The Approximated Change in Rate Differential Method of Establishing Base Rates for Insurance Rating Plans: 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 14, pp. 271-275.

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

Problem S5-114-1. Insurance Company Λ has a rating plan that involves a $120 fixed policy fee, a "keychain" rating variable K - where insureds who own more than five keychains receive a surcharge factor of 1.04, whereas all others receive a factor of 1 - and a "computer game" discount G, where an insured receives an 11% discount for playing computer games regularly. The company's current average premium is $400. The company wishes to set a base rate so as to increase average premium by 6%. With this rate revision, it is also decreasing the keychain surcharge factor to 1.03 and decreasing the computer game discount to 9%.

The company's book of business is distributed by premium in the following ways:

$31,513,000 of premium is paid by insureds who own five or more keychains.

$134,246,000 of premium is paid by insureds who own fewer than five keychains.

$123,124,000 of premium is paid by insureds who play computer games.

$42,635,000 of premium is paid by insureds who do not play computer games.

Use the approximated change in average rate differential method to find the approximated change in average rate differential (1 + Δ_S) pertaining to this rating change.

Solution S5-114-1. This question is based on the discussion in Werner and Modlin, pp. 271-273.

We first need to find the proposed average differential change for each of the two variables. We consider variable K.

The class of insureds who own 5 or more keychains is receiving a differential change of 1.03/1.04 = 0.9903846154.

The class of insureds who own fewer than 5 keychains is receiving a differential change of 1 (since the factor for this class is not changing).

The average rate differential change for this variable would be the differential changes for each class, weighted by current premium per class:

(31513000*0.9903846154 + 134246000*1)/(31513000 + 134246000) = 0.9981719869.

We consider variable G.

The class of insureds who play computer games is receiving a differential change of

(1-0.09)/(1-0.11) = 1.02247191.

The class of insureds who do not play computer games is receiving a differential change of 1 (since the factor for this class is not changing).

The average rate differential change for this variable would be the differential changes for each class, weighted by current premium per class:

(123124000*1.02247191 + 42635000*1)/(123124000 + 42635000) = 1.016691893.

The total approximated change in the average rate differential is the product of the variable-specific rate differential changes: 0.9981719869*1.016691893 = (1 + Δ_S) = 1.014833367.

Problem S5-114-2. Insurance Company Λ has a rating plan that involves a $120 fixed policy fee, a "keychain" rating variable K - where insureds who own more than five keychains receive a surcharge factor of 1.04, whereas all others receive a factor of 1 - and a "computer game" discount G, where an insured receives an 11% discount for playing computer games regularly. The company's current average premium is $400. The company wishes to set a base rate so as to increase average premium by 6%. With this rate revision, it is also decreasing the keychain surcharge factor to 1.03 and decreasing the computer game discount to 9%.

The company's book of business is distributed by premium in the following ways:

$31,513,000 of premium is paid by insureds who own five or more keychains.

$134,246,000 of premium is paid by insureds who own fewer than five keychains.

$123,124,000 of premium is paid by insureds who play computer games.

$42,635,000 of premium is paid by insureds who do not play computer games.

Use the approximated change in average rate differential method to find the change in the base rate (1 + Δ_B) needed to achieve the desired average premium change.

Solution S5-114-2. This question is based on the discussion in Werner and Modlin, pp. 271-273.

We use the formula

(1 + Δ_B) = ((1 + Δ)*P^-_C - A_P)/((P^-_C - A_C)*(1 + Δ_S)), where Δ is the overall desired average premium change, P^-_C is the current average premium, A_C is the current policy fee, A_P is the proposed policy fee, and (1 + Δ_S) is the approximated average change in rate differential.

Here, A_C = A_P = 120, Δ = 0.06, P^-_C = 400, and (1 + Δ_S) = 1.014833367 (from Solution S5-114-1). Thus, (1 + Δ_B) = (1.06*400 - 120)/((400-120)*1.014833367) = (1 + Δ_B) = 1.069844884.

Problem S5-114-3. Insurance Company Ξ has a rating plan that involves a $65 fixed policy fee, a "color of roof" rating variable C - where a surcharge factor of 1.2 is used for a green roof, whereas all other roofs receive a factor of 1 - and a "raspberry jam" discount J, where an insured receives a 20% discount for eating raspberry jam. The company's current average premium is $300. The company wishes to set a base rate so as to increase average premium by 3.33333333%. With this rating change, the company is also proposing to increase the surcharge factor for green roofs to 1.3, to decrease the factor for all other roofs to 0.8, and to increase the raspberry jam discount to 25%.

The following information is known about the insurer's book of business:
There are 23146 insureds who have a green roof.

There are 135600 insureds who have a non-green roof.

There are 31519 insureds who eat raspberry jam.

There are 127227 insureds who do not eat raspberry jam.

Use the approximated change in average rate differential method to find the approximated change in average rate differential (1 + Δ_S) pertaining to this rating change.

Solution S5-114-3. This question is based on the discussion in Werner and Modlin, pp. 273-275.

Here, we have an exposure distribution rather than a premium distribution. But the essential method is the same as if we had a premium distribution, except the differentials for each variable are weighted by exposures.

We consider variable C.

For the class of insureds who have a green roof, the rate differential change is 1.3/1.2 = 1.083333333.

For the class of insureds who do not have a green roof, the rate differential change is 0.8/1 = 0.8.

The exposure-weighted average rate differential change for variable C is thus (1.083333333*23146 + 0.8*135600)/(23146 + 135600) = 0.8413114871.

We consider variable J.

For the class of insureds who eat raspberry jam, the rate differential change is (1-0.25)/(1-0.2) = 0.9375.

For the class of insureds who eat raspberry jam, the differential change is 1 (since the factor for this class is not changing).

The exposure-weighted average rate differential change for variable J is thus

(0.9375*31519 + 1*127227)/(31519 + 127227) = 0.9875906322.

The total approximated change in the average rate differential is the product of the variable-specific rate differential changes: 0.8413114871*0.9875906322 = (1 + Δ_S) = 0.8308713434.

Problem S5-114-4. Insurance Company Ξ has a rating plan that involves a $65 fixed policy fee, a "color of roof" rating variable C - where a surcharge factor of 1.2 is used for a green roof, whereas all other roofs receive a factor of 1 - and a "raspberry jam" discount J, where an insured receives a 20% discount for eating raspberry jam. The company's current average premium is $300. The company wishes to set a base rate so as to increase average premium by 3.33333333%. With this rating change, the company is also proposing to increase the surcharge factor for green roofs to 1.3, to decrease the factor for all other roofs to 0.8, and to increase the raspberry jam discount to 25%.

The following information is known about the insurer's book of business:
There are 23146 insureds who have a green roof.

There are 135600 insureds who have a non-green roof.

There are 31519 insureds who eat raspberry jam.

There are 127227 insureds who do not eat raspberry jam.

Use the approximated change in average rate differential method to find the change in the base rate (1 + Δ_B) needed to achieve the desired average premium change.

Solution S5-114-4. This question is based on the discussion in Werner and Modlin, pp. 273-275.

We use the formula

(1 + Δ_B) = ((1 + Δ)*P^-_C - A_P)/((P^-_C - A_C)*(1 + Δ_S)), where Δ is the overall desired average premium change, P^-_C is the current average premium, A_C is the current policy fee, A_P is the proposed policy fee, and (1 + Δ_S) is the approximated average change in rate differential.

Here, A_C = A_P = 65, Δ = 0.033333333, P^-_C = 300, and (1 + Δ_S) = 0.8308713434 (from Solution S5-114-1). Thus, (1 + Δ_B) = (1.03333333*300 - 65)/((300 - 65)*0.8308713434) = (1 + Δ_B) = 1.25477091.

Problem S5-114-5. Insurance Company Φ has a book of business that contains five insureds. The company imposes a minimum premium requirement of $100. Without the minimum premium requirement, the following are the amounts of premium the insureds would pay under the company's current rating structure: $240, $31, $512, $89, $800. The company does not charge any policy fees.

By what offset factor would any base rate adjustment made by the company need to be multiplied in order to take into account the effect of the minimum premium requirement?

Solution S5-114-5. This question is based on the discussion in Werner and Modlin, p. 275.

The company's total premium without a minimum premium requirement would be (240 + 31 + 512 + 89 + 800) = 1672. The company's total premium with a minimum premium requirement would be (240 + 100 + 512 + 100 + 800) = 1752. Because the minimum premium requirement increases the premium from what would otherwise be collected, the required offset factor would need to reverse the effect of that increase when base rate changes are considered. Thus, the offset factor would be 1672/1752 = 0.9543378995.

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