Full integration and risk premium

Normally, risk premium = return on global market - risk-free. With full integration, risk premium = Beta x (return on global market - risk-free) = correlation i,m x (S.D asset i / S.D market) = correlation i,m x S.D asset i x Sharpe ratio Why does risk premium with full integration include beta, when normal risk premium = return on global - risk-free?

Your beta is still there in the first equation. It’s just equal to 1