I previously wrote a post in which I backtested a method of deriving large loss inflation directly from a large loss bordereux. This post is an extension of that work, so if you haven't already, it's probably worth going back and reading my original post. Relevant link:
www.lewiswalsh.net/blog/backtesting-inflation-modelling-median-of-top-x-losses
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In the original post I slipped in the caveat that that the method is only unbiased if the underlying exposure doesn’t changed over the time period being analysed. Unfortunately for the basic method, that is quite a common situation, but never fear, there is an extension to deal with the case of changing exposure.

Below I’ve written up my notes on the extended method, which doesn't suffer from this issue. Just to note, the only other reference I’m aware of is from the following, but if I've missed anyone out, apologies! [1]

St Paul's, London. Photo by Anthony DELANOIX

I was reviewing a pricing model recently when an interesting question came up relating to when to apply the policy features when modelling the contract. 

I wrote a quick script to backtest one particular method of deriving claims inflation from loss data. I first came across the method in 'Pricing in General Insurance' by Pietro Parodi [1], but I'm not sure if the method pre-dates the book or not.

In order to run the method all we require is a large loss bordereaux, which is useful from a data perspective. Unlike many methods which focus on fitting a curve through attritional loss ratios, or looking at ultimate attritional losses per unit of exposure over time, this method can easily produce a *large loss* inflation pick. Which is important as the two can often be materially different.

Source: Willis Building and Lloyd's building, @Colin, https://commons.wikimedia.org/wiki/User:Colin

Quota Share contracts generally deal with acquisition costs in one of two ways - premium is either ceded to the reinsurer on a ‘gross of acquisition cost’ basis and the reinsurer then pays a large ceding commission to cover acquisition costs and expenses, or premium is ceded on a ‘net of acquisition’ costs basis, in which case the reinsurer pays a smaller commission to the insurer, referred to as an overriding commission or ‘overrider’, which is intended to just cover internal expenses.

Another way of saying this is that premium is either ceded based on gross gross written premium, or gross net written premium.

I’ve been asked a few times over the years how to convert from a gross commission basis to the equivalent net commission basis, and vice versa. I've written up an explanation with the accompanying formulas  below.

There's some interesting literature from the world of forecasting and natural sciences on the best way to aggregate predictions from multiple models/sources.

For a well-written, moderately technical introduction, see the following by Jaime Sevilla:
forum.effectivealtruism.org/posts/sMjcjnnpoAQCcedL2/when-pooling-forecasts-use-the-geometric-mean-of-odds

Jaime’s article suggests a geometric mean of odds as the preferred method of aggregating predictions. I would argue however that when it comes to actuarial pricing, I'm more of a fan of the arithmetic mean, I'll explain why below.

The Cefor curves provide quite a lot of ancillary info, interestingly (and hopefully you agree since you're reading this blog), had we not been provided with the 'proportion of all losses which come from total losses', we could have derived it by analysing the difference between the two curves (the partial loss and the all claims curve)

Below I demonstrate how to go from the 'partial loss' curve and the share of total claims % to the 'all claims' curve, but you could solve for any one of the three pieces of info given two of them using the formulas below.
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I hadn't see this before, but Cefor (the Nordic association of Marine Insurers), publishes Exposure Curves for Ocean Hull risks. Pretty useful if you are looking to price Marine RI. I've included a quick comparison to some London Market curves below and the source links below.

This post is a follow up to two previous posts, which I would recommend reading first:
https://www.lewiswalsh.net/blog/german-flooding-tail-position
https://www.lewiswalsh.net/blog/german-flooding-tail-position-update
 
Since our last post, the loss creep for the July 2021 German flooding has continued, sources are now talking about a EUR 8bn (\$9.3bn) insured loss. [1] This figure is just in respect of Germany, not including Belgium, France, etc., and up from \$8.3bn previously. 

But interestingly (and bear with me, I promise these is something interesting about this) when we compare this \$9.3bn loss to the OEP table in our previous modelling, it puts the flooding at just past a 1-in-200 level. 

Here are two events that you might think were linked:

Every year around the month of May, the National Oceanic and Atmospheric Administration (NOAA) releases their predictions on the severity of the forthcoming Atlantic Hurricane season.

Around the same time, US insurers will be busy negotiating their upcoming 1st June or 1st July annual reinsurance renewals with their reinsurance panel. At the renewal (for a price to be negotiated) they will purchase reinsurance which will in effect offload a portion of their North American windstorm risk.

You might reasonably think – ‘if there is an expectation that windstorms will be particularly severe this year, then more risk is being transferred and so the price should be higher’. And if the NOAA predicts an above average season, shouldn’t we expect more windstorms? In which case, wouldn't it make sense if the pricing zig-zags up and down in line with the NOAA predictions for the year?

Well in practice, no, it just doesn’t really happen like that. 

This post is a follow up to a previous post, which I would recommend reading first if you haven't already:
https://www.lewiswalsh.net/blog/german-flooding-tail-position

In our previous modelling, in order to assess how extreme the 2021 German floods were, we compared the consensus estimate at the time for the floods (\$6bn insured loss) against a distribution parameterised using historic flood losses in Germany from 1994-2020. Since I posted that modelling however, as often happens in these cases, the consensus estimate has changed. The insurance press is now reporting a value of around \$8.3 bn [1]. So what does that do for our modelling and our conclusions from last time?
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As I’m sure you are aware July 2021 saw some of the worst flooding in Germany in living memory. Die Welt currently has the death toll for Germany at 166 [1].

Obviously this is a very sad time for Germany, but one aspect of the reporting that caught my attention was how much emphasis was placed on climate change when reporting on the floods. For example, the BBC [2], the Guardian [3], and even the Telegraph [4] all bring up the role that climate change played in the contributing to the severity of the flooding.

The question that came to my mind, is can we really infer the presence of climate change just from this one event? The flooding has been described as a ‘1-in-100 year event’ [5], but does this bear out when we analyse the data, and how strong evidence is this of the presence of climate change? 
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David Mackay includes an interesting Bayesian exercise in one of his books [1]. It’s introduced as a situation where a Bayesian approach is much easier and more natural than equivalent frequentist methods. After mulling it over for a while, I thought it was interesting that Mackay only gives a passing reference to what I would consider the obvious ‘actuarial’ approach to this problem, which doesn’t really fit into either category – curve fitting via maximum likelihood estimation.

On reflection, I think the Bayesian method is still superior to the actuarial method, but it’s interesting that we can still get a decent answer out of the curve fitting approach.

The book is available free online (link at the end of the post), so I’m just going to paste the full text of the question below rather than rehashing Mackay’s writing:

I received an email from a reader recently asking the following (which for the sake of brevity and anonymity I’ve paraphrased quite liberally)

I’ve been reading about the Poisson Distribution recently and I understand that it is often used to model claims frequency, I’ve also read that the Poisson Distribution assumes that events occur independently. However, isn’t this a bit of a contradiction given the policyholders within a given risk profile are clearly dependent on each other?

It’s a good question; our intrepid reader is definitely on to something here. Let’s talk through the issue and see if we can gain some clarity.

The term exposure inflation can refer to a couple of different phenomena within insurance. A friend mentioned a couple of weeks ago that he was looking up the term in the context of pricing a property cat layer and he stumbled on one of my blog posts where I use the term. Apparently my blog post was one of the top search results, and there wasn’t really much other useful info, but I was actually talking about a different type of exposure inflation, so it wasn’t really helpful for him.
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So as a public service announcement, for all those people Googling the term in the future, here are my thoughts on two types of exposure inflation:

I had to solve an interesting problem yesterday relating to pricing an excess layer which was contained in another layer which we knew the price for – I didn’t price the initial layer, and I did not have a gross loss model. All I had to go on was the overall price and a severity curve which I thought was reasonably accurate. The specific layers in this case were a 9m xs 1m, and I was interested in what we would charge for a 6m xs 4m.

Just to put some concrete numbers to this, let’s say the 9m xs 1m cost \$10m

The xs 1m severity curve was as follows: