LO 1.1: Estimate VaR using a historical simulation approach.

Estimating VaR with a historical simulation approach is by far the simplest and most straightforward VaR method. To make this calculation, you simply order return observations from largest to smallest. The observation that follows the threshold loss level denotes the VaR limit. We are essentially searching for the observation that separates the tail from the body of the distribution. More generally, the observation that determines VaR for n observations at the (1 a) confidence level would be: (a x n) + 1.

Professors Note: Recall that the confidence level, (1 a), is typically a large value (e.g., 95% ) whereas the significance level, usually denoted as a , is much smaller (e.g., 5%).

To illustrate this VaR method, assume you have gathered 1,000 monthly returns for a security and produced the distribution shown in Figure 1. You decide that you want to compute the monthly VaR for this security at a confidence level of 93%. A ta95% confidence level, the lower tail displays the lowest 3% of the underlying distributions returns. For this distribution, the value associated with a 95% confidence level is a return of

15.5%. If you have $1,000,000 invested in this security, the one-month VaR is $155,000 (-15.5% x $1,000,000).

Figure 1: Histogram of Monthly Returns – u C a j 3 crv U – i

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2018 Kaplan, Inc.

Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3

Example: Identifying the VaR limit

Identify the ordered observation in a sample of 1,000 data points that corresponds to VaR at a 93% confidence level.

Answer:

Since VaR is to be estimated at 93% confidence, this means that 5% (i.e., 50) of the ordered observations would fall in the tail of the distribution. Therefore, the 51st ordered loss observation would separate the 5% of largest losses from the remaining 95% of returns.

Professors Note: VaR is the quantile that separates the tail from the body of the distribution. With 1,000 observations at a 95% confidence level, there is a certain level o f arbitrariness in how the ordered observations relate to VaR. In other words, should VaR be the 50th observation (i.e., a x n), the 51st observation [i.e., (a x n) + 1], or some combination o f these observations? In this example, using the 51st observation was the approximation for VaR, and the method used in the assigned reading. However, on past FRM exams, VaR using the historical simulation method has been calculated as just: (a x n), in this case, as the 50th observation.

Example: Computing VaR

A long history of profit/loss data closely approximates a standard normal distribution (mean equals zero; standard deviation equals one). Estimate the 5% VaR using the historical simulation approach.

Answer:

The VaR limit will be at the observation that separates the tail loss with area equal to 5% from the remainder of the distribution. Since the distribution is closely approximated by the standard normal distribution, the VaR is 1.65 (5% critical value from the stable). Recall that since VaR is a one-tailed test, the entire significance level of 5% is in the left tail of the returns distribution.

>From a practical perspective, the historical simulation approach is sensible only if you expect future performance to follow the same return generating process as in the past. Furthermore, this approach is unable to adjust for changing economic conditions or abrupt shifts in parameter values.

2018 Kaplan, Inc.

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Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3

Pa r a m e t r i c E s t i m a t i o n A p p r o a c h e s