# B.1. Retrospective Rating

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• Manual Premium = ∑(Payroll/100 * Manual Rate) for all class codes
• Guaranteed Cost Premium = SP * (1 - D); D reflects lower fixed expense for larger risks
2. 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
3. Retrospective rating properties
• 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
4. Retrospective rating without an occurrence limit
• R = (b + CA + PCV) * TH ≤ R ≤ G
• b = basic premium to cover profit, net insurance charge (for having max/min), non-LAE
• 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
5. 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)
6. Net Insurance charge
• H/G = (b + CAH/G)T, where AH/G are bound by aggregate limits
• I = [ɸ(rG) - Ψ(rH)] * E[A] = Exp Losses above AG - Exp shortfall below AH
• entry ratio = ratio of actual to expected loss
• rG = AG/E[A], rH = AH/E[A]
7. 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)
8. Limited Loss Table M
• R* = (bLLM + 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)
9. 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
10. Table L
• R* = (bL + 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
11. 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
12. 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
13. Table M horizontal methodology
• list all claims amounts & r in the first columns, including \$0 and the values for AG and AH
• 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(ri) = ɸ(ri+1) + (ri+1 - ri)(% risks above ri)
14. 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
15. 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
16. 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)
17. 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
18. 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
19. Net Insurance Charge
• E[L]/E[A] = 1 + Ψ(rH) - ɸ(rG) → 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
20. 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(rH) - ɸ(rG) = [(e + E[A])T - H]/[CE[A]T] = [(1 - D)P - H]/[CE[A]T]
• entry difference: rG - rH = [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
21. 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* = (bLLM + CL* + PCF)T where L* is bounded above and below by AH and AG
• balance equation is the same, except we use A* in the denominator
• bLLM = e - (C - 1)E[A] + CILLM; note that E[A] remains the same
22. 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
• ɸL(r) - ɸ(r) = ΨL(r) - Ψ(r) and ɸL(∞) = 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 ɸL(rH) and ɸL(rG)
23. 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
24. 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
• using countrywide data may be more credible than using statewide data
25. Why low impact in replacing death claims with average
• < 6% of total \$ comes from death claims
• most death losses are close to the average
26. 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
27. Expense Ratio Tables (4D)
• Type A tables used for stock companies, Type B for non-stock
• table shows the e expense ratio as well as ELR, premium discount range, T