Formulas for Competitive Comparisons and Measurements of Performance for Insurers: 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).

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 13, pp. 240-257.

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

Problem S5-98-1. You know the following information:

Customer A is paying a premium of $231 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $413.

Customer B is paying a premium of $151 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $123.

Customer C is paying a premium of $135 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $323.

Customer D is paying a premium of $500 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $400.

Customer E is paying a premium of $240 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $200.

Assume that these are the only customers whose characteristics are used to compare competitive positions among insurers. From the standpoint of Company Q, what is the company's percent competitive position for each customer with respect to Company Z, in terms of premium? Assume, for the purposes of this problem, that a higher percent competitive position is more favorable to Company Q.

Solution S5-98-1. This question is based on the discussion in Werner and Modlin, p. 243. The lower Company Q's overall premium is relative to Company Z, the greater Company Q's competitive position. We have two choices for the formula for percent competitive position:

Choice a: (Competitor Premium)/(Company Premium) - 1
Choice b: (Company Premium)/(Competitor Premium) - 1

Since a higher percent is more favorable to Company Q, it would follow that Competitor Premium should be in the numerator. Thus, we select the formula in Choice a.

Competitive position for Customer A: (Competitor Premium)/(Company Premium) - 1 = 413/231 - 1 = 0.7878787878 = +78.7878787878%.

Competitive position for Customer B: (Competitor Premium)/(Company Premium) - 1 = 123/151 - 1 = -0.1854304636 = -18.54304636%.

Competitive position for Customer C: (Competitor Premium)/(Company Premium) - 1 = 323/135 - 1 = 1.392592593 = +139.2592593%.

Competitive position for Customer D: (Competitor Premium)/(Company Premium) - 1 = 400/500 - 1 = -0.2 = -20%.

Competitive position for Customer E: (Competitor Premium)/(Company Premium) - 1 = 200/240 - 1 = -0.166666667 = -16.666666667%.

Problem S5-98-2. You know the following information:

Customer A is paying a premium of $231 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $413.

Customer B is paying a premium of $151 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $123.

Customer C is paying a premium of $135 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $323.

Customer D is paying a premium of $500 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $400.

Customer E is paying a premium of $240 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $200.

Assume that these are the only customers whose characteristics are used to compare competitive positions among insurers. From the standpoint of Company Q, what is the company's overall percent competitive position with respect to Company Z, in terms of premium? Assume, for the purposes of this problem, that a higher percent competitive position is more favorable to Company Q.

Solution S5-98-2. As in Solution S5-98-1, we use the formula

(Competitor Premium)/(Company Premium) - 1. However, this time, we compare the overall premium that each company would charge to all of these five customers:

(413 + 123 + 323 + 400 + 500)/(231 + 151 + 135 + 500 + 240) - 1 = 0.399363564 = +39.9363564%.

Problem S5-98-3. You know the following information:

Customer A is paying a premium of $231 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $413.

Customer B is paying a premium of $151 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $123.

Customer C is paying a premium of $135 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $323.

Customer D is paying a premium of $500 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $400.

Customer E is paying a premium of $240 for insurance with Company Q. If this customer were insured with Company Z, the customer's premium would be $200.

Assume that these are the only customers whose characteristics are used to compare competitive positions among insurers. Also assume that customers only care about the amount of premium, and each customer will choose the insurer that charges the lower premium to that customer.

(a)From the standpoint of Company Z, what is the company's win percentage with respect to Company Q?

(b) At the next renewal period, what will be Company Q's retention ratio, with respect to these five insureds?

Solution S5-98-3. This question is based on the discussion in Werner and Modlin, pp. 243, 247.

(a) We use the formula (% Win) =
(Number of Risks Meeting Criteria (e.g., Premium Lower than Competitor ))/(Total Number of Risks).

Here, there are 5 risks. Of these, only 2 (A and C) are charged a lower premium by Q than they would be charged by Z. The others (B, D, and E) would receive a lower premium from Z, and Z can expect to win them over. Thus, for Z, (% Win) = 3/5 = 60%.

(b) We use the formula (Retention Ratio) = (Number of Policies Renewed)/(Total Number of Potential Renewal Policies). Since only A and C are charged a lower premium by Q than they would be charged by Z, only those customers will renew with Q, and so Q's retention ratio will be 2/5 = 40%.

Problem S5-98-4. Insurer Y had 1350 policies at the beginning of year 2113. During 2113, the insurer issued 43120 quotes and had a close ratio of 20%. With respect to policies that were in force at the beginning of 2113, the insurer had a retention ratio of 60%. What percent policy growth was experienced by Insurer Y during 2113?

Solution S5-98-4. This question is based on the discussion in Werner and Modlin, pp. 246-249.

Three formulas are relevant:

(% Policy Growth) = (Policies at End of Period)/(Policies at Onset of Period) - 1

(Retention Ratio) = (Number of Policies Renewed)/(Total Number of Potential Renewal Policies)

(Close Ratio) = (Number of Accepted Quotes)/(Total Number of Quotes)

The original 1350 policies had a retention ratio of 60%, meaning that 60% of those policies, or 1350*0.6 = 810 policies were renewed.

Of the 43120 issued quotes, 20% or 0.2*43120 = 8624 quotes resulted in policies being issued. Thus, at the end of year 2113, the insurer had 810 + 8624 = 9434 policies, compared to 1350 policies at the beginning of 2113.

Thus, (% Policy Growth) = 9434/1350 - 1 = 5.988148148 = +598.8148148%.

Problem S5-98-5. An insurer performs a persistency analysis ("lifetime value analysis") for two types of insureds, K and L.

An insured of type K has an 80% probability of renewing after the first year, a 60% probability of renewing after the second year, and a 90% probability of renewing after the third year. The premium charged to this insured starts at $1000 during the first year and decreases by 10% each year. The expected loss cost for this insured starts at $500 during the first year and increases by 20% each year.

An insured of type L has an 90% probability of renewing after the first year, a 95% probability of renewing after the second year, and a 70% probability of renewing after the third year. The premium charged to this insured starts at $600 during the first year and decreases by 5% each year. The expected loss cost for this insured starts at $600 during the first year and decreases by 30% each year.

Assume that there are no fixed expenses for insurance policies and that the risk-free interest rate is 0%.

Over a four-year time period, what is the insurer's expected profit from each insured? Which insured can be expected to bring more profit to the insurer?

Solution S5-98-5. This question is based on the discussion in Werner and Modlin, pp. 253-254.

We consider the insured of type K.

During year 1, the insurer's expected profit from this insured is 1000 - 500 = $500.
During year 2, the insurer's expected profit from this insured is 1000*0.9 - 500*1.2 = $300.
During year 3, the insurer's expected profit from this insured is 1000*0.9² - 500*1.2² = $90.
During year 4, the insurer's expected profit from this insured is 1000*0.9³ - 500*1.2³ = -$135.

During the first year, we assume that the insured will remain with the insurer with a probability of 1.
After the first year, there is a 0.8 probability that the insured will remain with the insurer.
After the second year, there is a 0.8*0.6 = 0.48 probability that the insured will remain with the insurer.
After the third year, there is a 0.8*0.6*0.7 = 0.336 probability that the insured will remain with the insurer.

Thus, the expected profit from the insured of type K is

1*500 + 0.8*300 + 0.48*90 + 0.336*(-135) = $737.84.

We consider the insured of type L.

During year 1, the insurer's expected profit from this insured is 600 - 600 = $0.
During year 2, the insurer's expected profit from this insured is 600*0.95 - 600*0.7 = $150.
During year 3, the insurer's expected profit from this insured is 600*0.95² - 600*0.7² = $247.50.
During year 4, the insurer's expected profit from this insured is 600*0.95³ - 600*0.7³ = $308.625.

During the first year, we assume that the insured will remain with the insurer with a probability of 1.
After the first year, there is a 0.9 probability that the insured will remain with the insurer.
After the second year, there is a 0.9*0.95 = 0.855 probability that the insured will remain with the insurer.
After the third year, there is a 0.9*0.95*0.7 = 0.5985 probability that the insured will remain with the insurer.

Thus, the expected profit from the insured of type L is

1*0 + 0.9*150 + 0.855*247.50 + 0.5985*308.625 = 531.3245625 = $531.32.

An insured of type K will bring the insurer more profit over the four-year period.

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