Cefor Exposure Curves - follow up

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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Source: Niels Johannes  ​https://commons.wikimedia.org/wiki/File:Ocean_Countess_(2012).jpg

Deriving the Total Loss share

Let A = the set of all partial claims, B = set of all total claims.

Then, let's extend the Cefor notation slightly to use $g(x)$ to be the value of the 'partial loss' exposure curve (Cefor notation can be found in the appendix of their presentation [1]) 
$$\frac{\sum_{A} \text{min} ( C , Vx )}{\sum_{A} C} = g(x)$$
And $f(x)$ be the 'all losses' exposure curve:
$$\frac{\sum_{A + B} \text{min} ( C , Vx )}{\sum_{A + B} C} = f(x)$$
Splitting out the top sum of the 'all losses' curve:
$$\frac{\sum_{A} \text{min} ( C , Vx )}{\sum_{A + B} C} + \frac{\sum_{B} \text{min} ( C , Vx )}{\sum_{A + B} C} = f(x)$$
By definition each $C$ for the $B$ set is a total, hence we can drop the min and just replace with $xV$:
$$\frac{\sum_{A} \text{min} ( C , Vx )}{\sum_{A + B} C} + \frac{\sum_{B} Vx}{\sum_{A + B} C} = f(x)$$
Taking $x$ outside on the right sum, and replacing $c$ with $V$ as we know each of these is a total:
$$\frac{\sum_{A} \text{min} ( C , Vx )}{\sum_{A + B} C} + x \frac{\sum_{B} C}{\sum_{A + B} C} = f(x)$$
Finally, let's replace the denominator of the left sum with $\sum_{A}$:
$$\frac{\sum_{A} \text{min} ( C , Vx )}{\sum_{A} C} \frac{\sum_{A} C}{\sum_{A+B} C} + x \frac{\sum_{B} C}{\sum_{A + B} C} = f(x)$$
But now, the left most sum is simply $g(x)$, and $\frac{\sum_{A} C}{\sum_{A+B} C}$ is the proportion of partial losses as a fraction of all losses. Which we can call $PA\%$, then we end up with a simple linear formula linking $f(x)$ and $g(x)$:

$$g(x)PA\% + x (1-PA\%) = f(x)$$

Let's test this formula, at $x = 10%$, for the $0-5M$ curve, $f(0.1) = 0.54$, and $g(0.1)=0.64$, and $PA\% = 0.82$, plugging these values in:
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$$g(x)PA\% + x (1-PA\%)x = f(x)$$
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Becomes:

$$0.64 * 0.82 + x (1-0.82)*0.1 = 0.522 + 0.018 = 0.54$$

And voila, it works.



[1] ​ https://cefor.no/globalassets/documents/statistics/nomis/2020/2020-cefor-nomis---exposure-curves-ocean-hull.pdf​ - (link to report)