Actuarial Outpost Fitting severity distribution - commercial pricing
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#1
08-10-2018, 02:02 PM
 TDH Member CAS Non-Actuary Join Date: Dec 2016 Posts: 38
Fitting severity distribution - commercial pricing

I've inherited a model for management liability insurance, where the original data is unavailable. Due to the nature of the class, losses are quite rare and it's quite difficult to get data without first paying a consultancy (think security class action suits). I don't want to completely throw the model away as it has been fairly accurate thus far, but the pricing methodology is something I've not seen before.

Let's say the exposure measure is turnover, the severity distribution is then a lognormal with parameters mu and sigma. The "mu" is derived based on the exposure using the formula:

mu = a * ln(exposure) + b

Where a and b are constants which look to be derived from some sort of fitting routine. Has anyone seen anything like this before in practice? Most of the commercial lines models will have some base rates then an ILF (usually a lognormal) which is used to price high excess policies. In this model we have a full distribution of losses, but I just don't understand how the mu came about.

I've tested the outcome of the distribution and things move in the right way - exposure -> tail gets fatter and expected cost goes up; but I am not sure how the formula for mu came around. Any help would be great!
#2
08-10-2018, 03:40 PM
 NormalDan Member CAS Join Date: Dec 2016 Location: NJ Posts: 6,612

I'd imagine the parameters were fit based on the moments of some sample, perhaps with some tweaks to smooth it across the entire data set.
#3
08-10-2018, 05:12 PM
 TDH Member CAS Non-Actuary Join Date: Dec 2016 Posts: 38

Quote:
 Originally Posted by NormalDan I'd imagine the parameters were fit based on the moments of some sample, perhaps with some tweaks to smooth it across the entire data set.
Could you explain the reasoning behind this? I am not sure where the log comes into it.
#4
08-10-2018, 05:55 PM
 examsarehard Member CAS Join Date: May 2011 Posts: 551

That looks like a power transform. The expected value of a lognormal distribution is:
$E(x)=e^{\mu+\frac{\sigma^2}{2}}$

So if you do the substitution, you get something of the form:

E(x) = c*exposures^a

I suspect a is greater than 1, and whoever created the distribution wanted the losses to scale more than linearly with exposure, which is a reasonable assumption for Management Liability.
#5
08-11-2018, 12:43 PM
 TDH Member CAS Non-Actuary Join Date: Dec 2016 Posts: 38

Quote:
 Originally Posted by examsarehard That looks like a power transform. The expected value of a lognormal distribution is: $E(x)=e^{\mu+\frac{\sigma^2}{2}}$ So if you do the substitution, you get something of the form: E(x) = c*exposures^a I suspect a is greater than 1, and whoever created the distribution wanted the losses to scale more than linearly with exposure, which is a reasonable assumption for Management Liability.
Thanks - do you have any reading material on how this would be done in practice? E.g. if you had exposure and loss data, how would you come up with this approximation for c and a.
#6
08-12-2018, 02:33 PM
 examsarehard Member CAS Join Date: May 2011 Posts: 551

Quote:
 Originally Posted by TDH Thanks - do you have any reading material on how this would be done in practice? E.g. if you had exposure and loss data, how would you come up with this approximation for c and a.
Assuming you have sufficient data, the estimation methods are covered under Exam C. In practice you would use something like the fitdistrplus package in R or some other software to do this.

What will probably end up happening is that you will find you don't have enough data to fit a model that passes reasonability checks. In that case there are several approaches you can take, but they basically involve your own judgement.
#7
08-12-2018, 03:43 PM
 TDH Member CAS Non-Actuary Join Date: Dec 2016 Posts: 38

Quote:
 Originally Posted by examsarehard Assuming you have sufficient data, the estimation methods are covered under Exam C. In practice you would use something like the fitdistrplus package in R or some other software to do this. What will probably end up happening is that you will find you don't have enough data to fit a model that passes reasonability checks. In that case there are several approaches you can take, but they basically involve your own judgement.

I'm aware of the various methods of fitting for a specific distribution, e.g. lognormal, pareto etc - however, I am just curious on how they got the formula for mu. How do you fit to both claims and exposure?
#8
08-13-2018, 12:21 AM
 therealsylvos Member CAS Join Date: Mar 2014 Posts: 16,003

Quote:
 Originally Posted by TDH I unfortunately don't have access to the material under Exam C. I'm aware of the various methods of fitting for a specific distribution, e.g. lognormal, pareto etc - however, I am just curious on how they got the formula for mu. How do you fit to both claims and exposure?
Your average claims (mu) is known (after trending and developing them of course). And your exposures are known.

So you'll have some matrix that looks like

Code:
1 u1 exp1
2 u2 exp2
3 u3 exp3
etc.

There are various tools to then fit your parameters. R has nonlinear fit, SAS can do it. Even excel has a built it function (logest) but that's not the best.

If you were doing it in excel you could add a third column with a prediction.

Put some seed value in F1, that's your estimate of a. Put another seed value in G1, that's the value of b. You want to choose something close to the right numbers or else it may not converge, so use the current estimates as a starting point. Then in column d enter =$F$1*LN(C1)+$G$1

Then in column E you can calculate the Square Error of your prediction. Sum up the SEs. Use the solver add-in to minimize the cell with the SSE by changing the values in the cells F1:G1, and presto, estimates for a and b.
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Last edited by therealsylvos; 08-13-2018 at 12:25 AM..
#9
08-13-2018, 03:58 AM
 TDH Member CAS Non-Actuary Join Date: Dec 2016 Posts: 38

Quote:
 Originally Posted by therealsylvos Your average claims (mu) is known (after trending and developing them of course). And your exposures are known. So you'll have some matrix that looks like Code: 1 u1 exp1 2 u2 exp2 3 u3 exp3 etc. There are various tools to then fit your parameters. R has nonlinear fit, SAS can do it. Even excel has a built it function (logest) but that's not the best. If you were doing it in excel you could add a third column with a prediction. Put some seed value in F1, that's your estimate of a. Put another seed value in G1, that's the value of b. You want to choose something close to the right numbers or else it may not converge, so use the current estimates as a starting point. Then in column d enter =$F$1*LN(C1)+$G$1 Then in column E you can calculate the Square Error of your prediction. Sum up the SEs. Use the solver add-in to minimize the cell with the SSE by changing the values in the cells F1:G1, and presto, estimates for a and b.
Thanks, but this is slightly different no? Our mu here is not the trended losses, it's the parameter for the lognormal; and it's mu = a * log(exposure) + b.

We can repeat your procedure by first parameterising mu, then finding the expected cost using the lognormal e^(mu + 0.5 sigma ^2) and finding the squared difference, but my question is where is the intuition that mu itself is in the form mu = a * log(exposure) + b?
#10
08-13-2018, 08:48 AM
 examsarehard Member CAS Join Date: May 2011 Posts: 551

The intuition is that your exposure base doesn't scale correctly across the size of accounts and the losses you would see in the data. I suspect the power transform was used because it is easy to find a closed form solution to the model fit.

This would be a good chance to revisit that assumption, but I also find it very odd that your company would give the model to you without any guidance.