Question:

Let’s say we find a good non-normal, fat-tailed distribution that characterizes an asset’s returns.

(i) What is the meaning of the sampling distribution of the sample mean?
(ii) Is there anything we can say about it?
(iii) Why might this not necessarily help us, from a risk perspective?

Answer:

(i)
If we draw a sample (say, 10 random trials), and take the average, we produce (X bar) a single SAMPLE MEAN.
Then do it again, now we have TWO SAMPLE MEANS.
And again and again until, say, we have THIRTY SAMPLE MEANS.
(btw, that is about where a small sample becomes a large sample; it’s where the student’s t approximately the normal)
We can plot a historical (a distribution) of those sample means. This is the sampling distribution of the sample mean.

(ii)
The central limit theorem (CLT) says this sampling distribution of sample means converges toward (approximates as N increases) the normal distribution REGARDLESS OF THE POPULATION DISTRIBUTION. We have a non-normal population distribution; yet the sampling distribution is normal!

(iii)
Because this concerns central tendency - then tendency of the SAMPLE MEAN. We are typically concerned with losses in the tail, a fourth moment (kurtosis) function, if you like.

Hi David,

Could you elaborate a bit on no. iii)

rocky420,

Frankly, (iii) is not well-phrased. I meant to highlight the specific “finding” of the CLT: that the sample mean converges to a normal distribution. As that’s about the sample *mean* and not dispersion or the tails, CLT doesn’t give information about losses in the tail. I was really trying to get at the idea that a distribution has a body and tails, and knowing something about the body may not be particularly helpful if we are concerned about the tails which is typically our concern.

Let me put another way. In the wake of the credit crunch, we read a lot of bashing of the normal distribution (“how can the idiots use models that allow for 25 sigma events?”—well, that tells you the distribution is not normal). But to professional risk managers, this is a straw man. Normal is not necessarily de facto outside of specific market risk applications (and where the statistics allows, which is not often. Why, statistically? Because rarely do we know the population variance. Even for tests of the sample mean, owing that we do not know population variance, therefore, the student’s t is appropriate really and not the normal.).In our credit and opRisk readings, we encounter all variety of distribution except the normal. When we do see the normal, it is almost always knowingly borne of utter convenience with sober understanding of its stark limitations. The CLT explains the prevalence of the normal distribution, but it does not justify its use in risk measurement.

David