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Q: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.

Q: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.

Q: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

Q: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.

Q: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.

Q: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.

Q: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)).

Q: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.

Q: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.

Q: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.

Q: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.

Q: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.

Q: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.

Q: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.

Q: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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