What is Value at Risk (VaR)?

Value at risk is like driving on empty…

You know the gas gauge in your car isn’t a perfectly accurate instrument; it only gives you a rough idea of how much fuel is left. So, as you’re trying to get to the gas station before running out of fuel, you’re pretty sure you’ll get there. But there’s a nagging question in the back of your mind. What’s the worst-case scenario? As you contemplate walking a few miles with a gas can, you’re assessing your risk. If those miles were dollars lost, you’d be thinking about your potential financial losses. That’s similar to contemplating your value at risk.

Value at risk (VaR) is a measure of risk, indicating a reasonable expectation of potential losses during a certain period. Most commonly, analysts use a 99% or a 95% confidence level to determine the VaR. In effect, the measure describes a company’s financial strength by disregarding the most unlikely adverse outcomes and then reporting the worst-case scenario of the remaining possible futures.

For example, if a bank loans a business $1M for capital improvement, the absolute worst-case scenario is that the company goes bankrupt and defaults on the loan. In that case, the bank losses $1M. But perhaps the business has some assets and a strong brand. The likelihood of a complete loss is very low. Therefore, the bank doesn’t truly consider a $1M loss to be a risk. Ignoring the absolute worst-case scenario, which is unlikely to occur, the bank might consider $100,000 to be its value at risk in this transaction, with 95% confidence. In other words, the bank is saying there’s less than a 5% chance of losing more than $100,000 by making the loan.

Conditional value at risk (CVaR) — also known as expected shortfall, expected tail loss, or average value at risk — is an alternative risk measure to value at risk (VaR). VaR provides the worst remaining outcome after removing the tail of the distribution (that is, the unlikely results toward the end of the set of all possible outcomes). Conversely, CVaR is the weighted-average of all possible outcomes within the tail.

In other words, CVaR is the average result of an investment or portfolio, assuming that the outcome is within a particular part of the distribution. For instance, a 5% CVaR is the average value of an investment, given that the result is in the bottom 5% of all possible outcomes. If the bottom five of daily stock price movements out of the last 100 days were -10%, -8%, -3%, -1%, and 0%, the CVaR would be the average of those values (-4.4%).

Companies and regulators use value at risk (VaR) to ensure there are enough net assets to survive significant adverse market movements over a given time horizon. That’s important because if a company’s investments suffer substantial losses, it may suddenly become insolvent (have more liabilities than assets) — which could lead to bankruptcy. In the financial sector, the loss of a banking institution can lead to the failures of other banks and a general lack of credit in the monetary system. Therefore, ensuring that large banks have more net assets than their VaR is one way for regulators to reduce the chances of another financial meltdown.

Individual corporations may use VaR and other risk metrics as internal controls for risk management. This is especially true in venture capital (taking equity positions in high-risk businesses) and investment banking (banks focused on financing projects). Companies that take on a lot of risky projects have the potential for massive gains or large losses. A series of losses can easily bankrupt a company if that risk isn’t managed correctly. Consequently, VaR modeling can be an essential part of running a successful firm.

There are two standard methods for calculating the value at risk (VaR) of an investment, and one typical approach to VaR modeling for a portfolio.

The most direct approach is to sort a dataset, then locate the value at the desired confidence level. Say you were contemplating an investment in a company’s common stock. One approach to determining the risk associated with that purchase would be to look at the historical data for that stock. For instance, you could download the closing price of the last 100 trading days. Then, determine how much the stock’s price changed in each of those days. Next, arrange the daily returns from largest to smallest. Finally, identify the value that corresponds to your target risk level. For example, if you want to calculate the 99% VaR for one day, find the 99th value on the list (aka, exclude the bottom 1% of outcomes).

Rather than picking the VaR off an arranged list of values, you can fit a normal distribution (bell-shaped symmetrical curve) to approximate a dataset. Doing so provides a convenient calculation of any given probability — based on an empirical rule of the normal distribution, which states a value will be greater than 2.33 standard deviations from the mean 99% of the time.

That empirical rule allows you to calculate the VaR in a few easy steps:

Step 1: Measure the average change in value from the dataset (use the =AVERAGE() formula in Excel). Step 2: Determine the standard deviation (a measure of dispersion within the data) of those daily fluctuations (use =STDEV.S() in Excel). Step 3: Multiply the standard deviation by 2.33 Step 4: Subtract the value in Step 3 from the value in Step 1. That means that the formula for calculating VaR is: VaR = average – 2.33 * standard deviation

Calculating the VaR of a portfolio is a bit more complicated than doing so for a single investment. If the portfolio consists of just two or three assets, calculating VaR with an Excel template is not difficult. Doing so is called the variance-covariance method, and you just need to find normal distributions of each asset and the correlations between the investment risks.

The formula for doing this is a little cumbersome, but several standard templates are available. Financial institutions with large portfolios use VaR modeling, which is a more complex approach to risk measurement. Most often, that involves using some type of Monte Carlo simulation software. This Monte Carlo method can use several risk factors to assess market risk and the portfolio value.

Imagine you bought a one-year corporate bond for $1,000, expecting to receive $1,200 at maturity. For simplicity, assume that you determine there is a 10% chance of the company going bankrupt without paying you. In this scenario, there is a 10% chance of losing $1,000 and a 90% chance of earning $200 of profit (plus your $1,000 back). Your value at risk is the $1,000 of potential loss.

Now, imagine you can buy a separate bond from an unrelated company under the same terms and risks. There are now three potential outcomes:

1. Both bonds default
2. One bond fails, and one pays off
3. Both bonds pay off

If both bonds fail, you end up losing $2,000. But the probability of that happening is 1% (10% of 10%). Perhaps you think that outcome is too unlikely to worry about, so you don’t consider that outcome in your risk assessment. The VaR metric tells you the remaining worst-case scenario, excluding the most unlikely outcomes.

In this scenario, that leaves the risk of one bond failing and the other paying off. If that happens, you’d lose your $1,000 investment on one bond and make $200 of profit on the other. Therefore, your VaR on these bonds is $800. That’s the amount of possible loss from this combination of bonds, excluding the maximum loss scenario.

The major advantage of the value at risk metric is that it is widely used and easy to understand. When someone says that their 99% annual VaR is $1M, it means that they are 99% confident they won’t lose more than $1M during the year.

However, that value doesn’t say anything about what happens in that other 1% of cases. Unlikely events may result in $1.1M of losses. Or the absolute worst-case possibility might be $100B in losses. In either case, the VaR is the same. Critics of the VaR metric highlight this hidden level of risk and prefer other approaches that take into account less likely scenarios.