Triangles

Traditional Triangles

Most of the models used in actuarial science are designed to work with aggregated data, meaning that Bermuda does not assume any knowledge of losses at the level of individual claims. The actuarial community traditionally aggregates loss data into a form known as a “loss triangle”. An example of a traditional loss triangle is shown below:

Accident Year	AY+0	AY+1	AY+2	AY+3
1988	952000	1529000	2813000	3647000
1989	849000	1564000	2202000
1990	983000	2211000
1991	1657000

The etymology of the term “triangle” should be fairly obvious from this view – the set of observed cells is a triangular region in the upper left of the table. In some cases, we have historical losses from many accident years, but we don’t have any data past a fixed development lag cutoff. In such cases, the data in the table forms a trapezoid instead of a triangle, but the generic term triangle is commonly used to loss “trapezoids” as well.

Sometimes, loss triangles are presented in a format where the column headers are evaluation dates instead of development lags. In the example above, we could relabel the columns headers as 1988, 1989, 1990, 1991, and move the data so that it forms a triangle in the upper right instead of the upper left of the table. It should be readily apparent that we can convert from development lags to evaluation dates and back without any issues, and that both formats contain exactly the same information.

The example loss triangle above presents paid losses, but it should be clear that we could just as easily present another metric – perhaps incurred losses, case reserves or claim counts – in the same format. The tabular format above can only handle one metric at a time, so actuaries typically speak of a “paid loss triangle”, an “incurred loss triangle”, and so forth.

Tabular Triangles

The traditional triangular data format is clean and compact, but it doesn’t generalize easily. The table below presents a triangle in a tabular format. (This is not the data format that Bermuda uses internally, but it is a useful mental model for reasoning about triangles.)

Experience Start	Experience End	Evaluation Date	Paid Loss	Reported Loss
1988-01-01	1988-12-31	1988-12-31	952000	1722000
1988-01-01	1988-12-31	1989-12-31	1529000	3830000
1988-01-01	1988-12-31	1990-12-31	2813000	3603000
1988-01-01	1988-12-31	1991-12-31	3647000	3835000
1989-01-01	1989-12-31	1989-12-31	849000	1581000
1989-01-01	1989-12-31	1990-12-31	1564000	2192000
1989-01-01	1989-12-31	1991-12-31	2202000	2528000
1990-01-01	1990-12-31	1990-12-31	983000	1834000
1990-01-01	1990-12-31	1991-12-31	2211000	3009000
1991-01-01	1991-12-31	1991-12-31	1657000	2305000

This format offers a few advantages over the traditional format. First, the traditional format encourages sloppy thinking about data types. Each cell of a traditional loss triangle is defined by two coordinates: an experience period (which is an interval in time) and an evaluation date (which is a single point in time). In the traditional format, both experience period and evaluation date are represented with ambiguous labels like “1993”, which could refer to either an interval in time or some undefined point within that interval. Some authors refer to the interval [1998-01-01, 1998-12-31] as “AY1998” and the evaluation date 1998-12-31 as “CY1998”, but this sort of terminology encourages sloppy thinking. In the tabular format, we represent the experience period with the pair of values “Experience Start” and “Experience End”, and the evaluation date as a date. This may seem like useless pedantry at this point, but we will see shortly how some real-world triangle-like data makes this distinction important.

(As an aside, we define our coordinates with an experience period and evaluation date, whereas the traditional format uses experience period and development lag. As noted above, there is no substantive difference between evaluation date and development lag. However, there two competing standards for the definition of development lag. We use the convention that development lag is equal to the signed difference between the evaluation date and the end of the experience period. Other authors use the convention that development lag is the signed difference between the evaluation date and the start of the experience period. In this presentation, we use evaluation date mostly because standard conventions for representing intervals between dates unambiguously are not as robust as conventions for representing dates themselves.)

Another advantage is that the tabular format can easily be extended to accomodate multiple fields. In the example table above, we show both paid loss and reported loss. The traditional format would require two separate triangles to store the same information, whereas this format stores arbitrarily many attributes for each cell within the same table row.

We can extend this idea by adding columns to the tabular format with arbitrary metadata. Say we are modeling loss development for a private passenger auto portfolio, and our loss development model could vary by state and coverage. Rather than try to manage \(50 \times 12 = 600\) separate triangles, we can simply add State and Coverage as columns in our tabular format, and concatenate all 600 triangles into a single table.

We can take this idea further still by including metadata columns that represent whether the experience periods are accident-basis or policy basis; whether the losses are gross or net of reinsurance; whether individual claims are capped, and if so, at what value; whether the losses include DCC or LAE; and so forth. (Of course, for many of these metadata columns, constructing a model that appropriately handles mixed data may be non-trivial.)

Taxonomy of Triangle Data

The final advantage of the tabular format is that it is able to handle unusual loss development data much more naturally. The traditional triangular format carries with it a strong implication that every experience period will be the same length, the interval between every pair of successive development lags will be the same, and every cell in the upper-left portion of the table will be observed. Some actuaries believe that all triangles satisfy all of the above criteria. People who have done consulting work or have analyzed reinsurance submissions know that all of the above assumptions can be violated on a regular basis. Therefore, a format that can handle violations of the above assumptions is not a mere theoretical nicety, but an essential prerequisite to modeling real-world data.

Essentially all of the actuarial literature on loss development assumes triangles that satisfy all of the above criteria. It’s understandable why this is the case: most of the literature is focused on new techniques for loss development, and it’s easier to present a new technique as applied to clean data, rather than spending a great deal of effort handling edge cases. Since unusual loss development data is rarely discussed, there is no standardized terminology for describing unusual triangles.

We at Ledger Investing have developed a taxonomy for these unusual triangles, which we have found greatly facilitates communication. We describe our terms bloew.

A square triangle has the same amount of time between successive experience periods as between successive evaluation dates. In the examples shown above, both the experience periods and evaluation date intervals are annual. If this condition does not hold (for example, experience periods are annual but evaluation date intervals are quarterly), the triangle is non-square. (Yes, we are aware of the incongruity of discussing square triangles.)

A complete triangle has observations at every cell, except for cases where a cell’s evaluation date is greater than the maximum evaluation date in the triangle. An incomplete triangle has some cells with missing observations.

A regular triangle has the same length for every experience period, and the same interval of time between successive evaluation dates or development lags. A semi-regular triangle has every experience period the same length, but the interval of time between successive evaluation dates or development lags is inconsistent. For example, a semi-regular triangle may contain observations from the following set of evaluation dates: [2018-12-31, 2019-06-30, 2019-09-30, 2020-06-30, 2020-12-31, 2021-03-31].

An irregular triangle has evaluation periods with inconsistent lengths. For example, an irregular triangle may contain observations from the following set of experience periods: [(2018-01-01, 2018-09-30), (2018-10-01, 2019-12-31), (2020-01-01, 2020-06-30)]. It’s important to note that a triangle is considered irregular if it has inconsistent experience period lengths, even if its evaluation dates are consistent. This is because many loss development models are able to handle semi-regular triangles much more easily and naturally than irregular triangles.

An erratic triangle has evaluation periods that are not strictly disjoint. For example, an erratic triangle may contain observations from the following set of experience periods: [(2018-01-01, 2018-12-31), (2018-07-01, 2019-09-30), (2019-07-01, 2019-12-31)]. As with irregular triangles, the only criteria for a triangle qualifying as erratic is whether there is any overlap in experience periods. A triangle is erratic (and therefore not semi-regular or irregular) even if all of its experience periods are the same length and its evaluation dates follow a consistent pattern. The principal challange that erratic triangles pose compared to irregular triangles is that common statistical assumptions about independent and identically distributed (i.i.d.) noise terms go out the window.

Finally, there are a couple of terms that we at Ledger use in a way that is not entirely consistent with the broader actuarial community. First, as we noted earlier, traditional triangles can only hold one type of data – .e.g., paid losses or open claims. Given that tabular representations of triangles don’t have this restriction, we can speak of a single triangle with multiple attributes: we talk about a triangle with paid losses and open claims, rather than a paid loss triangle and an open claim triangle.

Second, the actuarial community often refers to the sum of paid losses and outstanding case reserves as “incurred” losses. The problem with this terminology is that it excludes IBNR, which is uncomfortable – after all, IBNR stands for incurred but not reported! IBNR therefore definitionally belongs in incurred losses. We use the term “reported” losses to refer to the sum of paid losses and cases reserves, and “incurred” losses to refer to the sum of reported losses and IBNR. These definitions imply the identity incurred - reported = IBNR, which happens to align very nicely with names of the terms.

As an aside, this also means that we rarely work with incurred losses directly as modeling inputs. Usually, one of our primary objectives in modeling is to develop an independent estimate of IBNR. We reserve the term “incurred losses” to refer specifically to the estimate of ultimate losses that was booked by an insurer at a specific point in time. We refer to our estimates of ultimate losses as “estimated ultimate losses”, to avoid confusion with insurer estimates of IBNR, which are ontologically privileged by virtue of the fact that they are reflected in the insurer’s financial statements.