Capital, Solvency & Risk Measures

Capital is the buffer that absorbs the tail of the aggregate loss distribution. Two risk measures dominate.

Value at Risk

$$\text{VaR}_{\alpha}(S)=\inf\{s:\,F_S(s)\ge \alpha\}$$

The alpha-quantile of aggregate loss, e.g. the 1-in-200 (99.5%) loss under Solvency II style calibration.

Tail Value at Risk (TVaR / CTE)

$$\text{TVaR}_{\alpha}(S)=E\big[S\mid S>\text{VaR}_{\alpha}(S)\big]$$

The average loss beyond the VaR, a coherent measure that captures tail severity, not just its threshold.

Economic capital

Required capital is broadly the tail measure of loss less the resources already held:

Economic capital

$$EC=\rho(S)-E[S]$$

rho is the chosen risk measure (VaR or TVaR); economic capital covers unexpected loss beyond the expected.

BCAR-style thinking

Rating agencies assess capital adequacy by stressing the balance sheet, net required capital for underwriting, reserving, asset and credit risk against available capital. Power Re manages to a capital buffer above these requirements, with growth limits, concentration limits and an admissible-asset policy that preserve risk-adjusted capital. See financial strength.

Capital, Solvency & Risk Measures

Economic capital

BCAR-style thinking

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