On why it doesn’t really make sense to fit a Pareto distribution with a method of moments.

I was sent some large loss modelling recently by another actuary for a UK motor book. In the modelling, they had taken the historic large losses, and fit a Pareto distribution using a method of moments. I thought about it for a while and realized that it didn't really like the approach for a couple of reasons which I'll go into in more detail below, but then when I thought about it some more I realised I'd actually seen the exact approach before ... in an IFoA exam paper. So even though the method has some shortcomings, it is actually a taught technique. [1] 

Following the theme from last time, of London's old vs new side by side. Here's a cool photo which shows the old royal naval college in Greenwich, with Canary Wharf in the background. Photo by Fas Khan

​Okay, that's a bit of an exaggeration, but there’s a quirky mathematical result related to these deals which means the target loss cost can often end up clustering in a certain range. Let’s set up a dummy deal and I’ll show you what I mean.

Source:  Jim Linwood, Petticoat Lane Market, https://www.flickr.com/photos/brighton/4765025392
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I found this photo online, and I think it's a cool combo - it's got the modern City of London (the Gherkhin), a 60s brutalist-style estate (which when I looked it up online has been described as "a poor man's Barbican Estate"), and a street market which dates back to Tudor times (Petticoat lane). 

As a rule of thumb, news outlets like the Guardian [1] or BBC News [2] don't typically report on the decisions of the Delaware Court of Chancery, a fairly niche 'court of equity' which decides matters of corporate law in the state of Delaware. That is of course, unless those decisions involve Elon Musk. Recently, the Delaware court handed down a judgement which voided a /$56bn pay-out which was due to Musk for his role as Tesla’s CEO. The reasoning behind striking it down is quite legal and technical, and not really my area of expertise but Matt Levine has a good write up for those interested. [3]

What I am interested in is thinking about how we would assess the fairness of the pay-out. Now fairness is a slippery concept, but I'm going to present one angle, which I've haven't seen discussed elsewhere yet, which I think is one possible way of framing the situation.

I received a chain ladder analysis a few days ago that looked roughly like the triangle below, but there's actually a bit of a problem with how the method is dealing with this particular triangle, have a look at see if you can spot the issue. 

I’ve had the textbook 'Modelling Extremal Events: For Insurance and Finance’ sat on my shelf for a while, and last week I finally got around to working through a couple of chapters. One thing I found interesting, just around how my own approach has developed over the years, is that even though it’s quite a maths heavy book my instinct was to immediately build some toy models and play around with the results. I recall earlier in my career, when I had just got out of a 4-year maths course, I was much more inclined to understand new topics via working through proofs step-by-step in long hand, pen to paper. 

In case it’s of interest to others, I thought I’d upload my Excel version I built of the classic ruin process. In particular I was interested in how the Cramer-Lundberg theorem fails for sub-exponential distributions (which includes the very common  Lognormal distribution). Therefore the Spreadsheet contains a comparison of this theorem against the correct answer, derived from monte carlo simulation.

The Speadsheet can be found here:​
https://github.com/Lewis-Walsh/RuinTheoryModel

The first tab uses an exponential distribution, and the second uses a Lognormal distribution. Screenshot below.

​​I also coded a similar model in Python via Jupyter Notebook, which you can read about below.

I was thinking more about the post I made last week, and I realised there’s another feature of the graphs that is kind of interesting. None of the graphs adequately isolates what we in insurance would term ‘severity’ inflation. That is, the increase in the average individual verdict over time.

You might think that the bottom graph of the three, tracking the ‘Median Corporate Nuclear Verdict’ does this. If verdicts are increasing on average year by year due to social inflation, then surely the median nuclear verdict should increase as well right?!?

​Actually, the answer to this is no. Let's see why.​

​If you have seen the below graphs before, it’s probably because they've cropped up quite a few times in various insurance articles recently.

They were used by IBNR Weekly [2], Insurance Insider [3], Munich Re [4], Lockton Re [5], and those were just the ones I could remember or which I found with a 5 minute search. The graphs themselves are originally from a report on US social inflation by an organisation called Marathon Strategies (MS for short in the rest of the post) [1].

It's an interesting way of analyzing social inflation, so I recreated their analysis, which lead me to realize MS may have been a little creative in how they’ve presented the data, let me explain.

We previously introduced a method of deriving large loss claims inflation from a large loss claims bordereaux, and we then spent some time understanding how robust the method is depending on how much data we have, and how volatile the data is. In this post we're finally going to play around with making the method more accurate, rather than just poking holes in it. To do this, we are once again going to simulate data with a baked-in inflation rate (set to 5% here), and then we are going to vary the metric we are using to extract an estimate of the inflation from the data. In particular, we are going to look at using the Nth largest loss by year, where we will vary N from 1 - 20.

Photo by Julian Dik. I was recently in Losbon, so here is a cool photo of the city. Not really related to the blog post, but to be honest it's hard thinking of photos with some link to inflation, so I'm just picking nice photos as this point!

We've been playing around in the last few posts with the 'Nth largest' method of analysing claims inflation. I promised previously that I would look at the effect of increasing the volatility of our severity distribution when using the method, so that's what we are going to look at today. Interestingly it does have an effect, but it's actually quite a subdued one as we'll see.

I'm running out of  ideas for photos relating to inflation, so here's a cool random photo of New York instead. Photo by Ronny Rondon

In the last few posts I’ve been writing about deriving claims inflation using an ‘N-th largest loss’ method. The thought popped into my head after posting, that I’d made use of a normal approximation when thinking about a 95% confidence interval, when actually I already had the full Monte Carlo output, so could have just looked at the percentiles of the estimated inflation values directly. 

Below I amend the code slightly to just output this range directly.

Continuing my inflation theme, here is another cool balloon shot from João Marta Sanfins

In my last couple of post on estimating claims inflation, I’ve been writing about a method of deriving large loss inflation by looking at the median of the top X losses over time. You can read the previous posts here:

Part 1:  www.lewiswalsh.net/blog/backtesting-inflation-modelling-median-of-top-x-losses
Part 2: www.lewiswalsh.net/blog/inflation-modelling-median-of-top-10-losses-under-exposure-growth
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One issue I alluded to is that the sampling error of the basic version of the method can often be so high as to basically make the method unusable. In this post I explore how this error varies with the number of years in our sample, and try to determine the point at which the method starts to become practical.

Photo by Jøn

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. 

​The Python library SciPy, contains a version of the Lomax distribution which it defines as:
$$f(x,c) = \frac{c}{(a+x)^{(c+1)}}$$
Whereas, the ‘standard’ specification is [1]:
$$f(x,c, \lambda) = \frac{c \lambda ^ c}{(a+x)^{(c+1)}}$$
Which is also the definition in the IFoA core reading [2]:

​​So, how can we use the SciPy version of the Lomax to simulate the standard version, given we are missing the $ \lambda ^c​$ term? ​

I noticed something surprising about the Maximum Likelihood Estimator (MLE) for a uniform distribution yesterday.

Suppose we’re given sample $X’ = {x_1, x_2, … x_n}$ from a uniform distribution $X$ with parameters $a,b$. Then the MLE estimator for $a = min(X’)$, and $b = max(X’)$. [1] All straight forward so far.

However, examining the estimators, we can also say with probability = 1 that $a < min(X’)$, and similarly that $b > max(X’)$. Isn't it strange that the MLE estimators are clearly less/more than the true values? 

So what can we do instead?