**This section of sample problems and solutions is a part of** **The Actuary’s Free Study Guide for Exam 6, authored by Mr. Stolyarov. This is Section 58 of the Study Guide. See an index of all sections by following the link in this paragraph.**

**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****.**