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WC expense program
 in experience rating, SP = MP * mod
 expenses assumed to be a flat % of P
 appropriate for small risks, but as P incr, expenses should make up a smaller portion of the higher layers
 can reflect this via discount factors
 Guaranteed Cost Premium = SP  Discount

Formula for retro premium
 H ≤ R = (b + cL)T ≤ G
 R/T = b + cL = b + crE

Retro  Basic premium
 insurance carrier services
 loss control services
 adjustment for limiting premium
 profit / contingencies
 b = e  (c  1)E + cI = expense component + cI
 e: provision for expense & profit as ratio of SP
 cI: converted insurance charge

Retro  Expense
 e = (1  D)/T  E
 T(e + E) = 1  D = Guaranteed Cost Premium

Retro  Tax multiplier
 tax: proportional to net premium (τ)
 assessment: proportional to losses (μ)
 0.2 = expense piece of retro P

Retro  Insurance charge
 quantifies the net impact to insr for application of min & max P
 cI = c(X_{G}  S_{H})E
 X_{G} = table M charge at entry ratio r_{G}
 S_{H} = table M savings at entry ratio r_{H}

Retro  Balance equations
 Value difference
 Entry difference
 circular ref of insurance charge
 if satisfy both, expected retro P = GCP
 if no min, S_{G} = [G/T  (e + E)] / cE

Constructing a Table M
 Calculate entry ratio. If no E given, use E(hat)
 # risks at LR
 # risks over LR
 losses over LR: start bottom, cumulative sum of previous
 losses over / max losses over = φ(r)
 Ψ(r) = φ(r) + r  1

Retro  Aggregate balance
 retro = (b + cL)T
 apply H and G
 check if sum = # risks * risk amt * GCP
 if yes  actuarial balance btwn retro & prospective rating


Continuous case for φ(r) and Ψ(r)

