Combining Expected Losses and Portfolio Tail Risk

Summary

The discussion distinguishes combining expected losses from estimating portfolio risk. If two factor returns have stated expected losses, their combined expected loss is determined by their portfolio weights and the linearity of expectation; correlation does not change that expected value. For equally weighted factors with the stated losses, the portfolio’s expected loss is their weighted average. The accepted answer also gives a matrix expression involving returns, weights, and a correlation matrix, but its notation is not explained clearly enough to establish that it computes expected loss; correlation matrices are generally used to combine variances or covariances.

Correlation matters for dispersion and tail outcomes, not for the mean of a weighted sum. Estimating portfolio tail risk requires more information about the joint distribution of factor returns, and correlation alone is generally insufficient. The response also cautions that simple returns do not add in the same way as log returns. The discussion does not specify a loss horizon, distribution, or portfolio weights beyond the equal-weight illustration, so it cannot yield a general tail-loss estimate.

Key ideas

• The expected value of a weighted sum is the weighted sum of expected values.
• Correlation does not alter expected portfolio loss, though it affects portfolio variability.
• Equal weighting makes the combined expected loss the average of the two expected losses.
• Tail-risk estimates require information about the joint return distribution beyond correlation alone.
• Simple returns and log returns have different aggregation properties.

Tags

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


# 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

## Answer by Suminda Sirinath S. Dharmasena (score 0, accepted)

https://quant.stackexchange.com/a/2948

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

## Answer by David Nehme (score 1)

https://quant.stackexchange.com/a/2951

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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