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Implementation

Hardboiled Notes

Expense Fees

  • Premium = multiplicative (variable) + additive (flat) portion

  • Additive portion = Fixed expense fee = Charges for fixed expenses adjusted for Variable UW expenses and profits

  • Fixed Expense Fee per exposure = \(\dfrac{\text{FE per exopsure}}{1 - V- Q}\)
  • Fixed Expense Fee per policy = \(\text{FE fee} \times \dfrac{\text{Exposures}}{\text{Policy}}\)

Off-balance factors

  • What is it?
    • Applied to base rate
    • To nullify the effect of rels change.
  • Four ways to calculate it? And the idea behind each one of them.
    1. \(\dfrac{OLEP}{PLEP}\)
    2. \(\dfrac{\text{Curr EWARF}}{\text{Prop EWARF}} =\dfrac{\dfrac{\text{Curr Rel}\ast\text{Exposures}}{\text{Exposures}}}{\dfrac{\text{Prop Rel}\ast\text{Exposures}}{\text{Exposures}}} \times \dfrac{\text{Base rate}}{\text{Base rate}} = \dfrac{OLEP}{PLEP}\)
    3. \(\left(\dfrac{\text{Rel chg Factors}\ast OLEP_{i}}{OLEP}\right)^{-1}\)
    4. Let \(X\) be current premium at base… \(\dfrac{\text{Curr XWARF}}{\text{Prop XWARF}}\), the common base rate gets cancelled in those premiums.
  • How to approximate it, if detailed exposure distribution isn't unavailable.
    • In such a case it would be 1 for some variables (which aren't changing)

Deriving new BASE RATES

  • PAF: Proposed Additive Fee
  • EWARF: Exposure Weighted Average Rating Factors
  • Extension of Exposures: Remove the effect of the rel changes on the overall rate change, from the proposed relativity. Apply the residual effect to the seed base rate.
    • \(\dfrac{\text{Prop Avg Prem}-PAF}{\text{Avg Prem @Seed + New Rates}- PAF} = \dfrac{1.2}{1.1}\)
    • We want overall rates to increase by 20%
    • If the Rels were revenue neutral, at Avg Prem at seed would be \(1\)… thus we remove the effect of relativity changes…

  • Approx. Avg Rate differential: Simply extract out the base rate by dividing by EWARF
    > To be used when we don't have exposure distribution

    • Approximate overall EWARF by multiplying by individual rating variables' EWARF
    • Approx. Prop Avg RF = \(1.175 \times 0.98 = 1.1515\) (Prop Territory EWARF \(\times\) Prop Discount EWARF)
    • Approx. Change in Avg Rate differential
    • Is the most generic… we are doing \(\dfrac{\text{prop avg prem}}{\text{curr avg prem}}\) (as we did if no Rels were changing) and multiply this to the current base rate.
    • Multiply the overall off-balance factor if exposure distribution is available.
    • Approximation is: For each changing rel, find the off balance factor and multiply it to the ratio

    Rate capping

  • Find the rel change factor \(\dfrac{\text{Prop Rel}}{\text{Curr Rel}}\)

  • New total premium (required)
  • Depending on…
    • Non-base being capped: Let Base rate change factor be \(X\)
    • Base being capped: Let Relativity Adjustment factor be \(X\)
  • Equation forming
    • Fix the increase for the capped variable (regardless of base or non-base)
      • (50,000)(1+15%)
      • (138,000)(1+20%)
    • For other variables
      • Base rate change factor X
        • You are adjusting base rate for all
        • 659,000X ← For the base rate
        • (PLEP)X ← For non base
      • Rel Adj factor X (15% cap on base rate)
        • You are adjusting the relativities for non-base
        • (20k)(1+15%)(Rel \(\Delta\) factor \(\times X\))
        • (30k)(1+15%)(Rel \(\Delta\) factor \(\times X\))
    • Finally, for the capped factor
      • Final level 3 rel = \(\dfrac{\text{Current}(1+15\%)}{\text{Base rate Adjustment}}\)
        • Think about it: You finally want (50,000)(1+15%) even after the base rate changes… so divide by base rate adjustment factor… and the rating factor should be just the \(\text{Current}\times 1.15\)

Notes

Derive new base rate

  • Without rating factor changes
    • \(\dfrac{\text{Prop Base Rate}\times \text{Prop. Exp-wtd RF}}{\text{Curr Base Rate}\times \text{Curr. Exp-wtd RF}}=\dfrac{\text{Prop Avg Prem}- \text{Prop Add Fee}}{\text{Curr Avg prem} - \text{Curr Add Fee}}\)
    • Prop base rate = \(\text{Curr Base Rate} \times \dfrac{\text{Prop Avg Prem}- \text{Prop Add Fee}}{\text{Curr Avg prem} - \text{Curr Add Fee}}\)
  • With rating factor changes
    • The relativities have changed
    • Have detailed data?
      • Extension of exposures
    • Don't?
      • Approximated avg rate differential
      • Approximated Change in avg rate differential

Extension of Exposures

\[ \text{Prop. base rate} = \text{SBR} \times \dfrac{\text{Prop. Avg Prem} - \text{Prop. Add Fee}}{P_{Base} - \text{Prop. Add Fee}} \]

Given,

  • Proposed Rating Factors
  • Proposed Additive Fee
  • Proposed Avg Premium

Assume,

  • SBR = Seed Base rate (\(\neq 0\))

Calculation,

  • \(P_{Base} =\) Avg Premium with Seed base \& new rates

Approximated Avg. rate Differential

\[ \text{Prop. Base Rate} = \dfrac{\text{Prop Avg Prem}- \text{Prop Add Fee}}{\text{Prop Avg Rating Factor}} \]

Given,

  • Proposed Average Premium
  • Proposed Rating Factors

Approximation (Proposed Avg Rating Factor)

  • Actual: all combinations of rating variables multiplied together with detailed exposures as weights
  • Approximation: \(\prod_{\text{Rating Variables}} \text{Prop. Avg Rating Factor}\)

A better approximation,

  • Instead of exposures as weights
  • Use Variable Premium at base

Approximated Change in Avg. rate Differential

Notes on OBF

Finally, the Off-balance Factor means just this

\[ OBF = \dfrac{\text{Premiums at Current rates}}{\text{Premiums at Proposed Rates}} = \dfrac{OLEP}{PLEP} \]

So that when we multiply this \(OBF\) with each of our newly proposed rates, the overall premium remains unchanged.