The Approximated Average 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. 267-271.

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

Problem S5-113-1. 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 wishes to set a base rate so as to achieve an average premium of $310.

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 average rate differential method to find the approximated average rate differential S^--_P for this book of business.

Solution S5-113-1. This problem is based on the discussion in Werner and Modlin, pp. 267-269. The approximated average rate differential is the product of the weighted-average rate differentials for each of the variables.

For variable C, the weighted-average rate differential is
(1.2*23146 + 1*135600)/(23146 + 135600) = 1.02916105.

For variable J (which is a discount - so the raspberry-jam-eating population gets assigned a factor of 1 - 0.2 = 0.8), the weighted-average rate differential is (0.8*31519 + 1*127227)/(31519 + 127227) = 0.9602900231.

S^--_P is the product of these weighted-average rate differentials for the individual variables:
1.02916105*0.9602900231 = S^--_P = 0.9882930884.

Problem S5-113-2. 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 wishes to set a base rate so as to achieve an average premium of $310.

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 average rate differential method to find the base rate B^-_P that the company would propose under this method.

Solution S5-113-2. This problem is based on the discussion in Werner and Modlin, pp. 267-269. We use the formula B^-_P = ((P^-_P - A_P)/S^--_P ), where P^-_P is the desired average premium, A_P is the fixed policy fee, S^--_P is the approximated average rate differential. Here, P^-_P = 310, A_P = 65, and S^--_P = 0.9882930884 (from Solution S5-113-1). Thus, B^-_P = ((310 - 65)/0.9882930884) = B^-_P = 247.9021688.

Problem S5-113-3. Why is the approximated average rate differential method less accurate than the extension of exposures method in determining a proposed base rate?

Solution S5-113-3. This problem is based on the discussion in Werner and Modlin, p. 270.

The approximated average rate differential method less accurate than the extension of exposures method in determining a proposed base rate because the average rate differential method fails to take into account the possible distributional bias between variables. Calculating an average rate differential for each variable and then multiplying such average rate differentials together assumes that the variables are completely independent from one another and that there is no interaction or exposure correlation between them. In reality, insureds from a particular class of variable X might also be more or less likely to be in a particular class of variable Y, and thus this assumption of independence may not reflect reality.

Problem S5-113-4. 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 wishes to set a base rate so as to achieve an average premium of $424. 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 average rate differential method to find the approximated average rate differential S^--_P for this book of business.

Solution S5-113-4. This problem is based on the discussion in Werner and Modlin, pp. 270-271.

The approximated average rate differential is the product of the weighted-average rate differentials for each of the variables.

Here, to find the weighted-average rate differentials for each variable, it is necessary to adjust the premium for each class of each variable to what the premium would be had that class been charged the base rate. This is accomplished by first dividing the premium for a particular class by the rate differential applicable to that class.

For variable K, the following facts hold:

For the class of insureds who own five or more keychains, the premium at the base rate is 31513000/1.04 = 30300961.54.

For the class of insureds who own fewer than five keychains, the premium is already at the base rate and so is $134,246,000.

Now we find the weighted-average rate differential for variable K, using the company's proposed differentials: (1.03*30300961.54 + 1*134246000)/(30300961.54 + 134246000) = 1.005524434.

For variable G, the following facts hold:

For the class of insureds who play computer games (and receive an 11% discount), the premium at the base rate is 123124000/0.89 =138341573.

For the class of insureds who do not play computer games, the premium is already at the base rate and so is $42,635,000.

Now we find the weighted-average rate differential for variable G, using the company's proposed differentials: (138341573*0.91 + 42635000*1)/(138341573 + 42635000) = 0.9312024682.

S^--_P is the product of these weighted-average rate differentials for the individual variables:
1.005524434*0.9312024682 = S^--_P = 0.9363468348.

Problem S5-113-5. 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 wishes to set a base rate so as to achieve an average premium of $424. 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 average rate differential method to find the base rate B^-_P that the company would propose under this method.

Solution S5-113-5. This problem is based on the discussion in Werner and Modlin, pp. 270-271. We use the formula B^-_P = ((P^-_P - A_P)/S^--_P ), where P^-_P is the desired average premium, A_P is the fixed policy fee, S^--_P is the approximated average rate differential. Here, P^-_P = 424, A_P = 120, and S^--_P = 0.9363468348 (from Solution S5-113-4). Thus, B^-_P = (424-120)/0.9363468348 = B^-_P = 324.6660198.

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