The Treynor Ratio is a risk-adjusted measure of return developed by Jack Treynor in 1965. It is used to assess how well a portfolio perform given its level of 'systematic' risk. Systematic risk of a portfolio is the portion of total risk related to the market performance which cannot be reduced by diversification. Unlike the Sharpe ratio which takes into account the portfolio's standard deviation, the Treynor Ratio calculates the returns separately for each component of the portfolio and identifies the extra returns that the portfolio may generate, adjusted for its systematic risk.

The Treynor Ratio is calculated as the return on the portfolio divided by the portfolio's beta (systematic risk) and is expressed as a percentage. For example, if the portfolio has an average return of 10% and a beta value of 1, then the Treynor Ratio is 10%.

In contrast to the Sharpe Ratio, the Treynor Ratio does not take into account any un-systematic risk. Un-systematic risk is the risk associated with individual assets in the portfolio and is reduced or eliminated through diversification. Thus, when comparing portfolios with different levels of un-systematic risk, Treynor Ratio is preferred over the Sharpe ratio because it provides a better measure of 'true' risk-adjusted returns.

The Treynor Ratio is most frequent;ly used to compare different portfolios and help investors to make decision on which one should be selected. It is particularly useful when selecting an investment whose performance is affected by the performance of the stock market. For instance, a portfolio that seeks absolute returns rather than relative returns (i.e has a beta of 0) would be a better investment according to the Treynor Ratio because its return is not affected by market fluctuations and therefore provides more consistent portfolio returns.

Overall, the Treynor Ratio is a useful ratio for investors who want to compare portfolios of similar risk levels and determine which one is the more suitable investment for their needs. While it does not take into account any un-systematic risk, its focus on systematic risk makes it an effective tool in risk-adjusted performance assessment.