A Fictional Workers' Compensation Insurance Experience 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. 289-291.

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 Φ writes workers' compensation insurance and employs an experience rating plan based on the following formula:

M = (Z_P*A_P + (1 - Z_P)*E_P + Z_E*A_E + (1 - Z_E)*E_E)/E, where

M is the multiplicative experience modification factor, applied to the manual premium;
A_P denotes the actual primary losses;
A_E denotes the actual excess losses;
E_P denotes the expected primary losses;
E_E denotes the expected excess losses;
E denotes the total expected losses (E = E_P + E_E);
Z_P denotes the credibility of actual primary loss data;
Z_E denotes the credibility of actual excess loss data.

Manuel's Manual Manual Manufacturing, a company, owned by Manuel, that employs workers to write study manuals by hand, is the insured under this program. The credibility of its primary loss data is 70%, and the credibility for its excess loss data is 25%.

Experience from the past two policy periods is used to determine any modification to the manual premium applicable to the policy. The current policy period is the entire year of 2102. For this year, the manual premium is $12,012, subject to any experience rating adjustments.

The primary loss limit is considered to be $4,000.

It is also known that the loss elimination ratio (LER) at the primary loss limit is 0.45.

In 2100, there were two workers' compensation claims pertaining to Manuel's Manual Manual Manufacturing. Claim A had reported losses of $31,000, and Claim B had reported losses of $3,125. In 2100, the payroll of Manuel's Manual Manual Manufacturing was $502,204, and the expected loss cost per $100 of payroll was 5.54.

In 2101, there were three workers' compensation claims pertaining to Manuel's Manual Manual Manufacturing. Claim C had reported losses of $5,312, Claim D had reported losses of $1,111, and Claim F had reported losses of $1,011. In 2100, the payroll of Manuel's Manual Manual Manufacturing was $505,506, and the expected loss cost per $100 of payroll was 3.31.

Problem S5-117-1. What are the total actual primary losses (A_P) and the total actual excess losses (A_E) for the historical experience period?

Solution S5-117-1. For each claim during the historical experience period, the primary loss is that portion of the loss which is less than $4,000.

For Claim A, the claim amount is $31,000, of which $4,000 is primary and $27,000 is excess.
For Claim B, the claim amount is $3,125, of which $3,125 is primary.
For Claim C, the claim amount is $5,312, of which $4,000 is primary and $1,312 is excess.
For Claim D, the claim amount is $1,111, of which $1,111 is primary.
For Claim F, the claim amount is $1,011, of which $1,011 is primary.

The total actual primary losses are 4000 + 3125 + 4000 + 1111 + 1011 = A_P = $13,247.

The total actual excess losses are 27000 + 1312 = A_E = $28,312.

Problem S5-117-2. What are the total expected losses (E) for the historical experience period?

Solution S5-117-2. Expected losses are determined by multiplying payroll by the expected loss cost per unit of payroll. We note that the unit of payroll is $100, so we will express payroll in hundred-dollars rather than in single dollars.

In 2100, payroll was 5022.04 hundred-dollars, and the expected loss cost per unit was 5.54.
Thus, expected losses for 2100 are 5022.04*5.54 = 27822.1016.

In 2101, payroll was 5055.06 hundred-dollars, and the expected loss cost per unit was 3.31.
Thus, expected losses for 2100 are 5055.06*3.31 = 16732.2486.

Thus, total expected losses for these two years are 27822.1016 + 16732.2486 = 44554.3502 = E = $44,554.35.

Problem S5-117-3. What are the total expected primary losses (E_P) and the total expected excess losses (E_E) for the historical experience period?

Solution S5-117-3. In Solution S5-117-2, we found that the total expected losses (E) for the historical experience period are 44554.3502. To split total expected losses into primary and excess components, we apply the loss elimination ratio (LER) at the primary loss limit, given as 0.45. Total expected losses, multiplied by this LER, give us the expected primary losses: 0.45*44554.3502 = E_P = 20049.45759. The remainder of total expected losses constitutes the expected excess losses: 44554.3502 - 20049.45759 = E_E = 24504.89261.

Thus, E_P = $20,049.46, and E_E = $24,504.89.

Problem S5-117-4. What is the experience modification factor pertaining to Manuel's Manual Manual Manufacturing?

Solution S5-117-4. We apply the formula M = (Z_P*A_P + (1 - Z_P)*E_P + Z_E*A_E + (1 - Z_E)*E_E)/E, where

A_P = 13247 (Solution S5-117-1);
A_E = 28312 (Solution S5-117-1);
E_P = 20049.45759 (Solution S5-117-3);
E_E = 24504.89261 (Solution S5-117-3);
E = 44554.3502 (Solution S5-117-2);
Z_P = 0.7 (given);
Z_E = 0.25 (given).

Thus, M = (0.7*13247 + 0.3*20049.45759 + 0.25*28312 + 0.75*24504.89261)/44554.3502 = M = 0.9144877336.

Problem S5-117-5. What is the premium that Manuel's Manual Manual Manufacturing will have to pay for the year 2102?

Solution S5-117-5. The actual premium is the manual premium, multiplied by the experience modification factor M, which we found in Solution S5-117-4 to be 0.9144877336. The manual premium is given as $12,012. Thus, the actual premium will be 12012*0.9144877336 = 10984.82666 = $10,984.83.

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