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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 - ( ...

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