Calculations Pertaining to Coinsurance Requirements and Penalties in Property Insurance: 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.

Let a be the coinsurance apportionment ratio.

Let c be the required coinsurance percentage.

Let e be the coinsurance penalty.

Let F be the amount of insurance selected by the insured, i.e., the face value of the insurance policy being purchased.

Let L be the amount of a loss, after the relevant deductible has been applied to it.

Let V be the value of the property being insured.

Then the following formulas hold:

Formula 90.1:

a = min(F/(cV), 1)

Formula 90.2:

I = L*F/(cV), where I ≤ F and I ≤ L.

Formula 90.3:

e = L - I if L ≤ F;

e = F - I if F < L < cV;

e = 0 if cV ≤ L.

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 11, pp. 203-208.

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

Problem S5-90-1. Assume that the value of a property being insured is $316,000 at the time of loss. The applicable insurance policy has a required coinsurance percentage of 65%. The insured has purchased $165,000 of insurance on the property. A $30,000 deductible applies to the policy. A loss of $165,000 occurs.

(a) What is the coinsurance apportionment ratio?

(b) What is the amount the insured will receive from the insurer as compensation for the loss?

(c) What is the coinsurance penalty applicable to this situation?

Solution S5-90-1.

(a) We use Formula 90.1: a = min(F/(cV), 1). Here, F = 165000, c = 0.65, and V = 316000. Thus, F/(cV) = 165000/(0.65*316000) = a = 0.8033106134.

(b) We use Formula 90.2: I = L*F/(cV). We already know that F/(cV) = 0.8033106134. Furthermore, L = 165000 - 30000 = 135000 (to account for the deductible). Thus, I = 135000*0.8033106134 = 108446.9328 = $108,446.93.

(c) We use Formula 90.3. Here, L < F, so e = L - I = 135000 - 108446.9328 = 26553.06719 = $26553.07.

Problem S5-90-2. Assume that the value of a property being insured is $316,000 at the time of loss. The applicable insurance policy has a required coinsurance percentage of 65%. The insured has purchased $165,000 of insurance on the property. A $30,000 deductible applies to the policy. A loss of $215,000 occurs.

(a) What is the amount the insured will receive from the insurer as compensation for the loss?

(b) What is the coinsurance penalty applicable to this situation?

Solution S5-90-2.

(a) We use Formula 90.2: I = L*F/(cV). We already know from Solution S5-90-1(a) that F/(cV) = 0.8033106134. Furthermore, L = 215000 - 30000 = 185000. Thus, I = 185000*0.8033106134 = 148612.4635 = $148,612.46.

(b) We use Formula 90.3. Here, cV = 0.65*316000 = $205,400, so F < L < cV, which means that e = F - I = 165000 - 148612.4635 = 16387.5365 = $16,387.54.

Problem S5-90-3. Assume that the value of a property being insured is $316,000 at the time of loss. The applicable insurance policy has a required coinsurance percentage of 65%. The insured has purchased $165,000 of insurance on the property. A $30,000 deductible applies to the policy. A loss of $285,000 occurs.

(a) What is the amount the insured will receive from the insurer as compensation for the loss?

(b) What is the coinsurance penalty applicable to this situation?

Solution S5-90-3.

(a) We use Formula 90.2: I = L*F/(cV). We already know from Solution S5-90-1(a) that F/(cV) = 0.8033106134. However, L = 285000 - 30000 = 255000, and I would at first glance appear to be 255000*0.8033106134 = 204844.2064. However, the amount the insured receives cannot be greater than the face value of the policy, which is $165,000 in this case, so the insured will receive $165,000.

(b) We use Formula 90.3. Here, L > cV, so e = 0. There is no coinsurance penalty.

Problem S5-90-4. Assume that the value of a property being insured is $316,000 at the time of loss. The applicable insurance policy has a required coinsurance percentage of 65%. The insured has purchased $165,000 of insurance on the property. A $30,000 deductible applies to the policy. What will be the loss amount for which the maximum magnitude of coinsurance penalty would apply?

Solution S5-90-4. This question is inspired by Table 11.16, displayed by Werner and Modlin on p. 208. The maximum magnitude of coinsurance penalty would apply if the otherwise covered amount of theloss (i.e., the actual loss minus the deductible) were equal to the face value of the policy. This would occur if an absolute loss amount of $165,000 + $30,000 = $195,000 occurred. For such a loss amount, the entire $165,000 of the covered loss would be multiplied by the coinsurance apportionment ratio of 0.8033106134 to arrive at the amount the insured will receive as compensation, leading to the greatest possible magnitude of coinsurance penalty. For higher loss amounts, the difference between the face value of the policy and the amount to which the insured is entitled will diminish in a linear fashion until reaching zero for a loss equal to cV = 0.65*316000 = $205,400.

Problem S5-90-5. Assume that the value of a house is $100,000, and possible loss amounts for that house are uniformly distributed from 0 to 100000. Assume that the frequency of losses is 4%.

(a) What would be the rate per $1000 of insurance if this house were insured to value?

(b) What would be the rate per $1000 of insurance if this house were insured to $60,000?

Solution S5-90-5.

(a) The mean severity of loss here is $50,000, based on the uniform distribution given. Thus, the pure premium is (Frequency)*(Severity) = 0.04*50000 = $2000. The amount of insurance is $100,000, so the rate per $1000 is (2000/100000)*1000 = $20.

(b) For every loss above $60,000, the insured would receive $60,000 in compensation. The mean of the truncated uniform loss distribution would be 0.6*((60000 - 0)/2) + 0.4*60000 = $42,000. Thus, the pure premium is (Frequency)*(Severity) = 0.04*42000 = $1680. The amount of insurance is $60,000, so the rate per $1000 is (1680/60000)*1000 = $28.

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