Investopedia/David R. Harper: An Introduction to Value at Risk (VAR)

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An Introduction to Value at Risk (VAR)

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By David R. Harper
Updated Feb 6, 2019

Value at risk (VAR or sometimes VaR) has been called the "new science of risk management," but you don't need to be a scientist to use VAR.

Here, in part 1 of this short series on the topic, we look at the idea behind VAR and the three basic methods of calculating it. In Part 2, we apply these methods to calculating VAR for a single stock or investment.
The Idea Behind VAR

The most popular and traditional measure of risk is volatility. The main problem with volatility, however, is that it does not care about the direction of an investment's movement: a stock can be volatile because it suddenly jumps higher. Of course, investors aren't distressed by gains (See The Limits and Uses of Volatility).

For investors, risk is about the odds of losing money, and VAR is based on that common-sense fact. By assuming investors care about the odds of a really big loss, VAR answers the question, "What is my worst-case scenario?" or "How much could I lose in a really bad month?"

Now let's get specific. A VAR statistic has three components: a time period, a confidence level and a loss amount (or loss percentage). Keep these three parts in mind as we give some examples of variations of the question that VAR answers:

    What is the most I can - with a 95% or 99% level of confidence - expect to lose in dollars over the next month?
    What is the maximum percentage I can - with 95% or 99% confidence - expect to lose over the next year?

You can see how the "VAR question" has three elements: a relatively high level of confidence (typically either 95% or 99%), a time period (a day, a month or a year) and an estimate of investment loss (expressed either in dollar or percentage terms).
Methods of Calculating VAR

Institutional investors use VAR to evaluate portfolio risk, but in this introduction we will use it to evaluate the risk of a single index that trades like a stock: the Nasdaq 100 Index, which is traded through the PowerShares QQQ Trust. The QQQ is a very popular index of the largest non-financial stocks that trade on the Nasdaq exchange.

There are three methods of calculating VAR: the historical method, the variance-covariance method and the Monte Carlo simulation.
1. Historical Method

The historical method simply re-organizes actual historical returns, putting them in order from worst to best. It then assumes that history will repeat itself, from a risk perspective.

As a historical example, let's look at the Nasdaq 100 ETF, which trades under the symbol QQQ (sometimes called the "cubes"), and which started trading in March of 1999. If we calculate each daily return, we produce a rich data set of more than 1,400 points. Let's put them in a histogram that compares the frequency of return "buckets." For example, at the highest point of the histogram (the highest bar), there were more than 250 days when the daily return was between 0% and 1%. At the far right, you can barely see a tiny bar at 13%; it represents the one single day (in Jan 2000) within a period of five-plus years when the daily return for the QQQ was a stunning 12.4%.

Notice the red bars that compose the "left tail" of the histogram. These are the lowest 5% of daily returns (since the returns are ordered from left to right, the worst are always the "left tail"). The red bars run from daily losses of 4% to 8%. Because these are the worst 5% of all daily returns, we can say with 95% confidence that the worst daily loss will not exceed 4%. Put another way, we expect with 95% confidence that our gain will exceed -4%. That is VAR in a nutshell. Let's re-phrase the statistic into both percentage and dollar terms:

    With 95% confidence, we expect that our worst daily loss will not exceed 4%.
    If we invest $100, we are 95% confident that our worst daily loss will not exceed $4 ($100 x -4%).

You can see that VAR indeed allows for an outcome that is worse than a return of -4%. It does not express absolute certainty but instead makes a probabilistic estimate. If we want to increase our confidence, we need only to "move to the left" on the same histogram, to where the first two red bars, at -8% and -7% represent the worst 1% of daily returns:

    With 99% confidence, we expect that the worst daily loss will not exceed 7%.
    Or, if we invest $100, we are 99% confident that our worst daily loss will not exceed $7.

2. The Variance-Covariance Method

This method assumes that stock returns are normally distributed. In other words, it requires that we estimate only two factors - an expected (or average) return and a standard deviation - which allow us to plot a normal distribution curve. Here we plot the normal curve against the same actual return data:

The idea behind the variance-covariance is similar to the ideas behind the historical method - except that we use the familiar curve instead of actual data. The advantage of the normal curve is that we automatically know where the worst 5% and 1% lie on the curve. They are a function of our desired confidence and the standard deviation (
VARstdeviation.gif

):
Confidence     # of Standard Deviations (σ)
95% (high)     - 1.65 x σ
99% (really high)     - 2.33 x σ

The blue curve above is based on the actual daily standard deviation of the QQQ, which is 2.64%. The average daily return happened to be fairly close to zero, so we will assume an average return of zero for illustrative purposes. Here are the results of plugging the actual standard deviation into the formulas above:
Confidence     # of σ     Calculation     Equals
95% (high)     - 1.65 x σ     - 1.65 x (2.64%) =     -4.36%
99% (really high)     - 2.33 x σ     - 2.33 x (2.64%) =     -6.15%
3. Monte Carlo Simulation

The third method involves developing a model for future stock price returns and running multiple hypothetical trials through the model. A Monte Carlo simulation refers to any method that randomly generates trials, but by itself does not tell us anything about the underlying methodology.

For most users, a Monte Carlo simulation amounts to a "black box" generator of random, probablistic outcomes. Without going into further details, we ran a Monte Carlo simulation on the QQQ based on its historical trading pattern. In our simulation, 100 trials were conducted. If we ran it again, we would get a different result--although it is highly likely that the differences would be narrow. Here is the result arranged into a histogram (please note that while the previous graphs have shown daily returns, this graph displays monthly returns):

To summarize, we ran 100 hypothetical trials of monthly returns for the QQQ. Among them, two outcomes were between -15% and -20%; and three were between -20% and 25%. That means the worst five outcomes (that is, the worst 5%) were less than -15%. The Monte Carlo simulation therefore leads to the following VAR-type conclusion: with 95% confidence, we do not expect to lose more than 15% during any given month.
The Bottom Line

Value at Risk (VAR) calculates the maximum loss expected (or worst case scenario) on an investment, over a given time period and given a specified degree of confidence. We looked at three methods commonly used to calculate VAR. But keep in mind that two of our methods calculated a daily VAR and the third method calculated monthly VAR. In Part 2 of this series we show you how to compare these different time horizons.

To read more on this subject, see Continuously Compound Interest.
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