Formulas B.1 Prospective Rating xperience Rating quity and Predictive Accuracy : Venter The formula that minimizes the expected squared error, subject to the linearly constraint A = Actual Loss = xpected Loss K = the credibility constant X = Loss amount Modification= A+K + K 1961 plan Primary loss amount X 10000 X +8000, X 2000 Modified xpected Loss (ML) in No-Split plan Z = Credibility factor Basic formula : ML=Z A+(1 Z ) if Z = +K K formula: ML= A+K +K Modified xpected Loss (ML) in Split plan ML=Z p A p +(1 Z p ) p +Z e A e +(1 Z e ) e.. p = for primary loss.. e = for excess loss Since there are two credibilities, two credibility constants are needed Z p = +B, Z = e + K then W = +B +K = Z e Z p ML=[ (A +W A +(1 W ) +B ) ] p e e (+B) transforming ML to
B = Ballast value W = Weighting WC R: What very Actuary Should Know : Gillam, W.R. The formula for the experience ration mod M = the risk modification M = (A p +W A e +(1 W ) e + B ) (+ B) A = the actual or rated loss for the risk = the expected loss for the risk.. p = for primary loss.. e = for excess loss W = weighting B = ballast value Computation of the value of = all classes i i = all classesi (Payroll i 100) LR i LR i = xpected loss rate for class i Computation for the value of p p = all classesi D i = D-ratio for class i D i i and e = p Computation for the value of A A= A n n A n : Individual loss indexed at n Computation of A p and A e A p = n A np and A e = A A p A np : Primary component of individual loss indexed at n Limited loss
SAL = SRP / 10 Multiple claim occurences limit = 2 * SAL Total disease loss for policy year limit = 3 * SAL + 1.2 * Risk's total expected loss for the experience period Primary component of a single loss = 5000$ Primary component of multiple claim occurences = 10000$ Primary component of disease loss for policy year = 10000$ + 0.4 * Risk's total expected primary loss for the experience period SRP = State reference point SAL = State accident limit Computation of the SRP SRP = 250 * SACC * Trend Trend Period rounded to nearest 5000$ SACC = State average cost per claim (no lim, 100k on L) Trend = Taken from the most recent countrywide Retrospective Rating xpected Loss Size Ranges update filing Computation of G G = SRP / 250000 G = Scale factor for credibilities varying by state B= 0.1 +2570G +700G B = Ballast value Computation of B = xpected loss of an insured C= 0.75 +203825G +5100G, subject to 7500$ minimum Computation for C, subject to 150000$ minimum C = intermediate value used for the calculation of W W = + B +C Computation of W, rounded to nearest 0.01
W = Weighting Fundamentals of Individual Risk Rating Part 1 : Gillam and Snader Analysis of the basic formula in the no split plan Z ( A ) M =1+ M = the experience modification A = the actual or rated loss for the risk = the expected loss for the risk Z = credibility Unity represents standard rate Second term represents a debit/credit moderated by Z/ The Surety Association Plans (no split) M =(1 Z )+ P = the standard premium (1-Z) = the premium modifier Z / P A P Z / [ / P] = the adjusted loss multiplier A / P = the adjusted loss ratio Fundamental expression for credibility in the no split plan Z = credibility = xpected loss Z= +K K = determined from the swing desired in the plan K Formula for xperience Rating in the no slit plan A = Actual loss M = A+K +K Loss free modification in the no split plan M = K +K =(1 Z )
M tilt = Modification when there is no loss Analysis of the experience modification formula in the spit plan M = the risk modification M =1+Z p (A p p ) +Z e ( A e e ) A = the actual or rated loss for the risk = the expected loss for the risk Z = credibility factor.. p = for primary loss.. e = for excess loss If Z p = +K and Z e = + J then M =1+ (A p p ) +K + ( A e e ) +J K = credibility constant for primary loss (B above) J = credibility constant for excess loss (K above) Loss free modification in the split plan M =1 Z p p Z e e M tilt = modification when there is no loss Perryman's 1 st Formula (Q < < S) derivation (same as above, different notation) W = Q S Q M = the risk modification and B=(1 W ) K then M = (A p +W A e +(1 W ) e + B ) (+ B) A = the actual or rated loss for the risk = the expected loss for the risk.. p = for primary loss.. e = for excess loss W = weighting
B = ballast value S = self-rating point for total loss (Z p = Z e = 1) Q = self-rating point for primary loss (Z p = 1) The 1940 multi-split plan formula for primary component N I A<( N +1) I then A = loss amount I = width of each increments (500$ in 1940) N = number of increments in the loss d = rate of discount (1/3 in 1940) A p = primary component of loss, max is I / d The 1961 split plan formula for primary component A p = A (I +C) A+C A = loss amount C = constant (8000$ in 1960) I = split point (2000$ in 1960) A p = primary component of loss, max is (I+C) The 1998 Adjustment to the R Plan : NCCI Comparison R Plan Manual for WC and L : NCCI Formulas can be found under Gillam. xperience and Schedule Rating Plan of GL : ISO Present Average Company Rate Method when Dramatic Change BL premium for the policybeing rated Average annual per occurrence BL company rate= xposures on the special u/w basis Annual BL loss cost =(xposures on special u/w basis) ( BL rates) (Co. LR) Company Subject Loss Cost = Annual BL Loss cost PAF 1 PAF 2 Detrend sublines 3 years PAF 1 = Bring lost cost to full occurrence level for CM PAF 2 = liminate loss cost related to midi-tail cov. for CM
Historical xposures at Present Company Rates Method when Dramatic Change Annual BL loss cost=( Historical xposures) ( BL rates) ( ILF) (Co. LR) ILF = Increased limit factor Computation under rule 4A Company Subject Loss Cost= BL Prem current LR PAF 1 PAF 2 Detrend sublines 3 years Actual limited losses+ ARULL AR= Company Subject Loss Cost M = AR R Z R ARULL= Company Subject Loss Cost R LDF sublines 3 years LR = Current expected loss ratio AR = Actual experience ratio ARULL = Adjustments to reflect Ultimate Loss Levels R = xpected experience ratio M = Modification factor Z = Credibility MSL = Maximum Single Loss LDF = for occurrence policies only Company xpense Variation Factor LRUnderlying the company manual premium VF= Actual LR for the risk VF = xpense variation factor to apply to premium Computation of included loss amount Loss amount included= Minimum(Minimum(L, Basic Limit )+ ALA, MSL) Do not include loss payable due to midi-tail under CM B.2 Retrospective Rating Table M Construction : Brosius Table M ntry ratio
r= AL L r = entry ratio AL = actual loss (ratio) L = expected loss (ratio) Fundamentals of Individual Risk Rating Part 2 : Gillam and Snader Basic Formula for Retrospective Rating H = the minimum premium H R=(b+C L) T G R = the retrospective premium b = basic premium (basic factor * standard premium) c = loss conversion factor for expenses that vary with loss L = actual loss incurred subject to limitation T = tax multiplier G = the maximum premium Minimum premium H =(b+c r H ) T =(b+c L H ) T r H = L H r H = entry ratio at minimum premium L H = loss that will result in minimum premium Maximum premium G=(b+c r G ) T =(b+c L G ) T r G = L G r G = entry ratio at maximum premium L G = loss that will result in maximum premium b = basic premium Basic premium computation b=e (c 1) +cl
e = the provision in the Guaranteed cost premium for total expense and profit, excluding taxes, expresses as a ratio to Standard Premium c = Loss conversion factor for expenses which vary with loss cl = the converted insurance charge Insurance Charge Computation cl=c ( X G S H ) X G = the table M charge at entry ratio r G S H = the table M savings at entry ratio r H Restated Basic Premium Formula b=e (c 1) +c ( X G S H ) xpense Ratio Computation e= (1 D) T Guaranteed Cost Premium computation e = the expense ratio T (e+ ) SP=(1 D) SP D = the applicable discount for a given risk T = the Tax Multiplier = xpected Loss (as % of standard premium) SP = Standard premium T = Tax Multiplier = permissible loss ratio Tax Multiplier Computation T = 0.2+ (1+μ) 1 0.2+ 1 τ τ=τ 1 +τ 2 +... μ=μ 1 +μ 2 +... τ = various taxes (premium-based levies) μ = assessments Table M entry difference
r G r H = G H c T Table M value difference e+ H /T X H X G = c For any entry ratio r, the savings are S = savings at entry ratio r X = charge at entry ratio r S r = X r +r 1 Insurance Charge Reflecting Loss Limitations (ICRLL) using a limited loss table H R=(b'+cF +c L ') T G F = LR r ' G r ' H = G H c ' T e+ H /T X ' H X ' G = c ' ' = 1+0.8 LR 1 LR = 1+0.8 F / 1 F / r' r = entry ratios of actual limited loss to expected limited loss X' r = limited insurance charge ' = expected limited loss RR Plan Manual for WC and mployers Liability : NCCI The Retrospective Premium Formula RP = [BP + CL] * TM BP = SP * BPF RP = Retrospective Premium BP = Basic Premium BPF = Basic Premium Factor SP = Standard Premium CL = Converted Loss TM = Tax Multiplier Computation of the elective element (1) xcess Loss Premium
LP = SP * LPF * LCF LP = xcess Loss Premium LPF = xcess Loss Premium Factor LCF = Loss Conversion Factor Computation of the second elective element (2) Retrospective Development Premium RDP = SP * RDPF * LCF RDP = Retrospective Development Premium RDPF = Retrospective Development Premium Factor LCF = Loss Conversion Factor The Retrospective Formula when the two elective elements are included RP = [BP + CL + LP + RDP] * TM The Mathematics of XOL Coverages and RR : Lee Description of the Table M charge and savings in term of functions A = Actual loss of the risk = [A], the expected loss ϕ(r)= ( y r)df ( y) r r ψ(r)= (r y)df ( y) 0 Y = A/, the actual loss in units of expected loss F(Y) = the cumulative distribution function of Y φ(r) = Table M charge (X r ) ψ(r) = Table M savings (S r ) Properties of these functions ϕ' (r)= G(r) ψ' (r)=f (r) ϕ' ' (r)= f (r) ψ' ' (r)= f (r) ϕ' ' (r)= f (r)=ψ ' ' (r) ψ(r)=ϕ(r)+r 1
r = entry ratio G(r) = 1 F(r) ' = first derivative '' = second derivative A More General result 1 Let L={r A r 2 A r 1 r 1 A r 2 r 2 A [ L]/ ψ(r 1 )+ϕ(r 2 )=1 [ L]= (1+ψ(r 1 ) ϕ(r 2 ))= I I = (ϕ(r 2 ) ψ(r 1 )) r 1 = entry ratio at lower bound r 2 = entry ratio at upper bound = expected loss [L] = expected ratable loss I = net insurance charge of the Table M Balance equation (1) : charge difference formula e = total expenses = expected loss H = minimum premium C = loss conversion factor (e+ ) H =C (ϕ(r H ) ϕ(r G )) r H = entry ratio at the minimum r G = entry ratio at the maximum Balance equation (2) : entry ratio difference formula G = maximum premium G H =C (r G r H ) Description of the Table L charge and savings in term of functions ϕ*(r)= ( y r)df *( y)+k r
[ A *] k= r ψ*(r)= (r y)df *( y) 0 A* = Actual limited loss of the risk A = Actual loss of the risk = [A], the expected loss k = loss elimination ratio Y = A*/, the entry ratio F*(Y) = the cumulative distribution function of Y φ(r) = Table M charge (X r ) ψ(r) = Table M savings (S r ) The California Table L : Skurnick Computation of the net Table L insurance charge I *= (ϕ *(r G ) ψ*(r H )) H A * L H Let L={L A * L H A* L G L G L G A * [ L]= I * I* = net Table L insurance charge A* = actual limited loss L H = actual limited loss that produces minimum premium L G = actual limited loss that produces maximum premium The expected value of loss portion of the retrospective premium when the plan is in balance RP = [L + I*] = [L] + I* = I* + I* = RP = Retrospective premium The expected value of the expense portion of the retrospective premium when the plan is in balance RP = { SP(1 D) C + (C 1)*(I* + L) } = SP(1 D) SP = standard premium
D = expense gradation in the plan as a ratio to SP C = loss conversion factor The incremental Charge on Table M due to superimposing a per case limit Table L Charge = Table M Charge + Incremental charge Incremental Charge=ϕ *(r) ϕ(r)=ψ *(r) ψ(r) The California Table L : Skurnick Discussion by Gillam Formula shift in Table M column to approximate a limited loss table of charges ' = xpected loss used to find the expected loss group Hazard Group Differential applied to update xpected Loss Size m(s /H )= CWS SWS m(s/h) = Hazard Group Differential CWS = Country Wide Total Severity (all Hazard Groups) SWS = State Wide Severity (a Hazard Group) The premium impact of an update to xpected Loss Size ranges in the retro rating plans RP = Retrospective premium ' [RP ] [RP ] Impact= [RP ] '[RP] = expectation according to the updated loss distribution B.3 Loss Sensitive Rating Fundamentals of Individual Risk Rating Part 3 : Gillam and Snader r = the retention k = the loss elimination ratio Straight deductible formula k= L r+(n n)r = [ x ;r ] L [x] N = total number of claims in the study
n = number of claims that do not exceed r L = total losses in the study L r = losses due to claims which do not exceed r [x;d] = expected loss limited at d Disappearing deductible formula L' = indemnification amount λ = claim amount r = retention lower bound R = retention upper bound 0 λ r L R '={(λ r)[ R r ] r<λ< R λ λ R Aggregate payment for losses between r and R R (L R rn R ) [ R r ] L R = total losses in the study between r and R N R = number of claims in the study between r and R Amount saved on losses between r and R R L R (L R rn R ) [ R r ] Loss elimination ratio R L r +L R (L R rn R ) [ R r ] LR=k = L All notation can be found above Discount Formula for Deductible Coverage Full Coverage Premium Formula P=P+nP+( A+T + p) P e=n+a P=( a)p+ep+( A+T + p) P
f = safety factor ( a) P+e P P= 1 A T p Deductible Coverage Premium P ' =(1 fk )( a) P+eP+(A+T + p) P ' (1 fk)( a) P+e P P ' = 1 A T p Discount for Deductible Coverage k = loss elimination ratio D=1 P ' P fk ( a) D= 1 A T p Notation for xpense and Profit Provisions n = provision for expense other than acquisition, taxes, profit and ALA e = provision for expense other than acquisition, taxes, profit. (e = n + a) A = provision for acquisition T = provision for taxes p = provision for profit Notation for Premium and Loss Components = xpected Loss ratio including ALA a = provision of ALA P = full coverage premium P' = deductible coverage premium D = discount for deductible coverage Discount Formulas for xcess Coverage Case 1 : inspection expense, ULA, home office expense vary with premium Full Coverage Premium Formula P=P+eP +( A+T + p+i+u+gh)p P+eP P= 1 A T p i u gh xcess Coverage Premium P ' =(1 fk ) P+eP+(A+T + p+i+u+gh) P '
(1 fk ) P+eP P ' = 1 A T p i u gh Discount for xcess Coverage D=1 P ' P fk D= 1 A T p i u gh i = inspection expense (vary with premium) u = ULA (vary with premium) gh = home office expense (vary with premium) e = provision for all other expenses Discount Formulas for xcess Coverage Case 2 : inspection expense, ULA, home office expense vary with excess losses end ALA Full Coverage Premium Formula P=P (1+i +u + gh )+ep +( A+T + p) P P= P(1+i +u +gh )+ep 1 A T p xcess Coverage Premium P ' =(1 fk ) P(1+i +u +gh )+ep+( A+T + p) P ' P ' = (1 fk) P (1+i +u + gh )+ep 1 A T p Discount for xcess Coverage D=1 P ' P D= fk(1+i +u +gh ) 1 A T p i = inspection expense (vary with XS losses and ALA) u = ULA (vary with XS losses and ALA) gh = home office expense (vary with XS losses and ALA) e = provision for all other expenses Discount Formula for Workers Compensation x-medical Coverage Full Coverage Premium Formula P=P+eP +( A+T ) P
P= +e P 1 A T = 1 A T e x-medical Premium P ' = k M +ep+( A+T ) P ' P ' = k M +e P 1 A T Discount for x-medical Coverage D=1 P ' P D= k M +ep Notation for xpense Provisions e = provision for expense other than acquisition, taxes A = provision for acquisition T = provision for taxes Notation for Premium and Loss Components = xpected Loss or Total pure premium excluding ALA M = the medical pure premium k = portion of medical pure premium eliminated (determined by judgement) P = full coverage premium P' = ex-medical premium D = ex-medical discount Loss Conversion Factor Adjustments Under Retrospective Rating J =c 1 J ' =c' 1 J (1 A T e) J ' =J =...= k M (1 D)(1 A T ) e J ' =c' 1>J =c 1 c = loss conversion factor for the ex-medical retro plan c' = adjusted loss conversion factor for the ex-medical retro plan
Pricing WC Large Deductible and xcess Ins : Teng Pricing Formula for LDD Premium [ L ( XL+ULA+ LBA)+SP (GO+CR) ] LDD Premium= 1 A T P L = xpected Total Loss (include ALA load if used when calculating the deductible) XL = xpected excess loss as a % of total loss (excess on a per occurrence or aggregate basis) LBA = Loss based assessment factor (for states requiring payment to 2 nd injury funds on losses below the deductible) ULA = Ratio of ULA to Loss SP = Standard Premium GO = Ratio of General Overhead to Standard Premium CR = Ratio of Credit Risk Compensation to Standard Premium A = Ratio of Acquisition xpense to Net LDD Premium T = Tax and Assessment Rate Ratio based on Net LDD Premium P = Ratio of Profit and Contingency to Net LDD Premium xcess Loss Cost for per occurrence deductible XSLC = xcess loss cost L = xpected total loss XSLC = L * LPPF LPPF = xcess loss pure premium factor (listed by deductible and by WC Hazard Group) Insurance charge for aggregate deductible IC = Insurance charge IC = XS / L XS = xpected amount of loss above an aggregate deductible level L = total loss Pricing Formula for xcess WC Premium
[ L XL (1+ULA )+SP GO ] xcesswc Premium= 1 A T P A = Ratio of Acquisition xpense to Net xcess WC Premium T = Tax and Assessment Rate Ratio based on Net xcess WC Premium P = Ratio of Profit and Contingency to Net xcess WC Pricing Aggregates on Deductible Policies : Fisher