Changes in Insurer Losses Due to Reinsurance and Changes in Coverage or Benefit Levels: 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 6, pp. 97-100.

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

Problem S5-40-1. X Insurance Company enters into a proportional reinsurance arrangement with Y Reinsurance Company, with the proportion of premium ceded being 30%. During policy year 2067, X Insurance Company collected $600,000 in premium and had $400,000 of losses. What is the difference between the premium ceded to Y Reinsurance Company and the loss ceded to Y Reinsurance Company under this arrangement?

Solution S5-40-1. This question is based on the discussion in Werner and Modlin, p. 98. Under a proportional reinsurance arrangement, the proportion of premium ceded is the same as the proportion of losses; here, this proportion is 30%. Thus, the difference between premium ceded and losses ceded is just 30% of the difference between premium and losses:

0.3*(600000 - 400000) = $60,000.

Problem S5-40-2. Θ Insurance Company enters into a catastrophe excess-of-loss reinsurance agreement with Λ Reinsurance Company, where Λ will cover 64% of total losses in excess of $500,000, up to $900,000. The reinsurance premium charged for this coverage is $200,000 per policy year. During policy year 3705, the following individual losses occurred within Θ's book of business:

$60,000
$360,000
$900,000
$76,800
$560,000

What is the difference between the amount of losses ceded to Λ and the amount of reinsurance premium paid to Λ by Θ? Did Θ make money on this reinsurance arrangement?

Solution S5-40-2. This question is based on the discussion of catastrophe excess-of-loss reinsurance in Werner and Modlin, p. 98.

Total losses for Θ's book of business during this policy year are

60000 + 360000 + 900000 + 76800 + 560000 = 1956800, clearly in excess of $900,000. Thus, reinsurance will pay 64% of $400,000, which is the entire difference between $900,000 and $500,000. That is, reinsurance will pay 0.64*400000 = 256000, meaning that the difference between losses ceded and reinsurance premium is 256000 - 200000 = $56,000. Thus, Θ made money on this reinsurance arrangement.

Problem S5-40-3. In an alternate universe, Θ Insurance Company enters into a per-risk excess-of-loss reinsurance agreement with Λ Reinsurance Company, where Λ will cover 32% of any individual loss in excess of $500,000, up to $900,000, for five specified risks. The reinsurance premium charged for this coverage is $200,000 per policy year. During policy year 3705, the following individual losses for the five risks in question occurred within Θ's book of business:

$60,000
$360,000
$900,000
$76,800
$560,000

What is the difference between the amount of losses ceded to Λ and the amount of reinsurance premium paid to Λ by Θ? Did Θ make money on this reinsurance arrangement?

Solution S5-40-3. This question is based on the discussion of per-risk excess-of-loss reinsurance in Werner and Modlin, p. 98. The only losses in excess of $500,000 here are the $900,000 loss and the $560,000 loss. Thus, the reinsurer will pay 32% of (900000 - 500000) + (560000 - 500000) = 460000. 0.32*460000 = 147200, meaning that the difference between losses ceded and reinsurance premium is 147200 - 200000 = -$52,800. Thus Θ lost money on this reinsurance arrangement.

Problem S5-40-4. In policy year 4530, Limitless Insurance Company did not have policy limits on its policies and paid out the following amounts in claims:

56,000 Golden Hexagons (GH)
45,000 GH
43,000 GH
34,000 GH
12,000 GH

On January 1, 4531, Limitless Insurance Company was acquired by Limits, Ltd., which promptly proceeded to impose a limit of 40,000 GH on each policy. Assuming that the underlying empirical loss distribution can be expected to remain the same in policy year 4531, what is the percentage effect of this change on the projected losses that would be paid out by the company?

Solution S5-40-4. This question is based on the discussion of changes in coverage limits in Werner and Modlin, p. 99.

In PY 4530, the total losses paid out by the company were 56000 + 45000 + 43000 + 34000 + 12000 = 190000. In PY 4531, the three largest losses above are capped at 40000, making the total payout 40000*3 + 34000 + 12000 = 166000. The percentage change is 100*(166000/190000 - 1) = -12.63157895%.

Problem S5-40-5. In a particular state, the statutorily mandated workers' compensation benefit is 57% of the pre-injury wage. A change in the law increases the benefit to 93% of the pre-injury wage. Actuary Q also estimated that this increase in the benefit would result in claimants being more reluctant to return to work after an injury, and that the effect of this increased reluctance would be to increase losses for the insurer by 6% per worker of what the loss amount would have been without the incentive change. Before the change in law, an insurer's workers' compensation losses were $235,000. What can this insurer's losses be expected to be after the change in the law, assuming that no other factors change?

Solution S5-40-5. This question is based on the discussion of workers' compensation benefit changes in coverage limits in Werner and Modlin, pp. 99-100. Without the incentive change, the change in law would only result in an increase in losses by a factor of 0.93/0.57 = 1.631578947. The added incentive not to return to work inflates losses by an additional factor of 1.06, leading to a total factor of 1.631578947*1.06 = 1.729473684, by which losses are inflated, leading to an expected loss of 235000*1.631578947 = $406,426.32.

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