An Alternative Pricing System through Bayesian Estimates and Method of Moments in a Bonus-Malus Framework for the Ghanaian Auto Insurance Market

Round 1

Reviewer 1 Report

Review for “Alternative Pricing System through Bayesian Estimate and Method of
Moments in Bonus-Malus Framework for the Ghanaian Auto Insurance Market”

This paper focuses on the relevance of BMS versus NCD systems in the particular case of Ghana. The work proposes two BMS under Poisson-exponential and Poisson-Gamma contexts obtaining rewards for good drivers and punishments for bad drivers. Results are interesting but, in my opinion, the paper has to address several major and minor points before publication.

This work is not very novel but can be useful for insurance companies and researchers, although some major points, in my book, must be clarified before its publication:

1. In subsections 3.3, 3.4 and 3.5 authors show the negative binomial model, the poisson exponential and the Poisson Gamma model. As authors surely know, negative binomial model is obtained from Poisson-Gamma mixture and, furthermore, Poisson Exponential model is a particular case of Poisson-Gamma mixture. I think authors should let only one subsection including all the significant information.
2. Delete figure 1 and 2.
3. In relation to the information criteria, include the DIC criterium (Deviance Information criterium).
4. I think the information about Considiring Claim Frequency and Severity information, it can be deleted from the paper. Authors finally do not use the severity information (I suppose because they do not have it) so it is not necessary to include this information in this paper.
5. The discusión only taks about Table 5. Why? I think results from Table 6 are interesting. Why don´t authors compare both tables? Table 6 seems less competitive and more profitable.
6. In the conclusions, authors said that “The BMS obtained using the models proposed in this work does not modify the discounts make in the absence of claims”. I dont agree with this afirmation.
7. In general, the format of the paper is a bit awkward: references, formulas,… are hard to read.
   1. References:
      1. Pg. 1. Awunyo-Vitor (2012) and Nii Anang Laryea (2016): put the “and” and the 2012 year correctly.
      2. Pg. 1. Renshaw (1994) is not found in the reference section.

• Pg. 1. Please, here and in the hole paper sort the reference from oldest to newest.

1. Pg. 1. Ibiwoye et al. (2011): the year (in all the paper).
2. Pg. 2. Boucher 2007 or 2006?
3. Pg. 2. Put Gómez-Déniz in this way in all the paper.

• In the first paragraph of Pg 2, you can talk about the sensitivity of the BMS (See Gómez-Déniz, 2002. Measuring sensitivity in a Bonus-malus system. Insurance: Mathematics and Economics, 105-113) and cite it.
• Bolancé in this way in all the paper.

1. Paris and Walhin or Walhin and Paris: in the paper sometimes appears in one way and sometimes in the other way.
2. Thuring (2011): the year.
3. Gouri”eroux is wrong.

• In all the paper, if you list several references, put “and” before the last one.
• Pg. 4: NIC, 2018: reference?
• Bawa (2017) is not in the reference section.

1. Pg. 8 and 9: Dorina et al.

• Pg 14: reference section:
  1. Bhoola: 2014 or 2016?
  2. Put “and” before the last author if needed.
  3. Lemaire reference are not sorted.
  4. In general, reference list do not follow the Chivago style guide. Please follow this guide.
  5. Ghana National insurance and the National Insurance reference are not found in the paper.
  6. Lydia L.B. is not found in the paper.

Minor points:

1. Pg 1. Line 2. Put the reference in brackets.
2. Pg 1. Line -6: Renshaw (1994),…. In brackets.
3. Pg 2: At the end of the second paragraph, “qfghjkl’”: ¿???
4. 3. Last paragraph before 2.0 Contemporary Practice in Ghana: The rest of the paper….. rise the “Section 3 explains….” sentence and put it after “in Ghana”.
5. 4: Pitrebois et al.; Varisco and Bawa in brackets.
6. The sample contains 101202 (don’t put any space between numbers)
7. 4, line -6: “101202 final samples. Sorting was done…”. Delete “With the 101 202 sample size.
8. 4 Line -4. Section 3.5 instead 3.4.
9. 5 Line 11: several authors “say premium may be reviewed” (delete the “s”).
10. L(i) formula format is awkward
11. 5. Line -4 P^(t+1)_j format is awkward.
12. Equation (2) format is awkward.
13. Pg 6: all the formulas and symbols format are awkward.
14. Pg 10. Line -8: table 2 instead table 4
15. Pg 11. variance instead varianve.
16. K=r=1.40 format is awkward.
17. First paragraph of Discussions is redundant. Delete it.
18. Pg 13. Line -5: find instead finds.
19. In general, put spaces after dots ot brackets.

Author Response

Dear Reviewer,

Thank you very much for your valuable comments. In this revision, we carefully studied your comments and tried our best to respond to the issues raised by you. Below we provide a point-to-point response to your comments and the revised portions are highlighted in the paper in BLUE COLOUR.

We warmly appreciate your handy comments with its explicit aim of improving and shaping the quality of our paper. It is our fervent hope that the above explanation will meet your sincere request.

Thank you.

Author Response File: Author Response.pdf

Reviewer 2 Report

Contents This paper proposed an alternative Bonus-Malus System
for the Ghana's auto insurance market.
General Comments The paper provides a nice contribution to the existing literature
on pricing system in auto insurance markets, in particular they
provide a new efficient method with respect to existing No-Claim
Discount in Ghana. The authors propose the Negative Binomial law
in order to deal with a heterogeneous and independent portfolio,
which is a good fit to this kind of data. As stated by the authors,
there are other distributions able to deal with these data;

I would suggest the authors to provide a more arguments and some
empirical evidence supporting the use of the Negative Binomial distribution.
Regarding this issue, I found a very extensive literature review on
insurance and Bonus-Malus systems, but a non exhaustive one on other
distributions and statistical tools used in this framework.
See for example: GE Willmot (1987), Omari et al. (2018) and references therein. Finally the paper can not be published in the current form as it
contains various typos and imprecise statement (see next section
for some examples). I suggest to revise carefully these aspects.   Other Comments p. 1, "Therein, because of their heterogeneous and random nature, a priori rating model on such longitudinal information needs to be dynamic, and hence the need for BMS." I would suggest to replace "Therein" with "Therefor". p. 2, "Mzxcvbnmoreover, we incorporate both the claims frequency and severity as priors to obtain optimal BMS which satisfies qfghjkl;’desirable transition rule.". In this sentence, there is a formatting error. p.4, "The working sample contains 101 202 policyholders which are mainly private cars merged from five major insurance companies in Ghana for a year.". There is a typo in writing the number, it is not very clear that there one number given the space. This typo is repeated several times in the paper. p. 5, the paragraph "Consistent Akaike and Bayesian information criteria" it is not well placed in the text. I would suggest to remove the paragraph and to rewrite this part, by introducing it better. p. 5, In Eq.1, I would make more clear that the probability at t+1 only depends on $L_{t}$. p.5, I would make more clear the difference between "small" $P$ and \textbf{P}. In general, I would suggest to write better al the equations. p. 10, "With the help of the above estimates, given that the premium charged for a given insurance policy at the beginning (t=0) is 100\% (without discount), then the risk premium for a policyholder combining his frequency and severity into effect at (t+1) would be equal;", this statement is not very clear. p. 11, "We therefore considers the method of moments for our model parameters estimated", should be "consider".
Action The paper provide an interesting contribution to the literature
on auto insurance and it is a more efficient and fair system as
the existing one in Ghana. Nevertheless, the paper should be
accepted for publication in the "Journal of Risk and Financial
Management", only after a careful revision which accounts for
the remarks presented in this report.

Author Response

Dear Reviewer,

Thank you very much for your valuable comments. In this revision, we carefully studied your comments and tried our best to respond to the issues raised by you. Below we provide a point-to-point response to your comments and the revised portions are highlighted in the paper in BLUE COLOUR.

We warmly appreciate your handy comments with its explicit aim of improving and shaping the quality of our paper. It is our fervent hope that the above explanation will meet your sincere request.

Thank you.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Authors have satisfied all the recommendations and changes. From my point of view, the work is publishable in this journal. Please, consider the next minor changes before publication:

• Authors cite many times in a wrong way. For example, at the begining of the introduction:

"The role of the insurance market for economic growth and development can never be underestimated Stojakovié and Jeremié (2016). As one of the driving forces for a country's economic growth and development, (Bhoola et al. 2014) states that the market needs to be well structured and established in the area of risk predictions to assure sustainability".

You need to put "; see Stojakovié and Jeremié (2016) for further details" or something like this. Furthermore, you write: "(Bhoola et al. 2014)" between brackets but the sentence continues "states that...". 

These mistakes continues several times in the rest of the paper. Authors have to cite in a better way.

• Pg. 5: (Varico 2002), Pitrebois et al. (2005) and (Bawa 2017). Varico and Bawa between brackets.... why?
• Pg. 203: very large subscripts.

Author Response

We warmly appreciate your handy comments that has improved and shaped the quality of our paper and making it publishable in Journal of Risk and Financial Management.

Thank you.

Author Response File: Author Response.docx

Reviewer 2 Report

none

Author Response

We warmly appreciate your handy comments that has improved and shaped the quality of our paper and making it publishable in Journal of Risk and Financial Management.

Thank you.