The claims-triangle

Earlier, I wrote a blog in Dutch about the claims triangle (or loss triangle). Now I am revisiting the subject in my English blogpost and that´s why I looked for a new article about the subject.  Although this article talks about SQL (a rather technical subject), it explains the nature of the claims triangle very well:–a-jab-at-sqlscripting

The claims triangle is a statistical instrument, used by actuaries (mathematicians specialised in risk issues – often insurance-related). It shows how claims have developed throughout the years and how quickly the final amount was settled.

For simple claims, like a broken window or a slightly damaged car, the settlement will be quick – most likely within the same year as the damage happening. For large numbers of more complex claims however the losses will not be immediately clear and it could take several years before the final payment goes out.

The difference between earnings (premiums paid) and costs (including payments because of claims) is not pure profit, because additional costs will come forward in the future. A trading company will know about future costs (like goods received without being invoiced yet) rather well, but for an insurance company it´s less easy to predict the future. That´s why a claims triangle is a useful instrument. Of course in reality the subject is much more complicated, but this explanation will do. For insurance see:

Why is it called a triangle? Well, that’s quite simple: claims of about ten years ago, have had ten years to mature, but for losses occurring only five years ago, only five years of data will be available. And losses from last year will provide two years of information at most. So the more recent the claim, the smaller the range of data and the table will look like a triangle, like the example below (values throughout the years are cumulates).






The blue values are all known in 2016, e.g. for 2012 the fourth year after the claim will be 2016 and for 2014 (being year zero itself) the second year is 2016.

The purpose of the triangle is to predict the future (the empty cells, that is), with the help of statistics. But wouldn’t it be nice to present this table in a spatial graph, like the ones generated by the VRBI-tools?

Good news: using the numbers in the claims-triangle from the article referred to at the beginning of this post (–a-jab-at-sqlscripting ) a nice 3D-graph was generated! The triangle from the article is repeated as a screenshot (see below).


The claims year was taken as the x-value, the amount as y-value and the number of the year after the initial claim-year was used as a z-value. Ball-size was set to 2. Labels were added to the balls for year “0” and a legend (ball-colours) was added to the z-axis. The real graph is a webpage in html, to be viewed in a browser. Click this link to see the 3D-graph (or click right to download it first) and manipulate it yourself!

To experience the full power of a spatial graph, click on the screen and then move your mouse to manipulate the graph: rotating and tilting is done easily. Translation is done by right-clicking and moving. Double clicking will change the origin. To return to the starting position, just refresh your screen!

Below a couple of screenshots are presented,  with some comments added.

First screenshowFrom the left to the right the purple balls in front present all the initial years of the claims. Towards the back of the graph the first, second, third – and so on – year is presented, with a different colour for every single year. The initial claims go up from year to year, probably because the company is growing: more customers is more income from premiums, but also higher claims. From front the back the lines go up as well, but this is because the cumulative payments go up until the claim is completed. That’s why the curves are bending down –clearly visible for 2005 and 2006.

screenshot of spatial graph VRBIIf we look from a different angle, from the present to the past, something strange hits the eye: although the purple balls show a steady increase, the most recent values for year 1 (red) and year 2 (brown) seem to jump upwards. This could mean the claims were processed quicker than in the past.

Looking from yet another angle, we cannot be so sure about the process becoming more efficient.

screenshot from spatial graph VRBIThe first and second row in front (2005), tend to bend down to the left. For the more recent years (now in the back), less data is available, but the curves seems to be steeper. So the red ball to the top at the right side, could be a messenger of ever higher costs coming up! The actuary will know this from calculations, but other experts will see and believe by exploring this 3D-graph!


Last time I promised the demo-package (full functionality, but only presenting up to five balls) would be available by now. Well, it is! Just ask for it by mailing or to get a free package.

Over AnRep3D

AnRep3D is the new company, founded after the handover of Scientassist (together with VRBI) to one of my sons. From now I will focus on three-dimensional graphs for the financial markets, showing the main figures from annual reports in comparison. As per 2021 a second product is available: EnRep3D. It is meant to visualise energy. Although the engine is the same, the texts, manual, website and examples (including blogposts) are focused at energy.
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