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Law of Large Numbers and Central Limit Theorem

The law of large numbers and the central limit theorem are stated, proved and applied to inference in the statistics course. This lesson is about what they do specifically in finance, and it starts from an observation that is easy to miss.

Finance averages in two directions. You average one asset over many days, and you average many assets over one day. Both are sums of random variables, both invoke the same two theorems, and they behave completely differently. Conflating them is the source of a surprising amount of bad risk argument.

Direction one: across time

Hold one asset for nn days. If daily returns are independent with variance σ2\sigma^2, variances add and the standard deviation grows as the square root:

σn-day=σn\sigma_{n\text{-day}} = \sigma\sqrt{n}

This is the basis of the square-root-of-time rule and of the 252\sqrt{252} that annualises a daily volatility.

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