The Investment Advisor Body of Knowledge + Test Bank. IMCA
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Название: The Investment Advisor Body of Knowledge + Test Bank

Автор: IMCA

Издательство: John Wiley & Sons Limited

Жанр: Зарубежная образовательная литература

Серия:

isbn: 9781118912348

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СКАЧАТЬ there is a high probability that the manager's expected return is greater than 10 percent (a good result), we have to say, “We reject the null hypothesis that the manager's returns are less than or equal to 10 percent at the x percent level.” This is very close to a double negative. Like a medical test where the good outcome is negative and the bad outcome is positive, we often find that the good outcome for a null hypothesis is rejection.

      To make matters more complicated, what happens if the portfolio manager doesn't seem to be that good? If we rejected the null hypothesis when there was a high probability that the portfolio manager's expected return was greater than 10 percent, should we accept the null hypothesis when there is a high probability that the returns are less than 10 percent? In the realm of statistics, outright acceptance seems too certain. In practice, we can do two things. First, we can state that the probability of rejecting the null hypothesis is low (e.g., “The probability of rejecting the null hypothesis is only 4.2 percent”). More often we say that we fail to reject the null hypothesis (e.g., “We fail to reject the null hypothesis at the 95.8 per- cent level”).

      Sample Problem

      Question:

      At the start of the year, you believed that the annualized volatility of XYZ Corporation's equity was 45 percent. At the end of the year, you have collected a year of daily returns, 256 business days' worth. You calculate the standard deviation, annualize it, and come up with a value of 48 percent. Can you reject the null hypothesis, H0: σ = 45 percent, at the 95 percent confidence level?

      Answer:

      The appropriate test statistic is:

      Notice that annualizing the standard deviation has no impact on the test statistic. The same factor would appear in the numerator and the denominator, leaving the ratio unchanged. For a chi-squared distribution with 255 degrees of freedom, 290.13 corresponds to a probability of 6.44 percent. We fail to reject the null hypothesis at the 95 percent confidence level.

      Application: VaR

      Value at risk (VaR) is one of the most widely used risk measures in finance. VaR was popularized by J.P. Morgan in the 1990s. The executives at J.P. Morgan wanted their risk managers to generate one statistic at the end of each day, which summarized the risk of the firm's entire portfolio. What they came up 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 СКАЧАТЬ