Risk Transfer Criteria that are not Ad Hoc: A Decrease in the Coefficient of Variation and Cost-Effective Pricing

1. INTRODUCTION

Reinsurance is, for the most part,^[1] a valuable tool that helps primary insurers cover risks that otherwise would entail too much potential loss for their balance sheets to support. However, there have been some abuses. For example, a reinsurer ostensibly covering all the payments in the runoff of a company’s loss reserves might actually be selling ‘time and distance.’ In effect, the reinsurer would agree to pay back the expected payout of the reserve in each calendar year, with interest discounting. Of course, if the payouts are higher than expected, the primary insurer must still pay them, but cannot seek coverage for the additional payments from the reinsurer. So the transaction involves the appearance of eliminating the loss reserve from the company’s books when the company still has all the risk. Another issue involves, say, a situation where commissions are limited but the company desires to provide additional incentives to its production sources. The company may allow the producers to form small reinsurers^[2] and purchase reinsurance from them on terms are overly profitable.

One of the results of these situations is that an accounting requirement that each treaty ‘transfer risk’ arose. However, as the saying goes, “The devil is in the details.” Specifically, the industry now had to figure out how to determine whether each treaty had ‘risk transfer.’ A number of formulas to assess this have been proposed , and some of them have been widely used. Currently, the most commonly used criteria appear to be an ad hoc ‘10/10 rule’ and the expected reinsurer deficit (“ERD”).

The first rule requires that there must be at least a 10% probability that the reinsurer will encounter a pre-expense loss of at least 10% of the ceded premium. For reference, just the cash flows between the primary insurer and reinsurer are used. The possible present values of the net cash outflows from the reinsurer to the cedant are subtracted from the corresponding present values of the cash the cedant pays the reinsurer in each scenario. This is done across a probability-based range of possible loss scenarios. Note also that the reinsurer’s expenses and profit are excluded from the calculation.

That rule is clearly ad hoc. For example, why not use 5/5, 20/20, or even 10/15? As such, it could potentially be ‘gamed.’ Further, any thoroughly proper excess-of-loss treaty that only has a 5% chance of suffering a loss would be classed, per the rule, as lacking risk transfer. This paper suggests a more natural, robust method for assessing whether a treaty properly transfers risk. It more directly requires that the treaty results in a lower coefficient of variation (or other risk measure) of the possible net losses compared to that of the possible losses when the treaty is excluded.

The second approach involves computing the expected value of the possible pre-expense losses to the reinsurer (frequency of a reinsurer pre-expense loss times the average severity of one when it occurs), then dividing that by the present value of the total reinsurance premium. Although to this point the calculation has been carefully and objectively proscribed, now one must decide what constitutes an acceptable value of the ERD. A commonly used value is 1%, which, considering that this approach accommodates a broader range of results (say, perhaps a 9% chance of an 11% loss) than the 10/10 rule, is a slightly weaker standard than the 10/10 rule. Thus, in the end this approach is also ad hoc and generates concerns similar to those associated with the 10/10 rule.

Those methods are designed to deal with whether treaties generate some risk reduction. That leaves the second concern mentioned earlier, whether the treaty is an appropriate use of the company’s funds. To resolve that concern, this paper posits reviewing the interest that would be required if the insurer were to forego the treaty and instead simply borrow additional surplus. Specifically, if the company could borrow enough surplus to make the company just as certain of paying all its claims as it would be if the treaty were in place, then one would compare the interest cost on that additional borrowed surplus to the net cost of the reinsurance it would replace. In other words, the cost of the additional interest would be compared to the reinsurer’s apparent post-discount expense and profit load. If the interest cost would be more than the “net cost” (the reinsurer’s apparent expense and profit) of the reinsurance, then the treaty could logically be considered a prudent purchase.

Both these criteria are “natural” in that they represent basic expressions of “risk reduction” and “cost-effectiveness.” Therefore, they avoid the arbitrary “10%” in the “10/10 rule” and “1%” in ERD".

2. THE DISTRIBUTIONS USED IN THE EXAMPLES

To illustrate how the new and existing approaches compare, some sample distributions will be used, one for specific excess-of-loss, one for various coverages on the aggregate loss:

• Distribution 1: A Pareto distribution with α=2.5 and a truncation point of $100,000 shifted all the way to zero. The shape of the curve is as follows:

Figure 1.pdf of Distribution 1 (Individual Claims)

The mean of this distribution is $66,667. The variance is 2.222E+10.

• Distribution 2: This is an aggregate loss distribution. Continuing the theme of distribution 1, each individual claim would be a sample from a Pareto distribution. In this case the distribution of individual claim severities has α=1.5 and a truncation point of $100,000 , shifted back to unity (1.00). The number of claims is Poisson distributed per the mean parameter λ =500. This roughly matches the distribution one might encounter at a midsized medical malpractice insurer. The corresponding annual earned premium selected is $115 million, which one may confirm suggests a loss ratio of 65.2%.

  • The approximate^[3] mean of this distribution is 98,855,192. The approximate variance is 5.778E+14.

  • The values of this curve could not readily be obtained analytically, so a curve was fit to 30,000 samples^[4] then the process in Boor 2022 was used to fit a smooth curve to the data. All values from the distribution that are included in this article were computed from the fitted curve in that article.

Figure 2.pdf of Distribution 2 (Aggregate Losses)

3. BASIC EXAMPLES

First, it is worth assessing how well this approach works under some fairly common situations where reinsurance is used. Examples involving specific excess-of-loss and aggregate excess-of-loss are provided.

3.1. One High Specific Excess Loss from Distribution 1

This example relates to the simplest possible scenario, one loss of unknown size with a high excess (say $100,000) attachment. In this case an excess loss given an excess loss still has α=2.5 but now has a truncation point of $200,000 shifted to $100,000. So, the mean of the excess loss (given a loss) is $233,333 using the standard Pareto formula for the mean of a shifted distribution

$$\left( \frac{\alpha T_{\min}}{\alpha - 1} - s \right).$$

Also, the probability of a loss reaching $100,000 is 17.6%. Then, the mean of the excess loss across all scenarios is ((1-17.6%)×$0 + 17.6%×233,333) = $41,066. Therefore, the mean of the remaining, or “retained” layer is $66,667-41066 = $25,601. Per simulation (for convenience), the variance is approximately 666 million. So, the coefficient of variation (CV) of the retained loss is approximately 101%. Since the CV of the full loss distribution is 223%, this clearly satisfies the CV requirement for risk transfer.

For reference, since the mean of the ceded loss computed earlier is approximately $41,066 its variance may be computed using the $E\left\lbrack X^{2} \right\rbrack - {E\lbrack X\rbrack}^{2\ }$ rule and the formula for the variance, shifted or not, of $\left( \frac{T_{\min}^{2}\alpha}{(\alpha - 1)^{2}(\alpha - 2)} \right)$.

Then $E\left[X^2\right]$ for the ceded loss =(1-17.6%)×0+17.6%×[8.888E+10+$233,333²] or roughly 1.433E+11. Using the mean $E\lbrack X\rbrack$ of the excess distribution of $41,066, the variance of the excess distribution is 1.433E+11-$41,066², or roughly 1.416E+11, so its CV is roughly 916%. In this case the 10/10 rule should be satisfied as long as the expense loading is not overly high. Considering the shape of the distribution, when no expense loading is included on loss, the 10th percentile payout in the excess layer is slightly over $150,000, somewhat beyond 110% of the mean of $41,066. Under most expense scenarios it is likely to pass the 10/10 test. Similarly, at a zero expense load the ERD is 11.1%×$166,667/$41,066, or approximately 45%, so it would very likely be accepted under the ERD criterion as well, even when expenses are included.

3.2. A Poisson Distributed Number of High Specific Excess Losses from Distribution 1

In this example, the number of claims is Poisson distributed per some value λ, but the attachment point, etc., from Section 3.1 is unchanged. So, there are a number of losses of unknown sizes against a say $100,000 attachment. As before the excess loss has α=2.5 and a truncation point of $200,000 shifted to $100,000. The collective risk equation for the variance given a Poisson distribution is straightforward. When the severity is represented by the variable “$S$,” the variance of the entire portfolio of claims is simply $\lambda E\lbrack S^{2}\rbrack$, and the mean is $\lambda E\lbrack S\rbrack$, so the CV is

$$\frac{\sqrt{E\lbrack S^{2}\rbrack}}{(\sqrt{\lambda}E\lbrack S\rbrack)}$$

and since the expected $S^{2}$ is simply the variance plus the square of the mean, everything needed to compute the CV, no matter how large or small $\lambda$ is available in the previous Subsection 1 and the definition of Distribution 1.

Say, for example, that $\lambda$ = 200. Then, across the whole, gross of reinsurance, distribution the mean is $66,667 and the variance is 2.222E+10. Then $E\left\lbrack S^{2} \right\rbrack$ is 2.222E+10+(66,667²) = 2.666E+10. So, the CV of the gross loss is $\sqrt{2.666 E+10} /(\sqrt{200} \times \$66,667)=17.3 \%$.

Similar calculations for the retained layer lead, in this case, to a CV of 0.5%. Therefore, in this situation the treaty would fulfill the CV criteria.

In this case the calculation of the 10/10 and ERD criteria are significantly hampered by the complexity of the aggregate distribution. While a simplified calculation could be made by fitting a lognormal distribution or something similar using the mean and variance, adhering to the original Compound Poisson distribution makes the calculations more complicated.

3.3. Ceding the Underlying Portion of Distribution 1 Instead of the Excess

A key issue to consider is what happens when a company cedes the predicable losses, for whatever reason, and elects to retain the riskier losses. Different companies have different levels of capital to support risky ventures such as this, but it would represent a “red flag” in many circumstances. So, it is worth analyzing what would happen in Subsection 3.1 if the losses up to $100,000 were reinsured, and the excess retained.

The first, and most obvious conclusion is that if the sole criterion for risk transfer is a lowering of the CV between the gross and retained losses, such an arrangement would not pass the risk transfer requirement. As specially noted in Subsection 3.1 the CV of the gross losses is 223%. As noted further on in that Subsection, the CV of the here-retained excess layer is roughly 916%, so the criterion would not be fulfilled.

However, there is something else to consider. What if the distribution of the excess layer dropped off quickly, or it had a low excess limit? Then, its CV might be lower than the CV of the gross loss distribution, but still higher than the CV of the underlying layer that is reinsured in this scenario. So, then the CV of the losses retained after reinsurance would be higher than the CV of the losses that were ceded. In other words, the more predictable part would be ceded, and the risky part would remain. Thus, there would seem to be a potential red flag^[5] from a CV standpoint. If this CV approach is to be used, the profession may wish to consider adding a criterion requiring that a progression through the various treaties perhaps should be done, adding treaties to the program one-by-one. One could choose to require that as each treaty is added to the mix, it consistently reduces the CV and consistently involves a higher ceded CV than retained-so-far CV as each consecutive treaty is analyzed.

3.4. Aggregate Excess with a Cap

Those examples deal with specific excess, but it is helpful to consider aggregate excess coverage. Thus, given that the aggregate losses are distributed according to Distribution 2, and that the corresponding premium is $200 million. Then, for example, one could imagine a treaty for 25% excess of a 60% loss ratio. That translates to losses between $120 million and $170 million.

The approximate mean and variance of the whole fitted Distribution 2 are already known to be 98,855,192 and 5.778E+14. So, the CV is approximately 24%. To compute the mean and variance of the retained losses, one must consider the losses below the retention, the lower portion of the losses above the retention, and the excess portion above the limit. Due to the nature of this fitted distribution, the calculations must be done numerically using the fitted data. For the retained losses, they indicate a mean of roughly $96 million, an approximate variance of 3.045E+14, and a CV of 18%. Thus, this treaty, with a fairly low yet meaningful limit, does reduce the CV.

To describe the situation completely, the ceded losses have an indicated mean of roughly $22 million, an approximate variance of 2.839E+14, and a CV of 77%. Thus, the ceded losses are “riskier” than the retained losses.

4. IS IT PRUDENT TO PURCHASE THE TREATY?

Even when the retained losses are less risky, there still may be concerns with the reinsurance. Since reinsurance transfers funds from one entity to another, regulators may be concerned that the treaty may be so costly that, CV reduction or not, it is still unfair to the cedent. So, it makes sense to require that the treaty be a prudent purchase for the cedant.

Then, some method to evaluate the appropriateness of the cost must be established. To do so, it seems logical to begin with the maxim for determining the attachment point of a reinsurance, that the net cost (average net present value of cash flows to the reinsurer minus the average net present value of payments to the cedant) of the reinsurance should be less than the cost of the maintaining enough additional capital that the reinsurance is not needed. In other words, the less expensive of the two approaches to providing solvency should logically be used.

Actuaries are accustomed to evaluating the net cost of reinsurance. It is currently an aspect of risk transfer evaluations under both the 10/10 rule and the ERD. However, a methodology for estimating the cost of additional capital may be helpful. One approach to the computations runs as follows:

• Define the solvency criterion. This might be a 99% probability that the company’s funds will be adequate to pay the claims (99% VaR threshold), a 95%TVaR, etc. within the company’s present reinsurance structure, including the treaty being analyzed. This would be the internal type of standard (VaR, TVaR, etc.) applied to the amount of funds in the company at present or expected to be there during the term of the reinsurance matched to the projected loss distribution of the company during the same term.

• Estimate the distribution of all possible net losses if the contract in question is removed from the reinsurance program.

• Apply the criterion from the first bullet to that revised distribution. For example, if the company funds could pay all claims 99% when the treaty is included, what is the 99th percentile of that revised distribution.

• Subtracting the company’s actual funds from the last bullet gives the amount of additional surplus or capital that is needed to replace the treaty.

• An interest rate for the additional surplus must be determined. In most situations, the going rate for a new surplus note would seem to be proper. However, if a company’s credit rating has dipped below normal, then perhaps documentation of the credit rating plus an adjusted interest rate (also with documentation) would likley be appropriate. In the similar situation where the greater leverage associated with funding the additional surplus would cause a reduction in the company’s credit rating, perhaps a similar process could be followed; and last,

• The product of the additional surplus or capital times that interest rate yields the cost of the additional surplus.

That provides a methodology. Note that all of the calculations are presently in the actuary’s toolkit. The funding percentile is in common use, and the determination of the interest rate uses familiar concepts. Determining the aggregate loss distribution is more challenging, but some actuaries do that type of calculation regularly when simulating the net cost of reinsurance treaties and in estimating survival probabilities for self-insurance funds^[6] or captives. So, although this would be a new step, the underlying components are already available.

5. EXAMPLE: TESTING THE PRUDENCE OF THE AGGREGATE EXCESS TREATY

To illustrate how the process works, one may apply it to the example in Subsection 3.4. In this case, the fitted distribution is discrete, and both the gross and net distributions increase uniformly. Further, no interest rate considerations requiring net present value are required. The 99th percentile occurs where the gross loss is $230 million. The net loss at that value is $180 million. Assuming that the reinsurer charges $30 million for its expected loss of $22 million, the net premium is $170 million. The cedant will also have expenses of say, $80 million, as well, leaving $90 million to pay claims. Thus, the current surplus at the 99th percentile would be $90 million.

The indicated direct surplus is $230 million less the direct premium (less expenses) of $120 million, or $110 million. Assuming a current surplus note rate of 6%, the cost of maintaining the additional capital is 6% times $20 million, or $1.2 million. The net cost of the reinsurance is (in this case, assuming no discount) $30 million-$22 million = $8 million. So, instead of purchasing this treaty, it appears to be more prudent to simply add more capital.

That process benefited from the discrete distribution that defines it. So, it is worthwhile to look at things another way. For example, if we treat the direct values as raw data, a lognormal distribution might fit them well. Matching the mean and variance of the data would provide estimated lognormal parameters. We know that the CV of roughly 24% = $\sqrt{{\text{exp}(\sigma}^{2}) - 1}$, so $\sigma^{2}$=.056. Then, µ must be approximately ln(99,000,000)-.056/2, or 18.38. There are many complex ways of assessing the value-at-risk (VaR) level, but all that is really needed is the 99th percentile of the direct loss distribution, Given µ and $\sigma^{2}$, standard spreadsheet software easily computes a lognormal inverse of .99 at roughly $167 million. This does notch the “actual” 99th percentile of $230 million, but if all one had was a limited amount of claims data, that might be the best estimate that is obtainable. Thus, the direct 99th percentile is $167 million, and the net 99th percentile is $120 million. On a direct basis the surplus need is $167 million -$120 million = $47 million. On a net basis the surplus need is $120 million - $90 million = $30 million. Hence the annual cost of the additional surplus is 6% × ($47 million-$30 million) = $1.020 million. The net cost of reinsurance requires knowing the value of the losses in the ceded layer (or at least the losses that the cedant expects when they purchase the layer — after all, it is their decision that should be prudent^[7]). The easiest way to compute this is by random simulation,^[8] resulting in claim costs ceded of approximately $3.072 million. Thus, the net cost of reinsurance is approximately $22 million - $3 million = $19 million under these distribution assumptions. In conclusion, this loose approximation suggests more of a gap from the cost of funding the direct layer to the net cost of the potential reinsurance. However, excepting the net cost of the reinsurance is much higher, this mismatched distribution yields results that are not totally dissimilar to those of the correct distribution. So, this shows that the calculations need not be inordinately difficult.

6. SPECIAL SITUATIONS

There are a few special situations that likely merit special handling that are discussed in the following Subsections.

6.1. Programs Requiring Special Expertise that are 100% Reinsured

There are some lines that require a significant specialized technical infrastructure in order for the line to be successful. For example, a company selling boiler and machinery insurance without boiler inspectors would most likely be unsuccessful. As this is written, cyber insurance requires more resources than some companies can deploy. Also, umbrella insurance requires special expertise in large claims, while some companies see very few large claims. In lines such as that, an insurer may form a relationship with special expertise in the line in question. Often, the insurer charges the premium the reinsurer recommends, and passes all the premium (net of a ceding commission) and much of the policy and claims processing to the reinsurer. The insurer’s business is enhanced by the additional coverage options offered to its policyholders, and the reinsurer can penetrate another market.

It should be obvious that reinsuring this will usually reduce the insurer’s CV. However, using the methodology of Sections 4 and 5, it may not seem to be a prudent purchase. On the other hand, being a prudent purchase generally means being less expensive than the alternatives and retaining these lines would require building the infrastructure needed for success in the line. Therefore, it seems to be logical that the cost of developing such an infrastructure should be added to the cost of the additional capital. There is one additional caveat. It seems logical that infrastructure should include activities of a technical nature. So, for example, special technical underwriting, inspection and claims adjusting activities would be included, but expense associated with the sale of the policies would not. Even with that enhancement, the valid special expertise treaties should pass the risk transfer criteria.

6.2. Quota Share and Other Treaties that are Designed to Reduce Total Risk and Enhance Surplus.

Anyone with experience with quota share treaties will quickly note that many of them will have difficulty reducing the CV of the retained losses. Many of them begin with, say, a 50% share of the business, and then add caps, loss corridors, etc. Consequently, the losses ceded are limited significantly. Thus, they may not pass the CV test.

However, many of these treaties are bought, nonetheless. On consideration, though, they are sold for a different purpose than excess of loss reinsurance. Excess of loss treaties are sold to make the retained business more predictable. In effect, it is sold to reduce the relative risk, or risk in relation to net earned premium or something similar. Quota share is generally bought because the cedant does not have enough surplus to cover the amount of business it writes. Thus, instead of reducing the relative risk, quota reduces the absolute aggregate risk of the company. So, for quota share and other treaty types (loss portfolio transfers, for example) that are solely designed to expand the amount of business a company may write, it makes sense to drop the CV requirement, but keep the prudence requirement.

6.3. New Insurance Programs

It is not unusual for an insurer to enter a new line of business, essentially copying the rates of a competitor, but then finding the reinsurance costs to be higher than those of the competitor. They may be so high that, under the competitor’s assumptions, the treaty would not pass muster under one or both parts of the approach in this article.

That last statement expresses the issue, but also explains why it happens. At present, the company has neither operational experience nor actual loss data of its own. So, there is more to the calculation of their CV than the competitor’s pricing assumptions. They would want to carefully review how their underwriting and claims practices may potentially turn out to differ from the competitors, and reflect that when they compute the variance and then the CV.

6.4. Insurers with a Large, Complex, Reinsurance Portfolio

Some insurers have a large number of programs, each of which has its own set of treaties, and often many treaties are shared between multiple reinsurers. Questions may arise regarding which treaty to test first and so forth. It makes sense to conclude, subject to basic two criteria of this article, that this approach could be administered the same way the 10/10 and ERD rules are presently administered.

7. COMPLIANCE WITH NAIC REQUIREMENTS

The use of risk transfer as a criterion for determining whether contracts qualify as insurance (and hence properly matching the future claim payouts with the premiums on the policies that generate them), began in earnest when the National Association of Insurance Commissioners (in the United States) required appropriate risk transfer as a criterion for treating given a policy as insurance. They implemented the criterion with the following approach:

• The reinsurer must assume significant insurance risk; and

• It must be reasonably possible that the reinsurer may suffer a significant loss from the transaction.

This is of course a verbal, and, as such, a slightly subjective threshold. However, a certain amount of logic will explain why this approach qualifies, even for quota share and similar treaties. For item 1, it is clear that the retained losses are less risky than the gross or direct losses. If one adds the criteria that the CV of the ceded losses must be more than that of the retained losses, it is then abundantly clear that the reinsurer is assuming significant insurance risk. However, that does not resolve the concern for quota share and other treaties that expand the insurer’s capacity to write business. In this case, the losses transferred are by nature losses associated with the expanded business the treaty allows the insurer to write, therefore the absolute risk must be significant.

For item 2, if the CV is reduced and the CV of the ceded losses is larger than that of the retained losses, then the reduction in variance would appear to require the prospect that the reinsurer suffer a significant loss (in relative terms). Further, when the quota share and other surplus expansion treaties are viewed in terms of the absolute rather than relative losses they can produce, and the alternative cost of capital is lower than that of the reinsurance, it appears that the reinsurer must be exposed to a significant loss of some sort.

Thus, a treaty compliant with all the requirements expressed in this article would qualify for risk transfer treatment on both a common sense and regulatory basis.

8. SUMMARY

As one may see, the combination of the CV requirement and the prudent purchase requirement creates what appears to be a very effective and reasonable approach to assessing risk transfer. Hopefully, actuaries will include this in their toolkits when performing those assessments.