Decision-aware reference

These functions are illustrative realized-consequence calculations. They are not pricing, reserving, treaty, solvency, or regulatory-capital models, and they do not supply governance thresholds or prospective validation.

ActEval's decision layer implements the architecture

predictive distribution -> explicit decision -> financial loss

It does not assert that one loss function is appropriate for every insurer, jurisdiction, product, or business objective.

Generic regret

Given model decision d_m, benchmark d_b, realized outcome y, financial loss L, and effective weights w:

model_loss     = weighted mean L(d_m, y)
benchmark_loss = weighted mean L(d_b, y)
regret         = model_loss - benchmark_loss

Negative regret means the model decision outperformed the supplied benchmark. Relative regret is only returned when benchmark loss is positive.

Pricing

premium_from_distribution() applies

mean loss * (1 + profit loading) / (1 - expense ratio)

pricing_regret() uses asymmetric absolute consequence:

c_under * max(y - premium, 0) + c_over * max(premium - y, 0)

These costs must represent the user's economic view. This simplified loss does not model demand elasticity, regulation, expenses that vary by policy, or multi-period customer behavior.

Loss ratio

sum w_i loss_i / sum w_i premium_i

The signed impact is realized ratio minus target. It is a portfolio consequence, not a proper statistical score.

Reserve and capital shortfall

Per observation:

max(realized loss - held amount, 0)

ActEval reports weighted aggregate and mean shortfall, weighted frequency, and conditional mean when shortfall occurs. Reserve and capital functions share the formula but retain different decision labels because their governance and time horizons differ.

Reinsurance

ReinsuranceOption represents a quoted excess-of-loss contract with retention r and premium pi. Ceded loss is max(loss-r, 0) and retained loss is min(loss, r).

Projected selection minimizes

pi + E[retained loss] + capital_cost_rate * rho(retained loss)

where rho is VaR or expected shortfall at an explicit quantile. This mirrors actuarial retention work that combines reinsurance premiums, retained losses, and tail risk measures, but it is only one possible business rule.

Realized option regret compares pi + min(loss, r) to the supplied benchmark quote. It excludes taxes, reinstatements, limits, counterparty default, commissions, and contract wording unless users incorporate them in a custom loss function.

References

• Cai, J. and Tan, K. S. (2007), Optimal Retention for a Stop-Loss Reinsurance under the VaR and CTE Risk Measures, ASTIN Bulletin 37(1), 93–112.
• Major, J. A. and Mildenhall, S. J., Introduction to Capital Modeling and Portfolio Management, Casualty Actuarial Society Monograph No. 15.
• Blanchet, J., Lam, H., Tang, Q. and Yuan, Z. (2016), Applied Robust Performance Analysis for Actuarial Applications.