Incremental Sharpe
Incremental Sharpe, or iSharpe, is a measure of the contribution of a position or sub-portfolio to the overall Sharpe ratio of a portfolio. iSharpe can be used in portfolio allocation to maximize the Sharpe ratio of a portfolio.
If the iSharpe of a sub-portfolio is positive, then increasing the allocation to that sub-portfolio slightly will increase the overall Sharpe ratio of the portfolio. Likewise, if the iSharpe of a sub-portfolio is negative, then increasing the allocation to that sub-portfolio slightly will decrease the overall Sharpe ratio of the portfolio.
Assume the ith position in a portfolio has exposure or weight, wi, and a Sharpe ratio of Si. Further, assume that the Sharpe ratio of the portfolio is Sp, and that the correlation between the portfolio and the ith position is ρ, then the iSharpe of that position relative to the portfolio would be:
If two positions are perfectly correlated, ρ = 1, then the Sharpe ratio of a position must be higher than the Sharpe ratio of the portfolio in order for it to be accretive to the Sharpe ratio of the portfolio. At the same time, adding to a position with a very low Sharpe ratio may increase the overall Sharpe ratio of a portfolio if the position is weakly or negatively correlated with the existing portfolio. Even if the Sharpe ratio of a position is low, if the iSharpe is positive, then adding to that position, at the margin, will increase the Sharpe ratio of the portfolio. As with delta and incremental value at risk, iSharpe is only valid for small changes.