Value at risk (VAR or sometimes VaR) has been called the "new science of risk management", but you do not need to be a scientist to use VAR. Here, in part 1 of this series, 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 are not 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 commonsense fact. By assuming investors care about the odds of a really big loss, VAR answers the question, "What is my worstcase 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:
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 trades under the ticker QQQQ. The QQQQ is a very popular index of the largest nonfinancial stocks that trade on the Nasdaq exchange.
There are three methods of calculating VAR: the historical method, the variancecovariance method and the Monte Carlo simulation.
1. Historical Method
The historical method simply reorganizes actual historical returns, putting them in order from worst to best. It then assumes that history will repeat itself, from a risk perspective.
The QQQ started trading in Mar 1999, and if we calculate each daily return, we produce a rich data set of almost 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 fiveplus 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 rephrase the statistic into both percentage and dollar terms:
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 variancecovariance 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 ():
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:
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 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 resultalthough 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 VARtype conclusion: with 95% confidence, we do not expect to lose more than 15% during any given month.
Summary
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.
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 are not 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 commonsense fact. By assuming investors care about the odds of a really big loss, VAR answers the question, "What is my worstcase 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?
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 trades under the ticker QQQQ. The QQQQ is a very popular index of the largest nonfinancial stocks that trade on the Nasdaq exchange.
There are three methods of calculating VAR: the historical method, the variancecovariance method and the Monte Carlo simulation.
1. Historical Method
The historical method simply reorganizes actual historical returns, putting them in order from worst to best. It then assumes that history will repeat itself, from a risk perspective.
The QQQ started trading in Mar 1999, and if we calculate each daily return, we produce a rich data set of almost 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 fiveplus years when the daily return for the QQQ was a stunning 12.4%!
 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%).
 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.
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 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 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 resultalthough 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):
Summary
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.