Actuarial Science, Insurance Capital & Solvency II Regulation
Solvency II BSCR Correlation Diversification & Coverage Formula
Calculate EIOPA Solvency II Standard-Formula Undiversified SCR Sum, Variance-Covariance Correlated Basic Solvency Capital Requirement SQRT(SCR_1^2 + SCR_2^2 + 2*rho*SCR_1*SCR_2), Diversification Benefit, and Solvency Coverage Ratio.
Aggregate insurance risk modules via EIOPA’s quadratic correlation matrix—never simple addition!
Under European Solvency II (99.5% 1-year Value-at-Risk Standard Formula), insurers do NOT simply add 1-in-200-year capital charges across Market Risk and Non-Life Underwriting Risk, because equity market crashes and windstorm underwriting losses rarely hit their 99.5th percentile simultaneously! EIOPA prescribes a variance-covariance aggregation matrix where Basic Solvency Capital Requirement BSCR = SQRT(SUM_i,j Corr_ij * SCR_i * SCR_j)—which for Market Risk (SCR_mkt) and Non-Life Underwriting Risk (SCR_nl) with statutory correlation rho = 0.25 reduces to SQRT(SCR_mkt^2 + SCR_nl^2 + 2 * rho * SCR_mkt * SCR_nl)! Subtracting BSCR from the Undiversified Sum quantifies the Capital Diversification Benefit, while Eligible_Own_Funds / BSCR measures regulatory Solvency Coverage (statutory minimum >= 1.00x / 100%)!
Verify on a Composite P&C Insurer—Market_Risk_SCR = 100.00 EUR M (B2), NonLife_Underwriting_SCR = 60.00 EUR M (B3), EIOPA_Correlation_Rho = 0.25 (B4), and Eligible_Tier1_Own_Funds = 200.00 EUR M (B5): Undiversified Sum is 100 + 60 = 160.00 EUR M (E2); Correlated BSCR is SQRT(10,000 + 3,600 + 2 * 0.25 * 6,000) = SQRT(16,600) = 128.84 EUR M (E3); Capital Diversification Benefit saves 160 - 128.84 = 31.16 EUR M (E4); and Solvency Coverage Ratio reaches 200 / 128.84 = 1.5523 (155.23% in E5)!
| Solvency II Capital Metric (Mkt=100M, Non-Life=60M, rho=0.25, EOF=200M) | Formula | Verified Output |
|---|---|---|
| Undiversified SCR Sum (EUR M) | SCR_Market + SCR_NonLife | 160.00 EUR M |
| Correlated BSCR Charge (EUR M) | SQRT(SCR_m^2 + SCR_n^2 + 2*rho*SCR_m*SCR_n) | 128.84 EUR M (160.00 @ rho=1) |
| Capital Diversification Benefit | Undiversified_Sum - Correlated_BSCR | 31.16 EUR M (0.00 @ rho=1) |
| Solvency II Coverage Ratio (x) | Eligible_Own_Funds / Correlated_BSCR | 1.5523 (155.23%; 1.2500 @ rho=1) |
Solvency II Variance-Covariance BSCR Aggregation Formula
Validate non-negative module SCRs (with SCR_m + SCR_n > 0), valid correlation -1 <= rho <= 1, and Eligible_Own_Funds > 0!
=LET(sc_m, B2, sc_n, B3, rh_o, B4, eo_f, B5, IF(OR(sc_m < 0, sc_n < 0, sc_m + sc_n <= 0, rh_o < -1, rh_o > 1, eo_f <= 0), "Invalid Solvency II Input", ROUND(SQRT(sc_m ^ 2 + sc_n ^ 2 + 2 * rh_o * sc_m * sc_n), 2)))Perfect-Correlation Stress Check (Set Correlation rho = 1.00)
In a shared ORSA stress-testing tab, set Correlation B4 = 1.00: Correlated BSCR E3 rises to the full 160.00 EUR M undiversified sum, Diversification Benefit E4 drops to 0.00, and Solvency Coverage E5 tightens to 200 / 160 = 1.2500 (125.00%)!
=IF(ISTEXT(E3), E3, ROUND(B5 / E3, 4))Try the example
Use the original values or change them to test your understanding.