# Robust sum of non-independent random variables

What approach could be used to sum non-independent variables?

I have probability distributions of stock prices and want to calculate the probability distribution of the portfolio price (sum of some of those stocks).

Monte Carlo can't be used, because variables are non-independent and it requires joint probabilities. I want to avoid dealing with joint probabilities or correlation, it's hard to calculate and estimate the errors, and it changes with time (in normal times correlation could be ~0, in crisis suddenly correlation could jump to ~1 as all stocks go down).

So, what else could be used? (Ideally something from robust statistics).

There are following inputs:

• Historical prices for individual stocks.
• Historical prices for all the possible portfolios (we just sum up historical prices of stocks to get it).
• Some black box algorithm that uses those historical data and predicts probabilities of future prices for individual stock.
• (if it helps) The final distributions assumed to be Pareto-distribution.

How can we use these inputs to estimate the performance of the portfolio? Ideally get the probability distribution for the price of portfolio, or if it's not possible get at least some other estimate.

P.S.

I guess the task is similar to the river flow estimation. You need to estimate the flow of river to guard against possible water overflow. In mountains river has lots of inflows. During the year they are very small and non-correlated (because rain in one area not affecting inflows in other areas). But then spring comes and everywhere in mountains ice start melting and suddenly all the inflows during all the length of a river flood tons of water in a correlated manner rising river level disproportionally. I wonder what tools used to model those things?

• Since the distribution of the sum will depend on the joint distribution you cannot avoid dealing with joint probabilities. Whatever approach you choose you will make an assumption about the joint distribution. This assumption may be explicit or just implicit somewhere but it always will be there.
– g g
Dec 7, 2019 at 16:00
• If you have good historical data on inflow from streams, you probably also have good data on total river flow, and you’d model the total directly. Dec 7, 2019 at 19:20
• @MattF. yes, there's historical data for all possible portfolio combinations. The thing is - there's also fundamental data for individual stocks. Modelling portfolio directly on past data will rely on prices only. While predictions for individual stocks rely on prices + fundamentals and maybe can be more accurate. The problem is - I can predict individual prices, but then can't sum it up into portfolio. Or - I can use the direct approach and use past portfolio prices, but then I don't know how to utilise the fundamental data for individual stock. Dec 8, 2019 at 4:00
• @MattF. I reformulated the question, please check it out stats.stackexchange.com/questions/439822/… Dec 8, 2019 at 4:37
• The question is and was clear, but 1) most natural answers are ruled out by the refusal to consider joint probabilities or correlations; and 2) the PS does not actually provide a good parallel. Dec 8, 2019 at 4:44