# Practical Use of Copulas and Sample Generation

I'm currently studying Copulas. However, i did not understand something. The very basic phases of Copula fitting is as follows i assume;

1. Model each samples distribution with a parametric(or non-parametric) pdf.
2. By using(or assuming) the model CDF, convert each RV to the [0,1] range.
3. Fit a parametric copula to the converted RVs.

However, on the 4th step, all the examples on the Internet evaluates a regeneration phase from the fitted Copula. They basically sample new values from copula function and by using Inverse-CDF, they obtain real values in the domain of each original marginals.

What i do not understand is what is the practical reason for re-sampling phase? I already have some observations and by fitting a copula i can say that either "these two distributions are dependent and this(copula) is the model of their dependency" or "they seem to be independent (could not find any suitable copula)". However, what is the reason for re-generating some non-observed values as they are actually observed ? Where should I use these "synthetic" values in real life ?

• You should use them to simulate scenarios with some degree of tail dependency between the chosen financial instruments. Let you have just two instruments and a copula: now you know how to give a probability to each pair of instruments values (after the 4th step), even if that pair has never been observed in the past. That could be considered a non trivial advantage. – Lisa Ann May 8 '19 at 9:42
• So what if I want to obtain a probabilistic sum of two RVs?. Is regenerating sample is related with this purpose? For example if i want to obtain the sum of them with %99 probability what should i do basically? – Levent Bekdemir May 8 '19 at 10:29
• What you're getting from any point of your copula function is the joint probability of two different events that might or might not have been observed in the past. To get that number, you take your copula surface (like this one), fix $x$ and $y$ coordinates and find the proper value on $z$. It's like calling the copula function $z=f(x,y)$. – Lisa Ann May 8 '19 at 13:33
• Ok, I got your point. "What is the sum of two value from two distribution will be with %99 probability ?": the answer is that you can get that number by generating a lot of random observations from your copula and taking the quantile with 99% probability. The simulated observations are paired (in case of bivariate copula) and ordered, so you can see what lies above or below some rank thresholds. However, you'll have to model somehow the marginal densities before (unless you want to work with pseudo observations, of course). – Lisa Ann May 8 '19 at 22:06
• VaR is the same, in fact you can use copula to estimate VaR for your portfolio of financial instruments. The approach of fitting a parametric copula can be overkill when you already have a lot of tail risk samples in your historical data (then VaR is basically an empirical copula); on the contrary, parametric copula is strongly advised when you have instruments which have never shown bloody plunges in their history. In such cases, you want to find a way to model what could happen in worst case scenarios - think about an investment grade bond that suddenly defaults - and copula is the tool. – Lisa Ann May 9 '19 at 6:46