why do some problem just say (add if Sharpe is > existing portfolio) and other use the Sharpe * correlation formula?
I think they are trying to simplify the idea. Regardless of the correlation, if the new asset’s Sharpe ratio (Sn) is greater than the current portfolio’s Sharpe ratio (Sp), then you will increase the mean variance efficiency of the portfolio by including the new asset.
For a given Sn and Sp: Sn > Sp <==> Sn > Sp*corr(Rn,Rp) for all correlations, since correlations are bounded by -1 and 1.
However, if the Sn isn’t greater than Sp, you should use the formula of Sp*corr(Rn,Rp) to see if Sn exceeds that quantity.
Hi Danv, The way you want to approach these problems: Is the Sharpe ratio of the new investment greater than the correlation of the new invetsment to the existing port * the exisiting ports sharpe. IE: NEW SHARPE> OLD SHARPE * (CORREL NEW INVESTMENT W/ PORT) If given multiple investments, look to maximize the difference in NEW SHARPE- O LD SHARPE * (CORREL NEW INVESTMENT W/ PORT), that will be the best investment to add to the port.
I think the OP understood that, but he was asking why some problems don’t include the correlation, which is what you omitted.