24.4 Motivations for Using Copulas

Limitations of joint distributions

There are only a few joint distributions that are tractable to work with (normal, lognormal, exponential, etc)

  • We can’t do Weibull, Pareto or gamma

  • We can’t mix distributions like joining normal with exponential

Modeling all business together is not feasible due to inconsistent of mix of business over time

24.4.1 Joint Distribution Plots

3 ways to look at them to try and see if there are dependencies; Each box should have the same number of points if independent

  1. Straight plotting on \((x,y)\)

    • Draw lines at 25%, 50% and 75% and segment the plot into 16

    • If the 2 marginal distributions are independent \(\Rightarrow\) there should be about \(\frac{1}{16}\) of the points in each rectangles

    • This is useful to show us actual values, but

    • Might be difficult to see points in some of the rectangles

  2. Plot on log log scale \((\ln(x), \ln(y))\)

    • This alleviate the issue above and shows a clearer picture of the dependencies

  3. Plot the percentile from each marginal distn

    • This gives us a clean picture as all the rectangles are the same size since the axis are now the percentile

    • Only need the marginal distributions \(F(x)\) and \(G(y)\) to plot

24.4.2 Advantages of using copula to describe dependency with percentiles

  1. It is independent of the underlying distributions

    • Describe the relationship between the percentiles of 2 different distributions

  2. Can update the marginal distn without changing the dependency structure

  3. Can joint distn that is not the same