14.5 Feldblum’s Discussion

We do not develop premium directly because:

  • Estimate of ultimate inc’d losses can be obtained sooner

  • Retro premium depends on incurred loss

Both Fitzgibbon and PDLD are based on the retro rating formula (with some difference)

Both methods estimate the parameter on the data, downside:

  1. Unable to look up retro parameters since this is done for the whole book of business

  2. Max/min will be different for account

  3. \(TM\) is different across state lines

\(\therefore\) Easier to do a regression using empirical data to estimate the parameters

Fitzgibbons and Berry

  • Also create a linear relationship of premium to losses, but they forecast ultimate premium directly, rather than premium development \(\Rightarrow\) can lead to large deviations

  • Premium asset \(\propto\) expected unreported losses

  • Caveat: Does not respond if the actual premium to loss relationship is different than estimated

  • Berry’s solution is to give less weight to this method as the year matures and give more weight to his DR2 method

14.5.1 Improvements from Teng & Perkins

  1. Slope of premium to loss changes (reduces) as the year matures

    • Large losses that pierce the claim cap tend to have their development later

    • No future development one aggregate max is hit

  2. Forecast only future premium development

    • Projected premium asset is based on projected unreported losses, does not consider losses to date

    • Errors in projecting premium to date are automatically corrected for since it projects premium development and not ultimate premium

14.5.2 Teng & Perkins Assumptions

  • Premium responsiveness \(\dfrac{\Delta P}{\Delta L}\) in a period is \(\perp\!\!\!\!\perp\) of responsiveness at prior adjustments

  • Premium responsiveness depends on the maturity, not the beginning loss ratio or beginning retro premium ratio

  • This is superior to Fitzgibbons since it is projecting only future development

14.5.3 Enhancement

\(PDLD_1\) should separate the basic premium component from the component \(\propto\) losses

Simple adjustment, subtract out the fixed component of \(CPDLD_1\)

Fixed component of \(CPDLD_1\):

\[\begin{equation} \dfrac{P_{fixed}}{\operatorname{E}[L]} = \underbrace{\dfrac{BP}{SP} \cdot TM}_{\text{Basic Prem Charge}} \Big / ELR \tag{14.9} \end{equation}\]
  • Based on \(P_{fixed} = BP \cdot TM\) and divided by \(\mathrm{E}[L]\)

We calculate \(P\) as the fixed component \(\times\) \(\mathrm{E}[L]\) and add the variable component \(\times\) \(L\)

  • Fixed component and variable components are what?

This can help better understand changes in the \(PDLD_1\) ratio