Making the square-root formula compatible with capital allocation

Modern regulatory capital standards, such as the Solvency II standard formula, employ a correlation based approach for risk aggregation. The so-called "square-root formula" uses correlation parameters between, for example, market risk, non-life insurance risk and default risk to determine the company's aggregate capital requirement. To support decision-making, companies will allocate the required capital back to business segments and risk drivers. We demonstrate that capital allocations based on the square-root formula can substantially differ from those based on the true risk distribution if correlations are viewed as Pearson or tail correlations. An EVA-maximizing insurer receives misleading steering signals which can induce mispricing of risk and a default probability substantially above the desired level. To make the square-root formula feasible for business steering, we propose partial-derivative-implied correlations which reflect how marginal exposure changes impact the aggregate capital requirement. We show that the square-root formula in combination with partial-derivative-implied correlations provides capital allocations in line with the true risk distribution.