All About Value at Risk

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 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 inderval, say 1 day. This is a probabalistic statement, and therefore VaR is a statistical measure of risk exposure. The calculation of VaR requires the application of statistical theory. [top]

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 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 enviornment, where profit and loss are equivalent to changes in value, EaR and VaR should be the same. [top]

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 (capital "a") also stands for "Value Added Reseller" and "Vector Auto Regression." [top]

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, becuase the confidence interval tells you how precise is your estimate. [top]

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. [top]

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 [top]

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 forecsting 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)). [top]

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. [top]

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. [top]

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. [top]

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. [top]

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. [top]

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 forthe 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. [top]

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 nonlinaer positions. [top]

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. [top]

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. [top]

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. [top]

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-preduction is expected. If losses exceeding VaR have a 1 in 100 chance of occuring, 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.

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. [top]

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. [top]

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. [top]

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. [top]

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. [top]

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). [top]

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. [top]

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. [top]

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. [top]

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