The Pure Premium Method and the Loss Ratio Method in Insurance Ratemaking: 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.

The Pure Premium Method

Under the Pure Premium Method, the following equation can be used to determine the indicated average rate per exposure:

Formula 75.1:

Indicated Average Rate =

((Pure Premium (including LAE)) + (Fixed Underwriting Expense Per Exposure))/
(1 - Variable Expense Ratio - Target Profit Percentage).

The symbolic expression of this formula is as follows :

Formula 75.2:
P^-_I = (L^- + E^-_L + E^-_F)/(1 - V - Q_T)

We can also express this result as Formula 75.3:

P^-_I = ((L + E_L + E_F)/X)/(1 - V - Q_T).

Definitions of variables:
E_F: Total fixed expense
E^-_F: Fixed expense per exposure
E_L: Total loss adjustment expense
E^-_L: Average loss adjustment expense per exposure
L: Total incurred losses
L^-: Average loss per exposure
P^-_I: Indicated average rate
Q_T: Company-selected profit provision as a fraction of premium
V: Variable expense as a fraction of premium
X: Number of exposures

The Loss Ratio Method

Under the Loss Ratio Method, the following equation can be used to determine the indicated rate change:

Formula 75.3:

(Indicated Change Factor) = (Loss & LAE Ratio + Fixed Expense Ratio)/
(1 - Variable Expense Ratio - Target Underwriting Profit %).

The symbolic expression of this formula is as follows:

Formula 75.4:

(Indicated Change Factor) = ((L + E_L)/P_C + F)/(1 - V - Q_T)

Formula 75.5:

(Indicated Change) = ((L + E_L)/P_C + F)/(1 - V - Q_T) - 1

Definitions of variables:
E_L: Total loss adjustment expense
F: Fixed expense ratio
L: Total incurred losses
P_C: Total earned premium
Q_T: Company-selected profit provision as a fraction of premium
V: Variable expense as a fraction of premium

Note that the Loss Ratio Method can only be used to evaluate experience for an insurance program that is already in place.

Under the Loss Ratio Method, Q_C, the expected profit percentage, assuming current rates, can be expressed as follows:

Formula 75.6:

Q_C = 1 - ((L + E_L) + E_F)/P_C) - V.

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 8, pp. 138-145.

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

Problem S5-75-1. In calendar year 2501, Insurance Company Ω had 67000 Golden Hexagons (GH) in incurred losses, 9600 GH in loss adjustment expenses, and 4300 GH in fixed expenses. The company selected a variable expense provision of 10% and a profit provision of 4%. The company had 3680 earned exposures, and all expenses are compared to earned exposures. Using the pure premium method, what is the indicated average rate?

Solution S5-75-1. We use Formula 75.3: P^-_I = ((L + E_L + E_F)/X)/(1 - V - Q_T), where
L = 67000, E_L = 9600, E_F = 4300, X = 3680, V = 0.1, and Q_T = 0.04. Thus,
P^-_I = ((67000 + 9600 + 4300)/3680)/(1 - 0.1 - 0.04) = P^-_I = 25.5624368 GH.

Problem S5-75-2. In calendar year 2310, Insurance Company Ψ had an average loss per exposure of 90 Golden Hexagons (GH). Loss adjustment expenses were 53 GH per exposure, and fixed expenses were 1 GH per exposure. The company selected a variable expense provision of 23% and a profit provision of 2%. Using the pure premium method, what is the indicated average rate?

Solution S5-75-2. We use the Formula 75.2: P^-_I = (L^- + E^-_L + E^-_F)/(1 - V - Q_T), where

L^- = 90, E^-_L = 53, E^-_F = 1, V = 0.23, and Q_T = 0.02. Thus,

P^-_I = (90 + 53 + 1)/(1 - 0.23 - 0.02) = P^-_I = 192 GH.

Problem S5-75-3. In calendar year 2344, Insurance Company Σ had a total of 54440 Golden Hexagons (GH) in losses and 2400 GH in loss adjustment expenses. The company also earned 80000 GH of premium; rate levels have not changed since that time. The company estimates that it fixed expense ratio is 4%, and that its variable expense ratio is 19%. The company selects a profit provision of 5%. Using the loss ratio method, find the following:

(a) The indicated rate change factor;

(b) The indicated rate change.

Solution S5-75-3.

(a) We use Formula 75.4: (Indicated Change Factor) = ((L + E_L)/P_C + F)/(1 - V - Q_T).
Here, L = 54440, E_L = 2400, P_C = 80000, F = 0.04, V = 0.19, and Q_T = 0.05. Thus,
(Indicated Change Factor) = ((54440 + 2400)/80000 + 0.04)/(1 - 0.19 - 0.05) = 0.7505/0.76 = 0.9875.

(b) The indicated rate change is, by Formula 75.5, (Indicated Change Factor) - 1 = 0.9875 - 1 = -0.0125 = -1.25%.

Problem S5-75-4. In calendar year 2550, Insurance Company Π had a total of 4270 Golden Hexagons GH in losses, 3260 GH in loss adjustment expenses, and 460 GH in fixed expenses. The company earned 10000 GH of premium. The company's variable expense provision is 20%. Assume that rate levels have not changed since 2550 and that all expenses are being compared to earned premium for the purpose of determining expense ratios.

(a) What is the company's profit percentage for calendar year 2550?

(b) What is the indicated rate change, if the company seeks to earn 5% profit?

Solution S5-75-4.

(a) We use Formula 75.6: Q_C = 1 - ((L + E_L) + E_F)/P_C) - V. Here, L = 4270, E_L = 3260, E_F = 460, P_C = 10000, and V = 0.2. Thus, Q_C = 1 - (4270 + 3260 + 460)/10000 - 0.2 = Q_C = 0.001 = 0.1%.

(b) We use Formula 75.5: (Indicated Change) = ((L + E_L)/P_C + F)/(1 - V - Q_T) - 1, where Q_T = 0.05. Here, F is the fixed expense ratio, or 460/10000.

Thus, (Indicated Change) = ((4270 + 3260 + 460)/10000)/(1 - 0.2 - 0.05) - 1 = 0.0653333333 = a +6.53333333% increase.

Problem S5-75-5. The following questions pertain to comparisons between the Pure Premium Method and the Loss Ratio Method:

(a) Which of these methods requires premium to be at current rate levels?

(b) Which of these methods always produces a higher rate indication than the other?

(c) Which of these methods requires clearly defined exposures?

(d) Which of these methods works better if data regarding premium are not available?

(e) Which of these methods is preferable for rating most commercial general liability (CGL) insurance products?

(f) Which of these methods must be used for a new line of business (if the actuary does not wish to rely on judgment alone)?

Solution S5-75-5. This problem is based on the discussion in Werner and Modlin, pp. 143-144.

(a) The Loss Ratio Method requires premium to be at current rate levels.

(b) This is a trick question. Neither method produces a higher rate indication than the other, since the two methods are mathematically equivalent, if executed correctly using comparable, accurate, and consistent data.

(c) The Pure Premium Method requires clearly defined exposures.

(d) The Pure Premium Method works better if data regarding premium are not available.

(e) The Loss Ratio Method is preferable for rating most commercial general liability (CGL) insurance products, because the Pure Premium Method requires clearly defined exposures, and it is extremely difficult to identify what an exposure is for a CGL product, since such a broad variety of coverages is being provided.

(f) The Pure Premium Method must be used for a new line of business.

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