# Insurance: Mathematics and Economics

### Volume 34, Issue 1, Pages 1-125 (20 February 2004)

**Quantification of automobile insurance liability: a Bayesian failure time approach **

*Pages 1-21 *

David A. Stephens, Martin J. Crowder and Petros Dellaportas

This paper models the payout of the current inventory of claims. The paper is based on Greek auto data where claims have to be reported in three working days, so the number of claims is known very quickly. The paper sets up a three stage Bayesian hierarchical model: number of claims in a period (to allow for modeling future claims), when do the claims close, and amount of each claim. The unique feature of the author's model is to regard claim closure as a failure/survival time analysis and model time to closure using a hazard function. Thus the model makes no assumption that claims will be closed in a cer-tain length of time, as is typically implicit in triangle based methods. The authors de-scribe MCMC-based simulations from their model and apply it to their Greek auto loss data.

The paper is mathematically and notationally challenging, and readers not familiar with MCMC simulation techniques may be confused about how to replicate the simulation re-sults shown in the paper. It would be helpful to have WinBUGS code to do this if there are any volunteers. The restriction to known claims is a limitation of the model for US actuaries.

The author's abstract is available on the journal's web site.

**Modelling zeros in stochastic reserving models**

*Pages 23-35 *

Michael Kunkler

Kunkler's paper extends the Zehnwirth-type regression models to allow zero observa-tions. Recall that most regression based development models work with incremental data and use a log transformation, so incremental payments are lognormal. A weakness of this approach is that incremental changes less than or equal to zero are not allowed. Here the author uses a mixture approach to allow zero (but not negative) claims. The method is clearly described and is easy to understand. Incremental losses for each evaluation are zero with probability p and lognormal with probability 1-p. Thus, there is one extra pa-rameter p to estimate over the standard model. The author explains a Bayesian approach to parameterization.

Kunkler applies his enhanced method to the reinsurance loss data from Mack's 1994 IME paper (also published in the CAS Forum). The original data actually contained negative incrementals, which he replaces with zeros. He shows that correctly accounting for the possibility of zero incremental payments results in a lower ultimate estimate than simply assigning missing values to the non-positive incrementals.

The author's abstract is available on the journal's web site.

**A seemingly unrelated regression model in a credibility framework**

*Pages 37-54 *

Georgios Pitselis

Paper Abstract: There are various formulations for the randomly varying behavior of re-gression coefficients, which can be applied to credibility estimation. In this paper two different credibility estimation models are proposed, the credibility model of seemingly unrelated regressions (SUR) with fixed coefficients and of SUR model with random coef-ficients. For each model, unbiased credibility estimators are established, asymptotic op-timality is proved and an application to real data is presented.

Pitselis' paper generalizes Hachemeister's regression credibility model to allow contracts to be subdivided into subunits, each with certain attributes such as geographic location or industry class. The author uses a method called Seemingly Unrelated Regressions (SUR) introduced in the 1960's to econometrics by Zellner. Pitselis claims SURs are "consid-ered as one of the most successful and lasting innovations in the history of economet-rics." SURs are described briefly (one half-page) as a system of regression equations with a block variance-covariance matrix structure. The methods described are applied to Greek personal auto data: premiums are estimated using different regression equations for new vs. old cars. The example appears at the end of the paper, after the author has introduced a considerable amount of theory, so paper has a very steep learning curve.

The author's abstract is available on the journal's web site.

**Pricing of arithmetic basket options by conditioning**

*Pages 55-77 *

G. Deelstra, J. Liinev and M. Vanmaele

A basket option is a derivative contract where the underlying asset is a portfolio of stocks rather than a single stock. Black Scholes, and other similar models, use a geometric Brownian motion model of stock returns and so produce a lognormal price distribution. The value of a portfolio is a weighted average of the value of several securities and so is no longer lognormal under the standard models. This means there is no closed form solu-tion for pricing basket options. This paper determines some easily computed upper and lower bounds on the price of such options.

The paper begins by introducing upper and lower bounds on the distribution of a sum of random variables whose marginal distributions are known but whose joint distribution is not known. These techniques are of general interest and apply to P/C work as well as op-tion pricing. The upper bound is easy to determine: it is called the comonotonic upper bound. The comonotonic upper bound has the same distribution as summing percentiles of the marginals---it represents the "worst" possible correlation. The lower bound is more difficult to determine. The paper shows that if the marginals are known conditional on some explanatory variable L then the lower bound, in convex order sense (cheapest ex-cess loss costs) is given by E(X_1+…+X_n | L). The obvious P/C application is to a mul-tivariate mixed compound Poisson, where L would be the value of the frequency mixing variable. In the paper L is based on the sum of the returns of the underlying assets, which is known since each asset is assumed to have a normal return.

The author next shows how to break the price of a basket option into two pieces, one of which can be computed exactly. This is accomplished through the use of the conditioning variable L. The exact component accounts for about 90% of the total price of the option, giving a quick way of determining approximate prices. The author then determines upper and lower bounds for the price of the option combining the price decomposition with the general theory on upper and lower bounds introduced in the first section. The author also considers method of moment based approximations to the price of basket options. Again these use conditioning variables. The paper ends with an extensive numerical test of the various upper and lower bounds.

The ideas in this paper could potentially be applied to derive upper and lower bounds on the price of an aggregate excess cover on multiple lines of business where losses are modeled using a multivariate mixed compound Poisson distribution.

The author's abstract is available on the journal's web site.

*Stephen Mildenhall*