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Let me give a try....

1. Beta=r(p,m)*Sigma(P)/Sigma(M)

2. If portfolio P is well diversified, the correlation r(p,m) between portfolio P and the martket equals 1.

3. therefore, Beta(P)=1*Sigma(P)/Sigma(M)=Sigma(P)/Sigma(M)

Does it make sense?

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