Treatment of Expenses in Ratemaking and the All Variable Expense Method: 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.

Equation 72.1 is the fundamental insurance equation, transformed to determine average premium:

P^- = (L^- + E^-_L + E^-_F)/(1 - V - Q_T)

Definitions of variables:
E^-_F: Fixed expense per policy
E^-_L: Average expected loss adjustment expense per policy
L^-: Average expected loss per policy
P^-: Average premium per policy
Q_T: Company-selected profit provision as a fraction of premium
V: Variable expense as a fraction of premium

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 7, pp. 122-127.

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

Problem S5-72-1. An insurance company has determined the following:

The average expected loss per policy is $700.
The average expected loss adjustment expense per policy is $78.
The fixed expense per policy is $30.
For each dollar of premium collected, 24 cents cover expenses that vary with the premium.
The company has selected a profit provision of 2% of premium.

Using this information and the fundamental insurance equation, determine the average amount of premium that the company will need to charge.

Solution S5-72-1. We use Equation 72.1: P^- = (L^- + E^-_L + E^-_F)/(1 - V - Q_T).

Here, L^- = 700, E^-_L = 78, E^-_F = 30, V = 0.24, and Q_T = 0.02. Thus, our answer is

P^- = (700 + 78 + 30)/(1 - 0.24 - 0.02) = 1091.891892 = $1091.89.

Problem S5-72-2. Werner and Modlin, p. 123, discuss the following categories of underwriting expense:

1. Commissions and brokerage
2. Other acquisition expenses
3. Taxes, licenses, and fees
4. General expenses.

Match each of the following items to the category within which it would be included. It is also possible that some of these items belong to category 5. None of the above.

(a) Costs of media advertisements
(b) Investment income expenses
(c) Premium taxes
(d) Actuarial salaries
(e) Payments to agents based on predetermined volume goals
(f) Building maintenance
(g) Costs of mailings to prospective insureds
(h) Federal income taxes
(i) Payments to brokers based on loss ratios of business generated
(j) Licensing fees
(k) Salaries of non-commissioned sales employees

Solution S5-72-2.

The following items belong to category 1. Commissions and brokerage:

(e) Payments to agents based on predetermined volume goals
(i) Payments to brokers based on loss ratios of business generated

The following items belong to category 2. Other acquisition expenses:

(a) Costs of media advertisements
(g) Costs of mailings to prospective insureds
(k) Salaries of non-commissioned sales employees

The following items belong to category 3. Taxes, licenses, and fees:

(c) Premium taxes
(j) Licensing fees

The following items belong to category 4. General expenses:

(d) Actuarial salaries
(f) Building maintenance

The following items belong to category 5. None of the above:

(b) Investment income expenses
(h) Federal income taxes

Problem S5-72-3. You know the following data about a large commercial insurer's book of business:

In the year 2350, expenses were 53600 Golden Hexagons (GH), and written premium was 360000 GH.
In the year 2351, expenses were 46000 GH, and written premium was 400000 GH.
In the year 2352, expenses were 50000 GH, and written premium was 380000 GH.

Use the All Variable Expense Method to determine the following:

(a) The variable expense percentage for each year.

(b) The selected variable expense percentage, based on a 3-year average of variable expenses.

(c) The estimated expenses in the year 2353, if written premium is 560000 GH.

Solution S5-72-3.

(a) The All Variable Expense Method assumes that all expenses are variable expenses. The variable expense percentage for each year is (Expenses)/(Written Premium).

For 2350, this percentage is 53600/360000 = 14.88888889%.
For 2351, this percentage is 46000/400000 = 11.5%.
For 2352, this percentage is 50000/380000 = 13.15789474%.

(b) The three-year average variable expense percentage is equal to

(Sum of expenses during the three years)/(Sum of written premium during the three years) =
(53600 + 46000 + 50000)/(360000 + 400000 + 380000) = 13.12280702%.

(c) The estimated expenses in the year 2353, if written premium is 560000 GH, are 13.12280702% of the written premium, or 560000*0.1312280702 = 73487.7193 GH.

Problem S5-72-4. This question pertains to the All Variable Expense Method.

(a) What determines whether written premium or earned premium is used in the denominator of the variable expense percentage calculation?

(b) When will the use of written premium versus earned premium have a material impact on the variable expense percentage calculation?

Solution S5-72-4. This question is based on the discussion in Werner and Modlin, pp. 124-125.

(a) If expenses are mostly incurred at the beginning of the policy - as is the case for expenses such as commissions - then written premium should be used. If expenses are mostly incurred gradually over the lifetime of the policy - as is the case for overhead and salaries - then earned premium should be used.

(b) If the company is substantially growing or shrinking, then the use of written premium versus earned premium have a material impact on the variable expense percentage calculation. If the company is growing, written premium will be higher than earned premium. If the company is shrinking, written premium will be lower than earned premium.

Problem S5-72-5. An insurance company has determined the following:

The average expected loss per policy is $700.
The average expected loss adjustment expense per policy is $78.
The fixed expense per policy is $30.
For each dollar of premium collected, 24 cents cover expenses that vary with the premium.
The company has selected a profit provision of 2% of premium.

Now consider a policy for which the true loss and loss adjustment expense cost, known to the company, is $500.

(a) What would be the premium on this policy if it were rated correctly?

(b) What would be the premium on this policy if it were rated according to the All Variable Expense Method?

(c) By what percentage and in what direction is the answer in (b) different from the answer in (a)?

Solution S5-72-5.

(a) We use Equation 72.1, applied to the particular policy: P^- = (L + E^-_L + E^-_F)/(1 - V - Q_T).
Here, L+ E_L = 500, E^-_F = 30, V = 0.24, and Q_T = 0.02. Thus, our answer is
P^- = (500 + 30)/(1 - 0.24 - 0.02) = 716.2162162 = $716.22.

(b) From Solution S5-72-1, we know that the average premium for this book of business is 1091.891892. The percentage of this that the fixed expense constitutes is 30/1091.891892 = 2.74752475%. This gets incorporated into the variable expense under the All Variable Expense Method, leading the variable expense percentage to be 24% + 2.74752475% = 26.74752475%.

Now we can find the premium: P^- = (L + E_L)/(1 - V - Q_T) = (500)/(1-0.2674752475 - 0.02) = 701.7300076 = $701.73.

(c) The premium in part (b) is too low. The percentage difference from the premium in part (a) is 701.7300076/716.2162162 - 1 = -0.0202260271= -2.02260271%.

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