Pinto and Gogol: Logic of the Paper

Reinsurers need loss development factors by layer of loss, since they cover excess layers of loss. The goal in this paper is to derive loss development factors by layer of loss.

We don’t have the data which we need. That is, we don’t have data by layer of loss for sufficiently high layers and for sufficiently long development. Even where we do have data, the data are sparse, and we would have credibility problems for plain empirical loss development factors.

Why do we need to examine loss development factors by layer of loss? Why do we think that loss development factors are higher for higher layers of loss? Answer: If loss development factors are greater than unity, then they will increase by layer of loss. The reasons for this are the same as the reasons that loss cost trend has a greater effect on excess limits losses than on basic limits losses. There is a "frequency" reason and a "severity" reason. The frequency reason is that as losses below the retention increase by development above the retention, they have little effect on basic limits loss development but an "infinite" effect (for those losses) on excess limits loss development. The severity reason is that as losses which already exceed the retention increase even more because of development, the effect on basic limits loss development is zero but the effect on excess limits loss development is great (more than the effect on total limits loss development).

This is entirely different from the proposition that trend is greater for larger losses, or that "trend varies by size of loss." What we are saying is that for trend which affects all losses equally, the effect is greater for higher layers of loss. Similarly, for development which affects all losses equally, the effect is greater for higher layers of loss. Make sure this distinction is clear to you.

Pinto and Gogol say this on page 229: Although ISO has used higher trend for higher layers in determining increased limits factors, we did not find support for this procedure in the data provided. Higher trend for higher layers would produce a trend towards smaller maximum likelihood estimates of the Pareto parameter, but this is not the case, as shown in Reichle and Yonkunas [2].

We have two sets of data: RAA data and ISO data. These data sets are described on pages 228 and 229. Pinto and Gogol use a round-about technique because they don’t have the standard data sets. That is, they don’t have loss triangles in the form that they need, so they derive the loss development factors by an ingenious method.

The RAA data is in the standard form, and it shows that excess layers (reinsured losses from excess-of-loss treaties) have significantly higher loss development factors. But we can’t use the RAA data, for two reasons. (1) The data are for various sub-lines of general liability combined, and (2) the data are for various retentions combined. Read carefully the first full paragraph on page 228, especially the sentence: "In that study, however, excess losses in various layers are all grouped together, so the data does not indicate the development patterns by line for various individual layers." Also see the line on the top of page 239: "The subline mix underlying the ‘General Liability Excluding Asbestos’ experience is also difficult to determine."

Instead, we will use the ISO data, not the RAA data. The ISO data have the sub-line break-down for general liability which we need; the RAA data do not. But the ISO data is of size of loss distributions. A major insight of Pinto and Gogol is that a different loss development factors by layer of loss means changes in the size of loss distribution by development period. In fact, they can model the differences in loss development factors by layer of loss by modeling the change in the parameters of the size of loss distribution. The mathematics of this is complex and will probably not be tested. Bear’s discussion deals with the mathematics of this.

The key is that ISO has size-of-loss distributions by line of business and by maturity date. That is, they have an OL&T size-of-loss distribution at 27 months, at 39 months, and so forth. They use these size-of-loss distributions to compute excess development factors. See the first line of page 234: "The excess development factors shown were all derived directly from the underlying size-of-loss distributions."

What "factors" are they talking about? Answer: They refer to the factors in Exhibit 1. These are link ratios (age to age factors) for losses excess of certain retentions. These factors are in a four dimensional array. The dimensions are (1) retention, (2) months of development, (3) line of business, and (4) treatment of ALAE. The treatment of ALAE is discussed clearly on page 230; you should have no trouble with that. They use three lines of business: OL&T, M&C, and Products. Everything is bodily injury only. For months of development, they go out to 99 months only. For retentions, they show 8 retention levels, one of which is $0 (i.e., ground-up losses).

"How do they derive these factors?" We know how to derive loss development factors from historical loss triangles, but how does one derive loss development factors from changed size-of-loss distributions? This is not explained in the Pinto and Gogol paper, because the ISO actuaries who deal with these size-of-loss distributions already know how to do this.

Pinto and Gogol talk about size-of-loss distributions, but they don’t seem to say what they do with them. That is correct; they assume that the reader knows how to derive loss development factors from changing size-of-loss distributions.

Conceive of a matrix that has development period as the horizontal axis and retention as the vertical axis. We have data for the lower left hand corner of the matrix: that is, for low to moderate retentions and for early development period. We first extend the empirical data upwards, by fitting curves to the experience data. We then extend the columns of data to the right, again by fitting curves. Conceptually, this is what Pinto and Gogol are doing, though they just show the mathematics.

This is summarized in the second paragraph on page 234: "For each development interval, a curve is estimated to fit the excess loss development factors as a function of retention. These curves are then fitted to a smoothly progressing series of curves."

The exact procedure is shown on the rest of page 234 and 235, and an illustration is shown in Exhibit 4 on page 235. You have to read this slowly, since Pinto and Gogol jump from "a" to "a-prime" to "a-double-prime." In the formula y=a_nx^bn, the "n" subscript refers to the maturity and the "x" refers to the retention. The third paragraph on page 234 says that "x is the retention divided by $10,000." The fifth paragraph on page 234 shows the relation between the "n" subscript and the maturity.

We must combine excess loss development factors from different valuation dates. The excess loss development factors depend on the retention. That is, the excess loss development factor for a retention of $100,000 is not equal to the excess loss development factor for a retention of $250,000. However, we can not simply compare excess loss development factors for a retention of $100,000 at different valuation dates, since the real value of the retention changes with monetary inflation.

The technique for properly comparing retentions from different valuation dates is not theoretically difficult. Three papers do this: Pinto and Gogol, Berquist-Sherman, and Siewert. The Pinto and Gogol paper does it correctly; one deflates all nominal figures by observed inflation. The other two papers use an approximation that is not correct; they use a selected trend figure for all years. Siewert notes that he really ought to use the Pinto and Gogol procedure. The Pinto and Gogol paper has a further complication, since the deflated retention must be matched with one of the 118 buckets in the size-of-loss distribution.

We have completed the derivation of the excess loss development factors, and we haven’t used the RAA data at all. Where is it used in the paper? Answer: We derived the excess loss development factors in a round-about fashion. We want to test whether our results are reasonable. ISO doesn’t have the empirical excess loss development factors; only RAA has such empirical factors. We must compare the factors derived from the ISO data with the empirical factors from the RAA data.

But the ISO data and the RAA data are not comparable. We mentioned two differences earlier: the RAA data is for all retentions combined and it doesn’t separate the general liability sublines. We have a third difference as well: ISO is policy year data and RAA is accident year data. We need some assumptions and approximations to compare the derived factors with the empirical factors. The list on page 239 shows the assumptions that Pinto and Gogol use. Memorize this list. Note especially the third item in the list. Work out by pencil and paper why Pinto and Gogol use the 25%–75% weighting for the policy year to accident year adjustment.

Page 240 notes a slight difference between ISO derived factors and RAA empirical factors. This is an additional complication, and you should know the proposed enhancement in the second paragraph. However, this is a side item.

Once we have the reported loss development factors, we use the paid to incurred ratios by development period to determine paid loss development factors. See question 56 from the Fall 1998 Part 7 examination for an example of this:

Pinto and Gogol provide a formula for computing the loss development factors by layer of loss. The formula is straight arithmetic. Levine proves the formula algebraically.