# Pricing Squad

## Welcome back to Pricing Squad!

Pricing Squad is a newsletter for fellow pricing practitioners and actuaries in general insurance.

Today's issue is on measuring strength of pricing factors.

You can also read a review of *Non-Life Insurance Pricing with Generalized Linear Models*
by Esbjörn Ohlsson and Bjorn Johansson.

## Factor strength

A typical pricing model has between 10 and 30 rating factors. This is plenty so I like to have a way to rank and quantify these rating factors.

One cool method is called "factor strength".

Factor strength has an intuitive interpretation. Strength 0% means the factor does nothing. Strength 30% means that on average the factor influences premiums by +/- 30%. And so on.

For example, if a portfolio is equally distributed across three regions rated -15%, 0% and +15% respectively then factor strength for "region" would be (15% + 0% + 15%) / 3 = 10%.

Here is how you calculate factor strength in five simple steps.

- For each level of this pricing factor take average premium with and without this pricing factor and then calculate their ratio.
- Calculate exposure weighted average of this ratio.
- For each level, normalise the ratio by its exposure weighted average - to allow us not to worry about the choice of base level.
- For each level, calculate a percentage strength as the absolute value of the normalised ratio minus one.
- Calculate exposure weighted average of this percentage strength.

Easy, right?

You can do this for linear and non-linear models, combined or by peril.

Here are typical rating strengths for a few pricing factor in private motor.

- age of driver - 30%
- no claim discount - 26%
- credit score - 24%
- area - 20%
- car group - 9%
- marital status - 5%
- driver's occupation - 0.5%

You would be surprised how much time companies spend being unfocussed or analysing the wrong parts of their models.

If your company is like that too, now you can help them.

## Book review: *Non-Life Insurance Pricing with Generalized Linear Models* by Esbjörn Ohlsson

Non-Life Insurance Pricing with Generalized Linear Models is a great pricing book for three reasons.

Firstly, it congruently covers basic pricing, credibility models, GLMs, GAMs and splines. All essential aspects are covered. This is done clearly, exhaustively and using only 133 pages. There are just enough examples to succinctly demonstrate the various theories. The authors must have spent plenty of time organising the content in the best possible way so that you do not have to.

Secondly, the book offers a great explanation of methods which are more advanced than standard GLMs: GAMs (Generalized Additive Models), splines, combining GLMs with credibility theory and more. For instance, one can learn from it how GAMs select the optimal function f(x) from a certain class of functions that will best predict risk of a numeric risk factor x. GAMs are powerful and allow things like automated 2D- smoothing of geographic factors. The book does a fine job of explaining the potential and maths of GAMs.

I welcomed the section on Method of Marginal Totals which I used many times as a heuristic unaware that it had a proper name and a theoretical framework. Method of Marginal Totals allows fitting pricing factors to historical data and while relying on goodness of fit by one-way cuts only. The Pricing Squad from January 2017 shows one ways of implementing this approach.

Thirdly, the book maintains good mathematical rigour and elegance rare for actuarial textbooks. The narrative includes numbered theorems and lemmas with proofs. About 20% of the book consists of appendices which diligently cover the probability theory and the algebra underlying the methods covered is earlier chapters.

"Non-Life Insurance Pricing with Generalized Linear Models" is a great book for the actuary who wants to understand the workings for GLM, GAM and other advanced methods. It is also a must read for anyone programming these methods because it comprehensively covers the relevant algebra.

## Do you need support?

If you need access to pricing tools to radically simplify your work and deliver reduced loss ratio quickly, or if you are simply looking for an actuarial contractor, get in touch.

Thank you for reading, and have a great day,

*Jan Iwanik, FIA PhD*

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