Monetary risk measures are studied here in terms of acceptable outcomes, normalized non-negative prices, and resulting shortfalls. At center stage stand convex analysis, saddle functions and associated max-min formulae. The latter comply with common sense and established theory.

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S. D. Flåm. “Risk Measures, Convexity, and Max-Min Shortfalls.” Journal of Convex Analysis 22 (2015), No. 3, 603–626.