Asymptotic Behaviour of Unexpected Losses and Risk Ratios for Co-Monotonic Alternatives

Abstract

The aggregation of individual risks in large credit and insurance portfolios is guided by diversification and the law of large numbers, which formalizes the convergence of sample averages to their means. At the same time, regulatory capital requirements and insurance premia are designed to provide a capital buffer or risk margin above the mean. The resulting excess, given by the difference between the nonlinear valuation of the aggregate loss and the corresponding mean, reflects the idea of protection against unexpected losses in the sense of banking and insurance regulation. This paper studies the asymptotic behaviour of this excess for large weighted portfolios. The main result shows that, for monotone cash-additive risk measures on Banach-lattice-valued Orlicz spaces, convergence along weighted averages satisfying a weak law of large numbers together with a uniform integrability condition is equivalent to scalar continuity at the origin. If the risk measure is positively homogeneous, this continuity condition is automatically satisfied, and we prove that the unexpected losses of large weighted portfolios are of order $o(n\overlineλ_n)$, where $\overlineλ_n$ denotes the average weight assigned to the first $n$ random variables. We establish analogous asymptotic results for Choquet insurance premia. Finally, we derive risk-ratio limits that quantify the potential underestimation arising when diversified portfolios are compared with co-monotonic alternatives.