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| All About Value-at-RiskTM |
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| By Barry
Schachter. About
the author (Courtesy: www.gloriamundi.org.
Reproduced
with permission) |
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Value at Risk FAQ
I am not sure what are the most frequently asked questions
about
VaR, but I have made a stab at it here. If you have any suggestions
for additions to this list, I would welcome them.
If you have a question, suggest a Q for the FAQ by clicking here
What is VaR?
What's the difference between EaR,
VaR, and EVE?
Should the "a" be capitalized in VaR (VAR)?
What's statistics got to do with it?
What's the tail of a loss distribution?
What does VaR say about the "tail" of the loss
distribution?
What does VaR assume as a risk measure?
What's market risk?
What is VaR used for?
How is VaR calculated?
What is Monte Carlo?
What is Historical Simulation?
What is RiskMetrics?
What is the Variance/Covariance Matrix or Parametric method?
What is a "linear" exposure?
What is a "non linear" exposure?
What is Stress Testing?
What is Backtesting?
What
do regulators think of VaR?
What is a lookback period?
What does VaR have to do with maximization of shareholder
wealth?
What's the difference between a confidence level and a
confidence interval?
How does VaR for corporates differ from VAR for financial
firms?
What is the most accurate VaR method?
What is the minimum number of simulations that should be run
in Monte Carlo VAR?
What confidence level should I use?; Is 99% more
conservative than 95%?
What time horizon should I use?
What is CVaR, Conditional Value at Risk?
What is the proper relation between the VaR in a portfolio
and the amount of capital that should be held against it?
What
is VaR?
I think of Value at Risk as a measure of potential loss from an unlikely,
adverse event in a normal, everyday market environment. VaR is denominated
in units of a currency, e.g., US dollars. To get more concrete, VaR is an
amount, say D dollars, where the chance of losing more than D dollars is,
say, 1 in 100 over some future time interval, say 1 day. This is a
probabilistic statement, and therefore VaR is a statistical measure of
risk exposure. The calculation of VaR requires the application of
statistical theory.
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What's the difference between EaR, VaR, and EVE?
Earnings at Risk typically looks only at potential changes in cash
flows/earnings over the forecast horizon. Value at risk looks at the
change in the entire value over the forecast horizon. Economic Value of
Equity also looks at value change, but typically over a longer forecast
horizon than VAR (up to 1 year). In a trading environment, where profit
and loss are equivalent to changes in value, EaR and VaR should be the
same.
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Should
the "a" be capitalized in VaR (VAR)?
When at work I capitalize, but in less formal situations I don't.
Actually, it's less confusing if you don't, since VAR also stands for
Value Added Reseller and Vector Auto Regression.
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What's
statistics got to do with it?
VaR is actually a piece of information about the distribution of possible
future losses on a portfolio. The actual gain or loss won't be known until
it happens. Until then it's uncertain; a random variable. Information
about the behavior of a random variable is called a statistic. As you may
guess, there are many statistics about a portfolio returns, for example
the expected return. The VaR is a very useful statistic for risk managers,
but it's unlikely that it's the only statistic that has some usefulness.
Nevertheless, it is the statistic focused on almost exclusively. Now for
the tricky stuff. VaR itself is a random variable, because not only is the
portfolio's future return unknown, but the distribution of the portfolio's
return must be guessed at by inference from observable data. That means
that the calculated VaR is really itself just an estimate of the true VaR.
So you could estimate a VaR of the distribution of the VaR! Most people
are content with estimating confidence intervals for any estimated
parameter, because the confidence interval tells you how precise is your
estimate.
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What's
the tail of a loss distribution?
The tail is that portion of the loss distribution that contains the
outlying (i.e., bad) events). Problem is, no one seems to know exactly
where the tail begins. Which can be a problem for some measures of risk.
However, a lot of research is going on in this area, so stay tuned.
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What
does VaR say about the "tail" of the loss distribution?
Not much really. If your VaR model assumes some shape for the entire
distribution of portfolio return, then everything you need to know about
the tail is embedded in that assumption
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What
does VaR assume as a risk measure?
Just about every VaR model assumes that the portfolio under consideration
doesn't change over the forecast horizon. This is a fiction, especially
for trading portfolios, but trying to incorporate forecasts of position
changes into a model forecasting returns is very complicated. VaR models
also assume that the historical data used to construct the VaR estimate
contains information useful in forecasting the loss distribution. Some VaR
models go further and assume that the historical data themselves follow a
specific distribution (e.g., a "normal distribution" in
RiskMetrics(TM)).
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What's
market risk?
Market risk is usually defined as the risk to loss in a financial
instrument from an adverse movement in market prices or rates. What's
adverse? Well, it depends. If you own a bond, then a rise in interest
rates is adverse, but if you have lent/sold a bond, it is a fall in rates
that is adverse. Generally people classify sources of market risk into
four categories, interest rates, equities, foreign exchange and
commodities.
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What
is VaR used for?
Not a food additive, yet. Originally VaR was used as an information tool.
I.e., it was used to communicate to management a feeling for the exposure
to changes in market prices. Then market risk was incorporated into the
actual risk control structure. I.e., trading limits were based on VaR
calculations. Now it is commonly used in the incentive structure as well.
I.e., VaR is a component determining risk-adjusted performance and
compensation. Interestingly, the theory of VaR has not kept pace. While we
understand it's usefulness as an information tool, it's not clear how it
fits into the shareholder wealth maximization paradigm of modern financial
theory.
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How
is VaR calculated?
It depends on the method used, variance/covariance, Monte Carlo,
historical simulation. Generally, it involves using historical data on
market prices and rates, the current portfolio positions, and models
(e.g., option models, bond models) for pricing those positions. These
inputs are then combined in different ways, depending on the method, to
derive an estimate of a particular percentile of the loss distribution,
typically the 99th percentile loss.
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What
is Monte Carlo?
A small principality in Europe. But you knew that. It is a simulation
technique. First make some assumptions about the distribution of changes
in market prices and rates (for example, by assuming they are normally
distributed), then collecting data to estimate the parameters of the
distribution). The Monte Carlo then uses those assumptions to give
successive sets of possible future realizations of changes in those rates.
For each set, the portfolio is revalued. When done, you've got a set of
portfolio revaluations corresponding to the set of possible realizations
of rates. From that distribution you take the 99th percentile loss as the
VaR.
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What
is Historical Simulation?
Like Monte Carlo, it is a simulation technique, but it skips the step of
making assumptions about the distribution of changes in market prices and
rates (usually). Instead, it assumes that whatever the realizations of
those changes in prices and rates were in the past is what they can be
over the forecast horizon. It takes those actual changes, applies them to
the current set of rates, then uses those to revalue the portfolio. When
done, you've got a set of portfolio revaluations corresponding to the set
of possible realizations of rates. From that distribution you take the
99th percentile loss as the VaR.
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What
is RiskMetrics?
It is a particular implementation of the Variance/Covariance approach to
calculating VaR. It is particular, not general, because it assumes a
particular structure for the evolution of market prices and rates through
time, and because it translates all portfolio positions into their
component cash flows (or "equivalent") and performs the VaR
computation on those. It is really responsible for popularizing VaR, and
is a perfectly reasonable approach, especially for portfolios without a
lot of nonliearity.
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What
is the Variance/Covariance Matrix or Parametric method?
This is a very simplified and speedy approach to VaR computation. It is
so, because it assumes a particular distribution for both the changes in
market prices and rates and the changes in portfolio value. Usually, this
is the "normal" distribution. The neat thing about the normal is
that a lot is known about it, including how to readily obtain an estimate
of any percentile once you know the variances and covariances of all
changes in position values. These are normally estimated directly from
historical data. In this method the VaR of the portfolio, is a simple
transformation of the estimated variance/covariance matrix. So simple that
it doesn't really work well for nonlinear positions.
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What
is a "linear" exposure?
A linear risk is one where the change in the value of a position in
response to a change in a market price or rate is a constant proportion of
the change in the price or rate.
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What
is a "non linear" exposure?
Everything that's not linear. For example, options are thought of as
nonlinear exposures, because they respond differently to changes in the
value of the underlying instrument depending on whether they are
in-the-money, at-the-money, or out-of-the-money.
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What
is Stress Testing?
I think of stress testing as measure of risk exposure that's complementary
to VaR. Stress testing is a measure of potential loss as a result of a
plausible event in an abnormal market environment. Two types of stress
testing are popular. The first is based on economic scenarios. Pretend
your portfolio experiences the 1987 or 1997 stock market crash again. The
second is "matrix" based. Change a bunch of assumptions about
correlations and variances and see what happens. Neither is statistical in
nature, in contrast to VaR. That is, you don't know the probability of any
particular scenario.
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What
is Backtesting?
Backtesting is a statistical process for validating the accuracy of a VaR
model. Banking regulators require backtesting for banks that use VaR for
regulatory capital. It involves a comparison between the number of times
the VaR model under-predicts the subsequent day's loss, versus the number
of time such an under-prediction is expected. If losses exceeding VaR have
a 1 in 100 chance of ocurring, then we expect to see 2 or 3 of those in a
year. There is a lot of debate about whether backtesting is meaningful,
because it is difficult to validate a model based on a few extreme events
- not enough data.
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What
do regulators think of VaR?
Love-hate, I think. Love first. Banking regulators internationally have
agreed to allow banks to use VaR models to calculate regulatory capital.
Don't ask why banks have minimum capital set by regulators, as that is a
different FAQ. In the USA, the securities regulator allows corporates to
use VaR to express their exposure to market risk in their annual and
quarterly regulatory public financial filings. Now hate. Regulators aren't
sure that VaR is the "right" measure of risk? Nor are they sure
how much weight should be given to it in risk management. They really
aren't sure whether VaR should be extended to the measurement of other
kinds of risk, such as credit risk.
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What
is a lookback period?
It is the period of history that is used to collect data used in the
computation of VaR. This is important, because if the data is
inappropriate for the forecast, the forecast is no good.
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What
does VaR have to do with maximization of shareholder wealth?
Excellent question. A precise answer isn't possible yet. This is
troublesome, as one hopes that it is being used in ways that are
consistent with the concept of shareholder wealth maximization, but we
can't be sure.
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What's
the difference between a confidence level and a confidence interval?
Pet peeve coming up. While VaR is an estimate of a percentile of the loss
distribution, it is commonly referred to as a confidence level. This is
because we say, we are 99% confident that the loss will not exceed $XX.
It's more exact to refer to VaR as a percentile estimate. Because VaR is a
statistical estimate, it is an uncertain amount itself, and that
uncertainty can be encapsulated in a statistical concept called a
confidence interval - I am 95% sure that the VaR actually lies between $AA
and $BB. It's too confusing to talk about a confidence interval around a
confidence level.
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How
does VaR for corporates differ from VAR for financial firms?
The use of VaR for non-financial firms is still evolving. Currently it is
mostly focused on either VaR for derivatives and hedging instruments only
or VaR for cash flows, i.e., EaR.
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What
is the most accurate VaR method?
Accuracy is in the eye of the beholder. A general answer to this question
is not possible, because it will depend on the nature of the portfolio and
the data used in the estimation of VaR. Several studies comparing
methodologies were conducted a few years back, typically with linear
portfolios, either equities or fx. These tended to show that the
variance-covariance approach was better when short histories of market
prices were used, because Monte Carlo and Historical Simulation would
under estimate the 99th percentile. With longer histories MC and HC were
equal to or better than VCV. But I don't recommend you generalizing from
these studies, because of their limited scope. Because of this, it is very
important to have an estimate of precision for every VaR estimate (A
confidence interval).
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What
is the minimum number of simulations that should be run in Monte Carlo
VAR?
I know that one major bank uses 500 simulations for its Monte Carlo VaR.
Again, the answer depends on the complexity of the portfolio. Linear
instruments, fewer simulations. But MC has its own peculiarities that
affect accuracy. For example, some MC routines use "variance
reduction." These are "tricks" used to improve accuracy for
a given simulation size. With variance reduction techniques (e.g.,
Antithetic Variates), the fewer simulations needed for a given accuracy.
Remember that underlying every MC is some distribution from which
observations of market rates are sampled. So assumptions about the
distribution and shortcuts taken to reduce the "dimensionality"
of the distribution will also have a cost in accuracy which should require
more simulations for a given level of accuracy.
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What
confidence level should I use?; Is 99% more conservative than 95%?
I know this is really two questions. The underlying question really is,
what percentile of the return distribution gives me better information
about risk exposure? If the portfolio return distribution were normally
distributed, it wouldn't matter, because every percentile is expressible
as a constant times the standard deviation of the return, the standard
deviation being the only real information you need for risk assessment in
the normal distribution case. The trade-off between choice of percentiles
in the real world in which we live is really about accuracy. It is more
difficult to accurately estimate a point farther out in the tail of the
distribution of returns, because there is less observable data to use in
the estimation. However, you may wish to look farther out in the tail if
you believe that your portfolio return distribution is more
"fat-tailed" (you may think of it as when the ratio: 99
percentile/95 percentile is greater than if the ratio were calculated for
a normally distributed return distribution). If there's more going on out
there in the tail, you may want to focus on it more. However, simply
because 99th percentile VaR yields a bigger VaR does not mean that using a
99th percentile rather than a 95th percentile VaR is a more conservative
of a measure of risk. All it means is that you are looking at a point
farther out in the tail and calling that your risk exposure. Whether you
use 95 or 99, you are generating an estimate of risk from the same
distribution of returns.
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What
time horizon should I use?
The standard time horizon (that period over which the VaR forecast is
made) is one day for most financial businesses with active trading
portfolios. The logic for this horizon is that it would take less than one
day to either exit or hedge out all the market risk in any position, so
that's really how long is the exposure. This reasoning suggests that the
horizon should be tuned to the interval to close out the market exposure.
This is a bit simplified, because it ignores liquidity issues (large
positions may take longer to exit, simply because they are large),
differences among portfolio instruments (it is not reasonable to employ a
one day horizon for some positions and a multi-day horizon for others, and
then to aggregate them for portfolio VaR calculations), and consistency
with credit VaR calculations (typically using a much longer horizon,
thereby making aggregation complicated - ignoring all the other
theoretical issues in aggregating credit and market risk). These two
problems have no completely satisfactory solutions. So, it may be best to
identify a singly horizon that best fits the portfolio's characteristics
and use that for everything when calculating VaR.
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What
is CVaR, Conditional Value at Risk?
Unfortunately, the term is not used consistently by all authors.
Conditional value at risk (cvar) is most often used to refer to a measure
of the risk of loss beyond the VaR. I.e., if the VaR of a portfolio is DM
5,000, then what is the expected loss beyond DM 5,000 (or "mean
excess loss"), given that an observed loss is greater than the
VaR. However, some use the term to mean the estimation of VaR from
"conditional" asset return distributions (a conditional
distribution is one that takes into account changes in the shape of the
distributions through time).
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What
is the proper relation between the VaR in a portfolio and the amount of
capital that should be held against it?
There are many considerations, if capital is to be based on VaR. VaR
doesn't tell you how big your losses could be on a bad day, it only
defines what distingishes a bad day from other days. If you have two
portfolios with exposures to risks of different markets, but the
portfolios nevertheless have the same VaR, then it may be wrong to keep
the same capital against each portfolio, because one may have much worse
performance given a VaR exceedance day. Also, since VaR looks at only a
particular forecast horizon, and a bad economic environment may extend
beyond that horizon, the relationship between VaR and a
business-continuity-threatening type of market event is murky at best.
Finally, the relationship between the amount of risk taken and the amount
of capital to be held may come down to the nature of the trading and the
risk appetite of the "owners" of the capital. If the portfolio
has significant nonlinear risks, then the relation between the capital and
VaR is even more difficult to judge, as it is sometimes the case that the
nonlinearities are greatest beyond the VaR (e.g., in a trading book, with
a portfolio of barrier options, where the barriers are not hit within the
set of market moves resulting in the VaR). I could go on, for example, the
relationship between VaR and the cost of capital under the investment rule
of shareholder wealth maximization is not clear - whereas if it were
clear, then we could deduce the amount of capital just sufficient to
support a given level of risk. And, the impact of VaR-based capital
requirements on the incentives of those taking the risks is not all all
clear. Having said all that, which should be pretty discouraging, I will
hazard that a one day VaR equal to about 3% of the trading capital is a
pretty good sized risk in a normal environment.
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Page authored by: Barry Schachter.
(Courtesy: www.gloriamundi.org.
Reproduced with permission. All trademarks belong to the author)
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