# Any known approximations of summing quantiles from joint (bernoulli / lognormal) distributions

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.

• Much too vague to solve as I read this. Please give a mathematical formula for a version of this problem as an example. How many risks? Do the all have the same lognormal dist'n. Do the Bernoulli's all have the same 'success' probability? // From what you say this might be a 'random sum of random variables', but that is unclear. – BruceET Sep 9 '20 at 5:40
• What are the probabilities and lognormal parameters for the three most likely risks? And what is the sum of the probabilities over all the risks? Those would be good starting points for finding a reasonable approximation. – Matt F. Sep 9 '20 at 6:06
• Are these assumed identically distributed, or in several homogeneous classes, or does each risk have its own $p, \mu, \sigma$? There's a bit of work on the first case in the actuarial literature. – Glen_b Sep 9 '20 at 6:50
• Even if you found an approximation, why bother? Simulation of your $B*L$ loss distributions is dirt cheap. You can create gazillions of samples in a millisecond and 95% is really not that far out. Furthermore, as Glen_b points out, such approximations always require simplifying assumptions such as identical distribution, which are untenable for messy real life data. If you really need to speed up your simulation (why? How much?) you might want to check out importance sampling. – g g Sep 9 '20 at 6:59