A Fictional Commercial Insurance Large Deductible 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. 297-299.

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 Θ writes commercial general liability insurance under a large deductible plan. The insured is Interactive Interfaces International, which has a deductible of $250,000 per occurrence.

Insurance Company Θ will pay for all loss adjustment expenses and will handle all claims, including claims that are entirely below the deductible. Allocated loss adjustment expenses (ALAE) are estimated at 23% of all losses. The cost to process losses that are below the deductible is estimated to be 5% of such losses.

The fixed expenses of this plan for the insurer are $100,000. Also, the insurer has variable expenses equal to 19% of premium and a profit provision of 4% of premium for a policy with no deductible. There is also a risk margin of 20% of excess losses for policies with a $250,000 deductible. The credit risk associated with the deductible payments from the insured is 3% of the expected deductible payments.

The loss elimination ratio associated with a loss limit of $250,000 is 76%.

During the policy period, it is estimated that the total ground-up losses of Interactive Interfaces International will be $711,240, without consideration of any deductibles.

Problem S5-119-1. What are the estimated losses of Interactive Interfaces International above the deductible?

Solution S5-119-1. We are given that the loss elimination ratio (LER) associated with a loss limit of $250,000 is 76%. We can calculate the excess ratio, which is equal to 1 - LER = 1 - 0.76 = 0.24. The excess ratio is the proportion of total estimated losses that are estimated to be above the deductible. Thus, the total magnitude of such losses is (Estimated total ground-up losses)*(Excess ratio) = 711240*0.24 = $170,697.60.

Problem S5-119-2. What are the estimated allocated loss adjustment expenses for the insurer?

Solution S5-119-2. ALAE in this case are a flat percentage (23%) of all losses, which are estimated to be $711,240. ALAE are thus estimated at 711240*0.23 = $163,585.20.

Problem S5-119-3. What are the estimated expenses to the insurer due to deductible processing?

Solution S5-119-3. It is given that the cost to process losses that are below the deductible is estimated to be 5% of such losses. Based on the given LER, the losses below the deductible of $250,000 are 0.76 of total losses. Thus, the estimated cost to process such losses is 0.05*0.76*711240 = $27,027.12.

Problem S5-119-4. Separately calculate the estimated dollar amount of credit risk posed by this policy to the insurer and the dollar amount corresponding to the insurer's risk margin.

Solution S5-119-4. The credit risk associated with the deductible payments from the insured is 3% of the expected deductible payments. Based on the given LER, the losses below the deductible of $250,000 are 0.76 of total losses (these are the expected deductible payments that the insured would need to refund to the insurer). Thus, the estimated dollar amount of credit risk is 0.03*0.76*711240 = 16216.272 = $16,216.27.

The risk margin is 20% of just the estimated excess losses. In Solution S5-119-1, we found the estimated excess losses to be $170,697.60. Thus, the dollar risk margin is 170697.60*0.2 = $34,139.52.

Problem S5-119-5. Based on the information given, what should be the premium charged for this policy?

Solution S5-119-5. We use the formula given by Werner and Modlin, p. 298:

Premium = (Losses above Deductible + ALAE + Fixed Expense + Credit Risk + Risk Margin)/
(1 - Variable Expense Provision - Profit Provision).

Here, Losses above Deductible = $170,697.60 (from Solution S5-119-1);
ALAE = $163,585.20 (from Solution S5-119-2);
Fixed Expense has two components: the $100,000 given fixed expense and the $27,027.12 estimated cost of processing losses below the deductible (from Solution S5-119-3);
Credit Risk = $16,216.27 (from Solution S5-119-4);
Risk Margin = $34,139.52 (from Solution S5-119-4);
Variable Expense Provision = 0.19 (given);
Profit Provision = 0.04 (given).

Thus, Premium =
(170697.60 + 163585.20 + 100000 + 27027.12 + 16216.272 + 34139.52)/(1 - 0.19 - 0.04) = 664500.9247 = $664,500.92.

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