8.1 Introduction

We have seen a variety of examples that involve functions of random variables. In this chapter, we relate those functions to insurance applications. Throughout this chapter, we assume that all random variables have support on all or a subset of the nonnegative real numbers. At times in this chapter and later in the text, we need to distinguish between a random variable that measures the payment per loss (so zero is a possibility, taking place when there is a loss without a payment) and a variable that measures the payment per payment (the random variable is not defined when there is no payment). For notation, a per-loss variable is denoted and a per-payment variable is denoted . When the distinction is not material (e.g. setting a maximum payment does not create a difference), the superscript is omitted.

8.2 Deductibles

Insurance policies are often sold with a per-loss deductible of d. When the loss, x, is at or below d, the insurance pays nothing. When the loss is above d, the insurance pays . In the language of Chapter 3, such a deductible can be defined as follows.

This concept has already been introduced along with formulas for determining its moments. The per-payment variable is

while the per-loss variable is

Note that the per-payment variable . That is, the per-payment variable is the per-loss variable conditioned on the loss being positive. For the excess loss/per-payment variable, the density function is

(8.1)

noting that for a discrete distribution, the density function need only be replaced by the probability function. Other key functions are

Note that as a per-payment variable, the excess loss variable places no probability at zero.

The left censored and shifted variable has discrete probability at zero of , representing the probability that a payment of zero is made because the loss did not exceed d. Above zero, the density function is

(8.2)

while the other key functions are¹ (for )

It is important to recognize that when counting claims on a per-payment basis, changing the deductible will change the frequency with which payments are made (while the frequency of losses will be unchanged). The nature of these changes is discussed in Section 8.6.

An alternative to the ordinary deductible is the franchise deductible. This deductible differs from the ordinary deductible in that, when the loss exceeds the deductible, the loss is paid in full. One example is in disability insurance where, for example, if a disability lasts seven or fewer days, no benefits are paid. However, if the disability lasts more than seven days, daily benefits are paid retroactively to the onset of the disability.

The terms left censored and shifted and excess loss are not used here. Because this modification is unique to insurance applications, we use per-payment and per-loss terminology. The per-loss variable is

while the per-payment variable is

Note that, as usual, the per-payment variable is a conditional random variable. The related functions are now

for the per-loss variable and

for the per-payment variable.

Expected costs for the two types of deductible may also be calculated.

8.2.1 Exercises

1. 8.1 Perform the calculations in Example 8.1 for the following distribution, using an ordinary deductible of 5,000:
   
2. 8.2 Repeat Exercise 8.1 for a franchise deductible.
3. 8.3 Repeat Example 8.3 for the model in Exercise 8.1 and a 5,000 deductible.
4. 8.4 (*) Risk 1 has a Pareto distribution with parameters and . Risk 2 has a Pareto distribution with parameters and . Each risk is covered by a separate policy, each with an ordinary deductible of k. Determine the expected cost per loss for risk 1. Determine the limit as k goes to infinity of the ratio of the expected cost per loss for risk 2 to the expected cost per loss for risk 1.
5. 8.5 (*) Losses (prior to any deductibles being applied) have a distribution as reflected in Table 8.1. There is a per-loss ordinary deductible of 10,000. The deductible is then raised so that half the number of losses exceed the new deductible as exceeded the old deductible. Determine the percentage change in the expected cost per payment when the deductible is raised.

x
10,000	0.60	6,000
15,000	0.70	7,700
22,500	0.80	9,500
32,500	0.90	11,000
	1.00	20,000

8.3 The Loss Elimination Ratio and the Effect of Inflation for Ordinary Deductibles

A ratio that can be meaningful in evaluating the effect of a deductible is the loss elimination ratio (LER).

While many types of coverage modifications can decrease the expected payment, the term loss elimination ratio is reserved for the effect of changing the deductible. Without the deductible, the expected payment is . With the deductible, the expected payment (from Theorem 8.3) is . Therefore, the loss elimination ratio is

provided that exists.

Inflation increases costs, but it turns out that when there is a deductible, the effect of inflation is magnified. First, some events that formerly produced losses below the deductible will now lead to payments. Second, the relative effect of inflation is magnified because the deductible is subtracted after inflation. For example, suppose that an event formerly produced a loss of 600. With a 500 deductible, the payment is 100. Inflation at 10% will increase the loss to 660 and the payment to 160, a 60% increase in the cost to the insurer.

8.3.1 Exercises

1. 8.6 Determine the loss elimination ratio for the distribution given here, with an ordinary deductible of 5,000. This is the same model as used in Exercise 8.1.
   
2. 8.7 Determine the effect of inflation at 10% on an ordinary deductible of 5,000 applied to the distribution in Exercise 8.6.
3. 8.8 (*) Losses have a lognormal distribution with and . There is a deductible of 2,000, and 10 losses are expected each year. Determine the loss elimination ratio. If there is uniform inflation of 20% but the deductible remains at 2,000, how many payments will be expected?
4. 8.9 (*) Losses have a Pareto distribution with and . There is an ordinary deductible of 2k. Determine the loss elimination ratio before and after 100% inflation.
5. 8.10 (*) Losses have an exponential distribution with a mean of 1,000. There is a deductible of 500. Determine the amount by which the deductible would have to be raised to double the loss elimination ratio.
6. 8.11 (*) The values in Table 8.2 are available for a random variable X. There is a deductible of 15,000 per loss and no policy limit. Determine the expected cost per payment using X and then assuming 50% inflation (with the deductible remaining at 15,000).
7. 8.12 (*) Losses have a lognormal distribution with and . Determine the ratio of the loss elimination ratio at 10,000 to the loss elimination ratio at 1,000. Then determine the percentage increase in the number of losses that exceed 1,000 if all losses are increased by 10%.
8. 8.13 (*) Losses have a mean of 2,000. With a deductible of 1,000, the loss elimination ratio is 0.3. The probability of a loss being greater than 1,000 is 0.4. Determine the average size of a loss, given that it is less than or equal to 1,000.

x
10,000	0.60	6,000
15,000	0.70	7,700
22,500	0.80	9,500
	1.00	20,000

8.4 Policy Limits

The opposite of a deductible is a policy limit. The typical policy limit arises in a contract where for losses below u the insurance pays the full loss but for losses above u the insurance pays only u. The effect of the limit is to produce a right censored random variable. It will have a mixed distribution with distribution and density function given by (where Y is the random variable after the limit has been imposed)

and

The effect of inflation can be calculated as follows.

For policy limits, the concepts of per payment and per loss are not relevant. All losses that produced payments prior to imposing the limit will produce payments after the limit is imposed.

A policy limit and an ordinary deductible go together in the sense that, whichever applies to the insurance company's payments, the other applies to the policyholder's payments. For example, when the policy has a deductible of 500, the cost per loss to the policyholder is a random variable that is right censored at 500. When the policy has a limit of 3,000, the policyholder's payments are a variable that is left truncated and shifted (as in an ordinary deductible). The opposite of the franchise deductible is a coverage that right truncates any losses (see Exercise 3.12). This coverage is rarely, if ever, sold. (Would you buy a policy that pays you nothing if your loss exceeds u?)

8.5 Coinsurance, Deductibles, and Limits

The final common coverage modification is coinsurance. In this case, the insurance company pays a proportion, , of the loss and the policyholder pays the remaining fraction. If coinsurance is the only modification, this changes the loss variable X to the payment variable, . The effect of multiplication has already been covered. When all four items covered in this chapter are present (ordinary deductible, limit, coinsurance, and inflation), we create the following per-loss random variable:

For this definition, the quantities are applied in a particular order. In particular, the coinsurance is applied last. For the illustrated contract, the policy limit is , the maximum amount payable. In this definition, u is the loss above which no additional benefits are paid and is called the maximum covered loss. For the per-payment variable, is undefined for .

Previous results can be combined to produce the following theorem, presented without proof.

Higher moments are more difficult. Theorem 8.8 gives the formula for the second moment. The variance can then be obtained by subtracting the square of the mean.

8.6 The Impact of Deductibles on Claim Frequency

An important component in analyzing the effect of policy modifications pertains to the change in the frequency distribution of payments when the deductible (ordinary or franchise) is imposed or changed. When a deductible is imposed or increased, there will be fewer payments per period, while if a deductible is lowered, there will be more payments.

We can quantify this process if it can be assumed that the imposition of coverage modifications does not affect the process that produces losses or the type of individual who will purchase insurance. For example, those who buy a 250 deductible on an automobile property damage coverage may (correctly) view themselves as less likely to be involved in an accident than those who buy full coverage. Similarly, an employer may find that the rate of permanent disability declines when reduced benefits are provided to employees in the first few years of employment.

To begin, suppose that , the severity, represents the ground-up loss on the jth such loss and there are no coverage modifications. Let denote the number of losses. Now consider a coverage modification such that v is the probability that a loss will result in a payment. For example, if there is a deductible of d, . Next, define the indicator random variable by if the jth loss results in a payment and otherwise. Then, has a Bernoulli distribution with parameter v and the pgf of is . Then, represents the number of payments. If are mutually independent and are also independent of , then has a compound distribution with as the primary distribution and a Bernoulli secondary distribution. Thus

In the important special case where the distribution of depends on a parameter such that

where is functionally independent of (as in Theorem 7.4), then

This result implies that and are both from the same parametric family and only the parameter need be changed.

This result may be generalized for zero-modified and zero-truncated distributions. Suppose that depends on parameters and such that

(8.3)

Note that and so is the modified probability at zero. It is also the case that, if is itself a pgf, then the pgf given in (8.3) is that for the corresponding zero-modified distribution. However, it is not necessary for to be a pgf in order for as given in (8.3) to be a pgf. In particular, yields the ZM logarithmic distribution, even though there is no distribution with as its pgf. Similarly, for yields the ETNB distribution. A few algebraic steps reveal (see Exercise 8.35) that for (8.3)

where . It is expected that the imposition of a deductible will increase the value of because periods with no payments will become more likely. In particular, if is zero truncated, will be zero modified.

In applications, it may be the case that we want to determine the distribution of from that of . For example, data may have been collected on the number of payments in the presence of a deductible and from that data the parameters of can be estimated. We may then want to know the distribution of payments if the deductible is removed. Arguing as before,

This result implies that the formulas derived previously hold with v replaced by . However, it is possible that the resulting pgf for is not valid. In this case, one of the modeling assumptions is invalid (e.g. the assumption that changing the deductible does not change claim-related behavior). It is instructive to note that the resulting pgf for is guaranteed to be a valid pgf [regardless of the value of ] if and only if the pgf for is of mixed Poisson form (for a proof of this fact, see Grandell [47, pp. 25–26]). This provides yet another strong motivation for the use of mixed Poisson models for insurance claim counts. The following two examples illustrate this point.

All members of the and classes meet the conditions of this section. Table 8.3 indicates how the parameters change when moving from to . If has a compound distribution, then we can write and therefore

Thus will also have a compound distribution with the secondary distribution modified as indicated. If the secondary distribution has an distribution, then it can modified as in Table 8.3. The following example indicates the adjustment to be made if the secondary distribution has an distribution.

	Parameters for
Poisson
ZM Poisson
Binomial
ZM binomial
ZM binomial
Negative binomial
ZM negative binomial
ZM binomial
ZM logarithmic
ZM binomial

The results can be further generalized to an increase or decrease in the deductible. Let be the frequency when the deductible is d and let be the frequency when the deductible is . Let , and then Table 8.3 can be used to move from the parameters of to the parameters of . As long as , we will have and the formulas will lead to a legitimate distribution for . This includes the special case of that was used at the start of this section. If , then and there is no assurance that a legitimate distribution will result. This includes the special case (removal of a deductible) covered earlier.

It should be noted that policy limits have no effect on the frequency distribution. Imposing, removing, or changing a limit will not change the number of payments made.

Finally, it is important to restate that all the results in this section depend on an assumption that may not hold. The assumption is that changing the deductible (or other coverage modifications) does not change the number of loss-producing incidents or the amount of losses.