## Calculating Value at Risk (VaR) - Historical, Analytical, and MCS Methods

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. In our first post on VaR we broke down what it is, how to interpret it, and some of its key limitations.

In this post we'll discuss the three core ways we can calculate VaR. The calculations and the pros/cons of using each approach are highly testable for the CFA L3 exam.

## How to calculate Value at Risk (VaR)

The CFA Level 3 curriculum gives us three primary methods to calculate value at risk. They are:

- The Analytical method - a formula/calculaton based approach
- The Historical method - a percentile or ranking-based system
- Monte Carlo Simulation

### The Analytical Method for Calculating VaR

The Analytical method assumes a normal distribution of returns and uses a one-tailed confidence interval (e.g. we only care about downside risk). It is calculated as:

The analytical method basically spits out a dollar value at a desired level of significance. So if you're using a 5% VaR, the dollar value is going to tell you that there's a 5% probability that the loss for that given time period will exceed the stated dollar value.

**Tricks for Calculating VaR Using the Analytical Method**

On the L3 exam, a couple of things are really important aboout the analytical method.

- First, if you are asked to compute this number for periods of less than one year, you must divide the annual return by the time period. So if you are calculating a quarterly VaR, you must divide the annual return by four.
- Second, you also need to adjust the standard deviation for the time period. To do this divide the annual standard deviation that you are given by the square root of time
- Third, if you are asked to compute a daily value at risk number, just assume that the expected return of the portfolio is zero.
- Finally you see those given levels of significance there for the Z values under the formula? They could be given on the exam, but if I were you I would prepare to know the Z-values at 5% and 1% VaR, (1.65 and 2.33)

#### Benefits and Drawbacks to using the Analytical Method

The analytical method is a strong way to calculate value at risk.

It's easy to do. It's understandable. You can really look at the correlation of risk and you can also use it across different time periods.

On the downside, because it uses a Z-score, *the analytical method assumes a normal distribution*. So if your underlying assets don't have normal distributions, then it's inappropriate to use it. This can come up when talking about options, hedge funds, or emerging markets. If you see those on the exam that should key you in that the analytical method is not appropriate.

To summarize the pros and cons:

### The Historical Method for Calculating VaR

With the historical method, what we're doing is, first, stack ranking kind of historical returns from low to high and then calculating the lowest 5% of returns, and then we use the highest value of that lowest 5%, and that becomes kind of our 5% value at risk for that time period, which is usually daily. So if you have 40 observations the lowest 5% of that will be the 2 lowest observations. To calculate the VaR you take the higher of those two lowest observations multiply that by the portfolio value, and then you've got your dollar value at risk. On the CFA Level 3 exam this would require you to look at/interpret the values from a table. The key here is just to normalize the # of observations to figure out which specific observation (by stack ranking) you need to use.

#### Benefits and Drawbacks to using the Historical Method

On the plus side in terms of using the historical method, it's easy, it doesn't assume normal distributions, and it is quite versatile.

The main downside obviously is that it assumes historical return pattern. So if you see an exam problem where returns in recent time periods are quite different than they were in the past, or the question talks about non-stationary data, you should be tipped off that the historical method is inappropriate.

### Monte Carlo Simulation and Calculating VaR

As a L3 candidate, Monte Carlo Simulation, is something you are very familiar with. Here you are basically running thousands and thousands of models and coming up with a VaR number from that distribution.

The advantage of using MCS are its versatility and the fact that it does not assume a normal distribution.

On the downside, it's expensive, it gives you the sense of false precision, it does, of course, rely on inputs, and it's a bit one-sided.

### Summarizing Approaches to Calculating VaR for the CFA L3 Exam

The three methods: analytical, historical, and MCS are all testable, both in terms of approach or calculations but especially for a compare/contrast question asking you to determine which measure might be most appropriate in a given situation. In other words, memorize the pro/con tables that you see above.

While VaR is a key component of the risk management section in the CFA curriculum it is not the only one. Our next post will discuss the various extensions to VaR.