It follows that the Treynor ratio is equal to the Sharpe ratio multiplied by:

Since the Sharpe ratio is given by Tport -risk-free Oport and the Treynor ratio is given by T'port - Prisk-free Bport it follows that the Treynor ratio is equal to the Sharpe ratio multiplied by: O Pport,market O market Omarket Pport market o Pport, market Omarket