A model may be fine within certain time horizons or other limits, but it is definitely worth realising that at some point all models will stray from reality

Modelling is very much in vogue in the reinsurance industry. Starting with his company’s corporate image, Richard King takes a light-hearted look at modelling and suggests that it’s neither that new nor that precise.

I’m sure many will be aware of the dog, “Churchill”, who encourages us Brits to save on our motor insurance premiums. He often appears as one of those annoying toys that many of my countrymen put in the back of their cars – the nodding dog. A year or so ago, I recall, the anecdote appeared in the trade press that, in his company, he was known with the first letter of that description amended to create an adjectival expletive indicating, at best, irritation. I sometimes think that my colleagues have a similar attitude to rabbits.

The reason for this derives from our corporate logo, which is a Fibonacci spiral .

This (or at least some of the derivatives thereof), is a naturally occurring phenomenon but was originally discovered by a thirteenth century Italian, Leonardo of Pisa, otherwise known as Fibonacci. So how did this come about?

The curve is formed on an infinite sequence of numbers by starting with 1, then 1 and then each succeeding term is the sum of the two previous : 1, 1, 2, 3, 5, 8, 13, 21 and so on

You might also have heard of the golden ratio (1.618, approximately), which appears in architecture and elsewhere (overall height to height of the navel is but one example). Its true accuracy and relevance is the subject of much debate but it is the limit of the ratio of successive Fibonacci numbers.

As an aside, you can do a little work on an Excel spreadsheet . Make up the series itself in one column (start with “1” in the first two rows then put in the addition formula adding the two cells above and copy down until you have the first 100 numbers or so, to taste). In the second cell down in the next column enter the division formula to create the ratio of that number to the previous one (I suggest 6 decimal places) and copy down so that you have the ratios for all the numbers you have created. You will note that the ratio is erratic for the first four calculations (1, 2, 1_, 1_) but then very quickly settles down to approaching 1.618034. In fact Excel will tell you it’s precisely this from the 38th calculation onwards. It isn’t; but Excel will tell you it is.

Look now at the actual Fibonacci numbers. Look at the 72nd and 73rd that you have calculated. Now we know the 74th is the sum of these two. But the 74th ends in a “0”, not the “7” it should. The computer is rounding things off to 15 significant figures. Probably fair enough, but just a demonstration that you shouldn’t put absolute faith in everything you read in the newspapers or on the computer screen. Or, perhaps, as the output of models; but we’ll come back to that.

Anyway, how did Fibonacci come up with his strange sequence?

Well, it is actually a model for the breeding of rabbits. And these are prolific creatures, as we know. The problem has been stated in many forms since Leonardo first considered it but one version is :

How many rabbits will be produced in a year, beginning with a single pair, if every month each pair produces a new pair, which becomes productive two months after birth?

In this source, the first term of the classic sequence is the beginning of the period and thus we require the thirteenth term, being the end of the twelfth month. The answer is 233.

So what? Lots of pie, cat food and, presumably, dressing for the vegetable patch. Mr McGregor had it easy.

But this lot have got going now. After 24 months there will be just over 75,000 bunnies. (One can understand why Australians perhaps rue the day their forefathers introduced the little furry chaps to the antipodes and why they built the rabbit proof fence).

At this rate, after just five years, there are over two and one half trillion of the little blighters. Now if the average rabbit weighs two kilos, and the average human fifty, that gives a totalmass of rabbits seventeen times that of the total human population. Or, to put it another way, everyone can have sixteen hundredweight of rabbit. Approximately.

Maybe this model has its deficiencies!

It certainly does. If I haven’t sent you to sleep yet, you will have been screaming three fairly obvious objections to this model. Firstly, none of these rabbits ever die; secondly, they all stem from one pair which suggests that we are making very optimistic assumptions about their ability to survive the problems of inbreeding and concentration of the gene pool and finally, they maintain fertility irrespective of any strains they put on their environment.

Now, Fibonacci never “sold” this as a model to a practical problem. It was a numerical puzzle that happened, as these things do from time to time , to have applications (or more accurately, resonances) in other areas of study. Pretty eerie resonances, some would claim, but this is certainly not intentional modelling. The three objections above make it “unsellable” as such.

Let me now take a separate, equally simple model that does address the first two objections above. Consider the human population of planet Earth. Currently there is enough genetic diversity for that not to be an issue and net growth rates assume inbuilt mortality, clearly.

Now we can extrapolate that, from current growth rates, it will take about another 2,000 years (only a re-run of the Christian era) until the entire mass of the earth is composed of people. This is clearly way beyond the sensible theoretical limit imposed by available necessary trace elements and other constraints (not least, the water necessary to support populations). I am not going to go down the obvious road of exploring the impact of the third objection above but it’s a worrying thought, clearly.

What then, is the point in this piece?

Simply that a model may be fine within certain time horizons or other limits, but it is definitely worth realising that at some point all models will stray from reality – sometimes by alarming amounts.

So, is modelling worthwhile? Well, I would say, yes.

Let’s go back twenty years (or more). The so called Lloyd’s first loss scale was then widely available and many of us used that as a guide. I always thought there was a glitch in it but I smoothed that out and carried on. After all, the business that provided the statistical base for the Lloyd’s scale was unlikely to be directly correlated with whatever I was doing, but it would almost certainly be a much better first estimation than anything else that I would have available. A good guess? It would be difficult to demur.

However, what this assumption (that the available first loss scale was a reasonable approximation to all similar scenarios) did provide was a basis of consistency. Every rate that we generated (on a risk Excess of Loss, for example) was consistent with the previous price, absent any compelling need to amend the pricing model. Whether we actually quoted that price depended upon other commercial pressures.

Similar approaches could be made for a number of other risk types – assuming that a casualty excess of loss layer should be priced at a certain percentage of the underlying, for example. That was a reasonable model and was tested against experience. I suspect that most would argue that (on the basis of results) the model was found wanting. I would suggest however that such a model depends upon two factors: the rate for the first excess being correct and the decay rate (ratio of one tranche to the underlying) also being correct. If only the first is correct then at least one layer (the first) will be right. If that is wrong (and you may not know that at the time) then you have to hope that a compensating error will correct matters - but in this type of model it will still only be “correct” at one point being either higher or lower (“expensive” or “cheaper”) on either side of this point. In casualty of course there are many other complicating factors such as price inflation, superimposed social inflation changing litigation practice and so on.

Presently, modelling’s highest profile is in catastrophe reinsurance. This has become very complicated. Geology, Meteorology and Civil Engineering meet higher Mathematics. All are relatively new to the field where, I fear, they are labouring under the unrealistic expectations of the end user. They will never get it right but they will get better. Many factors can only really be tested in reality. Observance of building codes is just one such example.

So my message is: don’t shoot the modeller - he’s doing his best. Try and understand his simplifying assumptions and the weaknesses they might introduce into the models you use. But even with those assumptions might give you an idea of what can happen. Maybe a greater acquaintance with our thirteenth century hero would have kept rabbits out of Australia!

This Special item appeared in issue 108 of JTW News - September 2006

Author: Richard King - Helix UK