Calculation of Cumulative Premium Development to Loss Development Ratios and Premium Assets for Retrospectively Rated Insurance Policies: Practice Questions and Solutions

Source:
Teng, M.T.S.; and Perkins, M.E., "Estimating the Premium Asset on Retrospectively Rated Policies," PCAS LXXXIII, 1996, pp. 611-647, excluding Section 5.

Original Problems and Solutions from The Actuary's Free Study Guide

The following conditions apply to all the problems in this section:
A particular book of retrospectively rated insurance policies has the following characteristics:

First Retro Adjustment
Emerged Losses (Incremental): 66600
Booked Premium (Incremental): 124000

Second Retro Adjustment
Emerged Losses (Incremental): 15500
Booked Premium (Incremental): 10300

Third Retro Adjustment
Emerged Losses (Incremental): 12220
Booked Premium (Incremental): 10000

Fourth Retro Adjustment
Emerged Losses (Incremental): 8000
Booked Premium (Incremental): 2000

There are no subsequent retro adjustments.

Problem S6-58-1. On the basis of the given information, calculate the empirical premium development to loss development (PDLD) ratios for each retro adjustment.

Solution S6-58-1. The PDLD ratio for each retro adjustment (in the absence of any additional data) is best estimated by dividing incremental booked premium by incremental emerged losses at each retro adjustment.

First Retro Adjustment
PDLD = 124000/66600 = 1.861861862

Second Retro Adjustment
PDLD = 10300/15500 = 0.664516129

Third Retro Adjustment
PDLD = 10000/12220 = 0.818330606

Fourth Retro Adjustment
PDLD = 2000/8000 = 0.25

Problem S6-58-2. As of each retro adjustment, calculate the incremental percent of the total losses emerged.

Solution S6-58-2. As of each retro adjustment, the percent of total losses emerged is the incremental emerged losses for that adjustment, divided by the ultimate emerged losses.

First Retro Adjustment
% Losses Emerged = 66600/(66600 + 15500 + 12220 + 8000) = 0.65089914 = 65.089914%.

Second Retro Adjustment
% Losses Emerged = 15500/(66600 + 15500 + 12220 + 8000) = 0.151485536 = 15.1485536%.

Third Retro Adjustment
% Losses Emerged = 12220/(66600 + 15500 + 12220 + 8000) = 0.119429242 = 11.9429242%.

Fourth Retro Adjustment
% Losses Emerged = 8000/(66600 + 15500 + 12220 + 8000) = 0.078186083 = 7.8186083%.

Problem S6-58-3. As of each retro adjustment, calculate the cumulative premium development to loss development (CPDLD) ratio.

Solution S6-58-3. Calculating the CPDLD ratio from the PDLD ratio and the percentage of losses emerged as of each retro adjustment takes four steps:

(1) For each retro adjustment, multiply the PDLD ratio by the incremental percentage of losses emerged.
(2) At each retro adjustment, calculate the cumulative ratio based on the results in step (1) (i.e., the sum of the ratio from step (1) for that adjustment and all subsequent adjustments).
(3) At each retro adjustment, calculate the cumulative percentage of loss emerging either at that retro adjustment or at subsequent retro adjustments.
(4) Divide the ratios from step (2) by the respective percentages from step (3) to get the CPDLD ratios.

In the displayed calculations below, it is intended that step (1) be performed first for each retro adjustment, followed by step (2) for each retro adjustment, etc.

First Retro Adjustment
(1) PDLD*(% Losses Emerged) = 1.861861862*65.089914% = 1.211884285
(2) Cumulative (PDLD*(% Losses Emerged)) = 1.211884285 + 0.100664582 + 0.097732604 + 0.019546521= 1.429827992
(3) Cumulative % Losses to Emerge = 100%
(4) CPDLD = (2)/(3) = 1.429827992/100% = 1.429827992

Second Retro Adjustment
(1) PDLD*(% Losses Emerged) = 0.664516129*15.1485536% = 0.100664582
(2) Cumulative (PDLD*(% Losses Emerged)) = 0.100664582 + 0.097732604 + 0.019546521= 0.217943707
(3) Cumulative % Losses to Emerge = 100% - 65.089914% = 34.910086%
(4) CPDLD = (2)/(3) = 0.217943707/34.910086% = 0.624300115

Third Retro Adjustment
(1) PDLD*(% Losses Emerged) = 0.818330606*11.9429242%=0.097732604
(2) Cumulative (PDLD*(% Losses Emerged)) = 0.097732604 + 0.019546521= 0.117279125
(3) Cumulative % Losses to Emerge = 100% - 65.089914% - 15.1485536% = 19.7615324%
(4) CPDLD = (2)/(3) = 0.117279125/19.7615324% = 0.593471815

Fourth Retro Adjustment
(1) PDLD*(% Losses Emerged) = 0.25*7.8186083% = 0.019546521
(2) Cumulative (PDLD*(% Losses Emerged)) = 0.019546521
(3) Cumulative % Losses to Emerge = 7.8186083%
(4) CPDLD = (2)/(3) = 0.019546521/7.8186083% = 0.25

Problem S6-58-4. Now you apply the CPDLD ratios derived in Solution S6-58-3 to the following related book of business with similar characteristics:

Policies at First Retro Adjustment
Expected future loss emergence of 40660
Premiums booked from prior adjustments of 0
Booked premium as of the present: 80210

Policies at Second Retro Adjustment
Expected future loss emergence of 22030
Premiums booked from prior adjustments of 60560
Booked premium as of the present: 90800

Policies at Third Retro Adjustment
Expected future loss emergence of 12300
Premiums booked from prior adjustments of 86000
Booked premium as of the present: 72040

Policies at Fourth Retro Adjustment
Expected future loss emergence of 6900
Premiums booked from prior adjustments of 96000
Booked premium as of the present: 80200

Find the estimated total (ultimate) premium for each set of policies above.

Solution S6-58-4. To find the estimated total (ultimate) premium for each set of policies above, we take the following steps:

(1) Multiply the expected future loss emergence by the CPDLD ratio to get the expected future premium.
(2) Add the premium booked from prior adjustments to the expected future premium to get the estimated total (ultimate) premium.

Policies at First Retro Adjustment
(1) Expected Future Premium = (Expected Future Loss Emergence)*CPDLD = 40660*1.429827992 = 58136.80615
(2) Estimated Total Premium = Expected Future Premium + Premium Booked from Prior Adjustments = 58136.80615 + 0 = circa 58136.81

Policies at Second Retro Adjustment
(1) Expected Future Premium = (Expected Future Loss Emergence)*CPDLD = 22030*0.624300115 = 13753.33153
(2) Estimated Total Premium = Expected Future Premium + Premium Booked from Prior Adjustments = 13753.33153 + 60560 = circa 74313.33

Policies at Third Retro Adjustment
(1) Expected Future Premium = (Expected Future Loss Emergence)*CPDLD = 12300*0.593471815 = 7299.703325
(2) Estimated Total Premium = Expected Future Premium + Premium Booked from Prior Adjustments = 7299.703325 + 86000 = circa 93299.70

Policies at Fourth Retro Adjustment
(1) Expected Future Premium = (Expected Future Loss Emergence)*CPDLD = 6900*0.25 = 1725
(2) Estimated Total Premium = Expected Future Premium + Premium Booked from Prior Adjustments = 1725 + 96000 = 97725

Problem S6-58-5. Using the information in Problem S6-58-4, estimate the premium asset for this book of business, for each set of policies and in total.

Solution S6-58-5. The premium asset is the difference between the estimated total premium and the premium booked as of the present.

Policies at First Retro Adjustment
Premium Asset = Estimated Total Premium - Premium Booked at Present = 58136.81 - 80210 = -22073.19

Policies at Second Retro Adjustment
Premium Asset = Estimated Total Premium - Premium Booked at Present = 74313.33 - 90800 = -16486.67

Policies at Third Retro Adjustment
Premium Asset = Estimated Total Premium - Premium Booked at Present = 93299.70 - 72040 = 21259.70

Policies at Fourth Retro Adjustment
Premium Asset = Estimated Total Premium - Premium Booked at Present = 97725 - 80200 = 17525

Total Premium Asset: -22073.19 - 16486.67 + 21259.70 + 17525 = 224.84.

See other sections of The Actuary's Free Study Guide for Exam 6.