Fictional Commercial Insurance Composite Rating Plan: Practice Questions and Solutions

This section of the study guide is intended to provide practice problems and solutions to accompany the pages of Basic Ratemaking, cited below. Students are encouraged to read these pages before attempting the problems. This study guide is entirely an independent effort by Mr. Stolyarov and is not affiliated with any organization(s) to whose textbooks it refers, nor does it represent such organization(s).

Some of the questions here ask for short written answers based on the reading. This is meant to give the student practice in answering questions of the format that will appear on Exam 5. Students are encouraged to type their own answers first and then to compare these answers with the solutions given here. Please note that the solutions provided here are not necessarily the only possible ones.

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 15, pp. 291-297.

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

All of the questions in this section apply to the following hypothetical example:

Insurance Company Δ writes commercial general liability insurance and on large, complex risks that are loss-rated via a composite rating plan. Marcus's Macroscopic Microscopes, a company, owned by Marcus, that manufactures giant microscopes, is the insured under this plan.

The experience period on the basis of which the rate is developed is the prior two years. The current policy period is the entire year of 2102.

For 2100, Marcus's Macroscopic Microscopes had reported bodily injury liability losses and allocated loss adjustment expenses of $315,125 and reported property damage liability losses and allocated loss adjustment expenses of $125,160. Total receipts from operations in 2100 were $31,124,160.

For 2101, Marcus's Macroscopic Microscopes had reported bodily injury liability losses and allocated loss adjustment expenses of $123,929 and reported property damage liability losses and allocated loss adjustment expenses of $231,250. Total receipts from operations in 2101 were $25,612,000.

All historical data are evaluated as of January 1, 2102, and all losses for a given year are assumed to occur at a point in time on January 1 of that year (for instance, all 2101 losses are assumed to occur on January 1, 2101).

The annual trend for losses and allocated loss adjustment expenses (ALAE) is -4%.

The annual trend for loss exposures is +6%. The exposure unit is $1000 of receipts from operations.

The expected loss and ALAE ratio is 65%.

The 12-month-to-ultimate bodily injury loss development factor is 1.45.

The 12-month-to-ultimate property damage loss development factor is 1.15.

The 24-month-to-ultimate bodily injury loss development factor is 1.23.

The 24-month-to-ultimate property damage loss development factor is 1.03.

Problem S5-118-1. For the historical experience period, what are the total trended ultimate looses and ALAE for Marcus's Macroscopic Microscopes?

Solution S5-118-1. The historical incurred losses and ALAE must be developed to ultimate and trended to current levels. The annual loss trend factor is 1 - 0.04 = 0.96. For each year, we multiply the reported losses for bodily injury by the applicable bodily injury loss development factor; then we add this sum to the product of the reported losses for property damage and the applicable property damage loss development factor. We multiple this sum by the trend factor taken to the power of the number of years to January 1, 2102.

For 2100, the total trended ultimate looses and ALAE are thus

(315125*1.23 + 125160*1.03)*0.96² = 476023.4957.

For 2101, the total trended ultimate looses and ALAE are thus

(123929*1.45 + 231250*1.15)*0.96 = 427809.168.

Total trended ultimate looses and ALAE for the two years are thus 476023.4957 + 427809.168 = 903832.6637 = $903,832.66.

Problem S5-118-2. For the historical experience period, what are the total trended ultimate exposures for Marcus's Macroscopic Microscopes?

Solution S5-118-2. To get the total trended ultimate exposures, we first need to trend the exposures from each year by multiplying them by the trend factor (here, 1 + 0.06 = 1.06), taken to the power of the number of years to January 1, 2102. Each $1000 of trended ultimate receipts from operations constitutes one exposure unit.

2100 trended ultimate exposures are thus (31124160/1000)*1.06² = 34971.10618.

2101 trended ultimate exposures are thus (25612000/1000)*1.06 = 27148.72.

Total trended ultimate exposures are thus 34971.10618 + 27148.72 = 62119.82618.

Problem S5-118-3. What is the adjusted premium for the historical period based on the data from the entire historical experience period?

Solution S5-118-3. The adjusted premium is equal to

(Trended Ultimate Loss & ALAE)/(Expected Loss & ALAE Ratio). Here, Trended Ultimate Loss & ALAE = 903832.6637 (from Solution S5-118-1), and Expected Loss & ALAE Ratio is given as 0.65. Thus, the adjusted premium is 903832.6637/0.65 = $1,390,511.79.

Problem S5-118-4. What is the rate per exposure unit, based on the data from the historical experience period?

Solution S5-118-4. The rate per exposure unit is equal to (Adjusted Premium)/(Trended Ultimate Exposures). Here, Adjusted Premium = 1390511.79 (from Solution S5-118-3), and Ultimate Exposures = 62119.82618 (from Solution S5-118-2). Thus, the rate per exposure unit is

1390511.79/62119.82618 = 22.38434773.

Problem S5-118-5. It is estimated that total receipts of Marcus's Macroscopic Microscopes for the year 2102 will be $31,130,103. What will be the deposit premium that Marcus's Macroscopic Microscopes will pay under this plan for 2102?

Solution S5-118-5. The deposit premium is equal to (Estimated Exposure Units)*(Rate Per Exposure Unit). Here, since the exposure unit is $1000 of receipts, there are 31130.103 estimated exposure units. From Solution S5-118-4, Rate Per Exposure Unit = 22.38434773. Thus, the deposit premium will be 31130.103*22.38434773 = 696827.0505. = $696,827.05.

See other sections of The Actuary's Free Study Guide for Exam 5.