
the past is a good indicator of the near-future or, in other itself. And we continue until we have reached the beginning
words, that the recent past will reproduce itself in the near- of the period. In this example, we will have had 264 simula-
future. As you might guess, this assumption will reach its tions.
limits for instruments trading in very volatile markets or Step 3 – Sort the series of the portfolio-simulated P&L from
during troubled times as we have experienced this year. the lowest to the highest value.
The below algorithm illustrates the straightforwardness of After applying these price changes to the assets 264 times,
this methodology. It is called Full Valuation because we will we end up with 264 simulated values for the portfolio and
re-price the asset or the portfolio after every run. This differs thus P&Ls. Since VaR calculates the worst expected loss
DECEMBER 2008 EDITION — 8

5. RISK MANAGEMENT
over a given horizon at a given confidence level under APPLICATIONS OF HISTORICAL SIMULATIONS VAR
normal market conditions, we need to sort these 264 values
Let us compute VaR using historical simulations for one asset
from the lowest to the highest as VaR focuses on the tail of
and then for a portfolio of assets to illustrate the algorithm.
the distribution.
Example 1 – Historical Simulations VaR for one asset
Step 4 – Read the simulated value that corresponds to the
desired confidence level. The first step is to calculate the return of the asset price
between each time interval. This is done in column D in
The last step is to determine the confidence level we are
Table 1. Then we create a column of simulated prices based
interested in – let us choose 99% for this example. One can
on the current market value of the asset (1,000,000 as
read the corresponding value in the series of the sorted
shown in cell C3) and each return which this asset has expe-
simulated P&Ls of the portfolio at the desired confidence
rienced over the period under consideration. Thus, we have
level and then take it away from the mean of the series of
100 x (-1.93%) = -19,313.95. In Step 3, we simply sort all
simulated P&Ls. In other words, the VaR at 99% confidence
the simulated values of the asset (based on the past
level is the mean of the simulated P&Ls minus the 1%
returns). Finally, in Step 4, we read the simulated value in
lowest value in the series of the simulated values. This can
column G which corresponds to the 1% worst loss. As there
be formulated as follows:
is no value that corresponds to 99%, we interpolate the
VaR1-α = μ(R) – Rα (1) surrounding values around 99.24% and 98.86%. That gives
where us -54,711.55.
• VaR1-α is the estimated VaR at the confi-
Table 1 – Calculating Historical Simulations VaR for one asset
dence level 100 × ( 1-α )%.
• μ(R) is the mean of the series of simu-
lated returns or P&Ls of the portfolio.
• Rα is the α th worst return of the series of
simulated P&Ls of the portfolio or in other
words the return of the series of simulated
P&Ls that corresponds to the level of
significance α .
We may need to proceed to some interpo-
lation since there will be no chance to get
a value at 99% in our example. Indeed, if
we use 265 days, each return calculated at
every time interval will have a weight of
1/264 = 0.00379. If we want to look at the
value that has a cumulative weight of
99%, we will see that there is no value that
matches exactly 1% (since we have
divided the series into 264 time intervals
and not a multiple of 100). Considering
that there is very little chance that the tail
of the empirical distribution is linear, Asset Price for one asset
proceeding to a linear interpolation to get 140
the 99% VaR between the two successive 130
120
time intervals that surround the 99th 110
percentile will result in an estimation of 100
the actual VaR. This would be a pity 90
80
considering we did all that we could to use
7
7
8
8
8
8
8
8
8
8
08
8
8
/0
/0
/0
/0
/0
/0
0
0
/0
/0
/0
/0
the empirical distribution of returns,
1/
1/
1/
/1
/1
1
1
1
1
1
1
/1
/1
1/
2/
3/
4/
5/
6/
7/
8/
9/
11
12
10
11
wouldn’t it? Nevertheless, even a linear
interpolation may give you a good esti-
Histogram of Returns for one asset
mate of your VaR. For those who are more
eager to obtain the exact VaR, the Extreme
Value Theory (EVT) could be the right tool 99% VaR
for you. We will explain in another article 55,745
5.57%
how to use EVT when computing VaR. It is
rather mathematically demanding and
would require us to spend more time to
explain this method. -6% -5% -4% -3% -2% -1% 0% 1% 2% 3% 4% 5% 6% 7%
DECEMBER 2008 EDITION — 9

6. RISK MANAGEMENT
This number does not take into account the mean, which is As you can see, we simply add a couple of columns to repli-
1,033.21. As the 99% VaR is the distance from the mean of cate the intermediary steps for the second asset. In this
the first percentile (1% worst loss), we need to subtract the example, each asset represents 50% of the portfolio. After
number we just calculated from the mean to obtain the each run, we re-value the portfolio by simply adding up the
actual 99% VaR. In this example, the VaR of this asset is simulated P&L of each asset. This gives us the simulated P&Ls
thus 1,033.21 – (-54,711.55) = 55,744.76. In order to for the portfolio (column J).
express VaR in percentage, we can divide the 99% VaR This straightforward step of simply re-composing the portfolio
amount by the current value of the asset (1,000,000), which after every run is one of the reasons behind the popularity of
yields 5.57%. this methodology. Indeed, we do not need to handle sizeable
Example 2 – Historical Simulations VaR for one portfolio Variance-Covariance matrices. We apply the calculated returns
Computing VaR on one asset is relatively easy, but how do of every asset to their current price and re-value the portfolio.
the historical simulations account for any correlations As we have noted, correlations are embedded in the price
between assets if the portfolio holds more than one asset? changes. In this example, the 99% VaR of the first asset is
The answer is also simple: correlations are already 55,744.76 (or 5.57%) and the 99% VaR of the second asset is
embedded in the price changes of the assets. Therefore, 54,209.71 (or 5.42%). We know that VaR is a sub-additive risk
there is no need to calculate a Variance-Covariance matrix measure – if we add the VaR of two assets, we will not get the
when running historical simulations. Let us look at another VaR of the portfolio. In this case, the 99% VaR of the portfolio
example with a portfolio composed of two assets. only represents 3.67% of the current marked-to-market value
of the portfolio. That difference represents the diversification
Table 2 – Calculating Historical Simulations VaR for a portfolio of two assets
Portfolio Unit price Portfolio Histogram of Returns
270
250
230 99% VaR
73,422
210 3.67%
190
170
7
7
08
08
08
08
08
08
08
8
08
8
08
/0
/0
0
/0
1/
1/
1/
1/
1/
1/
1/
1/
1/
1/
/1
/1
/1
1/
2/
3/
4/
5/
6/
7/
8/
9/
/
-5% -4% -3% -2% -1% 0% 1% 2% 3% 4% 5%
11
12
10
11
DECEMBER 2008 EDITION — 10

7. RISK MANAGEMENT
effect. Having a portfolio invested in these two assets makes Lastly, a minimum of history is required to use this method-
the risk lower than investing in any of these two assets solely. ology. Using a period of time that is too short (less than 3-6
The reason is that the gains on one asset sometimes offset months of daily returns) may lead to a biased and inaccurate
the losses on the other asset (rows 10, 12, 13, 17-20, 23, estimation of VaR. As a rule of thumb, we should utilize at
26-28, 30, 32 in Table 2. Over the 265 days, this happened least four years of data in order to run 1,000 historical simula-
127 times with different magnitude. But in the end, this bene- tions. That said, round numbers like 1,000 may have
fited the overall risk profile of the portfolio as the 99% VaR of absolutely no relevance whatsoever to your exact portfolio.
the portfolio is only 3.67%. Security prices, like commodities, move through economic
cycles; for example, natural gas prices are usually more
ADVANTAGES OF HISTORICAL SIMULATIONS VAR volatile in the winter than in the summer. Depending on the
Computing VaR using the Historical Simulations methodology composition of the portfolio and on the objectives you are
has several advantages. First, there is no need to formulate attempting to achieve when computing VaR, you may need to
any assumption about the return distribution of the assets in think like an economist in addition to a risk manager in order
the portfolio. Second, there is also no need to estimate the to take into account the various idiosyncrasies of each instru-
volatilities and correlations between the various assets. ment and market. Also, bear in mind that VaR estimates need
Indeed, as we showed with these two simple examples, they to rely on a stable set of assumptions in order to keep a
are implicitly captured by the actual daily realizations of the consistent and comparable meaning when they are monitored
assets. Third, the fat tails of the distribution and other over a certain period of time.
extreme events are captured as long as they are contained in In order to increase the accuracy of Historical Simulations
the dataset. Fourth, the aggregation across markets is VaR, one can also decide to weight more heavily the recent
straightforward. observations compared to the furthest since the latter may
not give much information about where the prices would go
DISADVANTAGES OF HISTORICAL SIMULATIONS today. We will cover these more advanced VaR models in
VAR another article.
The Historical Simulations VaR methodology may be very intu-
itive and easy to understand, but it still has a few drawbacks. CONCLUSION
First, it relies completely on a particular historical dataset and Despite these disadvantages, many financial institutions have
its idiosyncrasies. For instance, if we run a Historical chosen historical simulations as their favored methodology to
Simulations VaR in a bull market, VaR may be underestimated. compute VaR. To many, working with the actual empirical
Similarly, if we run a Historical Simulations VaR just after a distribution is the “real deal.”
crash, the falling returns which the portfolio has experienced However, obtaining an accurate and reliable VaR estimate has
recently may distort VaR. Second, it cannot accommodate little value without a proper back testing and stress testing
changes in the market structure, such as the introduction of program. VaR is simply a number whose value relies on a
the Euro in January 1999. Third, this methodology may not sound methodology, a set of realistic assumptions and a
always be computationally efficient when the portfolio rigorous discipline when conducting the exercise. The real
contains complex securities or a very large number of instru- benefit of VaR lies in its essential property of capturing with
ments. Mapping the instruments to fundamental risk factors one single number the risk profile of a complex or diversified
is the most efficient way to reduce the computational time to portfolio. VaR remains a tool that should be validated through
calculate VaR by preserving the behavior of the portfolio successive reconciliation with realized P&Ls (back testing)
almost intact. Fourth, Historical Simulations VaR cannot and used to gain insight into what would happen to the port-
handle sensitivity analyses easily. folio if one or more assets would move adversely to the
investment strategy (stress testing).
DECEMBER 2008 EDITION — 11
Digitally signed by Sreehari Menon
Signature Not Verified
Sreehari Menon DN: cn=Sreehari Menon, c=IN
Date: 2010.09.25 07:23:34 Z

8. RISK MANAGEMENT
AN OVERVIEW OF VALUE-AT-RISK:
PART III – MONTE CARLO SIMULATIONS VAR
by Romain Berry
J.P. Morgan Investment Analytics and Consulting
romain.p.berry@jpmorgan.com
This article is the fourth in a series of articles exploring risk management for institutional investors.
The last (and most complex) of the three main methodologies pricing we find in the financial markets. This process is called
used to compute the Value-at-Risk (VaR) of a portfolio of discretization, whereby we approximate a continuous
financial instruments employs Monte Carlo Simulations. phenomenon by a large number of discrete intervals.
Monte Carlo Simulations correspond to an algorithm that Step 2 – Draw a random number from a random number
generates random numbers that are used to compute a generator and update the price of the asset at the end of the
formula that does not have a closed (analytical) form – this first time increment.
means that we need to proceed to some trial and error in
It is possible to generate random returns or prices. In most
picking up random numbers/events and assess what the
cases, the generator of random numbers will follow a
formula yields to approximate the solution. Drawing random
specific theoretical distribution. This may be a weakness of
numbers over a large number of times (a few hundred to a
the Monte Carlo Simulations compared to Historical
few million depending on the problem at stake) will give a
Simulations, which uses the empirical distribution. When
good indication of what the output of the formula should be.
simulating random numbers, we generally use the normal
It is believed actually that the name of this method stems
distribution.
from the fact that the uncle of one of the researchers (the
Polish mathematician Stanislaw Ulam) who popularized this In this paper, we use the standard stock price model to
algorithm used to gamble in the Monte Carlo casino and/or simulate the path of a stock price from the ith day as defined
that the randomness involved in this recurring methodology by:
can be compared to the game of roulette. (1)
In this article, we present the algorithm, and apply it to where
compute the VaR for a sample stock. We also discuss the is the return of the stock on the ith day
pros and cons of the Monte Carlo Simulations methodology is the stock price on the ith day
compared to Analytical VaR and Historical Simulations VaR. is the stock price on the i+1th day
METHODOLOGY is the sample mean of the stock price
is the timestep
Computing VaR with Monte Carlo Simulations follows a
is the sample volatility (standard deviation) of the
similar algorithm to the one we used for Historical
stock price
Simulations in our previous issue. The main difference lies
is a random number generated from a normal
in the first step of the algorithm – instead of picking up a
distribution
return (or a price) in the historical series of the asset and
assuming that this return (or price) can re-occur in the next At the end of this step/day ( = 1 day), we have drawn a
time interval, we generate a random number that will be random number and determined by applying (1) since
used to estimate the return (or price) of the asset at the end all other parameters can be determined or estimated.
of the analysis horizon. Step 3 – Repeat Step 2 until reaching the end of the analysis
Step 1 – Determine the length T of the analysis horizon and horizon T by walking along the N time intervals.
divide it equally into a large number N of small time At the next step/day ( = 2), we draw another random
increments Δt (i.e. Δt = T/N). number and apply (1) to determine from .
For illustration, we will compute a monthly VaR consisting of We repeat this procedure until we reach T and can
twenty-two trading days. Therefore N = 22 days and Δt = 1 determine . In our example, represents the
day. In order to calculate daily VaR, one may divide each day estimated (terminal) stock price in one month time of the
per the number of minutes or seconds comprised in one day sample share.
– the more, the merrier. The main guideline here is to ensure Step 4 – Repeat Steps 2 and 3 a large number M of times to
that Δt is large enough to approximate the continuous generate M different paths for the stock over T.
MARCH 2009 EDITION — 4

9. RISK MANAGEMENT
So far, we have generated one path for this stock (from i to Exhibit 1: Historical prices for one stock from
i+22). Running Monte Carlo Simulations means that we 01/22/08 to 01/20/09
build a large number M of paths to take account of a broader
50
universe of possible ways the stock price can take over a
period of one month from its current value ( ) to an 45
estimated terminal price . Indeed, there is no unique 40
way for the stock to go from to . Moreover, is
only one possible terminal price for the stock amongst an 35
infinity. Indeed, for a stock price being defined on + (set 30
of positive real numbers), there is an infinity of possible
25
paths from to (see footnote 1).
20
It is an industry standard to run at least 10,000 simulations
even if 1,000 simulations provide an efficient estimator of 15
Jan 08 Mar 08 Apr 08 Jun 08 Jul 08 Sep 08 Nov 08 Dec 08
the terminal price of most assets. In this paper, we ran
1,000 simulations for illustration purposes.
20th of January 2009 was $18.09. We want to compute the
Step 5 – Rank the M terminal stock prices from the smallest
monthly VaR on the 20th of January 2009. This means we
to the largest, read the simulated value in this series that
will jump in the future by 22 trading days and look at the
corresponds to the desired (1- )% confidence level (95%
estimated prices for the stock on the 19th of February 2009.
or 99% generally) and deduce the relevant VaR, which is
the difference between and the th lowest terminal Since we decided to use the standard stock price model to
stock price. draw 1,000 paths until T (19th of February 2009), we will
need to estimate the expected return (also called drift rate)
Let us assume that we want the VaR with a 99% confidence
and the volatility of the share on that day.
interval. In order to obtain it, we will need first to rank the M
terminal stock prices from the lowest to the highest. We can estimate the drift by:
Then we read the 1% lowest percentile in this series. This
estimated terminal price, 1%
means that there is a 1% (2)
chance that the current stock price could fall to 1%
or The volatility of the share can be estimated by:
less over the period in consideration and under normal
market conditions. If 1%
is smaller than (which is the
case most of the time), then - 1%
will corresponds to (3)
a loss. This loss represents the VaR with a 99% confidence Note that since we chose = 1 day, these two estimators
interval. will equal the sample mean and sample standard deviation.
APPLICATIONS Based on these two estimators, we generate from
by re-arranging (1) as:
Let us compute VaR using Monte Carlo Simulations for one
share to illustrate the algorithm.
We apply the algorithm to compute the monthly VaR for one (4)
stock. Historical prices are charted in Exhibit 1. We will only and simulate 1,000 paths for the share.
consider the share price and thus work with the assumption The last step can be summarized in Exhibit 2. We sort the
we have only one share in our portfolio. Therefore the value 1,000 terminal stock prices from the lowest to the highest
of the portfolio corresponds to the value of one share. and read the price which corresponds to the desired
From the series of historical prices, we calculated the confidence level. For instance, if we want to get the VaR at a
sample return mean (-0.17%) and sample return standard 99% confidence level, we will read the 1% lowest stock
deviation (5.51%). The current price ( ) at the end of the price, which is $15.7530. On January 20th, the stock price
1
This is the reason why we used the discretized form (1) of the standard stock price model so that Monte Carlo Simulations can be handled more easily without losing too much
information. Thus, the higher N and M are, the more accurate the estimates of the terminal stock prices will be, but the longer the simulations will take to run.
MARCH 2009 EDITION — 5

10. RISK MANAGEMENT
was $18.09. Therefore, there is a 1% likelihood that the The main benefit of running time-consuming Monte Carlo
share falls to $15.7530 or below. If that happens, we will Simulations is that they can model instruments with non-
experience a loss of at least $18.09 – $15.7530 = $2.5170. linear and path-dependent payoff functions, especially
This loss is our monthly VaR estimate at a 99% confidence complex derivatives. Moreover, Monte Carlo Simulations
level for one share calculated on the 20th of January 2009. VaR is not affected as much as Historical Simulations VaR by
extreme events, and in reality provides in-depth details of
these rare events that may occur beyond VaR. Finally, we
Exhibit 2: Reading for one share
may use any statistical distribution to simulate the returns
as far as we feel comfortable with the underlying
assumptions that justify the use of a particular distribution.
DISADVANTAGES
The main disadvantage of Monte Carlo Simulations VaR is
the computer power that is required to perform all the
simulations, and thus the time it takes to run the
simulations. If we have a portfolio of 1,000 assets and want
to run 1,000 simulations on each asset, we will need to run
1 million simulations (without accounting for any eventual
simulations that may be required to price some of these
assets – like for options and mortgages, for instance).
Moreover, all these simulations increase the likelihood of
model risk.
Consequently, another drawback is the cost associated with
developing a VaR engine that can perform Monte Carlo
Simulations. Buying a commercial solution off-the-shelf or
outsourcing to an experienced third party are two options
worth considering. The latter approach will reinforce the
independence of the computations and therefore reliance of
its accuracy and non-manipulation.
CONCLUSION
Estimating the VaR for a portfolio of assets using Monte Carlo
Simulations has become the standard in the industry. Its
strengths overcome its weaknesses by far.
Despite the time and effort required to estimate the VaR for a
portfolio, this task only represents half of the time a risk
manager should spend on VaR. Indeed, the other half should
be spent on checking that the model(s) used to calculate VaR
ADVANTAGES is (are) still appropriate for the assets that composed the
Monte Carlo Simulations present some advantages over the portfolio and still provide credible estimate of VaR (back
Analytical and Historical Simulations methodologies to testing), and on analyzing how the portfolio reacts to extreme
compute VaR. events which occur every now and then in the financial
markets (stress testing).
MARCH 2009 EDITION — 6
Digitally signed by Sreehari
Signature Not Verified
Sreehari Menon Menon
DN: cn=Sreehari Menon, c=IN
Date: 2010.09.25 07:24:39 Z