Christopher Klein, Global Head of Business Intelligence

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**Case 1: Base Case**

A hypothetical ceding company contracts for the USD200 million casualty cover from a single Aaa rated reinsurer. The probability of a default is 1 percent, with a loss of USD200 million, while the probability of no default is 99 percent and would carry no loss.

A default means that there is a loss or a no recovery of USD200 million. Finally, the expected loss is equal to the sum of the product of the probability multiplied by the loss.

Expected Loss = [(.99*0) + (.01*2000)] = USD2 million

**Case 2: Two Aaa Rated Reinsurers**

To begin to see the effects of diversification, we move to the case where the ceding company splits its program evenly between two Aaa rated reinsurers. The outcome options and probabilities are captured in the grid below. In the left-most cell of the second row, RA is meant to stand for Reinsurer A, which has a 99 percent probability of no default. Moving to the right, the next cell in row two indicates that Reinsurer A has a 1 percent chance of defaulting-likewise for Reinsurer B. Because we have stated that the default events are not correlated (i.e., they are independent), the probability of each event described in the first row is the product of the individual probabilities within the appropriate column. Therefore, the probability of no default in this two-reinsurer program is .99*.99 = 98.01 percent.

The table of outcomes is then as follows:

With two Aaa rated reinsurers, the no recovery probability is 1-in-10,000. This is a tremendous reduction from the base case, where it was 1-in-100. Even this simple diversification has achieved an important reduction in the chances of an extremely adverse outcome. The trade-off for this is fairly obvious. First, the probability of a full recovery (no default) is slightly less than in the one reinsurer case or, conversely, there is an increased probability of some loss. However, the expected loss remains at USD2 million, because both reinsurers are rated Aaa. Though we have changed the probabilities of extreme events with the diversification, we have not eliminated (or even altered) the credit risk to which the cedent is exposed.

**Case 3: Adding a Lower-Rated Reinsurer to the Program**

We can introduce a credit distinction to the example above by looking at a program that is placed in equal portions with two reinsurers, one Aaa rated (Reinsurer A) and the other Aa (Reinsurer B). The grid below lays out possible combinations and probabilities:

This produces an outcomes table reflective of the new credit quality mix.

Not surprisingly, the substitution increases the probability of no recovery when compared with the two Aaa reinsurers case. It is still a substantial improvement over the single Aaa reinsurer example, as the no recovery probability with the Aaa/Aa split is less than 3-in-10,000. The weaker credit results in a decrease in the chance of full recovery, and the corollary is an increase in the probability of some loss. Further, the expected loss has increased to USD3.8 million.

**Case 4: Three Aaa Rated Reinsurers, Evenly Split Program**

What happens when we diversify further, with a third reinsurer? The answer should be predictable. If we split the program between three Aaa rated reinsurers, the grid of possible combinations and probabilities is expanded a bit, as there are now eight possibilities.

The outcome table for this program is below. Because the reinsurers are each rated the same, we can summarize the “only one default” and “only two default” events in the table by adding the individual probabilities.

When three Aaa rated reinsurers split the program, the probability of no recovery has been reduced dramatically, to 1-in-1 million. Remember; we have assumed that default probabilities are not correlated (i.e., completely independent). Relaxing this assumption, however, has a noticeable affect on the chance of no recovery. We have further reduced the probability of full recovery relative to either the single Aaa case or the case of the program split between two Aaa rated reinsurers. The probability of full recovery with the three Aaa participants is greater than the Aaa/Aa split program. Finally, the expected loss of this program is as it must be for an Aaa program: USD2 million.

**Case 5: Mixed Credit Ratings: Split Evenly Among Aaa, Aa, and A**

This example explores further the impact of a program involving reinsurers with mixed credit qualities. The possibilities grid has the same number of cells as that above.

This outcome table is developed as before, adding the probabilities on the single defaults and two defaults to get the following:

Clearly, the credit profile has resulted in a further decline in the probability of a full recovery, although the probability of ruin that comes with a no recovery outcome remains extremely remote, at 1-in-100,000. The expected loss, however, has increased quite a bit to USD4.8 million.

USD4.8 million = (.9296*0) + (.0689*66.7) + (.000297*133.3) + (.00001*200)

Expected loss probably can be managed by providing a larger share of the program to the Aaa company and a smaller share to the A rated reinsurer. For instance, if the Aaa had a 45 percent participation, the Aa 35 percent, and the A had a 20 percent share, the expected loss would be reduced by about 12 percent to USD4.2 million.

**Case 6: Correlation Among Reinsurers, Program Split Between Aaa and Aa**

Until this point, we have made the convenient (but perhaps unrealistic) assumption that the default experience among reinsurers is uncorrelated, making the probabilities independent. Correlation changes things, and some correlation in default experience is likely, as the conditions that might conspire to lead to a default (such as a catastrophic event or asbestos-like liability phenomenon or a crash of securities markets) are apt to have a common impact on a number of reinsurers.

Let’s assume that the ceding company splits the program evenly between an Aaa and an Aa reinsurer. Also, assume that the probability of default for Reinsurer A and Reinsurer B together is higher than in the independent case: A and B are correlated. The 1 percent probability of default for the Reinsurer A (the Aaa reinsurer) is made up of two components, a probability of 0.4 percent that Reinsurer A defaults but not Reinsurer B, and a probability of 0.6 percent that Reinsurer A and Reinsurer B default together. For Reinsurer B (Aa rated), the probability of a joint default with Reinsurer A is, logically, the same 0.6 percent, while the probability that B defaults alone is 2.2 percent. Given the individual and joint probabilities of default, the no default scenario must be equal to 100 percent minus the sum of the individual and joint probabilities.

The outcome table appears as follows:

The probability of no recovery is higher than that for the independent case (Table 3) but remains below that of the non-diversified Aaa base case. Diversification helps reduce the risk of ruin or no recovery even when default experience is correlated. The full recovery (no default) outcome is slightly better than the Table 3 case of independent reinsurers. Importantly, the expected loss is USD3.8 million, the same as for the Table 3 case because we have the same credit quality and the same program split. Again, we could manage the expected loss by allocating the program differently than a 50/50 split.

Correlation does diminish the impact of diversification on the probability of no recovery, and the consideration of correlation may lead to the placement of a program with reinsurers that have differing geographic, business, and market exposures as a way to limit correlation among reinsurance participants. Correlation may also be an issue if a ceding company’s fortunes and primary insurance portfolio are correlated with potential default events that could affect its reinsurers. A ceding company is effectively making an investment in the reinsurance recoverable asset, and attention to correlation is useful in that avoiding a panel of reinsurers with highly correlated default exposures or reinsurers with risks that are highly correlated with those of the ceding company are steps to preserve the value of this asset.

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Additional Contributor

- Sean Mooney, Chief Economist