## Defining and interpreting Value at Risk (VaR)

We started risk management on the CFA Level 3 curriculum with a disucssion of the different types of risk that we might look to hedge, whether those be financial or non-financial. From there we briefly surveyed centralized and decentralized risk management systems and hit on a couple of approaches on how to mitigate risk more generally.

All of those topics of course depend on the ability to model the risk you actually face. Enter the value at risk, or VAR, measure.

## What is Value at risk (VaR)?

Value at risk, or VaR, is always tested on the CFA Level 3 exam. Be prepared.

Value at risk is really concerned with measuring the given probability of loss within a specific investment portfolio over a defined period of time.

More formally **VaR is a single aggregate number which measures the probability that a portfolio's return is going to fall below a certain level over a specified period of time**.

In other words we are estimating the dollar loss that we expect to be exceeded within a given level of probability over a given time period.

### Interpreting a Value at Risk number

Let's say we have a $10 million portfolio, which has a 4.3% VaR at a 5% probability level for one month. What does that value at risk number mean?

What we're saying is that over the next month there's a 5% probability that the portfolio could lose more than 4.3% of its value. Put differently, we are also saying that there's a 95% chance that the portfolio will lose less than 4.3%.

VaR could also be given in dollar terms. Take a portfolio whose daily VaR is $1 million. That daily VAR can be read as: "we are 95% sure that our daily losses are not going to exceed $1 million in one day," or that "5% of the time portfolio losses will exceed $1 million in a day.

**Basic Implications of VaR**

There are a couple additional facts worth noting about value at risk that should be readily apparent from its definition.

- A 1% VaR has greater risk in dollar terms than a 5% VaR; it is further from the mean
- The longer the time period we include the greater the VaR will be. We cannot compare different VaRs unless they have the same time period

### Limitations to using Value at Risk (VAR)

The biggest drawback of value at risk, and it's a big one, is that **it doesn't tell you anything about the magnitude of the loss**.

In other words, VaR doesn't tell you how much worse than your specified threshold it could be. Take the above example. We just finished saying, "Okay, there's a 95% chance at a $1 million daily VaR that the portfolio losses won't exceed $1 million." Okay, fine, but, in that 5% probability that it DOES exceed $1 million we don't know if that daily loss is going to be $100 million when it happens. VaR says nothing about that.

Then the secondary drawback, which is not nearly as testable as the first, is that VaR doesn't consider liquidity risk at all. **How do we mitigate not knowing about the magnitude of potential loss?**

So to mitigate this, we look a the left tail, or the fat tails of the distribution to see just how serious a problem it is or how bad those losses could get. And then we also look at kind of a portfolio's general correlation with risk factors and the asset classes in it to kind of get another sense of risk there.

In our next post we review the three ways of calculating value at risk that are introduced in the CFA Level 3 curriculum (analytical, historical, and monte carlo).