Policy Limit
Mathematical models for insurance payments – part 3 – other modifications
This post is a continuation of the discussion on models of insurance payments initiated in two previous posts. Part 1 focuses on the models of insurance payments in which the insurance policy imposes a policy limit. Part 2 continues the discussion by introducing models in which the insurance policy imposes an ordinary deductible. In each of these two previous posts, the insurance coverage has only one coverage modification. A more interesting and more realistic scenario would be insurance coverage that contains a combination of several coverage modifications. This post is to examine the effects on the insurance payments as a result of having some or all of these coverage modifications – policy limit, ordinary deductible, franchise deductible and inflation. Additional topics: expected payment per loss versus expected payment per payment and loss elimination ratio.
Ordinary Deductible and Policy Limit
The previous two posts discuss the expectations and . The first is the expected insurance payment when the coverage has a policy limit . The second is the expected insurance payment when the coverage has an ordinary deductible . They are the expected values of the following two variables.
It is easy to verify that . Buying a coverage with an ordinary deductible and another coverage with policy limit equals full coverage. Thus we have the following relation.
The limited expectation has expressions in closed form in some cases or has expressions in terms of familiar functions (e.g. gamma function) in other cases. Thus the expectation can be computed by knowing the original expected value and the limited expectation .
Inflation
Suppose that losses (or claims) in the next period are expected to increase uniformly by %. For example, means 10%. What would be the effect of inflation on the expectations and ?
First, the effect on the limited expectation . As usual, is the random loss and is the policy limit. With inflation rate , the loss variable for the next period would be . One approach is to derive the distribution for the inflated loss variable and use the new distribution to calculate . Another approach is to express it in terms of the limited expectation of the preinflated loss . The following is the expectation , assuming that there is no change in the policy limit.
Relation (2) relates the limited expectation of the inflated variable to the limited expectation of the preinflated loss . It says that the limited expectation of the inflated variable is obtained by inflating the limited expectation of the preinflated loss but at a smaller policy limit .
The following is the expectation .
Similarly, (3) expresses the expected payment on the inflated loss in terms of the expected payment of the preinflated loss. It says that when the loss is inflated, the expected payment per loss is obtained by inflating the expected payment on the preinflated loss but at a smaller deductible.
Insurance Payment Per Loss versus Per Payment
The previous post (Part 2) shows how to evaluate the average amount paid to the insured when the coverage has an ordinary deductible. The average payment discussed in Part 2 is the average per loss (over all losses). As a simple illustration, let’s say the amounts of losses in a given period for an insured are 7, 4, 33 and 17 subject to an ordinary deductible of 5. Then the insurance payments are: 2, 0, 28 and 12. The average payment per loss would be (2+0+28+12)/4 = 10.5. If we only count the losses that require a payment, the average is (2+28+12)/3 = 14. Thus the average payment per payment is greater than the average payment per loss since only the losses exceeding the deductible are counted in the average payment per payment. In the calculation discussed here, the average payment per payment is obtained by dividing the average payment per loss by the probability that the loss exceeds the deductible. Note that 10.5/0.75 = 14.
Suppose that the random variable is the size of the loss (if a loss occurs). Under an insurance policy with an ordinary deductible , the first dollars of a loss is responsible by the insured and the amount of the loss in excess of is paid by the insurer. Under such an arrangement, a certain number of losses are not paid by the insurer, precisely those losses that are less than or equal to . If we only count the losses that are paid by the insurer, the payment amount is the conditional random variable . The expected value of this conditional random variable is denoted by or if the loss is understood.
Given that , the variable is called the excess loss variable. Its expected value is called the mean excess loss function. Other names are mean residual life and complete expectation of life (when the context is that of a mortality study).
For the discussion in this post and other posts in the same series, we use to denote . The P stands for payment so that its expected value would be average insurance payment per payment, i.e. the expected amount paid given that the loss exceeds the deductible. When the random variable is the age at death, would be the expected remaining time until death given that the life has survived to age .
The expected value is thus the expected payment per payment (or expected cost per payment) under an ordinary deductible. In contrast, the expected value is the expected payment per loss (or expected cost per loss), which is discussed in the previous post. The two expected values are related. The calculation of one will give the other. The following compares the two calculation. Let and be the PDF and CDF of , respectively.
Note that is calculated using a conditional density function. As a result, can be obtained by dividing the expected payment per loss divided by the probability that there is a payment. Thus the expected payment per payment is the expected payment per loss divided by the probability that there is a payment. This is described in the following relation.
Distributions of Insurance Payment Variables
Relation (1) and Relation (4) are used for the calculation of expected insurance payment when the coverage has an ordinary deductible. For deriving other information about insurance payment in the presence of an ordinary deductible, it is helpful to know the distributions of the insurance payment (per loss and per payment).
Let (payment per loss) and let (payment per payment). We now discuss the distribution for . First, the following gives the PDF, CDF and the survival function of .
Payment Per Loss  

CDF  
Survival Function 
Note that the above functions have a point mass at to account for the losses that are not paid. For , there is no point mass at . We only need to consider . Normalizing the function would give the PDF of . Thus the following gives the PDF, the survival function and the CDF of .
Payment Per Payment  

CDF  
Survival Function 
The following calculates the two averages using the respective PDFs, as a result deriving the same relationship between the two expected values.
Relation (1) and Relation (7) are identical. The former is calculated using the distribution of the original loss and the latter is calculated using the distribution of the payment per loss variable. Similarly, compare Relation (5) and Relation (8). The former is computed from the distribution of the unmodified loss and the latter is computed using the distribution of the payment per payment variable.
Note that Relation (7) and Relation (8) also lead to Relation (6), which relates the expected payment and the expected payment .
With the PDFs and CDFs for the payment per loss and the payment per payment developed, other distributional quantities can be derived, e.g. hazard rate, variance, skewness and kurtosis.
Loss Elimination Ratio
When the insurance coverage has an ordinary deductible, a natural question is (from the insurance company’s perspective), what is the impact of the deductible on the payment that is made to the insured? More specifically, on average what is the reduction of the payment? The loss elimination ratio is the proportion of the expected loss that is not paid to the insured by the insurer. For example, if the expected loss before application of the deductible is 45 (of which 36 is expected to be paid by the insurer), the loss elimination ratio is 9/45 = 0.20 (20%). In this example, the insurer has reduced its obligation by 20%. More formally, the loss elimination ratio (LER) is defined as:
LER is the ratio of the expected reduction in payment as a result of imposing the ordinary deductible to the expected payment without the deductible. Though it is possible to define LER as the ratio of the reduction in expected payment as a result of a coverage modification to the expected payment without the modification for a modification other than an ordinary deductible, we do not attempt to further generalize this concept.
Policy with a Limit and a Deductible
In part 1, expected insurance payment under a policy limit is developed. In part 2, expected insurance payment under a policy with an ordinary deductible is developed. We now combine both provisions in the same insurance policy. First, let’s define two terms. A policy limit is the maximum amount that will be paid by a policy. For example, if the policy limit is 10,000, the policy will paid at most 10,000 per loss. If the actual loss is 15,000, then the policy will paid 10,000 and the insured will have to be responsible for the remaining 5,000. On the other hand, maximum covered loss is the level above which no loss will be paid. For example, Suppose that the policy covers up to 10,000 per loss subject to a deductible of 1,000. If the actual loss is 20,000, then the maximum covered loss is 10,000 with the policy limit being 9,000. This is because the policy only covers the first 10,000 with the first 1,000 paid by the insured.
Suppose that an insurance policy has an ordinary deductible , a maximum covered loss with and no other coverage modifications. Any loss below is not paid by the insurer. For any loss exceeding , the insurer pays the loss amount in excess of up to the maximum covered loss , with the policy limit being . The following describes the payment rule more explicitly.
Under such a policy, the maximum amount paid (policy limit) is , which is the maximum covered loss minus the deductible. The policy limit is reached when . When , the payment is and the payment is . Then the expected payment per loss under such a policy is:
There is not special notation for the expected payment. In words, it is the limited expected value at (the maximum covered loss) minus the limited expected value at the ordinary deductible . The higher moments can be derived by evaluating using the PDF of the loss where .
The payment is on a per loss basis. It is also possible to consider payment or payment by removing the point mass at zero. The expected payment per payment is obtained by dividing (9) by the probability of a positive payment.
Franchise Deductible
An alternative to the ordinary deductible is the franchise deductible. It works like an ordinary deductible except that when the loss exceeds the deductible, the policy pays the loss in full. The following gives the payment rule.
Note that when the loss exceeds the deductible (), the policy with a franchise deductible pays more than a policy with the same ordinary deductible. By how much? By the amount . Thus the expected payment per loss under a franchise deductible is . The addition of reflects the additional benefit when . Instead of deriving calculation specific to franchise deductible, we can derive the payments under franchise deductible by adding the additional benefit appropriately.
Benefit Tables
The previous two posts and this post discuss coverage with two types of deductible (ordinary and franchise) as well as coverage that may include a limit. In order to keep the expected payments straight, the following table organizes the different combinations of coverage options discussed up to this point.
A = Ordinary Deductible Only
B = Franchise Deductible Only
C = Ordinary Deductible + Limit
D = Franchise Deductible + Limit
Expected Cost Per Loss  Expected Cost Per Payment  

A  
B  
C  
D 
Row A shows the expected payment in a coverage with an ordinary deductible for both per loss and per payment. The expected per loss payment in Row A is the subject of the previous post and the expected payment per payment is discussed earlier in this post. Row B is for expected payments in a coverage with a franchise deductible. Note that the coverage with a franchise deductible pays more than a coverage with an identical ordinary deductible. Thus Row B is Row A plus the added benefit, which is the deductible .
Row C is for the coverage of an ordinary deductible with a policy limit. The per loss expected payment is . The per payment expected value is a conditional one, conditional on the loss greater than the deductible. Thus the per payment expected value is the per loss expected value divided by .
Note that Row D is Row C plus the amount of , the additional benefit as a result of having a franchise deductible instead of an ordinary deductible.
Rather than memorizing these formulas, focus on the general structure of the table. For example, understand the payments involving ordinary deductible (Row A and Row C). Then the payments for franchise deductible are obtained by adding an appropriate additional benefit.
The following table shows the expected payments under the influence of claim cost inflation. The maximum covered loss and the deductible are identical to the above benefit table. The losses are subject to a % inflation in the next exposure period. Note that and are the modified maximum covered loss and deductible that will make the formulas work.
E = Ordinary Deductible Only
F = Franchise Deductible Only
G = Ordinary Deductible + Limit
H = Franchise Deductible + Limit
Expected Cost Per Loss (with Inflation)  Expected Cost Per Payment (with Inflation)  

E  
F  
G  
H 
There is really no need to mathematically derive the formulas for the second insurance benefit table (Rows E through H). Recall the effect of inflation on the expected payments and (see Relation (2) and Relation (3)). Then apply the inflation effect on the first insurance benefit table. Note that the second table is obtained by inflating the first table by but with smaller upper limit and deductible . Also note the relation between Row E and Row F (same relation between Row A and Row B) and the relation between Row G and Row H (same relation between Row C and Row D).
Calculation
One approach of modeling insurance payments is to assume that the unmodified insurance losses are from a catalog of parametric distributions. For certain parametric distributions, the limited expectations have convenient forms. Examples are: exponential distribution, Pareto distribution and lognormal distribution. Other distributions do not close form for but the limited expectation can be expressed as a function that can be numerically calculated. One example is the gamma function.
The table in this link is an inventory of distributions that may be useful in modeling insurance losses. Included in the table are the limited expectations for various distributions. The following table shows the limited expectations for exponential, Pareto and lognormal.
Limited Expectation  

Exponential  
Pareto  , 
Lognormal 
See the table in this link for the limited expectations of the distributed not shown in the above table. Once is computed or estimated, the expected payment for various types of coverage provisions can be derived.
Examples
The first example demonstrates the four categories of expected payments in the table in the preceding section.
Example 1
Suppose that the loss distribution is described by the PDF with support . Walk through all the expected payments discussed in the above table (Rows A through D) assuming and .
First, the basic calculation.
Four categories of expected payments are derived and are shown in the following table.
A = Ordinary Deductible Only
B = Franchise Deductible Only
C = Ordinary Deductible + Limit
D = Franchise Deductible + Limit
Expected Cost Per Loss  Expected Cost Per Payment  

A  
B  
C  
D 
Example 2
Suppose that a coverage has an ordinary deductible and suppose that the CDF of the insurance payment per loss is given by:
Determine the expected value and the variance of the insurance cost per loss.
Note that there is a jump in the CDF at . Thus is a point mass. The following is the PDF of the insurance payment.
It is clear that where is the PDF of the exponential distribution with mean 100. Thus is the PDF of the insurance payment per loss when the coverage has an ordinary deductible of 20 where the loss has exponential distribution with mean 100. We can use to calculate the mean and variance of . Using , the mean and variance are:
Note that the calculation for and is done by multiplying by mean and second moment of the exponential distribution with mean 100.
Example 3
You are given the following:
 Losses follow a Pareto distribution with parameters and .
 The coverage has an ordinary deductible of 100.
Determine the PDF and CDF of the payment to the insured per payment. Determine the expected cost per payment.
Note that the setting of this example is identical to Example 3 in this previous post. Since this only concerns the insurance when the loss exceeds the deductible, the insurance payment is the conditional distribution where is the Pareto loss distribution.
The following gives the PDF and CDF and other information of the loss .
The following shows the PDF and CDF for the payment per payment variable .
Note that these PDF and CDF are for a Pareto distribution with and . Thus the expected payment per payment and the variance are:
Example 4
Suppose that losses follow a Pareto distribution with shape parameter and scale parameter . Suppose that an insurance coverage pays each loss subject to an ordinary deductible . Derive a formula for the expected payment per payment .
Let be the payment per payment. In Example 3, we see that the CDF and PDF of are also Pareto but with shape parameter (same as for ) and scale parameter (the original scale parameter plus the deductible). Thus the following gives the expected payment per payment.
Note that in this case is an increasing linear function of the deductible . The higher the deductible, the larger the expected payment. This is a clear sign that the Pareto distribution is a heavy tailed distribution.
Example 5
Suppose that losses follow a lognormal distribution with parameters and . An insurance coverage has an ordinary deductible of 100. Compute the expected payment per loss. Compute the expected payment per loss if the deductible is a franchise deductible.
The following calculates and .
As a result, the expected payment per loss is . If the deductible of 100 is a franchise deductible, the expected payment per loss is obtained by adding an additional insurance payment, which is . The following is the added payment.
Thus the expected payment per loss with a franchise deductible of 250 is 84.70 + 74.54 = 159.24.
Practice Problems
The following practice problem sets are to reinforce the concepts discussed here.
Practice Problem Set 7 
Practice Problem Set 8 
Practice Problem Set 9 
models of insurance payment
ordinary deductible, franchise deductible
maximum covered loss
Daniel Ma Math
Daniel Ma Mathematics
Dan Ma Actuarial
2017 – Dan Ma
Mathematical models for insurance payments – part 1 – policy limit
Suppose an individual (or an entity) faces a random loss and the loss is modeled by the random variable . The individual purchases an insurance coverage to cover this loss. If the policy pays the loss in full (if a loss occurs), then the expected insurance payment is mean loss . Usually the expected insurance payment is less than due to the presence of policy provisions such as a deductible and/or limit. This post is the first post in a series of posts in discussing the probability models of insurance payments.
Policy Limit
In this post, we assume that the random loss is a continuous random variable taking on the positive real numbers or numbers from an interval such as . The methodology can be adjusted to handle discrete loss variable (mostly replacing integrals with summation).
Suppose that the random variable is the size of the loss (if a loss occurs). Under an insurance policy with a limit , the insurer pays the loss in full if the loss is less than the limit. Furthermore if the loss is or greater, the insurance payment is capped at . Under this policy provision, what is the expected amount of insurance payment if there is a loss? The insurance payment in the presence of a limit is denoted by . This is called the limited loss random variable.
First, let’s consider the mean (called limited expectation). Let , and be the probability density function (PDF), cumulative distribution function (CDF) and the survival function of the random loss, respectively.
The first integral is based on the definition of the limited loss variable, summing up the payments in the interval and in the interval . The second integral is an alternative way to evaluate the first integral (see here for more explanation). The following gives the second moment, which can be used in evaluating the variance of insurance payment.
Once again, the second moment is also expressed using the alternative calculation. The following two examples demonstrate the evaluation of the first two moments of .
Basic Example
Example 1
Suppose that the loss distribution has PDF . Evaluate and
The CDF of the loss is .
The limited expected value is calculated in two ways to demonstrate that the two approaches are equivalent. Without the policy limit, the expected loss is 3.3333. With the benefit payment capped at 4, the expected amount paid by the insurance policy is 2.6133 (per loss). The difference is 3.3333 – 2.6133 = 0.72, which is the expected loss responsible by the insured.
Example 2
Continue with Example 1. Calculate the variance of the insurance payment .
.
To compare, the variance of is as calculated below.
The example demonstrates that policy provisions such as limit has a variance reducing effect. Had the insurer been liable to pay for the entire loss amount, there would be a greater fluctuation in the payment.
A Censored Variable
Mathematically, the random variable is an upper censored random variable. Any realized value of above the limit is known only known as the limit (as far as payment is concerned). On the other hand, applying a limit on the loss turns the continuous random variable into a mixed random variable. In the interval , is continuous while there is a point mass at the point with probability mass .
The policy provisions such as deductible and limit have the effect of turning the unmodified loss variable into censored or truncated variables. As a result, these variables are mixed random variables. The subsequent posts will further demonstrate this point.
Parametric Distributions
There is a vast inventory of parametric distributions that are potential candidates for models of random losses. For example, the table in this link is an inventory of distributions that may be useful in modeling insurance losses. Included in the table are the limited expectations for various distributions. The following table shows three of the distributions. They are listed here primarily because the calculation is tractable and is better suited for demonstration of the concept.
Limited Expectation 


For formulas for the higher moments, refer to the table. This table will be used throughout the discussion (in subsequent blog posts) on estimating insurance payments. Here’s some blog posts on these three distributions – exponential, Pareto and lognormal.
The following example shows how these formulas are used.
Example 3
Evaluate for the following loss distributions.
 Exponential: mean 1000, = 2000.
 Pareto: mean 5, variance 75, = 10.
 Lognormal: mean 177.682811, variance 13680.72152, = 250.
The exponential distribution with mean 1000 has parameter (the scale parameter). The following gives the limited expected value.
The Pareto distribution with mean 5 and variance 75 translates to the parameters and . The following gives the limited expected value.

.
The lognormal distribution with mean 177.682811 and variance 13680.72152 corresponds to and . The following gives the limited expected value.
Comments
The discussion here is just the beginning. The limited expected value is for a simple policy provision, having just one modification of loss. It is a building block for other payments under more complicated insurance payment rules.
2017 – Dan Ma