The standard math used to calculate stock market returns breaks if prices ever go negative.
March 31, 2026
Original Paper
What Is a Return? (Especially When Prices Can Be Negative) An Axiomatic Theory for Markets with Negative Prices
SSRN · 6380198
The Takeaway
Almost every financial formula, from risk measurements to performance ratios, relies on 'log-returns' which are mathematically impossible to calculate for negative numbers. As oil and electricity prices increasingly dip below zero, this paper proves there is only one specific mathematical function—the arcsinh—that can actually keep the financial system's calculations from crashing.
From the abstract
Every formula in quantitative finance-CAPM, Markowitz, VaR, Sharpe ratio, GARCH-takes returns as input. Yet the standard definitions of return fail when prices cross zero: log-returns are undefined, and simple returns produce sign errors. This is not hypothetical: WTI crude settled at-$37.63 on April 20, 2020, and European electricity markets routinely produce negative prices. We ask: what properties must a universal return function satisfy? We prove that six axiomsfive structural (additivity, s