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?

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.

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

In which we recreate the previous analysis, but in Python this time. And then add a new submission using Mean rather than median to impute missing values.





Source: https://somewan.design

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​In which we start a new Kaggle competition, submit a dummy attempt, and then build a very basic Excel model to establish a baseline for future progress.
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​Source: https://somewan.design

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.

Investopedia defines TBVPS to be:

"Tangible book value per share (TBVPS) is the value of a company’s tangible assets divided by its current outstanding shares." [1]

I'm pretty sure this is incorrect. or at best misleading!

Did you know about this cool tool, which allows you to download data from a Wikipedia table as a csv:

wikitable2csv.ggor.de/

Here's a useful trick that you might not have seen before. Suppose we have some data with  includes rows with values missing, then we can use the below formula to apply linear interpolate to fill in the missing datapoints, without having to laboriously type in the interpolation formula long hand (which I used to do all the time)

I wrote a python script which uses Selenium to scrape the predictions for Fed rate movements from the CME FedWatch tool.

www.cmegroup.com/trading/interest-rates/countdown-to-fomc.html#resources

The tool works by converting the price of a 30 day Fed Fund future into an implied probability of a given range of yields.

The CME website embeds the output in a something called an 'iframe', which I had never heard of before, and the iframe then contains a dashboard powered by something called Quikstrike. It took me a while to figure out how to switch focus to the iframe, as you can't simply reference elements inside the iframe without first switching focus.

The script below may not look too complicated, but believe me, it took a while to write.

Old Federal Reserve building Philadelphia, Source: https://commons.wikimedia.org/wiki/User:Beyond_My_Ken

The official Microsoft documentation for the Excel Forecast.ETS function is pretty weak [1]. Below I’ve written a few notes on how the function works, and the underlying formulas.