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Let us ignore the riskless rate for simplicity of the presentation. If you have (historical or simulated) return series $r_i$ for the portfolio and $r_i^M$ for the market, then the beta is the OLS regression beta: $$\beta = cov(r_i,r_i^M)/var(r_i^M).$$ Then if you write $r_i = \alpha + \beta r_i^M + \epsilon_i$ on the other hand $$\epsilon_i = r_i - ( ... 2 You can do 2 things: incremental risk: Calculate the volatility with the asset and with the asset replaced by cash. The difference gives you the (non-linear) incremental risk contribution of the asset. They don't sum up to \sigma. contributions to volatility (Euler allocation) As \sigma = \sigma^2/\sigma you can define risk contributions by$$ ...

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Your question is very important! In formal way to demonstrate it is very interesting ... but a bit complicated ... and boring for non mathematicians. We may move around this demonstration to explain most of portfolio theory. However, to give the idea, if we have N risky assets we obtain, as efficient frontier, a semi-parabola and the weights of the countless ...

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