The Sharpe-ratio function and its shape for some renowned portfolios
Abstract
The Sharpe-ratio is a popular financial indicator. This paper defines the Sharpe-ratio function, which explicitly illustrates the dynamic evolution of the risk premium (the portfolio’s return minus the risk-free income) in relation to risk (the portfolio’s variance). The Sharpe-ratio function requires knowledge of the composition of the efficient mean-variance frontier. We demonstrate that the Sharpe-ratio function is a strictly convex and decreasing function when the mean-variance frontier is differentiable, and (decreasing) linear when it is not. However, market portfolios may not include negative investments in a security (short selling is impossible), and the efficient frontier of the portfolios may not be differentiable at every point (for example, at the terminal point). We empirically test this over the 2015–2024 period, exhibiting the shapes of the mean-variance efficient frontier and the Sharpe-ratio functions for portfolios such as the DAX 40 and the BUX 5. Among our key findings, we observe that the DAX 40’s Sharpe-ratios for the wartime period (February 2022 – December 2024) are significantly higher than those for the 2015–2024 period. The BUX 5 comprises only half of the DAX 40’s Sharpe-ratios. However, during this latest period the BUX 5 exhibited a return twice as high that of the DAX 40.
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