Hello,I've been study the CFA books and don't understand the significance of the Central Limit Theorem. Basically, my understanding of CLT is that, given a population that is distributed in any shape/form (could be crazy), if you take a a random sample size of n from that population, and calculate the mean of that sample, and do this repeatedly, the distribution of the means of the samples of size n will approach a normal distribution as n increases in size. The mean of the sample distribution (distribution of sample means) will also approach the mean of the population as n increases.From what I understand, the CLT is used to predict the population mean from repeatedly taking samples of the population.What I don't get is what's the point of learning this and what's so cool about the CLT? If you have access to the population distribution, why not just calculate the mean? Why go through this CLT stuff and try to approximate it? If you didn't' have access to the population distribution, then whatever sampling mean you calculate probably won't be anywhere close to the actual population mean. I just don't get what's so significant about the CLT other than it's a cool property to know. I don't see any practical use for it.Thanks!

First of all, not "any shape/form" distribution, but just the ones with a finite variance. Generalised Central Limit Theorem applies also to those with infinite variance, but then the limit can be a different (than Gaussian) alpha-stable distribution.There are two reasons for which the CLT is very useful:1) the true population distribution can be complicated and deriving the sample mean distribution directly (via convolution) can be difficult - CLT says that it is approximately Gaussian2) sometimes you do not know the whole population distribution but can draw samples from it - using CLT you just need to calculate the mean and variance of the sample, and you know the distribution of the sample mean3) it produces funny scaling laws

An example of using CLT in finance is the (quick & dirty) "poxy integration" method of calculating tranche expected loss in Gaussian copula model for CDOs.

you're not going to be asked any in-depth questions on this on the CFA exam, L1 and L2 are 100% multiple choice these days. They may give you some calculations to run but you don't need to understand the CLT to do that.My advice would be to not worry about understanding the CLT

CLT is simple to use. CLT is used to produce results quickly when you don't know the distribution of the population (which is usually the case).The precision of the results calculated with CLT depends on how close your samples follow the assumptions of the theorem. Namely, to what extent your samples are i.i.d. (independent and identically distributed)When the samples follow the assumptions - you will get accurate estimates for mean and stdev.When not - you will introduce the error. The size of the error depends on the deviation of your samples from the assumptions.Sometimes, your samples deviate so much from the assumptions that estimates obtained with CLT are useless. Sometimes this useless results are used by other people to prove "their point". You have to recognise when you are being bullshitted.As long as you know what the restrictions are, it is fine.One example of the applicationof the CLT - calculation of the VaR (Value at Risk).Basel rules guide banks to calculate 10 day VaR.Banks calculate 1 day VaR and then extrapolate this 1day VaR to 10 days VaR using "square root of time rule", which is based on CLT.It works well in well-behaved markets and it does not work in turbulent markets, where the daily returns are very much not i.i.d.

QuoteI just don't get what's so significant about the CLT other than it's a cool property to know. I don't see any practical use for it.CLT is one of the most fundamental properties for finance. There are two ways in my mind why it is important. First, it's a justification for the most fundamental models in finance. Consider a model representing log of stock price as drift and brownian motion. Most of analytic results in derivatives pricing rely on mathematical tractablility of brownian motion and powerful theorems about it. Why is it brownian motion? Justification is that innovation in price is a sum of many shocks to the firm and it's stock -- which results in normal distribution of return. Further, in the area of econometrics, 90% of analytic results rely on CLT -- all asymptotic distributions that give asymptotic confidence levels use multivariate CLT. Basically, CLT is so important because sum is a very common function and normal distribution is a uniquely analytically tractable distribution.

This example might show you how it can be a powerful tool:Say you toss 400 fair coins. What is the (approx.) probability that you get at least 220 heads? First try using the true distribution, then use the CLT. Which one is easier?

QuoteOriginally posted by: zerdnaQuoteI just don't get what's so significant about the CLT other than it's a cool property to know. I don't see any practical use for it.CLT is one of the most fundamental properties for finance. There are two ways in my mind why it is important. First, it's a justification for the most fundamental models in finance. Consider a model representing log of stock price as drift and brownian motion. Most of analytic results in derivatives pricing rely on mathematical tractablility of brownian motion and powerful theorems about it. Why is it brownian motion? Justification is that innovation in price is a sum of many shocks to the firm and it's stock -- which results in normal distribution of return. Further, in the area of econometrics, 90% of analytic results rely on CLT -- all asymptotic distributions that give asymptotic confidence levels use multivariate CLT. Basically, CLT is so important because sum is a very common function and normal distribution is a uniquely analytically tractable distribution.Exactly!I would add that there are many (related) CLT's worth learning: CLT for local martingales, for example.That last one gives you, as a bonus, a CLT for any stochastic integral.

When you don't have access to the population distribution and you want to get an idea of the population mean, you will have to estimate it using a sample mean. How do you know whether this sample mean is good enough? To this end, you need to learn how to construct a confidence interval of a sample mean, which, in turn, depends on the approximate distribution of the sampling distribution. Since by CLT we know that sample means approximately follow a normal distribution as sample size gets large, we can use normal distribution to establish the confidence interval of a sample mean.

QuoteOriginally posted by: 60202What I don't get is what's the point of learning this and what's so cool about the CLT? basically, there's no brownian motion without CLT. CLT is one of the most used results in probability theory. what's the point in learning it for CFA? I don't know. maybe there isn't, given the population of those who take it.

QuoteOriginally posted by: BuranQuoteOriginally posted by: 60202What I don't get is what's the point of learning this and what's so cool about the CLT? basically, there's no brownian motion without CLT. CLT is one of the most used results in probability theory. what's the point in learning it for CFA? I don't know. maybe there isn't, given the population of those who take it.I don't remember needing the CLT in the construction of Brownian motion....

Isn't it that CLT ( assuming population that is SND) states that if the sample size is over 25 then you can assume a similar mean to to the population.But would this still hold if the difference between the sample and population is huge ? I find that unlikely.

I don't remember needing the CLT in the construction of Brownian motion....Functional CLT - also called Donsker's theorem - is needed to construct BM.

All the probability books start with "17 ways to construct Brownian motion" -- well, maybe I exaggerate.I usually skip that part -- but suspect there must be lots (some?) not requiring CLT? If not, post it as a Brainteaser.

Suppose I hire you, and ask you what is a typical monthly spread between two wheat futures months, and two corn futures months. Are you going to come back and say fuck if I know, the numbers are all over the place? I want to know where to implement my spread strategy wheat or corn. But I need to have some idea of the mean spread, or whether there is a mean spread. How are you going to express to me in an objective way whether there even is a useful mean? Are you just going to show me a plot and say look, these numbers are all over the place. I want to know, wheat or corn? Which one?