# How to calculate cumulative loss from two factors that have negative correlation?

I went through the most advanced books on statistics and still can't find an answer. There is a well known formula for combining volatility of two correlating variables, but what about adding the actual amounts, which are already known?

Here is a simply put problem:

We know that the anticipated loss from factor X = 30%, anticipated loss from factor Y = 50% It is also known, that the coefficient of correlation between factor X and Y = -0.6 What cumulative loss from both factors should we expect?

P.S. A citation to a book or publication would be super appreciated

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You have to give the weights. – Suminda Sirinath Salpitikorala Feb 10 '12 at 14:14
I don't get it, what do you mean "anticipated loss"? If you mean expected loss, the answer is simple, investing in both equally would lead to an expected loss of 40%. Expected returns are linear in the weights. – Tal Fishman Feb 10 '12 at 16:37

Assuming the above return a log return (Simple returns are not additive)

(r .* w) * p * w'

r - expected loss

p - correlations

w - weights

' - transpose

.* - element-wise multiplication

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Where did you get that? What does that formula even mean? – Tal Fishman Feb 10 '12 at 16:44
Thanks! I figured out the formula, it is really great, even more than I needed. But what do I have to do to get a published source of the formula from you? Maybe you could give me an author or even a broad field where it was published: econometrics, statistics? I would really appreciate it. – Arthur Tarasov Feb 10 '12 at 18:48
@Arthur Tarasov, this is a back of the envelop calculation. Just read up on copulars. I hope this helps. – Suminda Sirinath Salpitikorala Feb 11 '12 at 8:35

If the two factors are equally weighted, then the expected loss of the portfolio is just the average of the two expected losses. The expected value of the sum of random variables is always the sum of the expected values. If you want to compute tail risks of a portfolio, then you need more information (unless the correlation was -1).

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