Category: Actuarial fundamentals · Reviewed by Al Jabbar, Broker · Specialist Risks · Last reviewed
Premium calculation principle
A premium calculation principle (PCP) is a mathematical rule that determines the premium an insurer should charge for a given risk, expressed as a functional of the loss distribution. PCPs sit at the foundation of actuarial pricing theory.
Common principles
Expected value principle — P = (1 + θ) E[X], for loading θ > 0.
Standard deviation principle — P = E[X] + αSD(X).
Variance principle — P = E[X] + βVar(X).
Exponential principle — P = (1/α) ln E[exp(αX)], derived from utility theory with constant absolute risk aversion.
Wang transform / proportional hazards principle — distorts the survival function S(x) before computing the expected loss; the principle dominates modern catastrophe and reinsurance pricing literature.
Esscher principle — P = E[X exp(hX)] / E[exp(hX)].
Desirable properties
Bühlmann and others have proposed properties a sensible PCP should satisfy:
No rip-off — P ≥ E[X] (the premium is at least the expected loss).
Monotonicity — if X ≤ Y almost surely then P(X) ≤ P(Y).
A principle satisfying translation invariance, positive homogeneity, sub-additivity and monotonicity is a coherent risk measure (Artzner et al., 1999).
References
Bühlmann, H. (1970). Mathematical Methods in Risk Theory. Springer.
Wang, S.S. (1996). Premium Calculation by Transforming the Layer Premium Density. ASTIN Bulletin 26(1).
Artzner, P., Delbaen, F., Eber, J.M., Heath, D. (1999). Coherent Measures of Risk. Mathematical Finance 9(3).
