The Investment Advisor Body of Knowledge + Test Bank. IMCA
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with was VaR.
Figure 3.8 provides a graphical representation of VaR. If the 95 percent VaR of a portfolio is $100, then we expect the portfolio will lose $100 or less in 95 percent of the scenarios, and lose $100 or more in 5 percent of the scenarios. We can define VaR for any level of confidence, but 95 percent has become an extremely popular choice in finance. The time horizon also needs to be specified for VaR. On trading desks, with liquid portfolios, it is common to measure the one-day 95 percent VaR. In other settings, in which less liquid assets may be involved, time frames of up to one year are not uncommon. VaR is decidedly a one-tailed confidence interval.
FIGURE 3.8 Value at Risk Example
For a given confidence level, 1 – α, we can define value at risk more formally as:
(3.82)
where the random variable L is our loss.
Value at risk is often described as a confidence interval. As we saw earlier in this chapter, the term confidence interval is generally applied to the estimation of distribution parameters. In practice, when calculating VaR, the distribution is often taken as a given. Either way, the tools, concepts, and vocabulary are the same. So even though VaR may not technically be a confidence interval, we still refer to the confidence level of VaR.
Most practitioners reverse the sign of L when quoting VaR numbers. By this convention, a 95 percent VaR of $400 implies that there is a 5 percent probability that the portfolio will lose $400 or more. Because this represents a loss, others would say that the VaR is –$400. The former is more popular, and is the convention used throughout the rest of the book. In practice, it is often best to avoid any ambiguity by, for example, stating that the VaR is equal to a loss of $400.
If an actual loss exceeds the predicted VaR threshold, that event is known as an exceedance. Another assumption of VaR models is that exceedance events are uncorrelated with each other. In other words, if our VaR measure is set at a one-day 95 percent confidence level, and there is an exceedance event today, then the probability of an exceedance event tomorrow is still 5 percent. An exceedance event today has no impact on the probability of future exceedance events.
Sample Problem
Question:
The probability density function (PDF) for daily profits at Triangle Asset Management can be described by the following function:
Triangular Probability Density Function
What is the one-day 95 percent VaR for Triangle Asset Management?
Answer:
To find the 95 percent VaR, we need to find a, such that:
By inspection, half the distribution is below zero, so we need only bother with the first half of the function:
Using the quadratic formula, we can solve for a:
Because the distribution is not defined for π < –10, we can ignore the negative, giving us the final answer:
The one-day 95 percent VaR for Triangle Asset Management is a loss of approximately 6.84.
BACK-TESTING
An obvious concern when using VaR is choosing the appropriate confidence interval. As mentioned, 95 percent has become a very popular choice in risk management. In some settings there may be a natural choice for the confidence level, but most of the time the exact choice is arbitrary.
A common mistake for newcomers is to choose a confidence level that is too high. Naturally, a higher confidence level sounds more conservative. A risk manager who measures one-day VaR at the 95 percent confidence level will, on average, experience an exceedance event every 20 days. A risk manager who measures VaR at the 99.9 percent confidence level expects to see an exceedance only once every 1,000 days. Is an event that happens once every 20 days really something that we need to worry about? It is tempting to believe that the risk manager using the 99.9 percent confidence level is concerned with more serious, riskier outcomes, and is therefore doing a better job.
The problem is that, as we go further and further out into the tail of the distribution, we become less and less certain of the shape of the distribution. In most cases, the assumed distribution of returns for our portfolio will be based on historical data. If we have 1,000 data points, then there are 50 data points to back up our 95 percent confidence level, but only one to back up our 99.9 percent confidence level. As with any distribution parameter, the variance of our estimate of the parameter decreases with the sample size. One data point is hardly a good sample size on which to base a parameter estimate.
A related problem has to do with back-testing. Good risk managers should regularly back-test their models. Back-testing entails checking the predicted outcome of a model against actual data. Any model parameter can be back-tested.
In the case of VaR, back-testing is easy. Each period can be viewed as a Bernoulli trial. In the case of one-day 95 percent VaR, there is a 5 percent chance of an exceedance event each day, and a 95 percent chance that there is no exceedance. Because exceedance events are independent, over the course of n days, the distribution of exceedances follows a binomial distribution:
(3.83)
In this case, n is the number of periods that we are using to back-test, k is the number of exceedances, and (1 – p) is our confidence level.
Sample Problem
Question:
As a risk manager, you are tasked with calculating a daily 95 percent VaR statistic for a large fixed income portfolio. Over the past 100 days, there have been four exceedances. How many exceedances should you have expected? What was the probability of exactly four exceedances during this time? Four or less? Four or more?
Answer:
The probability of exactly four exceedances is 17.81 percent:
Remember, by convention, for a 95 percent VaR the probability of an exceedance is 5 percent, not 95 percent.
The probability of four or fewer exceedances is 43.60 percent. Here we simply do the same calculation as in the first part of the problem, but for zero, one, two, three, and four exceedances. It's important not to forget zero:
For the final result, we could use the brute force approach and calculate the probability for k = 4, 5, 6, … , 99, 100, a total of 97 calculations. Instead we realize that the sum of all probabilities from 0 to 100 must be 100 percent; therefore, if the probability of K ⩽ 4 is 43.60 percent, then the probability of K > 4 must be 100 percent – 43.60 percent = 56.40 percent. Be careful, though, as what we want is the probability for K ⩾ 4. To get this, we simply add the probability that K = 4, from the first part of our question, to get the final answer, 74.21 percent:
EXPECTED SHORTFALL
Another criticism of VaR is that it does not tell us anything about the tail of the distribution. Two portfolios could have the exact same 95 percent VaR, but very different distributions beyond the 95 percent confidence level.
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