What is the Central Limit Theorem and What are its Implications for Finance?
The distribution of the average of a lot of random numbers will be normal (also known as Gaussian) even when the individual numbers are not normally distributed.
Play a dice game where you win $10 if you throw a six, but lose $1 if you throw anything else. The distribution of your profit after one coin toss is clearly not normal, it's bimodal and skewed, but if you play the game thousands of times your total profit will be approximately normal.
Let Xi,X2,...,Xn be a sequence of random variables which are independent and identically distributed (i.i.d.), with finite mean, m and standard deviation s. The sum
has mean mn and standard deviation s^Jn. The Central Limit Theorem says that as n gets larger the distribution of Sn'tends to the normal distribution. More accurately, if we work with the scaled quantity
then the distribution of Sn converges to the normal distribution with zero mean and unit standard deviation as n tends to infinity. The cumulative distribution for Sn approaches that for the standardized normal distribution.
Figure 2.1: Probabilities in a simple coin-tossing experiment: one toss.
Figure 2.1 shows the distribution for the above coin-tossing experiment.
Now here's what your total profit will be like after one thousand tosses (Figure 2.2). Your expected profit after one toss is
Your variance is therefore
so a standard deviation of ^/605/54 sa 1.097. After one thousand tosses your expected profit is
Figure 2.2: Probabilities in a simple coin-tossing experiment: one thousand tosses.
and your standard deviation is
See how the standard deviation has grown much less than the expectation. That's because of the square-root rule.
In finance we often assume that equity returns are normally distributed. We could argue that this ought to be the case by saying that returns over any finite period, one day, say, are made up of many, many trades over smaller time periods, with the result that the returns over the finite timescale are normal thanks to the Central Limit Theorem. The same argument could be applied to the daily changes in exchange rate rates, or interest rates, or risk of default, etc. We find ourselves using the normal distribution quite naturally for many financial processes.
As often with mathematical 'laws' there is the 'legal' small print, in this case the conditions under which the Central Limit Theorem applies. These are as follows.
• The random numbers must all be drawn from the same distribution
• The draws must all be independent
• The distribution must have finite mean and standard deviation.
Of course, financial data may not satisfy all of these, or indeed, any. In particular, it turns out that if you try to fit equity returns data with non-normal distributions you often find that the best distribution is one that has infinite variance. Not only does it complicate the nice mathematics of normal distributions and the Central Limit Theorem, it also results in infinite volatility. This is appealing to those who want to produce the best models of financial reality but does rather spoil many decades of financial theory and practice based on volatility as a measure of risk, for example.
However, you can get around these three restrictions to some extent and still get the Central Limit Theorem, or something very much like it. For example, you don't need to have completely identical distributions. As long as none of the random variables has too much more impact on the average than the others then it still works. You are even allowed to have some weak dependence between the variables.
A generalization that is important in finance applies to distributions with infinite variance. If the tails of the individual distributions have a power-law decay, x —1—a with 0 <a < 2, then the average will tend to a stable Levy distribution.
If you add random numbers and get normal, what happens when you multiply them? To answer this question we must think in terms of logarithms of the random numbers.
Logarithms of random numbers are themselves random (let s stay with logarithms of strictly positive numbers). So if you add up lots of logarithms of random numbers you will get a normal distribution. But, of course, a sum of logarithms is just the logarithm of a product, therefore the logarithm of the product must be normal, and this is the definition of lognormal: the product of positive random numbers converges to lognormal.
This is important in finance because a stock price after a long period can be thought of as its value on some starting day multiplied by lots of random numbers, each representing a random return. So whatever the distribution of returns is, the logarithm of the stock price will be normally distributed. We tend to assume that equity returns are normally distributed, and equivalently, equities themselves are lognormally distributed.
References and Further Reading
Feller, W 1968 An Introduction to Probability Theory and Its Applications, third edition. John Wiley & Sons Inc, New York