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What is a mathematical definition of risk

What is a mathematical definition of risk?

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A starting point for a mathematical definition of risk is simply as standard deviation. It is sensible due to the results of the Central Limit Theorem (CLT), that when you add up a large number of investments what matters so far as the statistical properties of the portfolio are only the expected return and the standard deviation of individual investments, and the resulting portfolio returns are certainly normal distributed.

The normal distribution being symmetrical regarding the mean, the potential downside can be measured in order of the standard deviation.

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