The Investment Advisor Body of Knowledge + Test Bank. IMCA
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Here the numerator is simply the difference between the sample mean and the population mean, while the denominator is the sample standard deviation divided by the square root of the sample size. To see why this new variable follows a t-distribution, we simply need to divide both the numerator and the denominator by the population standard deviation. This creates a standard normal variable in the numerator, and the square root of a chi-square variable in the denominator with the appropriate constant. We know from discussions on distributions that this combination of random variables follows a t-distribution. This standardized version of the population mean is so frequently used that it is referred to as a t-statistic, or simply a t-stat.
Technically, this result requires that the underlying distribution be normally distributed. As was the case with the sample variance, the denominator may not follow a chi-squared distribution if the underlying distribution is nonnormal. Oddly enough, for large sample sizes the overall t-statistic still converges to a t-distribution. If the sample size is small and the data distribution is nonnormal, be aware that the t-statistic, as defined here, may not be well approximated by a t-distribution.
By looking up the appropriate values for the t-distribution, we can establish the probability that our t-statistic is contained within a certain range:
(3.77)
where xL and xU are constants, which, respectively, define the lower and upper bounds of the range within the t-distribution, and (1 – α) is the probability that our t-statistic will be found within that range. The right-hand side may seem a bit awkward, but, by convention, (1 – α) is called the confidence level, while α by itself is known as the significance level.
In practice, the population mean, μ, is often unknown. By rearranging the previous equation we come to an equation with a more interesting form:
(3.78)
Looked at this way, we are now giving the probability that the population mean will be contained within the defined range. When it is formulated this way, we call this range the confidence interval for the population mean. Confidence intervals are not limited to the population mean. Though it may not be as simple, in theory we can define a confidence level for any distribution parameter.
Hypothesis Testing
One problem with confidence intervals is that they require us to settle on an arbitrary confidence level. While 95 percent and 99 percent are common choices for the confidence level in risk management, there is nothing sacred about these numbers. It would be perfectly legitimate to construct a 74.92 percent confidence interval. At the same time, we are often concerned with the probability that a certain variable exceeds a threshold. For example, given the observed returns of a mutual fund, what is the probability that the standard deviation of those returns is less than 20 percent?
In a sense, we want to turn the confidence interval around. Rather than saying there is an x percent probability that the population mean is contained within a given interval, we want to know what the probability is that the population mean is greater than y. When we pose the question this way, we are in the realm of hypothesis testing.
Traditionally the question is put in the form of a null hypothesis. If we are interested in knowing if the expected return of a portfolio manager is greater than 10 percent, we would write:
(3.79)
where H0 is known as the null hypothesis. Even though the true population mean is unknown, for the hypothesis test we assume the population mean is 10 percent. In effect, we are asking, if the true population mean is 10 percent, what is the probability that we would see a given sample mean? With our null hypothesis in hand, we gather our data, calculate the sample mean, and form the appropriate t-statistic. In this case, the appropriate t-statistic is:
(3.80)
We can then look up the corresponding probability from the t-distribution.
In addition to the null hypothesis, we can offer an alternative hypothesis. In the previous example, where our null hypothesis is that the expected return is greater than 10 percent, the logical alternative would be that the expected return is less than or equal to 10 percent:
(3.81)
In principle, we could test any number of hypotheses. In practice, as long as the alternative is trivial, we tend to limit ourselves to stating the null hypothesis.
WHICH WAY TO TEST?
If we want to know if the expected return of a portfolio manager is greater than 10 percent, the obvious statement of the null hypothesis might seem to be μr > 10 percent. But there is no reason that we couldn't have started with the alternative hypothesis, that μr ≤ 10 percent. Finding that the first is true and finding that the second is false are logically equivalent.
Many practitioners construct the null hypothesis so that the desired result is false. If we are an investor trying to find good portfolio managers, then we would make the null hypothesis μr ≤ 10 percent. That we want the expected return to be greater than 10 percent but we are testing for the opposite makes us seem objective. Unfortunately, in the case where 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.