14.3 Formula Approach for PDLD

First method, based on retro rating parameters

Formula requires details about the policies written

  • Reasonable for individual policies but difficult for an entire book

14.3.1 Retro Premium Formulas

\[\begin{equation} P = \left[BP + \left(CL \times LCF\right)\right] \times TM \tag{14.1} \end{equation}\]

  • \(P\): Ultimate premium

  • \(BP\): Basic Premium

    Covers minimum premium, commissions and expenses; premium when there are no losses

  • \(CL\): Capped Losses

    Losses are capped from premium calculation, individually and in aggregate

  • \(LCF\): Loss Conversion Factor

    Variable expense loading

  • \(TM\): Tax Multiplier

    For premium tax

The above (\(P\), \(BP\), \(CL\)) can be expressed in \(\propto\) Standard Premium

  • Standard Premium = Manual Rate \(\times\) E-Mod \(\times\) Sch-Mod = \(\dfrac{\text{Expected Loss}}{\text{ELR}}\)

14.3.1.1 Premium Adjustment

Polices can allow adjustments up to 6 years

  • Typically 1st adjustment is based on losses 18 months after inception of the policy

  • Further adjustments every 12 months

Same formula (14.1) is used to calculate the cummulative premium due @ each adjustment but with the below:

  • \(P_n\): Cumulative premium @ nth adjustment

  • \(CL_n\): Capped losses @ nth adjustment

    (This is the loss to date with some capping on large losses)

Remark.

  • Disadvantage for the insurer since the client doesn’t finish paying their premium for several years

  • No loss development is baked into formula (14.1) so the premium estimate at the first several valuations are likely too low

14.3.2 Calculating PDLD

\[\begin{equation} PDLD = \dfrac{\Delta P}{\Delta L} \sim \dfrac{d P}{d L} \tag{14.2} \end{equation}\]

  • Relationship between premium development and loss development

  • Apply to expected loss development to determine the expected premium development

  • \(L\) here is not capped

\[\begin{equation} PDLD_n = \dfrac{P_n - P_{n-1}}{L_n - L_{n-1}} \tag{14.3} \end{equation}\]

14.3.2.1 Estimating First Period PDLD

First adjustment includes Basic Premium (\(BP\)) \(\Rightarrow\) Requires different treatment from the other adjustments

\[\begin{equation} PDLD_1 =\underbrace{\left(\frac{BP}{SP} \right)}_{\text{Basic Prem Factor}} \frac{TM}{ELR \cdot \%Loss_1} + \underbrace{\left( \frac{CL_1}{L_1}\right)}_{\text{Loss Capping Ratio}} \cdot LCF \cdot TM \tag{14.4} \end{equation}\]
  • Loss capping ratio is estimated base on the actual losses and the structure of the contract

The formula above (14.4) is too responsive to actual losses in the first period, as only a portion of the formula is related to the \(L_1\) as shown below

\[\begin{equation} P_1 = \underbrace{\left( \frac{BP}{SP} \right) \frac{TM}{ELR \cdot \%Loss_1}}_{\text{Not }\propto\text{ Loss}_1} \times \operatorname{E}[L_1] + \underbrace{\left( \frac{CL_1}{L_1} \right) \cdot LCF \cdot TM}_{\propto \text{ Loss}_1} \times L_1 \tag{14.5} \end{equation}\]

  • So technically only the 2nd part of the formula should be applied to \(L_1\) while the first part is in relation to the \(\mathrm{E}[L_1]\)

  • Note that \(SP = \dfrac{\text{Expected Loss}}{ELR}\)

  • Use this formula to address the issue of over responsive

Remark. Basic premium factor vs charge

  • Basic Premium Factor = \(\dfrac{BP}{SP}\)

    Used by Teng Perkins

  • Basic Premium Charge = \(\dfrac{BP}{SP} \cdot TM\)

    Used by Feldblum

14.3.2.2 Estimating Subsequent PDLD

\[\begin{equation} \begin{array}{cccl} PDLD_n &= &\left(\dfrac{CL_n - CL_{n-1}}{L_n - L_{n-1}}\right) &\cdot \: LCF \cdot TM \:\:\:\:\text{For }n>1\\ &= &\left(\dfrac{\Delta CL}{\Delta L}\right) &\cdot \: LCF \cdot TM\\ \end{array} \tag{14.6} \end{equation}\]

  • \(LCF\) and \(TM\) are known at policy inception so we only have to estimate the loss capping ratio \(\dfrac{\Delta CL}{\Delta L}\)

  • \(LCF\) is smaller at older ages (since more large losses developed and capped)

  • This equation is basically the 2nd part of (14.4)