How is the risk-adjusted performance measure calculated?

The risk-adjusted performance measure is calculated by taking into account the level of risk associated with an investment or portfolio and comparing it to a benchmark or a risk-free rate of return. There are several commonly used risk-adjusted performance measures, including the Sharpe ratio, the Treynor ratio, and the Jensen's alpha.

The Sharpe ratio is calculated by subtracting the risk-free rate of return from the average return of the investment or portfolio, and then dividing it by the standard deviation of the investment or portfolio's returns. This ratio measures the excess return earned per unit of risk taken.

The Treynor ratio is similar to the Sharpe ratio but uses the systematic risk, also known as beta, instead of the standard deviation. It is calculated by subtracting the risk-free rate of return from the average return of the investment or portfolio, and then dividing it by the beta of the investment or portfolio. This ratio measures the excess return earned per unit of systematic risk taken.

Jensen's alpha, also known as the Jensen index or the Jensen's measure, is calculated by subtracting the risk-free rate of return from the actual return of the investment or portfolio, and then subtracting the product of the portfolio's beta and the difference between the benchmark return and the risk-free rate of return. This measure represents the excess return earned by the investment or portfolio above what would be expected based on its systematic risk.

These risk-adjusted performance measures help investors and portfolio managers evaluate the performance of an investment or portfolio by considering the level of risk taken to achieve the returns. By comparing the risk-adjusted performance measures of different investments or portfolios, investors can make more informed decisions about their investment choices.