Risk loading

A premium equal to expected loss leads to ruin with probability one over an infinite horizon, a classical result of ruin theory: surplus then follows a driftless walk, and a driftless walk hits zero. The risk loading is the drift you add. Its economic justification is remuneration of the capital a shareholder ties up to absorb deviations, so its correct size depends on the capital the risk consumes, not on its expectation. Three calculation principles coexist and give different answers. A loading proportional to expectation is simple and blind to volatility, so it underprices dispersed risks. A loading proportional to standard deviation recognizes volatility but not dependence with the rest of the book. A loading based on the cost of allocated capital, applying a return rate to the capital the risk consumes after diversification, is the only one consistent with balance sheet management, and it is the one internal models have converged on. Risk loading is not to be confused with expense loadings, which cover spending rather than risk.

Treaty covering a risk with a 10M EUR expectation and a 6M EUR standard deviation, 2026 year. A 5% loading on expectation returns 500K EUR. A 10% loading on standard deviation returns 600K EUR. A cost of capital loading, with 18M EUR of allocated capital after diversification and a 9% internal return rate, returns 1.62M EUR. The three principles applied to the same risk spread the gross premium by more than two combined ratio points.

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