A Fictional Workers' Compensation Insurance Retrospective 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. 299-304.

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 Ψ writes workers' compensation insurance under a retrospective rating plan. The insured is Interactive Interfaces International.

Under the plan, the retrospective premium is determined by the following formula:

Retrospective Premium = (Basic Premium + Converted Losses)*(Tax Multiplier), subject to a maximum and minimum agreed upon between the insurer and the insured.

Basic Premium is determined by the following formula:

Basic Premium = (Expense Allowance - Expense Provided Through LCF + Net Insurance Charge)*(Standard Premium).

Converted Losses are determined by the following formula:

Converted Losses = (Reported Losses)*LCF.

LCF is the Loss Conversion Factor, which is intended to account for loss adjustment expenses in the premium calculation.

Expense Provided Through LCF is determined by the following formula:

Expense Provided Through LCF = (Expected Loss Ratio)*(LCF - 1).

The Net Insurance Charge is determined by the following formula:
Net Insurance Charge = (Insurance Charge - Insurance Savings)*(Expected Loss Ratio)*LCF.

The Standard Premium for the policy, before any retrospective adjustments are made, is $530,350. The insurer and the insured negotiate a minimum retrospective premium ratio (ratio of retrospective premium to standard premium) of 50% and a maximum retrospective premium ratio of 130%. They also negotiate an LCF of 1.20 and a per accident loss limitation of $200,000.

The insurer sets its expense allowance (not considering taxes) to 25% and its expected loss ratio to 60%. It also uses a tax multiplier of 1.06.

The insurance savings to the insurer because of the minimum premium are accounted for by a factor of 0.04. The insurance charge that the insurer requires in order to accommodate the maximum premium is accounted for by a factor of 0.28.

After a passage of a mutually agreed-upon amount of time, losses during the policy period are observed and valued at $400,000. This is the official figure used for reported losses.

Problem S5-120-1. What is the Net Insurance Charge factor?

Solution S5-120-1. We use the formula

Net Insurance Charge = (Insurance Charge - Insurance Savings)*(Expected Loss Ratio)*LCF.

Here, Insurance Charge = 0.28, Insurance Savings = 0.04, Expected Loss Ratio = 0.60, and LCF = 1.20. Thus, (0.28 - 0.04)*0.60*1.20 = Net Insurance Charge = 0.2592.

Problem S5-120-2. What is the factor representing Expense Provided Through LCF?

Solution S5-120-2. We use the formula

Expense Provided Through LCF = (Expected Loss Ratio)*(LCF - 1) = 0.60*(1.20 - 1) = Expense Provided Through LCF = 0.12.

Problem S5-120-3. What are the Converted Losses?

Solution S5-120-3. We use the formula Converted Losses = (Reported Losses)*LCF, where Reported Losses = $400,000, and LCF = 1.2. Thus, 400000*1.2 = Converted Losses = $480,000.

Problem S5-120-4. What is the Basic Premium?

Solution S5-120-4. We use the formula Basic Premium = (Expense Allowance - Expense Provided Through LCF + Net Insurance Charge)*(Standard Premium).

Here, Expense Allowance = 0.25 (given);

Expense Provided Through LCF = 0.12 (from Solution S5-120-2);

Net Insurance Charge = 0.2592 (from Solution S5-120-1);

Standard Premium = $530,350.

Thus, (0.25 - 0.12 + 0.2592)*530350 = Basic Premium = $206,412.22.

Problem S5-120-5. What is the Retrospective Premium? (This will be the premium that Interactive Interfaces International will actually end up paying.)

Solution S5-120-5. We use the formula

Retrospective Premium = (Basic Premium + Converted Losses)*(Tax Multiplier).

Here, Basic Premium = $206,412.22 (from Solution S5-120-4);

Converted Losses = $480,000 (from Solution S5-120-3);

Tax Multiplier = 1.06 (given).

Retrospective Premium = (206412.22 + 480000)*1.06 = 727596.9532.

However, we still need to check whether this amount exceeds the maximum premium of 130% of Standard Premium, which is 530350*1.30 = 689455. Since the maximum agreed-upon premium is less than the calculated retrospective premium, the actual retrospective premium is capped at $689,455.

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