Reinsurance Actuarial Premium

Burning Cost

Typically burning cost is the estimated cost of claims in the forthcoming insurance period, calculated from previous years’ experience adjusted for changes in the numbers insured, the nature of cover and medical inflation.

1. Historical (aggregate) data extraction
2. Adjustments to obtain 'as if' data:
   1. present value adjustment using actuarial rate, prices index,...
   2. base insurane premium correction,
   3. underwriting policy evolution,
3. clauses application 'as if' data, calcul of the 'as if' historical reinsurance indemnity,
4. Reinsurance pure premium rate computing,
5. add charges, taxes and reduction of treaty

'As if' is term used to describe the recalculation of prior years of loss experience to demonstrate what the underwriting results of a particular program would have been if the proposed program had been in force during that period.

Premium formulation

Let us note p the and f the deductible of XS or XL, with the limite l = p + f ( p XS f ).

The premium : E [ S N ] = E [ ∑ i = 1 N Y i ] = E [ N ] × E [ Y ]

where E [ Y ] = l P [ X > l ] − f × P [ X ≥ f ] + E [ X ∣ f ≥ x ≥ l ]

XS or XL premium formulation with Pareto

If l = ∞ and α ≠ 1  : E [ S N ] = λ t α α − 1 f 1 − α $

if l = ∞ and α = 1 il n'y a pas de solution.

If l < ∞ and α ≠ 1  : E [ S N ] = λ t α α − 1 ( f 1 − α − l 1 − α )

If l < ∞ and α = 1  : E [ S N ] = λ t ln ⁡ ( 1 f )

XS premium using Lognormal cost distribution

If X follows L N ( x m , μ , σ ) then X − x m follows L N ( μ , σ )

Then: P [ X > f ] = P [ X − x m > f − x m ] = 1 − Φ ( ln ⁡ ( f − x m ) − μ σ )

E [ X ∣ X > f ] = E [ X − x m ∣ X − x m > f − x m ] + x m P [ X > f ] = e m + σ 2 / 2 [ 1 − Φ ( ln ⁡ ( f − x m ) − ( μ + σ 2 ) σ ) ] + x m ( 1 − Φ ( ln ⁡ ( f − x m ) − μ σ ) )

With deductible and without limit :

E [ S N ] = λ ( E [ X − x m ∣ X − x m > f − x m ] + x m P [ X > f ] − f P [ X > f ] ) = λ ( e m + σ 2 / 2 [ 1 − Φ ( ln ⁡ ( f − x m ) − ( μ + σ 2 ) σ ) ] ) + λ ( x m − l ) ( 1 − Φ ( ln ⁡ ( f − x m ) − μ σ ) )

Regression Estimation

Template:This method uses data along the x-y axis to compute fitted values. It is actually based on the equation for a straight line, y=bx+a.(2)

Long-Term Indemnity Claims

Actuarial reserves modellisation.