Consider an XL treaty with unbounded layer and retention x, and N_R(t) denoting the number of claims for the reinsurer. The number of claims overshooting a retention x is at least equal to r if and only if the r largest order statistics overshoot the level x. Consequently

and

(which can already be found in Galambos [367]). Correspondingly, if N(t) comes from a Poisson process or Pascal process, so does N_R(t) and these formulas lead to simple expressions for the c.d.f. of order statistics in terms of the respective sums (see Franckx [358] and Benktander [112] for the Poisson case and Kupper [519] as well as Ciminelli [214] for the Pascal case).

Let us consider the general case with independent claim number and claim size processes, and use the notation

(10.1.2)

for the probability of having fewer than r claims up to time t. Then from (5.6.37) we can write

Let us denote q = 1 − p = F_X(x). Then by reversing the order of summation we easily find that

where the first term corresponds to the fact that not even r claims have turned up by time t. For the second term we use the well‐known equality

together with the definition (5.1.1) to obtain

(10.1.3)

where

If X is a continuous random variable with density f_X, the corresponding density part (apart from a jump of size Π_r(t) at the origin) is

We now look into the one‐dimensional limit distribution of the rth largest order statistics (where r is kept fixed). Assume that there exist deterministic functions d(t) > 0 and c(t) such that when

(10.1.4)

where V_r is a non‐degenerate random variable. It is natural to assume here that . In terms of the tail quantile function this means that for an extreme value index and some auxiliary function a(x) > 0,

cf. Definition 3.1. Take fixed. From (10.1.3),

where

When both the integrand and the lower limit should tend to a reasonable limit. For the integrand it therefore seems natural to assume that the claim number process is nearly mixed Poisson, as defined in Section 5.3.1. Note, however, that the integrand is actually independent of the quantity t if the claim number process is mixed Poisson.

The remaining convergence condition can then be formulated in the form that when

for a function ψ(x) to be determined. The latter condition can be identified with the extremal condition in terms of U if we choose ψ_t(0) = 1, c(t) = U(t) and d(t) = a(t) as in the condition on U. Under the two conditions we see that when

This result follows since for any fixed r, with while . Rewriting the extremal laws by an affine transformation, the result can be expressed in the following easier form:

For and {N(t);t ≥ 0} near mixed Poisson, one has

where one of the three following cases emerges necessarily:

1. γ > 0, c(t) = 0, d(t) = U(t) and ψ(y) = y^−1/γ, y ≥ 0
2. γ = 0, c(t) = U(t), d(t) = a(t) and ψ(y) = e^−y
3. and ψ(y) = |y|^1/|γ|, y ≤ 0.

Notice that for a deterministic N(t) this result reduces to the classical extreme value laws discussed in Chapter 3. For the case r = 1 and N(t) discrete, see Galambos [367]. In this form the result is a special case of a more general weak convergence result in Silvestrov et al. [701] where even the independence condition between claim number and claim size processes is weakened considerably. It seems possible to derive more explicit statements in the case where F_X is assumed to belong to C_γ with remainder conditions. In the mixed Poisson case particularly, no further complications should show up.

Moments.

Of course a full treatment of the behavior of the moments under a max‐domain of attraction is possible. We restrict our attention to a few direct results. Equation (10.1.3) can also be used to derive the consecutive moments of the order statistics. For we have

(10.1.5)

The easiest way to prove this is by noticing that for a non‐negative random variable W and β > 0,

and that by (10.1.3)

The appearance of the quantile function is of course pleasing. It indicates precisely what information on the underlying two processes is needed. For the claim number process we need while the quantity incorporates the information requested on the claim size distribution.

The asymptotic expression of the moments under the domain of attraction condition is then easy if the claim number process is also near mixed Poisson. Indeed if with γ > 0, then for β ≥ 0 and

When γ = 0 one can sharpen this result while using the full strength of the domain of attraction condition.

Combinations of order statistics.

In this section we state a versatile formula that will allow us later to derive almost all desired expressions as special cases. In (10.1.1) we see that the first part of this ordered sample contains the small claims while on the right‐hand side of the scale we find the large claims; they are separated by intermediate claims. Let us use the following two abbreviations:

and

Here Σ refers to small while Λ refers to large. We easily arrive at properties of the claim fragments through the joint Laplace transform

where u, v, w ≥ 0. Conditioning on the number of claims at the time epoch t and interpreting X_{r, N(t)} = 0 whenever r ≤ 0, one can derive

(10.1.6)

Many special cases can be derived from this formula by properly choosing s, u, v and w. We give a couple of examples.

• For one order statistic. Choose u = w = 0 and take s = r − 1. Then we fall back on the Laplace transform of the order statistic X_{N(t)−r+1, N(t)} as treated in (10.1.3).
• For all but the largest claims. Take now u = v = 0 and s = r − 1. Then
  
  For the special case where r = 1 we find the formula
  

  for the transform of the sum of all but the largest claim. In the case of the Poisson claim number process this result goes back to Ammeter [38].

• For the smallest claims. Take u = 0 and again s = r − 1. Then
  
  If we invert this with respect to v we obtain the hybrid expression containing the distribution of X_N(t)−r+1, N(t) and the transform of Σ_r−1(t). From the latter relation one can, for example, derive a further relation that gives information on the ratio between any of the claims and all the smaller ones. The resulting formula might be used to normalize that portion of the portfolio that corresponds to the smaller claims.

Distributional aspects.

Denote the reinsured amount by

on the set 1_{N(t)≥r}. By putting u = θ, s = r, and v = w = 0 in (10.1.6) we deduce

For the case of exponential claim sizes more explicit results can be obtained, as shown by Kremer in [505].

Weak limit.

The above formula can be used to obtain the general form for the limit in distribution for the appropriately normalized expression L_r(t) when . As is the case of the limit behavior for the extreme order statistics we will assume that for and that the counting process is mixed Poisson. A little reduction yields

where

It is now quite obvious what we should do. We choose c(t) = rU(t) and d(t) = a(t); then . Therefore the right‐hand side tends to the limit

Let us specialize a bit.

1. If γ > 0, then one can replace a(t)/U(t) by its limit γ, yielding the somewhat simpler result
   
   For r = 1 the right‐hand side can also easily be written as a Laplace transform. The resulting expression for the limit in distribution coincides with that of the maximum from the previous section.
2. For γ = 0, the inner integral reduces to
   
   Using the structure variable we find for
   

The mean.

From the above Laplace transform we can immediately deduce the first few moments. We restrict attention to the mean. An easy deduction yields

(10.1.7)

We link the above expression with our knowledge about the classes C_γ used in the treatment of one order statistic. First make the change of variable v = w/t. Then replace . One easily finds that

Taking limits for shows that we need to assume that γ < 1. It then follows that

Further comments.

• As can be expected, the calculation of premiums quickly runs into mathematically intractable formulas. This has been recognized by Benktander [112], who deals with a relation between XL and the largest claim situations. For the calculation of the pure premium, see Berglund [118] and references therein.
• Kupper [520] compares the pure premium for the XL cover at retention M with that of the largest claims cover at retention r. He specifically deals with the case where the claim size distribution is strict Pareto. For example, the effect of truncation on the largest claims depends strongly on the index of this claim, as shown in [519].
• Berliner [119] considers a set of interesting problems connected with the largest claims covers. He assumes the claim number process to be Poisson and derives the joint distribution of two large claims X_{N(t)−r+1, N(t)} and X_{N(t)−s+1, N(t)}, and computes their covariance for the case of a strict Pareto law, as well as Cov(L_r(t), S(t)).
• Kremer [501] gives crude upper bounds for the pure premium under a Pareto claim distribution. The asymptotic efficiency of the largest claims reinsurance treaty is discussed in Kremer [502].
• Some practical aspects of large claim distributions have been treated by Schnieper [682]. Albrecher et al. [26] studied the joint distribution of larger and smaller claims for regularly varying claim distributions (see also Ladoucette and Teugels [525] for an overview).

Distributional aspects.

As defined in Chapter 2, the reinsured amount in an ECOMOR treaty is

Again the expression of the reinsured amount equals 0 if N(t) ≤ r. The special choice s = r, w = 0 and v = −ru in (10.1.6) gives us the following expression for the Laplace transform of the reinsured amount:

Weak limit.

We use a procedure similar to the one for the largest claims reinsurance. So we start from the expression above where we normalize by the auxiliary function a(t) from the max‐domain of attraction. We have

where

Note that the very definition of E_r(t) makes further centering unnecessary. We again assume that and that the counting process is mixed Poisson. It then easily follows that

In general the resulting limit distribution seems very hard to recover. However, in the integrand we find for a fixed w a power of a Laplace transform

For 0 < γ ≤ 1 it is not difficult to show that then

which looks very much like a generalized Pareto distribution.

Still two values of γ seem to give something special.

1. When γ = 0 then we get a simple expression in that then
   
   which can be directly interpreted as the Laplace transform of a gamma distribution.
2. When γ = −1 (as for the uniform distribution on [0, 1]) a simple calculation yields
   
   For degenerate Λ the limit distribution is a product of independent exponentials.

The mean.

From the above relation we derive the expression for the first moment.

We again indicate what happens when the C_γ‐classes are in force. With the notation introduced for one order statistic, we see that we again need γ < 1. Then, as before,

illustrating the role played by the structure variable Λ.

Further comments.

As for the ECOMOR treaty, a few more explicit results can be found in the actuarial literature.

• Ammeter [39] points out how the exclusion of one or more of the largest claims has the result of reducing the expected amount of the remaining aggregate claim amount. In some cases even an infinite expectation becomes finite after such a reduction. He also points out how in a portfolio with Pareto distributed claim sizes there is a preponderance of small claims. See Albrecher et al. [26] for generalizations.
• For a general study of ECOMOR treaties, see Ladoucette and Teugels [523]. Crude bounds for the pure premium go back to Kremer [501]. For a study of LCR and ECOMOR treaties for more general claim number processes, see also Asimit and Jones [55].
• Here is another possibility for a large claims reinsurance treaty that imitates an ECOMOR treaty but that has a different kind of retention. Accumulating information on the largest claims, one can first get an estimate for the number of claims K_t(a) falling within a fixed distance a from the observed record claim within the time slot [0, t]. The average over this number of claims yields an alternative large claim reinsurance treaty. Information on the quantity K_t(a) can be found in Li et al. [540] and Hashorva [423]. For studies on the asymptotic behavior of the tail probability of E_r(t) for exponentially bounded claim sizes, see Jiang and Tang [465] and Hashorva and Li [426]. Peng [610] studied the joint tail of the reinsured amounts in ECOMOR and large claim treaties.
• For a bivariate version of ECOMOR, see Hashorva [424, 425].

Alternative carriers.

Next to self‐insurance (which can be both regulated and non‐regulated and is particularly popular in the USA), reinsurance pools act like a mutual, where each insurance company can cede its risk and its premiums of a specific risk class, for instance for very large risks. Risk‐retention groups are based on a similar concept of pooling for companies to get access to certain types of liability insurance. As another example, captives are popular vehicles to lower insurance premiums as well as transaction costs. Typically located in a tax‐friendly environment, captives are insurance or reinsurance companies which insure the risks of their parent company in a cost‐effective manner. In that way, access to the global reinsurance market can be obtained, which due to the larger diversification possibilities of reinsurers may lower premiums through reduced capital cost. In addition, this allows a certain degree of time diversification (captives are usually allowed to hold equalization reserves, whereas according to the current accounting standards insurance companies are not). For smaller captives, it is also popular to operate as a reinsurance captive, which insures the risk of its parent as a reinsurer of a first‐line insurance company, to which the risk was first transferred. A particular advantage of such a construction is that from a regulatory perspective it will then be treated as a reinsurer, for which different rules apply.

Particularly in times when traditional reinsurance premiums are very high or coverage is not available at all (which happened after Hurricane Katrina), an alternative are so‐called reinsurance side‐cars, where investors deposit funds and in turn participate in the premiums and claims of the insurance company (typically in the form of a QS‐type treaty for a line of business, with a retention for the company that ensures alignment of interests of the two parties). The deposit equals the reinsurance contract limit (so it is a fully collateralized reinsurance form where counterparty risk is eliminated), so the investors’ liability is limited to the amount of the deposit.

Sidecars are only one way for capital market investors to gain exposure to insurance risk. In general, the capital market is a major alternative risk carrier, particularly through insurance‐linked securities (see below).

Finite risk reinsurance.

If in addition to insurance risk transfer there is also a significant weight on other goals in a product, one speaks of structured reinsurance. A particular example is finite risk reinsurance, which is a combination of risk transfer and risk financing between an insurer and a reinsurer in a form that is tailored towards the concrete needs of the insurer. While a substantial goal of the transfer is to enhance the insurer’s financial results, for tax reasons it has to contain a (limited) amount of insurance risk transferred to the reinsurer in order to be classified as reinsurance. Such contracts have a duration of several years and combine loss experience and investment returns. They formalize a longer‐term relationship between the two parties which has a time diversification component, as the reinsurer can count on incoming premiums and the insurer on agreed coverage to known conditions over a longer time horizon.

There are retrospective and prospective variants. An example of the former is a loss portfolio transfer, where the insurer transfers outstanding claims from some long‐tailed business of previous years to the reinsurer, and in turn pays a premium consisting of the net present value of these claims plus fees. In that way he passes on risks related to the timing and amount of loss development. In adverse development covers the IBNR losses are also included. Here the claims reserves are not transferred to the reinsurer, but the reinsurer only covers losses that exceed the reserves which the insurer has already built up, and for this transfer a premium is paid (this can be set up like an XL or SL contract on the adverse loss development and is also very convenient in cases of mergers or take‐overs of a company).

Prospective variants of finite risk reinsurance include spread loss treaties, where for the transfer of specified losses (with annual and overall limits) the insurer pays premiums to the reinsurer onto an “experience account” and these premiums (minus fees and expenses) are then invested. At the end of the contract period, the balance is settled with the insurer, exposing the reinsurer to the counterparty risk that the insurer may not be able to pay a potential negative balance. Finally, finite quota share arrangements can include over‐ or undercompensation of claims over prespecified periods of time.

Integrated products.

In multi‐year/multi‐line products several business lines are bundled together and/or over a longer time horizon, which leads to a smoothing of the aggregate risk and hence to lower premiums. A disadvantage is counterparty risk for the insurer, and it can also be non‐trivial to cooperate across business lines within the insurance company.

Multi‐trigger products.

Here the reinsurer pays losses only contingent on a second event, which typically is correlated to the insurer’s financial result. For instance, the reinsurer will only pay if the losses exceed a certain threshold and at the same time a stock index, commodity price, exchange rate etc. is below a prespecified level. Such an arrangement will lead to considerably lower premiums, but still may serve the overall financial result of the insurer well. At the same time, the reinsurer can also benefit from this variant, as resulting capital needs will decrease, particularly if there are several such contracts with independent triggers in the portfolio.

Contingent capital.

This refers to the option for the insurance company to raise debt or equity capital for prespecified conditions, in case there is a severe aggregate insurance loss experience or another prespecified event occurs. It can be seen as a put option on the own shares with a predefined strike value. By setting up the conditions before financial distress, fresh capital can in such a case be acquired in a much cheaper way than on the market. In fact, this is a means of financing rather than a transfer of insurance risk. As a variant of this idea, in recent years contingent convertible bonds (“CoCo” bonds) have become increasingly popular (particularly since in many countries this is now considered as regulatory‐efficient capital), and recently products have been issued that as a trigger combine the occurrence of natural catastrophes and the solvency ratio of the company.

Industry loss warranties (ILW).

These contracts resemble reinsurance contracts, but the loss under consideration is the loss of the entire insurance industry arising from an event (measured through some index¹) rather than the individual loss experience. The market for such contracts has considerably grown over the last years, with reinsurance companies and hedge funds being typical protection providers. A disadvantage for the insurer is that the reinsured amount is not based on their own loss experience, even if the industry index will usually be reasonably correlated. However, the resulting discrepancy (referred to as basis risk) may lead to inefficiencies.

Insurance‐linked securities (ILS).

In broad terms, these are financial instruments whose values are driven by insurance loss events. They enable insurance risk to be (directly) placed on the capital market (insurance securitization). Among the most important examples are catastrophe bonds (CAT bonds), which are bonds with the additional feature that the investor will not receive the coupon (or even not the principal) if a certain trigger related to the occurrence of natural catastrophes is hit. In turn, the coupon in the absence of that trigger is substantially higher. Such bonds are traded over the counter and can increase the insurance capacity for risks for which it is difficult to find traditional reinsurance. The initiator of the CAT bond is the insurance company that seeks this protection. The trigger can be the individual loss experience of the issuer due to a catastrophe or (more often) an index representing the average catastrophe loss experienced by the insurance sector in a prespecified time interval and business line. Finally, it is very common nowadays to have parametric triggers, that is, a physical measurement (such as wind speed, magnitude of an earthquake etc.), as this is a reliable, “objective”, and usually easily accessible trigger for investors (whereas for a trigger linked to the individual loss experience of the issuer there are obvious moral hazard issues and the final settlement can take much longer). One then speaks of an index transaction (instead of an indemnity transaction). However, triggers based on an index or parametric triggers again introduce basis risk for the issuer (on the plus side, the insurer does not have to pass on the actual claim data to the outside in this case).

In practice, there is usually a special‐purpose vehicle (SPV) that acts as an intermediary (located in a tax‐friendly environment), which then issues a conventional reinsurance policy to the insurance company. The SPV uses the premium payments of the insurance company to pay the coupons to the investor, and if the event is triggered (or maturity of the bond is reached without the trigger), the amount in the SPV is paid out to the insurer and the investors according to the bond specifications. One particular advantage of the CAT bond is that – in contrast to traditional reinsurance – there is no counterparty risk for the insurer (unless the construction involves further parties like swap providers), since the reinsured amount is already available in the SPV (or the trust account). At the same time, for investors this can be an attractive product, since the underlying trigger event will typically be independent of other investments and the excess coupon can be considerable (in the long run, it will of course be determined by demand and supply, as well as the concrete loss experience of previous years). Conceptually, CAT bonds represent a secondary reinsurance market, where the investor (who takes the reinsurer’s role here) has more flexibility to leave or enter the “contract” along the way. In recent years the CAT bond market has increased considerably in size. Further ILS products (with, however, a much smaller market) are, for example, longevity swaps and products related to embedded‐value securitization and extreme mortality securitization. Altogether, the global ILS risk capital outstanding in 2016 exceeded 25 billion US$ (Source: Artemis).