14.4 Empirical Approach for PDLD

Advantages: Easier to do for an entire book of business as not every client will have the same parameters (e.g. \(\frac{BP}{SP}\), \(LCF\), \(TM\), \(LCF\))

Book should be separate into homogeneous groups

  • Size of account and type of rating plan

Need to assume a lag from the date used to value the losses and when the actual premium is collected

  • Typically 3 to 9 months e.g. if the loss age is 18, we want to look at the premium at 27 if we assume a 9 months lag

  • First evaluation for loss is typically at 18 months

  • e.g. \(PDLD_2 = \dfrac{P_2 - P_1}{L_2 - L_1} = \dfrac{^{39}P - {^{27}P}}{^{30}L - {^{18}L}}\)

14.4.1 PDLD Ratios Selection

Compile PDLD triangle for all effective date groupings and make a selection for each age

If you observe increasing trends in the ratios for a give age:

  • More losses are within the loss capping layer, can be due to:

    • Higher agg max or lower agg min or higher per claim limits

    • Improvement in loss experience (fewer large losses so larger portion is within the cap)

  • Higher basic limits (for \(PDLD_1\))

PDLD can be negative if \(\Delta CL\) is negative:
Losses > max increase while claims within retro limit have a reduction in reserves

14.4.2 Cumulative PDLD Ratios

Cumulative PDLD based on a weighted average with expected % future report for each future periods

\[\begin{equation} CPDLD_d = \dfrac{\sum_{i=d}^{\infty} PDLD_i \times \text{% Reported in period i}}{\sum_{i=d}^{\infty} \text{% Reported in period i}} \tag{14.7} \end{equation}\]
  • If you need the CPDLD for every PYs, it is best done starting with the oldest year and recursively up to the most recent year
\[\begin{equation} CPDLD_d = \dfrac{ CPDLD_{d-1} \times \text{% rpt'd in period}_{d-1} + PDLD_d \times \text{% rpt'd in period}_d }{\text{% rpt'd in period}_{d-1} + \text{% rpt'd in period}_d} \tag{14.8} \end{equation}\]
  • Or us calculator data table to calculate \(\dfrac{\sum xy}{\sum y}\) adding one pair at a time

14.4.3 Application

For each period:

  1. Use losses reported to date to estimate ultimate losses

  2. Subtract losses reported at prior retro adjustment date to get \(\Delta L\)

  3. Multiply by CPDLD to get \(\Delta P\)

For each quarter

  1. Determine reported losses @ prior adjustment (given)

  2. Estimate ultimate losses

  3. \(\Delta L\) = (2) - (1)

  4. \(\sum \limits_{qtr \in last \: adj \: age \: i} \Delta L_{qtr}\)

    • i.e. group the quarters by the age of next adjustments they’re waiting for

For each year

  1. Expected premium emergence = \(\Delta P = CPDLD \times \Delta L_{yr}\)

  2. Determine premiums booked through prior adjustment (given)

  3. Estimated total premium = (1) + (2)

  4. Premium asset = (3) - Premium booked as of current evaluation

Remark.

  • Premium booked as of current eval \(\neq\) premium booked through prior adjustment due to:

    • Timing of retro adjustments

    • Minor premium adjustments

    • Interim premium booking

  • Negative premium assets means returning premium to client

  • For the most recent periods need to adjust for only exposure period that is being earned out

    • Pro rate your ultimate losses by % earned premium

14.4.4 Further Issues

If loss plan includes ALAE, you should include ALAE in the losses for this as well

\(\Delta\) in mix of business will affect the sensitivity of premium to loss and thus the \(PDLD\)

For premium that is not secured, a provision for bad debt should be held