“If the Sharpe ratio of the new investment is greater than the current portfolio Sharpe ratio multiplied by the correlation of the new investment’s returns with the portfolio’s returns, adding the investment to the portfolio will improve the portfolio Sharpe ratio”

Can someone explain the logic behind this?

My reasoning is this but please correct me if I am wrong.

There are two factors to consider when deciding whether you should add a new asset to the portfolio: 1. it’s individual risk and return (sharp ratio) and 2. the asset’s correlation with the portfolio.

Now since it states that the individual asset’s sharp ratio must be larger than the portfolio’s sharpe ratio multiplied by the correlation of the asset with the portfolio, it is essentially saying that the individual assets return over risk must be higher than the effects that the correlation has to the portfolio when adding a new asset. So for instance, if an asset is uncorrelated with the portfolio, the correlation effects could bring down the total return of the portfolio, but tinsce the sharpe ratio is higher, you should select. Does this make sense?