Table 10.

Summary of risk-adjusted performance measures.

Application purpose	Method/data	Description	Paper
Portfolio performance	Sharpe ratio	Sharpe analyzed mutual fund performance, introducing the Sharpe ratio as a fundamental risk-adjusted performance metric. The Sharpe ratio continues to be crucial for assessing portfolio performance and plays a significant role in empirical asset pricing	Ferruz and Vicente, 2005; Aragon and Ferson, 2006; Samarakoon and Hasan, 2013
Portfolio performance	Treynor ratio	Treynor employed the market risk represented by beta stock. The Treynor ratio emphasizes systematic risk, ranks portfolio performance, and assesses diversification adequacy. Grounded in the Security Market Line, it compares the expected total return of a security or portfolio with that of a market portfolio	Hübner, 2005; Beer et al., 2011; Verma and Hirpara, 2016; Robiyanto, 2018
Portfolio performance	Jensen's Alpha ratio	Alpha are considered one of the most widely used traditional measures of investment performance. Thus, Jensen's Alpha is unique and considers systematic risks. A tool used for assessing the relative performance of a portfolio in comparison to benchmarks	Jensen, 1968; Hübner, 2005; Samarakoon and Hasan, 2013
Portfolio performance	Modigliani and Modigliani ratio	This metric presents an alternative risk measure, utilizing return volatility within the CAPM framework. The adjusted portfolio is constructed and managed as a blend of the managed portfolio and a risk-free asset, ensuring it matches the total risk of the market portfolio	Modigliani and Modigliani, 1997; Samarakoon and Hasan, 2022.
Portfolio performance	Multibeta Models	Multibeta models emerge when investors ideally hold combinations of a mean–variance efficient portfolio plus hedge portfolios for the other relevant risks. Most asset-pricing models describe the cross-section of expected returns regarding risk factor exposures, or betas	Sharpe, 1977; Velu and Zhou, 1999; Balduzzi and Robotti, 2008.
Portfolio performance	Weight-Based Performance Measures	A manager with investment ability raises the fund's exposure to securities or asset class before it performs well, or who expects and avoids losers	Aragon and Ferson, 2006; Ferson, 2013.
Appraisal ratio	Treynor–Black Appraisal Ratio	In this scenario, Treynor and Black (1973) calculate the mean–variance optimum portfolio and show that the optimal deviations from the benchmark holdings for each security are affected by the “Appraisal Ratio”. The ideal portfolio should include covered securities and index funds	Treynor and Black, 1973; Kahneman and Tversky, 2013.
Market Timing	Merton-Henriksson Market Timing Measure	This model allows the manager to monitor a private signal about the market's future performance. It changes the portfolio's market exposure or beta at the start of the period. However, the resulting convexity can be described with put or call options	Henriksson and Merton, 1981; Henriksson, 1984; Ferson, 2013.