Rate Calculations for Property Insurance and Considerations Pertaining to Insurance to Value: 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.

We define the following variables in the context of a property insurance policy:

f = frequency of loss;
F = face value of policy (amount of insurance purchased);
L = amount of loss after the deductible has been applied;
s(L) = probability of loss of a given size (severity distribution);
V = maximum possible loss (may be unlimited).

For a discrete distribution of losses, the following formula holds:

Formula 91.1:

Rate = (f*(_L=1^FΣ(L*s(L))) + F*(1 - _L=1^FΣ(s(L))))/F.

For a continuous distribution of losses, the following formula holds:

Formula 91.2:

Rate = (f*(₀^F∫(L*s(L)*dL + F*(1 - ₀^F∫(s(L)*dL)))/F.

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 11, pp. 208-211.

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

Problem S5-91-1. Assume that the face value of insurance under consideration is $200,000. The frequency of loss is 2%. The severity distribution is discrete, with the following characteristics, on the basis of observed data:

Pr(L = 5000) = 0.2;
Pr(L = 25000) = 0.2;
Pr(L = 50000) = 0.4;
Pr(L = 100000) = 0.1;
Pr(L = 250000) = 0.1.

What is the rate for this situation?

Solution S5-91-1. We use Formula 91.1:

Rate = (f*(_L=1^FΣ(L*s(L)) + F*(1 - _L=1^FΣ(s(L))))/F.

Here, f = 0.02, and F = 200000.

We find _L=1^FΣ(L*s(L)) = _L=1²⁰⁰⁰⁰⁰Σ(L*s(L)) = 5000*0.2 + 25000*0.2 + 50000*0.4 + 100000*0.1 = $36,000.

We find _L=1^FΣ(s(L)) = _L=1²⁰⁰⁰⁰⁰Σ(s(L)) =0.2 + 0.2 + 0.4 + 0.1 = 0.9.

Thus, F*(1 - _L=1^FΣ(s(L)))) = 200000*(1-0.9) = $20,000.

Thus, Rate = (0.02*(36000 + 20000)/200000) = Rate = $0.0056 per dollar of coverage.

Problem S5-91-2. Assume that the face value of insurance under consideration is $200,000. The frequency of loss is 2%. The severity distribution is exponential with mean 50000. What is the rate for this situation?

Relevant properties of exponential distributions: The probability density function is f(x) = (1/θ)e^-x/θ, where θ is the mean of the distribution. The survival function is S(x) = e^-x/θ.

Solution S5-91-2. We use Formula 91.2: Rate = (f*(₀^F∫(L*s(L))*dL + F*(1 - ₀^F∫(s(L)*dL)))/F.

Here, f = 0.02, and F = 200000.

1 - ₀^F∫(s(L)*dL) = 1 - ₀²⁰⁰⁰⁰⁰∫(s(L)*dL) is, here, the survival function at 200000 of the exponential distribution with mean 50000. This is S(200000) = e^{-200000/50000} = e^-4 = 0.0183156389. This is the proportion of possible losses that will be in excess of the policy's face value.

We also find ₀^F∫(L*s(L))*dL = ₀²⁰⁰⁰⁰⁰∫(x/50000)e^-x/50000*dx.

We use the Tabular Method of Integration by Parts:

Signs....... u.................. dv
+............... x................ (1/50000)e^-x/50000
- ...............1..................-e^-x/50000
+............... 0..................50000e^-x/50000

Thus, ₀²⁰⁰⁰⁰⁰∫(x/50000)e^-x/50000*dx = (-xe^-x/50000 - 50000e^-x/50000) │₀²⁰⁰⁰⁰⁰ = -200000e^-4 - 50000e^-4 + 50000. The numerator of our formula for the rate is thus

0.02*(-200000e^-4 - 50000e^-4 + 50000 + 200000e^-4)/200000 = (1000 - 1000e^-4)/200000 =

(1- e^-4)/200 = $0.0049084218per dollar of coverage.

Problem S5-91-3.
(a) Fill in the blanks in the following sentence, given the options suggested:
"For a right-skewed loss distribution, corresponding to a preponderance of large losses, as the policy face value increases, the rate will _______ (decrease, increase, stay constant) at a ________ (decreasing, increasing, constant) rate."

(b) Fill in the blanks in the following sentence, given the options suggested:
"For a left-skewed loss distribution, corresponding to a preponderance of small losses, as the policy face value increases, the rate will _______ (decrease, increase, stay constant) at a ________ (decreasing, increasing, constant) rate."

(c) Fill in the blanks in the following sentence, given the options suggested:
"For a uniform loss distribution, as the policy face value increases, the rate will _______ (decrease, increase, stay constant) at a ________ (decreasing, increasing, constant) rate."

Solution S5-91-3. This question is based on the discussion by Werner and Modlin, p. 209. The following answers are correct:

(a) "For a right-skewed loss distribution, corresponding to a preponderance of large losses, as the policy face value increases, the rate will decrease at an increasing rate."

(b) "For a left-skewed loss distribution, corresponding to a preponderance of small losses, as the policy face value increases, the rate will decrease at a decreasing rate."

(c) "For a uniform loss distribution, as the policy face value increases, the rate will decrease at a constant rate."

Problem S5-91-4. Briefly describe two insurer initiatives designed to encourage insurance to value.

Solution S5-91-4. The following insurer initiatives, designed to encourage insurance to value, are described by Werner and Modlin, p. 209:

1. Guaranteed replacement cost (GRC) coverage, which allows replacement cost to exceed the policy limit, but only if the property is fully insured to value. This is most often capped at some percentage greater than 100% of the policy limit (for instance 125% of the policy limit).

2. Sophisticated property estimation tools, which can consider more aspects of an insured's home than was previously possible.

Other valid answers may also be possible.

Problem S5-91-5. From the insurer's perspective (disregarding the insured's possible interest in having complete coverage in the event of a total loss to a property and disregarding higher premiums collected as a result of insurance to value), what is a motivation for encouraging insurance to value?

Solution S5-91-5. This question is based on the discussion in Werner and Modlin, p. 205.

If the insurer assumes that all properties are insured to value, but some properties are actually not insured to value, then the insurer will charge the underinsured properties a lower rate than would be adequate to cover the exposure. (Mathematically, it can be shown that the rate per monetary unit of coverage is higher for underinsured properties than for otherwise identical properties that are fully insured.) This is because the insurer will charge the underinsured properties the same rate as properties that are insured to value are charged. A coinsurance penalty is an attempt by many insurers to mitigate the effect of this rate inadequacy.

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