Teng & Perkins Flashcards
Retrospectively rated (retro) policy
Premium paid by the insured reflects its actual loss experience during the policy period
- A deposit premium is paid at the start of the policy period based on expected losses
- A few months after the policy has expired (six months is common), a premium adjustment is made
- Additional premium adjustments often occur annually thereafter, and stop after a pre-determined number of adjustments
At each adjustment, the premium paid by the insured is subject to maximum and minimum premium amounts
Often include a per-occurrence limit that caps individual large losses before they are included in the premium calculation
Advantages of Retro Policies
- They encourage loss control and loss management by returning premium to the insured for good loss experience - attracts preferred customers
- They offer a cash flow advantage to insureds by allowing them to pay premium as losses are reported or paid
- Since premium varies directly with the insured’s actual loss experience, it shifts a large portion of the risk to the insured
Retro Premium
P_n = [BP + (CL_n * LCF)] * TM
P_n = premium at nth adjustment
BP = basic premium covering fixed expenses, net insurance charge, and per-occurrence excess loss charge
CL_n = capped loss at nth adjustment based on per-occurrence limit and max/min aggregate limits corresponding to max/min premium
LCF = loss conversion factor that covers LAE
TM = tax multiplier
Broken down into components:
1. Ratable Losses & Associated LAE - CL_n * LCF * TM
2. Basic Premium - BP * TM
Premium Development to Loss Development (PDLD) ratio
PDLD_1 = P_1 / L_1 = (BP / L_1 * TM) + (CL_1 / L_1 * LCF * TM)
L_n = unlimited loss at the nth adjustment
L_1 can be approximated as SP * ELR * %Loss_1
L_n can be approximated as SP * ELR * %CumulativeLoss_n
SP = standard premium
ELR = unlimited expected loss ratio
%Loss_n = cumulative expected percentage of loss emerged for the nth adjustment
BP / SP is the basic premium factor
First term can be written as Basic Premium Factor / (ELR * %Loss_1) * TM
Capped Losses
CL_n / L_n is the cumulative loss capping ratio at the nth adjustment
Ratio decreases as the data becomes more mature, since an increasing portion of the loss development occurs outside of loss limitations
CL_n / L_n = 1 - Net Insurance Charge - LER
Net Insurance Charge = Table M insurance charge at the retro maximum –Table M insurance savings at the retro minimum
Second PDLD Ratio
Refers to the incremental premiums developed between the first and second retro adjustments divided by the incremental losses developed between these two adjustments
PDLD_2 = (P_2 - P_1) / (L_2 - L1) = (CL_2 - CL_1) / (L_2 - L_1) * LCF * TM = Inc loss capping ratio at 2nd adjustment * LCF * TM
(CL_2 - CL_1) / (L_2 - L_1) is the incremental loss capping ratio
Advantages & Disadvantages of the Formula Approach
Advantages:
- Responds to changes in the retro rating parameters that are sold - if retro parameters change significantly over time, more weight should be given to PDLD ratios derived from the formula than those derived from historical data
- It is modeled directly on the retro rating formula, making it easy to explain
- Its focus on premium sensitivity is similar to how loss sensitive contracts are handled under Risk-Based Capital
- More stable PDLD ratios than the empirical approach due to using average parameters
Disadvantages:
- Potential bias exists since the formula approach uses the average parameters for the LCF, tax multiplier, maximum, minimum, and per accident limitation - should check for bias by retrospectively testing PDLD ratios against actual emergence
Empirical Approach to Calculating PDLD Ratios
PDLD_1 = premium booked through 27 months / losses reported through 18 months
PDLD_2 = (prems. booked at 39 mos. − prems. booked at 27 mos.) / (losses reptd. at 30 mos. − losses reptd. at 18 mos.)
Make selections for each ratio
Premium lags losses by 3 - 9 months (Teng & Perkins assume 9 month lag)
Upward Trend in the PDLD Ratios
- More liberal retro rating parameters, such as a higher maximum, minimum, or per accident limitation
- Improvement in loss experience, resulting in a larger portion of loss being within the boundaries of the retro maximum and the per accident limitation
Fluctuation in Historical PDLD Ratios
- Premium and loss development on a few policies can drive total incremental development
- Negative PDLD ratios are possible – upward development in high loss layers (resulting in no additional premium) and downward development in layers within loss limitations (resulting in return premium)
- If large fluctuation exists, average as many historical points as possible OR use the formula approach
Historical vs. Formula PDLD Ratios
- Worse (better) than expected loss experience may have caused a larger portion of the loss to be outside (inside) the boundaries of the retro maximum and the per accident limitation than the formula approach predicted
- Average retro parameters may be changing over time
Cumulative PDLD Ratios
Average of the PDLD ratios in all subsequent retro periods (including the current adjustment period), weighted by the incremental percentage of losses to emerge in each period
Tells an insurer how much premium it can expect to collect for a dollar of loss that has yet to emerge
The CPDLD ratio at the first retro adjustment is normally greater than 1.0 for the following reasons:
- First retro premium computation includes the basic premium
- Only a small portion of loss is limited at this point
- The application of the loss conversion factor and the tax multiplier results in more than a dollar of premium per dollar of loss
Cumulative to Incremental Loss Capping Ratios
Change in CL_n / L_n = [(CL_n/L_n) * %Loss_n - (CL_n-1/L_n-1) * %Loss_n-1] / (%Loss_n - %Loss_n-1)
Premium Asset
The ultimate premium that the insurer expects to collect based on the expected ultimate loss experience, less the premium that the insurer has already booked
Ultimate Premium = CPDLD * expected future loss emergence + booked premiums from the prior adjustment
Premium asset = Ultimate Premium - Current Booked Premium
How to Calculate Premium Asset
- Calculate the expected FUTURE loss emergence by PY
- Calculate CPDLDs for each retro adjustment
- Calculate expected FUTURE premium development by expected future loss * CPDLD
- Calculate the ultimate premium for each PY
- Calculate the premium asset = ultimate premium - current booked premium
Currently Booked Premium
Currently booked premium may differ from the premium booked at prior adjustment
- Differences in timing of retro adjustments
- Minor premium adjustments
- Interim premium booking
Fitzgibbon’s Method
Assumes that premium is a linear function of the (unlimited) incurred losses
P = C + B * Losses
Retro Adjustment = A + B * Standard Loss Ratio
where A = C / SP - 1
Fails to consider emerging premium responsiveness and simply relates the ultimate retro premium ratio to the ultimate loss ratio
- Emerging premium responsiveness may differ from expected because A and B are estimates and large losses on individual policies
- If true B is less than estimated B, overstates retro premium/premium asset
- If true B is greater than estimated B, understates retro premium/premium asset
Estimating the Parameters in Fitzgibbon’s Method
Uses a linear regression to estimate the parameters 𝐴 and 𝐵
Difficulties:
- Regression performed on historical data may not apply to current policies due to changes in rating plan factors and aggregate loss ratios
- Premium on individual plans is not a simple linear function of total incurred losses due to loss capping
Slope 𝐵 depends on the swing:
- For plans with narrow swing (i.e., small accounts with low loss limits and low maximum premiums), 𝐵 < 1
- For plans with wide swing (i.e., large accounts with high loss limits and high maximum premiums), 𝐵 > 1
Berry’s Method
Assumes a linear relationship between the loss ratio and the retrospective premium ratio
Modifies Fitzgibbon’s method by gradually discarding the method and giving more weight to the actual experience of the book of business
Premium Responsiveness
- As a book of business matures, premium responsiveness on loss-sensitive contracts declines (greater percentage of policies are excluded from retro rating by the maximum premium and by the loss limit)
- At higher loss ratios, premium responsiveness on loss-sensitive contracts declines
Teng & Perkins Solution
Relationship between reported loss ratio and retro premium ratio is a series of lines with decreasing slopes = PDLD ratio for that adjustment period
- The premium responsiveness during subsequent adjustments is independent of the premium responsiveness during preceding adjustments (even if PDLD_1 is different than expected, we do not change PDLD_2)
- The slope of the line segment depends on the time period, not on the beginning loss ratio or the beginning retro premium ratio (when moving from the first line segment to the second line segment, we change at the first retro adjustment, regardless of where the loss ratio ended up)
Fixes Fitzgibbon’s method:
- Fitzgibbon relates the ultimate loss ratio to the ultimate retro premium ratio - if the actual premium experience differs from what is expected, we have no way to get back on track
- The PDLD ratio relates the reported loss ratio to the retro premium ratio - if the actual premium experience deviates from what is expected, the next line segment begins at a starting point that corresponds to actual experience
Feldblum’s Enhancement
Separates the BP ratio and PDLD ratio in the first line segment
Subtract the average basic premium charge as a ratio to the standard loss ratio (BP/SP / ELR) from the first CPDLD ratio (include TM in numerator)
- CPDLD_Enhanced = CPDLD_TP - (BP/SP) / ELR * TM
Subtract the average basic premium charge as a ratio to the standard loss ratio multiplied by the incremental loss emergence at the first adjustment ((BP/SP) / (ELR*%Loss_1)) from the first PDLD ratio (include TM in numerator)
Retro premium adjustment 1 is decomposed into 2 components:
1. Expected Future Premium from Loss = Expected Future Loss * Enhanced CPDLD
2. Basic Premium Portion for First Adjustment = BP * TM
Ultimate Premium = Expected Future Premium from Loss + BP Portion for First Adjustment + Premium Booked from Prior Adjustments
Why not use the CL approach on historical triangles of
either collected premium or billed premium
- Estimates of ultimate incurred losses can be obtained sooner
than estimates of retrospective premiums can be obtained - Retrospective premiums depend on incurred losses
- PDLD approach can be updated each quarter as new loss data
becomes available as opposed to annually for CL approach
