This is my first post to this site!
For an insurance-like scenario, I have several independent risks which I want to sum together and find a 95% percentile. Currently I do this by Monte Carlo but I would like to know if there are available approximations.
I am using a joint distribution Bernoulli / LogNormal to model the risks. In my world, the distribution (B) of each risk eventuating is a Bernoulli with known p, and the distribution (L) of the cost of that risk (given that it occurs) is a Log-normal distribution with known parameters. So the cost of each risk is the first variable times the second ie B * L.
In my world, both the distributions are heavily skewed: the p is small (less than 0.1) and the sigma of L is large. The risks are independent, but not the same as each other:
Here are some of my risks:
Risk 1: p=0.1, mu=15, sigma=2.1
Risk 2: p=0.3, mu=10, sigma=2.5
Risk 3: p=0.05, mu=17, sigma=3.1
I have about 20 such risks, of similar proportions.
I can use Monte Carlo simulation to calculate percentiles: and I am most interested in the tail ie. 95th percentile.
So currently I simulate the bernoulli choosing either 0 or 1 with the probability p, then I simulate a lognormal by simulating a normal(mu,sigma) and then applying exponent. Then I multiply these together.
I have good closed-form estimates of the 95th percentile of one risk, but the 95% quantile of a sum of these risks is where I want help. I discovered on this site that sums of log-normals are sometimes approximatable by a lognormal here, but my distributions include the Bernoulli component as well; also the skewedness doesn't help much.
Does anyone know any approximations to such problems, or any improvements to a straight Monte Carlo approach? I am open to changing the distribution types if it provides better tractability.