Section 58

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

G. Stolyarov II
October 5, 2010 - Republished July 11, 2014 This section is part of Mr. Stolyarov's Free Study Materials for the CAS Exam 7.

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:

Emerged Losses (Incremental): 66600

Emerged Losses (Incremental): 15500

Emerged Losses (Incremental): 12220

Emerged Losses (Incremental): 8000

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.

PDLD = 124000/66600 = 1.861861862

PDLD = 10300/15500 = 0.664516129

PDLD = 10000/12220 = 0.818330606

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.

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

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

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

% 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.

(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

(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

(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

(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:

Expected future loss emergence of 40660
Booked premium as of the present: 80210

Expected future loss emergence of 22030
Booked premium as of the present: 90800

Expected future loss emergence of 12300
Booked premium as of the present: 72040

Expected future loss emergence of 6900
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.

(1) Expected Future Premium = (Expected Future Loss Emergence)*CPDLD = 40660*1.429827992 = 58136.80615

(1) Expected Future Premium = (Expected Future Loss Emergence)*CPDLD = 22030*0.624300115 = 13753.33153

(1) Expected Future Premium = (Expected Future Loss Emergence)*CPDLD = 12300*0.593471815 = 7299.703325

(1) Expected Future Premium = (Expected Future Loss Emergence)*CPDLD = 6900*0.25 = 1725

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.

Premium Asset = Estimated Total Premium - Premium Booked at Present = 58136.81 - 80210 = -22073.19

Premium Asset = Estimated Total Premium - Premium Booked at Present = 74313.33 - 90800 = -16486.67

Premium Asset = Estimated Total Premium - Premium Booked at Present = 93299.70 - 72040 = 21259.70

Premium Asset = Estimated Total Premium - Premium Booked at Present = 97725 - 80200 = 17525

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

Gennady Stolyarov II (G. Stolyarov II) is an actuary, science-fiction novelist, independent philosophical essayist, poet, amateur mathematician, composer, and Editor-in-Chief of The Rational Argumentator, a magazine championing the principles of reason, rights, and progress.

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