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Types of premium calculation
 Manual Premium = ∑(Payroll/100 * Manual Rate) for all class codes
 Modified Premium = Manual Premium * Experience Mod Factor
 Standard Premium = Modified Premium * Schedule Modification Factor
 Guaranteed Cost Premium = SP * (1  D); D reflects lower fixed expense for larger risks

Retrospective rating theory
 uses an insured's current loss experience in determining their current policy premium
 charge an initial premium at start of term, then adjust based on reported loss experience
 adjustments are made around 6 months after expiry, and every 12 months as losses develop
 Premium = [UW Expenses & Profit excl taxes + (1 + Exp LAE/Exp Loss) * Actual Losses] * Tax
 this is a simplification, if policies were rated as such there would be no point in insurance

Retrospective rating properties
 retrorated policies have max and min premium (add capping charge to premium)
 can also apply limit on impact of individual accident (other charge, consider overlap)
 those limits are not coverage limits, but rather limits on losses used for retro rating

Retrospective rating without an occurrence limit
 R = (b + CA + PCV) * T; H ≤ R ≤ G
 b = basic premium to cover profit, net insurance charge (for having max/min), nonLAE
 A = reported losses (w or w/o ALAE)
 C = Loss Conversion Factor = 1 + Exp LAE / Loss or 1+ Exp ULAE / Loss & ALAE
 T = [.2 + PLR(1 + μ)]/[.2 + PLR] * [1/(1  𝝉)]
 μ = assessments, 𝝉 = premium based taxes
 V = optional component = retro development factor to stabilize the first 3 adjustments (18/30/40 months); V ≈ exp % of loss unreported = (1  1/CDF) * ELR

Balanced plan
 if a retrospective plan is balanced, expected retro premium = prospective (GCP)
 b = e  (C  1)E[A] + CI
 e = total expenses & profit excluding taxes
 e  (C  1)E[A] = expense component of the basic premium
 I = net insurance charge for max and min premium (CI = converted net insurance charge)

Net Insurance charge
 H/G = (b + CA_{H/G})T, where A_{H/G} are bound by aggregate limits
 I = [ɸ(rG)  Ψ(rH)] * E[A] = Exp Losses above A_{G}  Exp shortfall below A_{H}
 entry ratio = ratio of actual to expected loss
 r_{G} = A_{G}/E[A], r_{H} = A_{H}/E[A]

Retrospective Rating with an Occurrence Limit
 standard table M no longer appropriate
 the aggregate impacts Pr(hit max/min premium)
 can use Limited Loss Table M, Table M with ICRLL adjustment, or Table L (Cali)

Limited Loss Table M
 R^{*} = (b^{LLM} + CA^{*} + PCF + PCV)T
 A* = reported limited loss; A ≠ A^{*} + PF, but E[A] = E[A^{*}] + PF
 F = XS loss factor as % of Standard Prem
 V is different with an occurrence limit, since development is impacted by the limit
 Theoretically the correct approach, but it’s not used by NCCI (too many tables)

Table M with Insurance Charge Reflecting Loss Limitation (ICRLL) adjustment
 simulate / approximate the LL Table M by using standard Table M but changing the column used to be appropriate for a larger risk size
 factor applied on E[A] is (1 + 0.8k) / (1  k)
 per occurrence reduces skewness of entry ratio distribution, similar to larger risks
 compromises on accuracy for practicality

Table L
 R^{*} = (b^{L} + CA^{*} + PCV)T
 similar to LL Table M, but Table L combines occurrence charge with aggregate charge
 California has different hazard groups than NCCI, and V varies by occurrence limit

NCCI filing by state
 in full rate states, NCCI files ELFs (denoted by F in formulas)
 in loss cost states, NCCI files Excess Loss Pure Premium Factors (ELPPFs), so insurers must use their own ELR to calculate ELFs

Creating Table M methods
 if sample average ≠ expected value we can 1) use sample or 2) divide charges by ɸ(0)
 using vertical slices: look at each (limited) loss separately; the average difference between unlimited and limited entry ratio is the premium charge
 using horizontal slices: sum by loss layer, divided by expected losses

Table M horizontal methodology
 list all claims amounts & r in the first columns, including $0 and the values for A_{G} and A_{H}
 add # risks at each entry ratio, then risks above (upward sum not including current)
 calculate % risks above (previous column over total # risks)
 ɸ(r) = 0 for largest value of r
 ɸ(r_{i}) = ɸ(r_{i+1}) + (r_{i+1}  r_{i})(% risks above r_{i})

Advantages of using vertical slices
 more natural since it corresponds to the way the data is presented
 easier to understand
 quicker if you just need ɸ(r) for a few entry ratios

Advantages of using horizontal slices
 more efficient when calculating ɸ(r) for multiple values or r
 less time consuming and computationally intensive when there are many risks since vertical slices require dealing with each risk individually

Properties of Table M
 let Y = A/E[A], with F(y) as the CDF of Y
 for a given r, area below the horizontal line and above the curve is Ψ(r), and above the line below the curve is ɸ(r)
 Ψ(r) = ɸ(r) + r  1 (derived from rectangle below r line)
 continuous : ɸ(r) = ∫_{r to ∞} (y  r)dF(y); Ψ(r) = ∫_{0 to r} (r  y)dF(y)

Properties of ɸ(r) and Ψ(r)
 ɸ(0) = 1, ɸ(∞) = 0; ɸ’(r) = G(r) ≤ 0
 G(r) = 1  F(r) and ɸ’’(r) = f(r) ≥ 0
 Ψ(0) = 0, Ψ(∞) = ∞
 Ψ’(r) = F(r) ≥ 0 and Ψ’’(r) = f(r) ≤ 0

Table M Charges and Premium Size
 as we take a sufficiently large sample of large risks, variance → 0 and all risks have the same amount of loss, so ɸ(r) = 1  r if r ≤ 1, 0 if r > 1
 as premium size → 0, ɸ(r) → 1
 in the case where standard premium is not equal by risk, use SP as weight instead of # risks

Net Insurance Charge
 E[L]/E[A] = 1 + Ψ(r_{H})  ɸ(r_{G}) → E[L] = E[A]  I
 GCP = (e + E[A])T = (1  D)P
 b = e  (C  1)E[A] + CI = (1  D)P/T  CE[A] + CI

Table M Balance Equations
 b = e  (C  1)E[A] + CI, where I depends on max/min premium; however max/min premium also depend on b, so we need trial and error to determine correct Table M row
 charge difference: ɸ(r_{H})  ɸ(r_{G}) = [(e + E[A])T  H]/[CE[A]T] = [(1  D)P  H]/[CE[A]T]
 entry difference: r_{G}  r_{H} = [G  H]/[CE[A]T]
 once we have the 2 balance equations, we search for matching r and ɸ(r)
 tip: to save time, assume each entry ratio pair changes the charge difference by 0.01

Constructing a Limited Loss Table M
 process is the same as Table M, except we use limited losses with r = A^{*}/E[A^{*}]
 k = LER = 1  E[A^{*}]/E[A]
 R^{*} = (b^{LLM} + CL^{*} + PCF)T where L^{*} is bounded above and below by A_{H} and A_{G}
 balance equation is the same, except we use A^{*} in the denominator
 b^{LLM} = e  (C  1)E[A] + CI^{LLM}; note that E[A] remains the same

Constructing a Table L
 Table L charge includes the charge for the per occurrence limit with r = A*/E[A]
 implicit charge for the current limit varies between 0 and k
 ɸ^{L}(r)  ɸ(r) = Ψ^{L}(r)  Ψ(r) and ɸ^{L}(∞) = k
 continuous: ɸ(r) = ∫_{r to ∞} (y  r)dF(y) + k; Ψ(r) = ∫_{0 to r}(r  y)dF(y)
 balance equations are the same except we use ɸ^{L}(r_{H}) and ɸ^{L}(r_{G})

Advantages of using Table L over NCCI approach
 it is mathematically accurate in correcting for the overlap
 there is no need for a separate per occurrence charge
 is built on state data rather than countrywide so more accurate in California

Disadvantages of Table L over NCCI approach
 it can't be used for alternate loss limits since charge by limit is built in
 it would require a very large number of tables countrywide
 it needs regular updates for inflation, incremental charges and aggregate loss distribution
 using countrywide data may be more credible than using statewide data

Why low impact in replacing death claims with average
 < 6% of total $ comes from death claims
 most death losses are close to the average

Expected Loss Groups Tables (4B)
 ELG = risk size group baed on Adjusted Expected Losses
 AEL = E[A] * State Hazard Group Differential * (1 + 0.8k)/(1  k)
 (1 + 0.8k)/(1  k) = loss group adjustment factor
 the last portion is to approximate a Limited Loss Table M; if we have to use this factor, than we'll use the Limited Loss Table M balance equations for the Table M search
 NCCI updates these tables for inflation, so they don't need to update the Table M curve

Expense Ratio Tables (4D)
 Type A tables used for stock companies, Type B for nonstock
 table shows the e expense ratio as well as ELR, premium discount range, T

Alaska Special Rating Values
 Alaska has specially derived values
 Hazard Groups Differentials are used in the formula to obtain Expected Losses
 Alaska is a loss cost state, so ELPPFs are shown instead of ELFs

