Imagine tossing a fair coin 10 times. Sometimes you get 7 heads, and sometimes 3. Now, imagine tossing it 10,000 times. Chances are you’ll get very close to 5,000 heads.
That’s not a coincidence, that’s probability in action. More importantly, it’s the Central Limit Theorem (CLT) at work, a concept that sits at the core of quantitative finance.
If you’re building trading algorithms, measuring portfolio risk, or designing simulations, the CLT is your silent partner. In this post, we will simplify what it means, why it matters, its limitations, and how it appears in interviews and real-world quantitative problems.
What is the Central Limit Theorem (CLT)?
Let’s say you want to calculate the average height of all women in the United States. To do that, you randomly selected fifty women and calculated their mean height. Likewise, you selected many samples of 50 and calculated their average height. Now, the Central Limit Theorem states that when you plot these sample averages, these averages will form a bell-shaped curve (normal distribution), even though those samples may not be normally distributed.
More formally, given a population of any distribution (not necessarily normal), the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, provided the samples are independent and identically distributed (i.i.d.) and have a finite variance.
Mathematically, let x_{1}, x_{2}, x_{3},{…},x_{n} be independent and identically distributed random variables with mean \mu and variance \sigma^{2}, then the standardized sample mean Z_{n} = \frac{X_{n}-\mu}{\sigma/\sqrt{n}} converges in distribution to the standard normal distribution Z_{n} \xrightarrow{d} \mathcal{N}(0, 1) as n\to \infty.
Why Central Limit Theorem important in finance?
The most important reason is that it allows us to assume the aggregated behaviours, such as average returns, risk metrics, etc., as approximately normal even if individual returns are not normally distributed.
This justifies the use of normal distribution in portfolio theory, risk metrics like Value-at-Risk (VaR), hypothesis testing, and confidence intervals. It also underpins Monte-Carlo simulations and pricing models by ensuring that the average of many random outcomes behaves predictably.
Moreover, it also supports the idea that as we aggregate more data, like returns over time or across assets, the distribution becomes more stable and easier to analyse statistically.
However, the one obvious question you will ask is:
Why we want these aggregate behaviours to be normally distributed?
Aggregated behaviours being normally distributed allows us to use powerful, well-developed mathematical and statistical tools with confidence. For example, if daily returns are normally distributed, we can use hypothesis testing, run regression, calculate confidence intervals, estimate VaR, CVaR, etc. These well-developed mathematical and statistical tools help simplify modelling, inference, and decision-making in finance.
So, CLT helps us to assume these aggregated behaviours are normally distributed, such that we can use these mathematical and statistical tools confidently.
What if the Central Limit Theorem fails?
The key conditions to assume any distribution to be normal, based on CLT, are: samples must be i.i.d., variance must be finite, and sample size must be large enough (at least 30). If any of these conditions are not true for an aggregated behaviour, then CLT fails. It means we can assume that the aggregated behaviour is normally distributed.
In that case, we must adjust our methods to handle non-normality. For instance, for risk, instead of using Value-at-Risk (VaR), we use Expected Shortfall (a more robust risk measure), or we model tail risk using Extreme Value Theory (EVT).
In reality, the stock return might not be independent and identically distributed, as stock returns might have volatility clustering, autocorrelation (in some cases), non-constant variance or sometimes fat tails.
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