The Method of Fitted Curves for Finding the Complement of Credibility in Excess Insurance Ratemaking and Cost-Benefit Analyses for Changes in Insurance Rating Structures: 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 following formulas pertain to methods in excess insurance ratemaking for finding the complement of credibility C.

Method of Fitted Curves

Let f(x) be the probability density function of a continuous distribution of losses which is fitted to the observed loss experience.

Formula 97.1

% of Losses in Layer (A, A + L) = (_A^A+L∫(x*f(x)*dx) + _A+L^∞∫(A+L)*f(x)*dx)/(_-∞^∞∫x*f(x)*dx)

Note that (_-∞^∞∫x*f(x)*dx) is the expected value of the distribution with density function f(x).

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapters 12 and 13, pp. 228-238.

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

Problem S5-97-1. An excess insurance policy has an attachment point of $200,000, and the excess insurer's limit of liability under the policy is $100,000. Ground-up insurance data is available for losses; the data are fitted to an exponential distribution with mean $60,000. Total losses are $5,310,013. From that distribution, what is the percentage of expected losses that are in the layer from $200,000 to $300,000?

Relevant property of exponential distributions: f(x) = (1/θ)e^-x/θ, where θ is the mean of the distribution.

Solution S5-97-1. We use Formula 97.1:

% of Losses in Layer (A, A + L) = (_A^A+L∫(x*f(x)*dx) + _A+L^∞∫(A+L)*f(x)*dx)/(_-∞^∞∫x*f(x)*dx).

Here, A = $200,000, and L = $100,000. Moreover, f(x) = (1/60000)e^-x/60000, where (_-∞^∞∫x*f(x)*dx) = 60000 is the mean of the exponential distribution.

We find the numerator of the formula:
(_A^A+L∫(x*f(x)*dx) + _A+L^∞∫(A+L)*f(x)*dx) =

₂₀₀₀₀₀³⁰⁰⁰⁰⁰∫(x/60000)e^-x/60000*dx) + ₃₀₀₀₀₀^∞∫(300000/60000)*e^-x/60000*dx).

We find ₃₀₀₀₀₀^∞∫(300000/60000)*e^-x/60000*dx) = ₃₀₀₀₀₀^∞∫5*e^-x/60000*dx) = (-300000e^-x/60000)│₃₀₀₀₀₀^∞ = 300000e^-5.

We find ₂₀₀₀₀₀³⁰⁰⁰⁰⁰∫(x/60000)e^-x/60000*dx) using the Tabular Method of Integration by Parts:

Signs....... u.................. dv
+............... x................ (1/60000)e^-x/60000
- ...............1..................-e^-x/60000
+............... 0...................60000e^-x/60000

Thus, ₂₀₀₀₀₀³⁰⁰⁰⁰⁰∫(x/60000)e^-x/60000*dx) = (-xe^-x/60000 - 60000e^-x/60000)│ ₂₀₀₀₀₀³⁰⁰⁰⁰⁰ =

-300000e^-5 - 60000e^-5 + 200000e^-10/3 + 60000e^-10/3 = 260000e^-10/3 - 360000e^-5.

The sum of the two integrals we found is 300000e^-5 + 260000e^-10/3 - 360000e^-5 =

260000e^-10/3 - 60000e^-5 = 8870.96145.

The percentage of the losses in the layer (200000, 300000) is thus 8870.96145/60000 = 0.1478493575 = 14.78493575%.

Problem S5-97-2. An excess insurance policy has an attachment point of $200,000, and the excess insurer's limit of liability under the policy is $100,000. Ground-up insurance data is available for losses; the data are fitted to an exponential distribution with mean $60,000. Total losses are $5,310,013. From that distribution, what is the magnitude of expected losses that are in the layer from $200,000 to $300,000?

Solution S5-97-2. The magnitude of expected losses in the layer from $200,000 to $300,000 is equal to the total losses multiplied by the percentage of losses in that layer (which we found to be 14.78493575% in Solution S5-97-1): 5310013*0.1478493575 = 785082.0104 = $785,082.01. This would be our complement of credibility to the observed total loss data pertaining to the excess insurance policies in question.

Problem S5-97-3. Consider the following four approaches to finding complements of credibility in excess insurance ratemaking (the first three of which were discussed in Section 96):

1. The method of increased limit factors;
2. The method of lower limits analysis;
3. The method of limits analysis;
4. The method of fitted curves.

(a) Which of these methods typically requires the most computational complexity?

(b) Which of these methods typically produces a complement of credibility which is the most logically related to the excess layer data being analyzed?

(c) Which of these approaches requires the assumption that the loss ratio does not vary by the limit of insurance chosen?

(d) Which of these approaches is most biased toward losses that occur in smaller ground-up amounts than would be applicable to the excess insurance policy under consideration?

(e) Which of these approaches requires data that have not been truncated below the attachment point of the excess insurance?

(f) Which of these approaches is typically used by reinsurers who do not have access to an insurer's full loss distribution?

Solution S5-97-3. This question is based on the evaluation of the four methods by Werner and Modlin, pp. 229-232.

(a) The method of fitted curves typically requires the most computational complexity.

(b) The method of fitted curves typically produces a complement of credibility which is the most logically related to the excess layer data being analyzed.

(c) The method of limits analysis requires the assumption that the loss ratio does not vary by the limit of insurance chosen.

(d) The method of lower limits analysis is most biased toward losses that occur in smaller ground-up amounts than would be applicable to the excess insurance policy under consideration.

(e) The method of increased limit factors requires data that have not been truncated below the attachment point of the excess insurance.

(f) The method of limits analysis is typically used by reinsurers who do not have access to an insurer's full loss distribution.

Problem S5-97-4. Insurer Q is currently charging all insureds of Class X a premium of $361. The insurer is considering implementing a more refined classification system, which Class X being split into three subclasses: 1, 2, and 3.

There are currently 4126 risks of subclass 1, whose total projected losses and expenses are $1,320,320.
There are currently 8000 risks of subclass 2, whose total projected losses and expenses are $2,800,000.
There are currently 1333 risks of subclass 3, whose total projected losses and expenses are $533,200.

(a) What is the current percentage of profit the company is achieving on subclass 1?

(b) What is the current percentage of profit the company is achieving on subclass 2?

(c) What is the current percentage of profit the company is achieving on subclass 3?

(d) What is the current percentage of profit the company is achieving overall?

Solution S5-97-4.

(a) For subclass 1, per risk, the projected losses and expenses are $1,320,320/4126 = $320.

The current rate per risk is $361. The difference is the profit per risk: 361 - 320 = $41. As there are 4126 risks, this translates to a total profit of 41*4126 = $169,166. The total premium charged for this subclass is 361*4126 = $1,489,486, so the profit percentage is $169,166/$1,489,486 = 0.1135734072 = 11.35734072%.

(b) For subclass 2 per risk, the projected losses and expenses are $2,800,000/8000 = $350.

The current rate per risk is $361. The difference is the profit per risk: 361 - 350 = $11. As there are 8000 risks, this translates to a total profit of 11*8000 = $88,000. The total premium charged for this subclass is 361*8000 = $2,888,000, so the profit percentage is $88,000/$2,888,000 = 0.0304709141 = 3.04709141%.

(c) For subclass 3 per risk, the projected losses and expenses are $533,200/1333 = $400.

The current rate per risk is $361. The difference is the profit per risk: 361 - 400 = -$39. As there are 1333 risks, this translates to a total profit of -39*1333 = -$51,987, i.e., a loss. The total premium charged for this subclass is 361*1333 = $481,213, so the profit percentage is

-$51,987/$481,213 = -0.108033241 = -10.8033241%.

(d) The overall profit percentage is the sum of the profits for each risk class, divided by the sum of total premiums charged to each risk class:

($169,166 + $88,000 - $51,987)/($1,489,486 + $2,888,000 + $481,213) = 0.0422292058 =
4.22292058%.

Problem S5-97-5. Insurer Q is currently charging all insureds of Class X a premium of $361. The insurer is considering implementing a more refined classification system, which Class X being split into three subclasses: 1, 2, and 3.

There are currently 4126 risks of subclass 1, whose total projected losses and expenses are $1,320,320.
There are currently 8000 risks of subclass 2, whose total projected losses and expenses are $2,800,000.
There are currently 1333 risks of subclass 3, whose total projected losses and expenses are $533,200.

(a) The company wishes to change the overall premium charged to each subclass of risk so that it achieves the same profitability over each subclass and maintains the same historical percentage of profitability. What should be the premium charged per risk in each of the subclasses?

(b) Assume that this change results in a 20% increase in insureds of subclass 1, a 4% decrease in insureds of subclass 2, and a 15% decrease in insureds of subclass 3. What is the maximum transition cost the company could incur for this rating change while still at worst not losing any money on net from the change? (Assume that the risk-free interest rate is 0%, so the time value of money is not an issue.)

Solution S5-97-5.

(a) From Solution S5-97-4(d), the overall profit percentage is 4.22292058%. The rate charged to each subclass should be (Loss and Expense Cost)/(1- Profit Percentage).

For subclass 1, the rate charged should thus be 320/(1-0.0422292058) = 334.1091131 = $334.11.
For subclass 2, the rate charged should thus be 350/(1-0.0422292058) = 365.4318425 = $365.43.
For subclass 3, the rate charged should thus be 400/(1-0.0422292058) = 417.6363914 = $417.64.

(b) From Solution S5-97-4, the total profit currently being earned is ($169,166 + $88,000 - $51,987) = $205,179.

The new number of subclass 1 exposures will be 4126*1.20 = 4951.2, rounded down to 4951.
The new number of subclass 2 exposures will be 8000*(1-0.04) = 7680.
The new number of subclass 3 exposures will be 1333*(1-0.15) = 1133.05, rounded down to 1133.

The total premium collected will be 4951*334.11 + 7680*365.43 + 1133*417.64 = $4,933,867.13. Of that, 4.22292058% is profit, meaning that total profit will be $4,933,867.13*0.0422292058 = 208352.5612 = $208,352.56. For the company to at least break even, the maximum possible transition costs would be (Anticipated profit) - (Current profit) = $208,352.56 - $205,179 = 3173.561197 = $3173.56.

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