Calculations of Loss Elimination Ratios Using Continuous Loss Distributions and Considerations Pertaining to Expenses in Workers' Compensation 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.

If the underlying loss distribution f(x) is continuous, then the following formulas apply with respect to determining the loss eliminated by application of deductible D and the loss elimination ratio (LER) associated with D.

Formula 89.1:

Loss eliminated by application of deductible D = ₀^D∫x*f(x)*dx + D*_D^∞∫f(x)*dx.

Formula 89.2:

LER(D) = (₀^D∫x*f(x)*dx + D*_D^∞∫f(x)*dx)/(₀^∞∫x*f(x)*dx).

Note that the integral ₀^∞∫x*f(x)*dx corresponds to the unlimited expected loss, i.e., the mean of the distribution with function f(x).

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 11, pp. 199-203.

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

Problem S5-89-1. Losses follow an exponential distribution with mean 631. What would be the loss eliminated by the application of a deductible of 300?

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

Solution S5-89-1. We use Formula 89.1: Loss eliminated by application of deductible D = ₀^D∫x*f(x)*dx + D*_D^∞∫f(x)*dx. Here, D = 300, and θ = 631. Thus, we need to find

₀³⁰⁰∫x*(1/631)e^-x/631*dx + 300*₃₀₀^∞∫(1/631)e^-x/631*dx. We find 300*₃₀₀^∞∫(1/631)e^-x/631*dx =

₃₀₀^∞∫(300/631)e^-x/631*dx = (-300e^-x/631)│₃₀₀^∞ = 300e^-300/631. We can also find ₀³⁰⁰∫x*(1/631)e^-x/631*dx using the Tabular Method of Integration by Parts:

Signs....... u.................. dv
+............... x................ (1/631)e^-x/631
- ...............1..................-e^-x/631
+............... 0...................631e^-x/631

Thus, ₀³⁰⁰∫(x/631)e^-x/631*dx = (-xe^-x/631 - 631e^-x/631)│ ₀³⁰⁰ = -300e^-300/631 - 631e^-300/631 + 0 + 631 =

631 - 931e^-300/631. The sum of the two integrals is 631 - 931e^-300/631 + 300e^-300/631 = 631 - 631e^-300/631 = Loss eliminated = 238.7615105.

Problem S5-89-2. Losses follow an exponential distribution with mean 631. What would be the loss elimination ratio associated with the application of a deductible of 300?

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

Solution S5-89-2. We use Formula 89.2:

LER(D) = (₀^D∫x*f(x)*dx + D*_D^∞∫f(x)*dx)/(₀^∞∫x*f(x)*dx).

In Solution S5-89-1, we found that (₀^D∫x*f(x)*dx + D*_D^∞∫f(x)*dx), the loss eliminated, is equal to 238.7615105. (₀^∞∫x*f(x)*dx) is the mean of the loss distribution, i.e., 631. Thus, 238.7615105/631 =LER(200) = 0.378385912.

Problem S5-89-3. Insurer behaviors may differ on the basis of deductible amounts chosen. Describe two kinds of insurer behavior that a simple loss elimination ratio approach to analyzing the effects of deductibles will not take into account.

Solution S5-89-3. This question is based on the discussion in Werner and Modlin, p. 200. The following two behaviors are possible:
1. Insurers with higher deductibles may be less likely to report certain kinds of losses. For instance, an insured with a $5000 deductible will be less likely to report a $5100 loss than an insured with a $1000 deductible, because the insured with a $5000 deductible would receive a $100 payment for the loss, but would also likely receive a premium increase associated with having filed a claim.

2. Lower-risk insureds will more often choose higher deductibles. This can make high-deductible policies more profitable to insurers than a simple loss elimination ratio analysis would suggest.

Problem S5-89-4. When determining expense provisions, many commercial lines insurers use the All Variable Expense Approach, which assumes that all expenses vary with the premium. This approach tends to undercharge smaller accounts and overcharge larger accounts. What are three ways in which workers' compensation insurers have attempted to compensate for this deficiency in the All Variable Expense Approach?

Solution S5-89-4. This question is based on the discussion in Werner and Modlin, p. 201. The following three ways are mentioned there:

1. Calculating a variable expense provision that only applies to the first $X of standard premium, which is premium before premium discounts and expense constants are applied;

2. Charging an expense constant to all risks, which accounts for fixed costs.

3. Applying a premium discount to policies that have premium exceeding a certain threshold.

Problem S5-89-5. A workers' compensation insurer insures large accounts and small accounts. Large accounts have a earned premium of $531,013 and losses of $310,000. Small accounts have earned premium of $31,002 and losses of $26,504. There are 123 large accounts and 321 small accounts. The insurer wishes to charge an additional loss constant to each small account so that the target loss ratio for the small accounts is equal to the observed loss ratio for the large accounts. What should be the loss constant per small account?

Solution S5-89-5. This question is inspired by the example accompanied by Table 11.11 in Werner and Modlin, p. 201.

The loss ratio on the large accounts is (Losses)/(Earned Premium) = 310000/531013 = 0.5837898507.

To achieve this target loss ratio on the small accounts, a premium of

(Losses on Small Accounts)/(Target Loss Ratio) = 26504/0.5837898507 = 45399.89855 would need to be collected. The shortfall with respect to the current premium is 45399.89855 - 31002 = 14397.89855. Assuming that this amount is distributed evenly throughout the 321 small accounts, the loss constant per small account would be 14397.89855/321 = 44.85326653 = $44.85.

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