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 4year maths course, I was much more inclined to understand new topics via working through proofs stepbystep 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 CramerLundberg theorem fails for subexponential 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/LewisWalsh/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. Nuclear Verdicts and shenanigans with graphs18/10/2023 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 bakedin 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 ‘Nth 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/backtestinginflationmodellingmedianoftopxlosses Part 2: www.lewiswalsh.net/blog/inflationmodellingmedianoftop10lossesunderexposuregrowth 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/backtestinginflationmodellingmedianoftopxlosses 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
Capped or uncapped estimators9/6/2023 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. Source: Dall.E 2, Open AI.
I thought it would be fun to include an AI generated image which was linked to the title 'capped vs uncapped estimators'. After scrolling through tons of fairly creepy images of weird looking robots with caps on, I found the following, which is definitely my favourite  it's basically an image of a computer 'wearing' a cap. A 'capped' estimator... The Lomax pareto distribution in SciPy17/3/2023 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?
MLE of a Uniform Distribution28/2/2023 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? (Since Gauss did a lot of the early work on MLE, here's a portrait of him as a young man. )
Source: https://commons.wikimedia.org/wiki/File:Bendixen__Carl_Friedrich_Gau%C3%9F,_1828.jpg What is a 'Net' Quota Share?7/10/2022 I recently received an email from a reader asking a couple of questions : "I'm trying to understand Net vs Gross Quota shares in reinsurance. Is a 'Net Quota Share' always defining a treaty where the reinsurer will pay ceding commissions on the Net Written Premium? ... Are there some Net Quota Shares where the reinsurer caps certain risks (e.g. catastrophe)?" It's a reasonable question, and the answer is a little context dependent, full explanation given below. Source: https://unsplash.com/@laurachouette, London
(As an aside, in the last couple of weeks, the UK has lurched from what was a rather pleasant summer into a fairly chilly autumn, to mirror this, here's a photo of London looking a little on the grey side.) 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 predates 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
An Actuary learns Machine Learning  Part 13  Kaggle Tabular Playground Competition  June 221/7/2022 An Actuary learns Machine Learning  Part 12  Kaggle Tabular Playground Competition  June 2224/6/2022 
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