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Rank-Based Multivariate Sarmanov for Modeling Dependence

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Anas Abdallah discusses the rank-based methods using the Sarmanov distribution to adequately estimate the loss reserves and properly capture the dependence between lines of business.

Image courtesy of interviewee. January 22, 2024

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This video video discusses the context of loss reserves for non-life insurance companies, differentiating them from life insurance companies. Non-life insurance covers areas such as auto and home insurance, requiring companies to set aside funds for future claims and to handle claims that have occurred but not yet been reported. The importance of managing reserves effectively, particularly considering the independence and interdependence of risks (e.g., due to natural disasters or pandemics), is emphasized.

Two main methodologies are presented for estimating dependence between risks in loss reserving: a multivariate approach using non-parametric models and a parametric approach based on copulas, which capture dependencies between risks. The rank-based method for estimating dependence is highlighted as a more robust option compared to conventional simultaneous estimation methods.

The proposed methodology combines rank-based estimation with flexible distributions, showing that this approach captures risk interdependencies better, thereby leading to a reduction in required risk capital and enhanced diversification benefits for insurance companies. The results from applying this methodology to real datasets from U.S. and Canadian insurance companies demonstrate its effectiveness.

This video video concludes by noting the relevance of this model in light of international regulations (IFRS 17), emphasizing its potential for improving solvency risk assessments and contributing positively to the economy and society as a whole.

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So this is the context and the, loss reserve

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for the non-life insurance companies.

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So for insurance companies, we have life insurance companies

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and non-life insurance companies.

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So non-life insurance companies are everything,

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which is not life insurance companies.

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So we can think of auto insurance, home insurance,

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and, and, and so on.

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Commercial insurance farm, et cetera.

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So the idea here is for lost reserves.

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The idea is to be able to kind of set aside some money,

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basically for future payments, right?

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Basically is for payments that are actually being paid,

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but not enough, not not settled,

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or also for, claims that actually occurred,

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but not reported.

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So all for those claims, the, the company needs

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to set aside some, reserves

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and also some capital to be able, basically to be, to pay,

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those kind of, payments for their,

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their claimants and policy holders.

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So what happens here in insurance companies also in

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practice, that we assume independence between risks.

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So I mentioned some kind of lines of business such as,

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auto insurance or also, or commercial.

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Uh, but in practice, while calculating the reserve

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or also kind of setting those risk capital,

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it is assumed that those risks are kind of independent.

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However, we could observe that there is kind

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of interdependency between those risks,

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which can be explained by, several factors.

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Uh, we can think, for example, of natural disasters,

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or pandemics that might actually,

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impact different risks at the same time,

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we could also think of legal,

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or regulatory changes,

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which also might affect different risks at the same time.

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And this should be, taken into account, not

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to mention operational risks

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or, even more timely now, the climate change, right?

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So with an increase, for example, in the frequency

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and severity, of weather, related, events.

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Um, lastly, I could also mention, at a granular level.

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So at this, what I mentioned now is more macro,

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but we could also think of micro level.

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So for example, the same person might be kind

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of have a tendency to basically kind

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of claim in several kind of, coverages, right?

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So they could actually claim, in, in, in home insurance,

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but also have a tendency to claim in auto insurance.

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And this will also can lead to a kind

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of a dependence between the risks.

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And this dependence has

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to be taken into account while basically calculating the

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reserve, for insurance company,

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basically model dependence between the risks.

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There are two schools of thought or two main mainstreams.

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Uh, basically, so we have the more multivariate multi

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reserve in, which assume, a non-prem model,

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which is basically distribution free,

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which are basically extensions of the,

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so-called chain ladder method, which is the kind

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of the benchmark, or the MAC model.

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Uh, so a lot of work had been done in literature,

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For doing that.

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And we have also the parametric approach, which is,

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basically mainly, based on caps, which is basically,

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one of the, most, powerful statistical tools

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to capture dependencies between risks.

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Uh, so caps, have been done a lot in, in the literature.

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And also, we do have, the, well, the sermon

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of family of distributions

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that also have been actually introduced to capture,

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dependencies between risks.

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So those are the tools. Okay?

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Uh, so the, the, the sermon of distribution,

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will basically has been proven to be more flexible,

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for example, than capitalist on high

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or higher triple capitalis.

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Uh, so in terms of tools, the sermon

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of distribution is a flexible tool.

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So now in terms of methods

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or methodologies, so to estimate dependence between risks,

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so with, again, mentioned two main, well streams

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or, approaches.

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So we have the, what we call the simultaneous approach,

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where we're actually going to estimate the dependence,

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factor or depends parameter at the same

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time as the marginal.

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So here we have, let's assume two risks, so X one and X two.

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So if you want to basically estimate the dependence

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between X one and X two, we're going

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to use a maximum likelihood summation.

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And we're going to have a simultaneous estimation

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of the marginals.

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By marginal, I mean the parameters for the risk X one

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and X two, but also the dependence

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parameter between the two.

04:27.185 --> 04:31.885
Uh, so the, the issue of doing that, so we have first

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of all, well, a lot of parameters,

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which actually can lead to over fitting, but also,

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and more importantly, sometimes a misestimating

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of the marginals can lead to a mis misestimating

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of the dependent structure and vice versa, right?

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So if you misestimate the dependent structure,

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this can also mis, misestimate the, the marginals.

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And if you misestimate the marginals, it means, it means

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that it'll actually have a mises

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estimation to the reserves, right?

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Because the reserves is based on the,

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parameters estimated on the marginals.

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So, so then comm's, the, a more robust way, a robust way

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to kind of estimate the, dependence,

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parameter is well based on the rank based method,

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which has been actually, brought, earlier

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to the literature,

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but has been actually introduced

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to the loss reserving context in, about 2016.

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Uh, so, which actually, proved

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to be more robust method to estimate dependence.

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So here in this paper, what we're doing, we're using a more,

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that more robust estimation, which is the rank based method,

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and combine it with the flexible family

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of distributions, which is the sarov.

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And then we use actually the two,

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which is we have a flexible tool, which is the sarov

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and the rob more robs, estimation, method,

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which is basically the, the rank based method.

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And then we use them actually in this paper,

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to estimate the dependence between the risks.

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I insisted on the flexibility on the salmon of distribution,

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which also provides close form expressions, which is,

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a very, interesting result, which is not always

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provided what the kaa, methodologies

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and the sarov distribution was basically used in various

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areas in for the first applications, interestingly,

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was more in the medical, medical area, right?

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Uh, so then here we're bringing it more to the, to the,

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reserve in context.

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Uh, so that's for the sarov.

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And then we also use the, well,

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the GRM for the marginals.

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So the generalized, the neuro models.

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Uh, so here the cover rates when we work with lost reserve.

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So when we work with actually lost triangles,

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the cover rates or explanatory variables would be basically

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the accident years.

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So the rows, so that would be, cover rates

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and also the development period.

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So the column, so, so this is basically the cover rates

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that we use, for the lost triangle.

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And what happens is for this methodology, so we, first,

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it's a two step inference, right?

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So it's a two stage inference method.

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So first we estimate the marginals, okay?

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And then we get the parameters for the marginals.

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From there, we calculate the ILS right from those,

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marginals obtained by the GM.

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After that, we calculate their rank of renewals,

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and then we actually use the, so the likelihood

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to calculate, well, of course, using those rank of renewals

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to, calculate the dependence parameter of the summon

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of distribution for both bate or tra triver it.

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So also, one of the, nice feature of the, of the,

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of the model is it's easily extendable to,

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we could easily extend it

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to more than two lines of business.

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So we use that for three risks, for example,

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but that could be easily extended

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to more than, three risks.

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So in terms of the main results, so we actually applied,

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our methodologies to seven lines

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of business from a real dataset, from, both a major us,

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property and casualty insurance company,

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but also a large Canadian, insurance company.

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And the results were really consistent in a way.

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So two things in particular.

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So the first thing is all the kind of,

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rank based mol method are proved

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to actually better capture the dependence than

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the one stage inference.

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So recall when I actually mentioned those,

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two types of summations.

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So the simultaneous one,

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the one one step inference versus the rank based method,

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which is caught up a two step.

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So the two step or two stage inference basically captured

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better the dependence than the one step inference.

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So for example, if I give you a line of business,

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let's say one and two,

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and we use the approach classical approach one,

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which is the one step inference, it did actually, fail

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to capture the dependence.

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So basically what means

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that the dependence is not significant if we use that.

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But when we actually model the dependence

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between this very same lines of business one

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and two, using the two stage inference,

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we were able actually to kind of capture the dependence

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and the de de dependence ended up being, significant.

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So that's the first, finding.

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The second finding, earlier I was discussing,

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for the insurance company.

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So of course, the insurance company has an inverted

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production cycle, right?

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So the idea, because, what I was mentioning earlier

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is basically we, receive, the, the, the, the,

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the, the, actually the, the, the, the, the, the price

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before knowing the cost.

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So the price is the premium

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before knowing the cost, which is basically the claim.

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So the insurance company has actually to estimate

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what is the cost, and then after,

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after that, basically estimating, the price of that.

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So this is interesting for the insurance production cycle,

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because it is inverted, unlike most

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of the other production cycles, if we are selling a shoe,

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for example, we know exactly how much it did cost us, right?

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So, and then in addition to that,

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we calculate the profit, and then we have the price.

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So here it's inverted.

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So we need to calculate that risk,

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but also have a kind of an expected fall, which is a kind

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of a, a buffer for adverse scenarios.

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So, and for that, we need to calculate,

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risk measures measure, on also

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what we call the risk capital.

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And the method we suggested, proved that we actually

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are able to reduce that risk capital.

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So we, we reduce the risk capital using the rank based,

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multi-grade mov.

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And this has a direct consequence

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for the insurance companies

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because it actually,

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provides a better diversification benefits, right?

10:33.205 --> 10:36.045
So we have more gains in terms of the risk capital,

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which is, a tangible, consequence here

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for the insurance industry.

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We proved that the, the,

10:42.745 --> 10:46.645
the proposed model outperformed classical models,

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for both capture and dependence between the risks,

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and also in terms of providing, a,

10:52.645 --> 10:54.405
a greater diversification benefit.

10:54.705 --> 10:58.285
So that's, a, a very important, result.

10:58.585 --> 11:00.805
Uh, of course, each work can be

11:01.045 --> 11:02.245
extended and can be improved.

11:02.665 --> 11:05.805
Uh, so here we discussed two step inference.

11:05.865 --> 11:07.645
So one step was for the marginals.

11:07.905 --> 11:11.605
So we could basically, improve, the, for the,

11:11.605 --> 11:15.245
the marginals here we could, for example, so we use the GLM,

11:15.345 --> 11:17.805
but we could also use, use for example, generalized,

11:17.835 --> 11:19.005
partial linear models

11:19.065 --> 11:22.005
or the GPLM, which incorporates both linear

11:22.105 --> 11:23.245
and nonlinear components.

11:23.465 --> 11:25.765
So that could basically,

11:25.875 --> 11:27.805
improve the result for the marginals.

11:28.105 --> 11:30.365
Uh, and same for the dependent structure.

11:30.425 --> 11:34.125
So we could also think, for example, we use here the OV for,

11:34.545 --> 11:35.765
aggregated data,

11:36.025 --> 11:38.845
but we could also think, for example, to extend it

11:38.845 --> 11:40.005
to granular data.

11:40.145 --> 11:41.965
So here, which is basically instead

11:42.125 --> 11:46.325
of having a lost triangle, instead of having basically,

11:46.465 --> 11:49.165
to capture dependence between two lines

11:49.165 --> 11:50.205
of business, right?

11:50.345 --> 11:52.725
So using aggregate data, so we could basically

11:52.745 --> 11:55.165
Use it for individual data, right?

11:55.225 --> 11:59.445
So here we could, access more granular, information,

11:59.515 --> 12:00.725
more detailed information.

12:01.065 --> 12:03.805
And this also could reduce, overfitting

12:03.805 --> 12:06.165
because we have, access

12:06.225 --> 12:08.165
to more information enhanced more data.

12:08.545 --> 12:11.165
And of course, last

12:11.185 --> 12:14.245
but not least, for industry professionals,

12:14.315 --> 12:16.405
this is a very important,

12:16.405 --> 12:19.325
because we want the resource as well to be applicable,

12:19.585 --> 12:22.125
and, applied also in the industry.

12:22.545 --> 12:25.605
And one of the, benefits is that the,

12:25.605 --> 12:29.285
these model can be very valuable resource for the property

12:29.285 --> 12:30.805
and casualty insurance companies.

12:31.225 --> 12:33.445
Uh, and especially now with,

12:33.505 --> 12:37.965
the International Financial Reporting standards IFRS 17,

12:37.965 --> 12:39.245
this could be a very good tool

12:39.265 --> 12:42.765
to aid them in meeting those IRS 17, regulations.

12:43.065 --> 12:45.725
And also, of course, while, enhancing their,

12:45.725 --> 12:47.165
solvency risk assessment.

12:47.345 --> 12:50.165
And if this actually happens,

12:50.165 --> 12:53.685
this is actually very beneficial for the, economy,

12:53.785 --> 12:54.925
but also for the society,

12:54.925 --> 12:57.205
because it'll have a positive, economic

12:57.265 --> 13:00.645
and societal impact, by improving the insurance company.

13:00.985 --> 13:03.165
Uh, so solvency ratio.

00:00 - So this is the context and the, loss reserve for the non-life insurance companies. So for insurance companies, we have life insurance companies and non-life insurance companies. So non-life insurance companies are everything, which is not life insurance companies.

00:13 - So we can think of auto insurance, home insurance, and, and, and so on. Commercial insurance farm, et cetera. So the idea here is for lost reserves. The idea is to be able to kind of set aside some money, basically for future payments, right? Basically is for payments that are actually being paid, but not enough, not not settled, or also for, claims that actually occurred, but not reported.

00:42 - So all for those claims, the, the company needs to set aside some, reserves and also some capital to be able, basically to be, to pay, those kind of, payments for their, their claimants and policy holders. So what happens here in insurance companies also in practice, that we assume independence between risks. So I mentioned some kind of lines of business such as, auto insurance or also, or commercial. but in practice, while calculating the reserve

01:12 - or also kind of setting those risk capital, it is assumed that those risks are kind of independent. However, we could observe that there is kind of interdependency between those risks, which can be explained by, several factors. we can think, for example, of natural disasters, or pandemics that might actually, impact different risks at the same time, we could also think of legal, or regulatory changes,

01:41 - which also might affect different risks at the same time. And this should be, taken into account, not to mention operational risks or, even more timely now, the climate change, right? So with an increase, for example, in the frequency and severity, of weather, related, events. lastly, I could also mention, at a granular level. So at this, what I mentioned now is more macro, but we could also think of micro level. So for example, the same person might be kind

02:10 - of have a tendency to basically kind of claim in several kind of, coverages, right? So they could actually claim, in, in, in home insurance, but also have a tendency to claim in auto insurance. And this will also can lead to a kind of a dependence between the risks. And this dependence has to be taken into account while basically calculating the reserve, for insurance company, basically model dependence between the risks.

02:35 - There are two schools of thought or two main mainstreams. basically, so we have the more multivariate multi reserve in, which assume, a non-prem model, which is basically distribution free, which are basically extensions of the, so-called chain ladder method, which is the kind of the benchmark, or the MAC model. so a lot of work had been done in literature, For doing that. And we have also the parametric approach, which is,

03:03 - basically mainly, based on caps, which is basically, one of the, most, powerful statistical tools to capture dependencies between risks. so caps, have been done a lot in, in the literature. And also, we do have, the, well, the sermon of family of distributions that also have been actually introduced to capture, dependencies between risks. So those are the tools. Okay? so the, the, the sermon of distribution,

03:33 - will basically has been proven to be more flexible, for example, than capitalist on high or higher triple capitalis. so in terms of tools, the sermon of distribution is a flexible tool. So now in terms of methods or methodologies, so to estimate dependence between risks, so with, again, mentioned two main, well streams or, approaches. So we have the, what we call the simultaneous approach,

03:57 - where we're actually going to estimate the dependence, factor or depends parameter at the same time as the marginal. So here we have, let's assume two risks, so X one and X two. So if you want to basically estimate the dependence between X one and X two, we're going to use a maximum likelihood summation. And we're going to have a simultaneous estimation of the marginals. By marginal, I mean the parameters for the risk X one

04:23 - and X two, but also the dependence parameter between the two. so the, the issue of doing that, so we have first of all, well, a lot of parameters, which actually can lead to over fitting, but also, and more importantly, sometimes a misestimating of the marginals can lead to a mis misestimating of the dependent structure and vice versa, right? So if you misestimate the dependent structure, this can also mis, misestimate the, the marginals.

04:53 - And if you misestimate the marginals, it means, it means that it'll actually have a mises estimation to the reserves, right? Because the reserves is based on the, parameters estimated on the marginals. So, so then comm's, the, a more robust way, a robust way to kind of estimate the, dependence, parameter is well based on the rank based method, which has been actually, brought, earlier to the literature,

05:19 - but has been actually introduced to the loss reserving context in, about 2016. so, which actually, proved to be more robust method to estimate dependence. So here in this paper, what we're doing, we're using a more, that more robust estimation, which is the rank based method, and combine it with the flexible family of distributions, which is the sarov. And then we use actually the two, which is we have a flexible tool, which is the sarov

05:44 - and the rob more robs, estimation, method, which is basically the, the rank based method. And then we use them actually in this paper, to estimate the dependence between the risks. I insisted on the flexibility on the salmon of distribution, which also provides close form expressions, which is, a very, interesting result, which is not always provided what the kaa, methodologies and the sarov distribution was basically used in various areas in for the first applications, interestingly,

06:16 - was more in the medical, medical area, right? so then here we're bringing it more to the, to the, reserve in context. so that's for the sarov. And then we also use the, well, the GRM for the marginals. So the generalized, the neuro models. so here the cover rates when we work with lost reserve. So when we work with actually lost triangles, the cover rates or explanatory variables would be basically

06:42 - the accident years. So the rows, so that would be, cover rates and also the development period. So the column, so, so this is basically the cover rates that we use, for the lost triangle. And what happens is for this methodology, so we, first, it's a two step inference, right? So it's a two stage inference method. So first we estimate the marginals, okay? And then we get the parameters for the marginals.

07:06 - From there, we calculate the ILS right from those, marginals obtained by the GM. After that, we calculate their rank of renewals, and then we actually use the, so the likelihood to calculate, well, of course, using those rank of renewals to, calculate the dependence parameter of the summon of distribution for both bate or tra triver it. So also, one of the, nice feature of the, of the, of the model is it's easily extendable to, we could easily extend it

07:37 - to more than two lines of business. So we use that for three risks, for example, but that could be easily extended to more than, three risks. So in terms of the main results, so we actually applied, our methodologies to seven lines of business from a real dataset, from, both a major us, property and casualty insurance company, but also a large Canadian, insurance company. And the results were really consistent in a way.

08:05 - So two things in particular. So the first thing is all the kind of, rank based mol method are proved to actually better capture the dependence than the one stage inference. So recall when I actually mentioned those, two types of summations. So the simultaneous one, the one one step inference versus the rank based method, which is caught up a two step.

08:27 - So the two step or two stage inference basically captured better the dependence than the one step inference. So for example, if I give you a line of business, let's say one and two, and we use the approach classical approach one, which is the one step inference, it did actually, fail to capture the dependence. So basically what means that the dependence is not significant if we use that. But when we actually model the dependence

08:51 - between this very same lines of business one and two, using the two stage inference, we were able actually to kind of capture the dependence and the de de dependence ended up being, significant. So that's the first, finding. The second finding, earlier I was discussing, for the insurance company. So of course, the insurance company has an inverted production cycle, right? So the idea, because, what I was mentioning earlier

09:16 - is basically we, receive, the, the, the, the, the, the, actually the, the, the, the, the, the price before knowing the cost. So the price is the premium before knowing the cost, which is basically the claim. So the insurance company has actually to estimate what is the cost, and then after, after that, basically estimating, the price of that. So this is interesting for the insurance production cycle, because it is inverted, unlike most

09:45 - of the other production cycles, if we are selling a shoe, for example, we know exactly how much it did cost us, right? So, and then in addition to that, we calculate the profit, and then we have the price. So here it's inverted. So we need to calculate that risk, but also have a kind of an expected fall, which is a kind of a, a buffer for adverse scenarios. So, and for that, we need to calculate, risk measures measure, on also

10:10 - what we call the risk capital. And the method we suggested, proved that we actually are able to reduce that risk capital. So we, we reduce the risk capital using the rank based, multi-grade mov. And this has a direct consequence for the insurance companies because it actually, provides a better diversification benefits, right? So we have more gains in terms of the risk capital,

10:36 - which is, a tangible, consequence here for the insurance industry. We proved that the, the, the proposed model outperformed classical models, for both capture and dependence between the risks, and also in terms of providing, a, a greater diversification benefit. So that's, a, a very important, result. of course, each work can be extended and can be improved.

11:02 - so here we discussed two step inference. So one step was for the marginals. So we could basically, improve, the, for the, the marginals here we could, for example, so we use the GLM, but we could also use, use for example, generalized, partial linear models or the GPLM, which incorporates both linear and nonlinear components. So that could basically, improve the result for the marginals.

11:28 - and same for the dependent structure. So we could also think, for example, we use here the OV for, aggregated data, but we could also think, for example, to extend it to granular data. So here, which is basically instead of having a lost triangle, instead of having basically, to capture dependence between two lines of business, right? So using aggregate data, so we could basically

11:52 - Use it for individual data, right? So here we could, access more granular, information, more detailed information. And this also could reduce, overfitting because we have, access to more information enhanced more data. And of course, last but not least, for industry professionals, this is a very important, because we want the resource as well to be applicable,

12:19 - and, applied also in the industry. And one of the, benefits is that the, these model can be very valuable resource for the property and casualty insurance companies. and especially now with, the International Financial Reporting standards IFRS 17, this could be a very good tool to aid them in meeting those IRS 17, regulations. And also, of course, while, enhancing their, solvency risk assessment.

12:47 - And if this actually happens, this is actually very beneficial for the, economy, but also for the society, because it'll have a positive, economic and societal impact, by improving the insurance company. so solvency ratio.

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