What is VaR?

Following on from the previous simple question, here's another: "What is VaR?" Again, a simple question deserves a (relatively) simple answer, so that's what I've tried to do below.

Let's assume you have a portfolio of shares worth 1M USD. What's the maximum you could lose over the next day or month or year? Without an answer to this question, you can't tell whether the return you're receiving on your portfolio is appropriate compensation for the risk of holding it.

Value at Risk (VaR) is an attempt to answer that question. For a given portfolio, probability, and time horizon, VaR  is a group of related mathematical models that attempt to estimate the maximum mark-to-market loss that you're going to experience. So it's a way of measuring the (primarily market) risk of holding a portfolio of financial instruments.

If your aforementioned share portfolio currently worth $1M has a one-day VAR of 1%, you can state that in 3 ways, each of which has a subtly different tone:

• $10K is the most you're going to lose tomorrow, 99% of the time.
• $10K is the least you're going to lose tomorrow, 1% of the time.
• You'll lose more than $10K on 1 out of 100 days, or around 3 days in the coming year.  

So VaR essentially divides risk into 2 regimes:

• Peace-time, where the probability distribution curve towers over you and provides the illusion of safety.
• War-time, where the "fat tail" whacks you on the head and your losses can be catastrophic.

So much for the "simple" answer. If you want to go shopping, now's the time. If you want more of the sordid detail, read on.

In capital markets, VaR has several uses:

• Front-office risk management - helping a trading desk to understand the day-to-day market risk of its positions.
• Middle-office risk measurement - aggregating VaR across desks and time zones. But note that VaR typically isn't additive across asset classes because of correlation risk. And if you want to abandon completely the idea of a simple answer, you can point out that VAR is difficult to roll-up across the enterprise even for a single asset class, because it isn't sub-additive without some tweaking.
• Financial reporting.
• Computing regulatory capital.

For front-office risk management:

• VaR is a system rather than a number.
• The VaR system is run periodically, but increasingly moving towards real-time as computing resources become cheaper and faster.
• The computed number isn't adjusted for input errors, computation errors, delays, or market movements.
• Abnormal market conditions are ignored - there's no stress-testing or scenario analysis.
• To back-test, you compare hypothetical profit & loss (P&L) to actual P&L based on the actual price movements during the period for which the VaR system is being tested.
• A trading desk's VaR limit (either an absolute number or a risk metric) is used to decide what risks to allow today.
• The VaR system is best used for making short-term and tactical decisions today.

For middle-office risk measurement:

• VaR is a number rather than a system.
• The VaR number is calculated periodically, usually daily.
• The computed number can be adjusted after the fact to correct input and computation errors.
• Stress-testing and scenario analysis are used to account for abnormal market conditions.
• To back-test, you compare retroactively-computed VaR numbers on historic position/pricing data to actual price movements calculated from the same data.
• Each trading desk's VaR limit is used as an input to measure that desk's risk-adjusted return.
• The VaR numbers are best used to make medium-term and strategic decisions for the future.

VaR has many well-known issues, including:

• It's essentially a peace-time statistic, and can give you a false illusion of safety in war-time. As pointed out above, it's a bad risk measure for "extreme" tail risk events.
• It doesn't measure liquidity risk. The less liquid an asset (and note that most assets become less liquid in times of crisis), the worse VaR is as a risk measure.
• It typically only uses 2 years of history, and is then unable to capture some types of event.
• It fails to distinguish between leverage that comes from long-term fixed-rate debt such as bonds, and short-term variable-rate debt such as loans and mortgages.
• It can be gamed. For example, selling a bunch of CDS products will generate a lot of small and steady gains, but occasionally a huge loss. VaR ignores the slim probability of giant losses.

As before, the above answer is just a simplistic introduction to the concept of VaR. The details are much more messy and entertaining.