An Actuarial Model of Excess of Policy Limits Losses

1. Introduction

Excess of policy limits (XPL) losses is a phenomenon that presents challenges for the practicing actuary. For example, exposure rating, one of the standard actuarial methods for pricing reinsurance layers, seems to be completely unworkable for the challenge of pricing XPL losses; yet often an exposure rating approach to reinsurance pricing is the only method that the practicing actuary has at his disposal.

In this paper, I propose an approach that incorporates XPL into the classic actuarial framework of frequency, severity, and limited expected value (LEV) of claims. In this way, XPL will simply be part of a broader landscape of claims behavior, and can draw upon and seamlessly integrate with standard actuarial tools for incorporating the price of XPL losses into the pricing of reinsurance contracts. In addition, using the classic actuarial framework allows one to incorporate XPL losses into stochastic economic capital models that are used for insurer enterprise risk management (ERM) purposes.

1.1. Research context

The actuarial literature has very limited discussion of actuarial approaches to modeling of excess of policy limits losses. I have found only one paper by Braithwaite and Ware (1997), which remains a crucially important paper.

1.2. Objective

In this paper, I propose a framework that builds upon the work of Braithwaite and Ware (1997) yet differs in some ways.

There are two main reasons for this difference in approach. The first reason relates to aligning resources with need. XPL is an important actuarial problem but by no means the paramount problem typically facing actuaries. As a result, I would like to propose a reasonable methodology that is more practicable than the one proposed in Braithwaite and Ware (1997). Whereas Braithwaite’s model required the actuary to build an additional, freestanding size-of-loss curve to describe XPL, this paper proposes a methodology that simply extends one’s existing size-of-loss curve, greatly simplifying the implementation.

The second reason that the proposed approach differs from Braithwaite is the need to quantify XPL losses in the context of a broader insurance portfolio; one ought to model and price for XPL in conjunction with other non-XPL losses. Braithwaite, discussing clash reinsurance treaties, focuses entirely on XPL losses. Yet the practitioner actuary often desires to price for XPL losses in working layer reinsurance; only a small percentage of losses will be XPL whereas the majority of losses will be non-XPL. The task, then, is to price these reinsurance layers for the XPL losses in a framework that aligns with traditional actuarial pricing methods. Similarly, another situation that requires modeling of XPL losses together with non-XPL losses is enterprise risk management (ERM), in which one seeks to model all the insurance risk of the company. Modeling requires an integrated framework that covers XPL and non-XPL losses together, which will be facilitated by the proposed new approach.

1.3. Background on XPL losses

What is an XPL loss, in what situation does it arise, and what are its salient traits? Foundationally, the underlying situation arises when an injured third party makes a liability claim against the insured policyholder and the final judgment against the insured exceeds the amount of insurance coverage (Reinarz et al. 1990). In this situation, the usual outcome would be that the insurance company indemnifies the policyholder only up to the amount of the policy limit but no more. However, sometimes the insurer must indemnify the policyholder beyond the policy limit up to the full amount of the judgment for the injured third party. Why should the insurer pay beyond the policy limit? Typically, the rationale is that during the claim handling and litigation of the liability claim, the insurer could have settled the claim for an amount less than or equal to the policy limit; by choosing an aggressive and riskier strategy not to settle the claim within the policy limit, the insurer exposed the policyholder to additional downside risk. As a result, the insurer ought to be responsible for this additional loss amount.

We notice that the salient traits of an XPL loss are:

1. The underlying claim arises out of an actual liability to an injured third party.

2. The underlying claim is not excluded but rather is covered by the contractual wording of the policy; the only restriction is that the indemnification amount is sub-limited.

These two traits bolster the idea that an XPL claim is fundamentally a claim that arises organically within the insurance policy contract; while the final loss amount exceeds the policy limit, the underlying loss is not foreign to the nature of the coverage.

Extra-contractual obligations (ECO) losses are often discussed in conjunction with XPL losses. ECO losses are claims that arise from actions that are not covered by the contractual wording of the policy and thus are viewed as outside the insurance policy. The salient aspects of an ECO claim are:

1. The underlying claimant is not an injured third party but rather the policyholder; thus the underlying policy need not be a liability policy but could also be a property policy.

2. The policyholder alleges against the insurance company an injury arising out of a wrongful act by the insurance company that issued the policy; usually this allegation relates to wrongful handling of the claim.

These traits indicate that ECO losses could be considered significantly different from other traditional insurance claims.

The key traits of XPL and ECO indicate that they are not exactly the same as run-of-the-mill insurance claims. Yet we should view different categories of claims on a spectrum: both XPL and ECO have similarities and dissimilarities to traditional insurance claims. Moreover, XPL claims could be viewed as more closely related than ECO to typical insurance claims, because XPL losses arise from underlying claims that are fundamentally covered by the insurance policy. For this reason, the discussion in this paper will relate most directly to XPL claims, whereas extending the results to ECO claims, mutatis mutandis, could be considered a further extrapolation subject to additional caveats.

2. Actuarial model of size of loss distribution with extension to XPL

We begin with the classic actuarial framework for evaluating loss costs in layers with a focus on limited expected value (LEV). Following Clark (1996), we can write that

1. X = random variable for size of loss

2. F_X(x) = probability that random variable X, the size of loss, is less than or equal to x

3. f_X(x) = probability density function, first derivative of F(x)

4. E[X] = expected value or average unlimited loss

5. E[X;k] = expected value of loss capped at k

The expected value of loss capped at an amount k can be defined as follows:

$$\begin{aligned} \operatorname{LEV}(X, k) & =E[X ; k] \\ & =\int_{0}^{k} x f(x) d x+\int_{k}^{\infty} k f(x) d x . \end{aligned} \tag{2.1}$$

$$\begin{aligned} L E V(X, k) & =E[X ; k] \\ & =\int_{0}^{k} x f(x) d x+k[1-F(k)] \end{aligned} \tag{2.2}$$

2.1. Limited expected value (LEV)

Historically, actuaries needed to quantify the value of the average loss limited by the insurance policy; they adopted limited expected value (LEV) as the framework to calculate this value, under the assumption that a policy limit caps the insurance loss.

2.2. Incorporating XPL losses

In light of our knowledge of XPL losses, we should revisit whether LEV is the ideal way to measure losses to an insurance policy. Let’s describe the average loss accruing to an insurance policy as the policy limited expected value (PLEV). Until now, the implicit assumption has been that PLEV = LEV.

The phenomenon of XPL losses shows us, however, that the policy limit written in the insurance policy contract is not always potent in capping losses. Thus the identity function, PLEV = LEV, is not fully accurate.

What could be a paradigm for how to think about the phenomenon of XPL losses? I propose that we begin to think of the effectiveness of the policy limit as being subject to a random variable.

Let’s define a random variable Z, which follows a Bernoulli distribution. This random variable can have a value of 1, or “success,” with probability p, and can have a value of 0, “failure,” with probability 1 − p. When Z = 1 we have “success” and the policy limit caps the insurance loss; when Z = 0 we have “failure” and the policy limit does not cap the insurance loss and we have an XPL situation.

Now we can say that the policy limited expected value is:

$$\begin{aligned} P L E V(X, k, Z)= & \int_{0}^{k} x f(x) d x \\ & +P(Z=1) * \int_{k}^{\infty} k f(x \mid Z=1) d x \\ & +P(Z=0) * \int_{k}^{\infty} x f(x \mid Z=0) d x \end{aligned} \tag{2.3}$$

Recalling that the probability that Z = 1 is p and that Z = 0 is 1 − p, we write:

$$\begin{aligned} \operatorname{PLEV}(X, k, Z)= & \int_{0}^{k} x f(x) d x+p \int_{k}^{\infty} k f(x \mid Z=1) d x \\ & +(1-p) \int_{k}^{\infty} x f(x \mid Z=0) d x \end{aligned} \tag{2.4}$$

If we let x = k + (x − k) in the final integral, we can rewrite equation (2.4) as follows:

$$\begin{aligned} P L E V(X, k, Z)= & \int_{0}^{k} x f(x) d x+k[1-F(k)] \\ & +(1-p) \int_{k}^{\infty}(x-k) f(x \mid Z=0) d x . \end{aligned} \tag{2.5}$$

One can say that on a fundamental level, equation (2.5) captures the approach crystallized in Braithwaite (1997). The additional loss above and beyond the policy limit follows a different conditional probability density function than the initial size of loss distribution; as a result, the XPL loss component is a completely new entity that is grafted onto the non-XPL loss component.

3. A more practical model

How can we make this model more practical and easier to use? Let’s revisit equation (2.4) and make some simplifying assumptions.

Let’s assume that the probability density function above the policy limit is not conditional on whether or not an XPL scenario has been triggered. As explained in Braithwaite (1997), the XPL situation arises when the policyholder is found liable for actual damage to a third party; the only question is whether or not the insurance company’s conduct provides a basis for the courts to override the capping effect of the policy limit. Thus, this simplifying assumption should be reasonable for XPL (although perhaps not for extra-contractual obligations, ECO).

We can then substitute the unconditional f(x) into equation (2.4) by replacing the conditional f(x|Z = 0) and f(x|Z = 1) and rewrite equation (2.4) as follows:

$$\begin{aligned} \operatorname{PLEV}(X, k, Z)= & \int_{0}^{k} x f(x) d x+p \int_{k}^{\infty} k f(x) d x \\ & +(1-p) \int_{k}^{\infty} x f(x) d x \end{aligned} \tag{3.1}$$

Thus we simply say that if random variable Z = 1 we have a success and the policy limit caps the loss, and if Z = 0 we have a failure and the policy limit does not cap the loss. Unlike equation (2.5) and unlike the approach of Braithwaite (1997), the XPL loss is not a completely new entity; rather, the XPL loss is simply an extension of the standard size-of-loss distribution that occurs when the policy limit’s capping effect is ineffective. Such a framework would be much easier to work with when attempting to incorporate XPL losses.

3.1. Practical applications: Insurance risk modeling

How can we apply the proposed paradigm of equation (3.1) in a practical way to achieve a tangible result? One possibility would be in a simulation environment.

3.1.1. Simulation application #1: Collective risk model for insurance losses

Step #1: Define the size of loss distribution for an insurance policy or portfolio of policies on a gross of policy limit basis.

Step #2: Simulate individual losses and simulate the limit of the policy associated with each loss.

Step #3: For each loss, if the loss is greater than the policy limit, then simulate Z, a Bernoulli random variable. If Z = 1, then cap the simulated loss at the policy limit. If Z = 0, then do not cap the loss.

Notice that there is only one small new step here: rather than always capping the loss at the policy limit, let the capping be subject to the outcome of a random variable that reflects whether the policy limit will be effective at capping the loss or not.

3.1.2. Simulation application #2: Cat modeling

The software vendors for cat modeling typically employ several steps in their calculations of the losses to an insurance portfolio for a given simulated cat event. After the software simulates a catastrophic (“cat”) event, the software evaluates how the physical phenomenon affects the physical structures in its path. Then, in one of the final steps, the software overlays the insurance policy’s contractual terms to achieve the financial loss to the company. Within this simulation environment, the final step could evolve away from the current deterministic view of the policy limit and towards a stochastic view of the policy limit. Moreover, one could consider correlating the individual probabilities that the policy limits fail; the correlation could depend upon geographical location and legal jurisdiction, among other factors. An approach to cat modeling simulations that treats policy limit capping of losses as a probable but not definite outcome would be more realistic and would show more severe risk metric output than current models.

3.2. Reinsurance pricing

A second practical application of the proposed paradigm of equation (3.1) could be reinsurance pricing.

Recall that traditional exposure rating is viewed as not producing loss cost indications that encompass XPL. After all, XPL losses by definition exceed the policy limit and thus exceed the exposure; how could exposure rating possibly incorporate XPL within its framework?

Let’s revisit equation (3.1):

$$\begin{aligned} \operatorname{PLEV}(X, k, Z)= & \int_{0}^{k} x f(x) d x+p \int_{k}^{\infty} k f(x) d x \\ & +(1-p) \int_{k}^{\infty} x f(x) d x \end{aligned}\tag{3.1}$$

If we multiply the first term on the right side of equation (3.1) by 1 and let 1 = p + 1 − p and rearrange terms, we can rewrite equation (3.1) as follows:

$$\begin{aligned} \operatorname{PLEV}(X, k, Z)= & p\left[\int_{0}^{k} x f(x) d x+\int_{k}^{\infty} k f(x) d x\right] \\ & +(1-p)\left[\int_{0}^{k} x f(x) d x+\int_{k}^{\infty} x f(x) d x\right] \end{aligned} \tag{3.2}$$

This is also the same as the following:

$$\begin{aligned} \operatorname{PLEV}(X, k, Z)= & p\left[\int_{0}^{k} x f(x) d x+\int_{k}^{\infty} k f(x) d x\right] \\ & +(1-p)\left[\int_{0}^{\infty} x f(x) d x\right] \end{aligned} \tag{3.3}$$

And:

$$P L E V(X, k, Z)=p(L E V(X, k))+(1-p) E[X] \tag{3.4}$$

Equations (3.3) and (3.4) demonstrate that in the presence of XPL losses we have a loss severity that has probability p of being limited by the policy limit and probability (1 − p) of not being limited by the policy limit.

We can use this framework to calculate expected layer loss for excess-of-loss reinsurance exposure rating.

Following Clark (1996), for each policy we want to calculate the exposure factor, i.e., the percentage of the policy’s total loss that is covered by the reinsurance layer.

$$\text { Exposure Factor }=\frac{\text { layer loss }}{\text { total loss }} \tag{3.5}$$

Now let’s calculate the layer loss.

$$\begin{aligned} \text{Layer loss} = & \text{Loss limited at the top of the}\\ & \text{reinsurance layer} - \text{loss limited at the}\\ & \text{bottom of the reinsurance layer.} \end{aligned}\tag{3.6}$$

Here, we have a probability p that the policy limit will cap the loss and a 1 − p probability that the policy limit will not cap the loss. While these probabilities apply to the primary policy, we assume that they do not apply at all to the reinsurance limit and attachment point.

Thus, when estimating the loss limited by the top of the reinsurance layer, we have a probability p that the loss will be capped by the lesser of the policy limit and the top of the reinsurance layer; we also have a probability 1 − p that the loss will be capped solely by the top of reinsurance layer, with no application of the policy limit.

$$\text{Loss limited at top of reinsurance layer}\\ \begin{array}{c} =\mathrm{p} * \operatorname{LEV}\left(\mathrm{X}, \min \left(\begin{array}{c} \text { policy limit, } \\ \text { reinsurance exit point } \end{array}\right)\right) \\ +(1-\mathrm{p}) * \operatorname{LEV}(\mathrm{X}, \text { reinsurance exit point }) . \end{array} \tag{3.7}$$

Note: Reinsurance exit point = reinsurance attachment point + reinsurance limit.

Similarly, when estimating the loss limited by the bottom of the reinsurance layer, we have a probability p that the loss will be capped by the lesser of the policy limit and the bottom of the reinsurance layer; we also have a probability 1 − p that the loss will be capped solely by the bottom of reinsurance layer.

$$\text{Loss limited at bottom of reinsurance layer}\\ \begin{aligned} = & \mathrm{p} * \operatorname{LEV}\left(\mathrm{X}, \min \left(\begin{array}{c} \text { policy limit, reinsurance } \\ \text { attachment point } \end{array}\right)\right) \\ & +(1-\mathrm{p}) * \operatorname{LEV}\left(\begin{array}{c} \mathrm{X}, \text { reinsurance } \\ \text { attachment point } \end{array}\right) . \end{aligned} \tag{3.8}$$

Thus:

$$\text{Layer loss}\\ \begin{array}{l} =\mathrm{p} * \operatorname{LEV}\left(\mathrm{X}, \min \left(\begin{array}{c} \text { policy limit, } \\ \text { reinsurance exit point } \end{array}\right)\right) \\ +(1-\mathrm{p}) * \operatorname{LEV}(\mathrm{X}, \text { reinsurance exit point }) \\ -\left\{\mathrm{p} * \operatorname{LEV}\left(\mathrm{X}, \min \left(\begin{array}{c} \text { policy limit, } \\ \text { reinsurance } \\ \text { attachment point } \end{array}\right)\right)\right\} . \\ \end{array} \tag{3.9}$$

Thus:

$$\begin{aligned} \text{Layer loss} = & \mathrm{p} * \text{traditional exposure rating layer}\\ & \text{LEV subject to primary policy limit}\\ & + (1-\mathrm{p}) * \text{layer LEV not subject}\\ & \text{to primary policy limit.} \end{aligned}\tag{3.10}$$

Having calculated the layer loss, which is the numerator of the exposure factor, we now need to calculate the denominator, the policy’s total loss.

Recall that the exposure factor produces layer loss by multiplying the policy’s total loss; total loss is usually calibrated based on policy premium multiplied by an expected loss ratio (ELR). Therefore, whether or not the ELR was calculated to include a provision for XPL losses will affect how one ought to calculate the denominator of the exposure factor.

For our discussion, let’s proceed under the assumption that the ELR does not include a provision for XPL loss. As a result, when calculating the “total loss” for the denominator of the exposure factor, we will calculate it based only on non-XPL losses.

$$\begin{aligned} & \text{Denominator of Exposure Factor}\\ & \quad = \text{Same as traditional exposure rating}\\ & \quad = \text{Policy total loss excluding XPL}\\ & \quad = \mathrm{LEV}(\mathrm{X}, \text{policy limit}) \end{aligned}\tag{3.11}$$

Then, combining equations (3.9) and (3.11), we derive:

$$\begin{aligned} & \text{Exposure Factor}\\ & \quad = {[\mathrm{p} * \mathrm{LEV}(X, \min (\text { policy limit, reinsurance }} \\ & \qquad \text { exit point }))+(1-\mathrm{p}) * \operatorname{LEV}(\mathrm{X}, \text { reinsurance } \\ & \qquad \text { exit point })-\{\mathrm{p} * \mathrm{LEV}(\mathrm{X}, \min (\text { policy limit, } \\ & \qquad \text { reinsurance attachment point })) \\ & \qquad +(1-\mathrm{p}) * \mathrm{LEV}(\mathrm{X}, \text { reinsurance attachment } \\ & \qquad \text { point })\}] / \mathrm{LEV}(\mathrm{X}, \text { policy limit }) . \end{aligned} \tag{3.12}$$

Or, more simply, combining equations (3.10) and (3.11), we derive:

$$\begin{aligned} & \text{Exposure Factor}\\ & \quad =[\mathrm{p} * \text{traditional exposure rating layer LEV}\\ & \qquad \text{subject to primary policy limit}\\ & \qquad +(1-\mathrm{p}) * \text{layer LEV not subject to primary}\\ & \qquad \text{policy limit]/traditional exposure rating}\\ & \qquad \text{ground up LEV capped at policy limit.} \end{aligned}\tag{3.13}$$

3.2.1. Reinsurance pricing: Numerical example

Now let’s do a numerical example of the proposed algorithm. The goal is to generate layer loss costs via exposure rating that include a loss provision for XPL losses.

First, let’s stipulate some hypothetical numerical values for our policy limits distribution:

We also need values for our size-of-loss severity curve:

Finally, we need to input parameter values for probability p that a policy limit will successfully cap losses and 1 − p that the policy limit will not cap losses; the values may vary for each policy. Here we select a simple parameter structure in which all the policies in our limits table have the same value for p.

We now apply the proposed methodology to the numerical values to produce the following output in Exhibit 4.

Column 6 of Exhibit 4 shows the “loading factor” for each layer loss attributable to XPL. What is notable about this output is that choosing one simple value for p creates layer loading factors for XPL that are different for the various layers. Also, these loading factors for XPL would be different for other portfolios with different policy limits distributions, even with no change in the underlying value of the p parameters.^[1]

3.2.2. Discussion of how to estimate the p parameter

The exposure rating discussed above depends upon selecting a value for the parameters p and (1 p), where p represents the probability that the contractual policy limit caps the insurance loss and 1 p represents the probability that the contractual policy limit will not cap the insurance loss. One way to estimate the value of these parameters would be to look at claims data; commensurate with an exposure rating exercise, these data should be broadly based and tailored to the level of risk. Thus, one might look at claims on a statewide basis by line of business by capping threshold. For example, to determine the probability 1 p that a policy limit of $1M would not cap a loss, one would follow the following steps:

1. Collect all claims in state a, line of business b, that accrued to policies with limit c.

2. Filter out claims that settle for less than the policy limit and select only those claims that are at least equal to (or greater than) the policy limit.

3. Of the claims selected in step #2, select XPL claims by identifying claims that exceed the policy limit.

4. Divide claims selected in step #3 by claims selected in step #2; the result is an empirical estimate of the probability (1 − p) of the policy limit “failing” to cap the loss amount.

Note that this procedure can be done at any level of granularity. It can be done on industry data or company-specific data, for a specific state, and for a specific policy limit amount; or it could be done on a more general plane.

3.2.3. Discussion of how to deal with XPL claims in the experience data

So far we have discussed how to perform reinsurance exposure rating inclusive of XPL loss cost in order to deal with a situation in which no XPL claims data is available. How should one modify this approach when some XPL claims do in fact appear in the historical claims data?

If the historical claims data contains XPL losses, then this experience data should provide some credible information; yet it’s unlikely that the historical XPL claims data would have 100% credibility.

Fundamentally, then, the reinsurance actuary is faced with a situation very similar to traditional excess-of-loss analysis: how to generate both experience rating and exposure rating analyses and then how to blend them together.

In our situation, the reinsurance actuary confronts the open question: how much loss cost does XPL generate, both on a ground up basis but especially in the various excess layers? Using historical claims data of the ceding company to create experience rating analysis for XPL loss cost is fairly straightforward, given that the reinsurance actuary has historical XPL claims in the data set. Ideally, though, one would evaluate this experience rating analysis of XPL loss cost relative to an exposure rating analysis of XPL loss cost.

Yet until now, no published methodologies provided a practicable approach for generating an exposure rating approach to calculating XPL layer loss costs. With the new approach proposed in this paper, the reinsurance actuary can generate an exposure rating indication for XPL loss costs by layer, compare to experience rating loss costs by layer, analyze the dynamics that drive the similarities and differences between exposure and experience as manifest in various different layers, and finally use credibility to blend them together into final loss cost selections by layer.

4. Conclusions

In this paper, I propose an actuarial paradigm for describing excess of policy limits (XPL) losses. The central idea is that one can envision a random variable governing the application of the policy limit; most of the time the policy limit is enforced as it is written in the insurance contract, whereas other times the policy limit is superseded. This paradigm is quite parsimonious; therein lies its attractiveness. At the same time, this simple framework can generate nuanced, differentiated, useful, and non-obvious output information for practicing actuaries. One practical application would be to incorporate XPL losses into actuarial exposure rating estimates for casualty excess-of-loss reinsurance layers; the output values vary based on the attachment point and limit of the reinsurance layer being priced as well as the granular policy limits usage of the particular insurance portfolio under review. A second practical application would be to incorporate XPL losses in a simulation environment such as commercial software for estimating losses arising from natural catastrophes; envisioning policy limits as random variables can affect the cat modeling and thus the critical risk metrics of an insurer’s portfolio.