DeepTriangle: A Deep Learning Approach to Loss Reserving

Round 1

Reviewer 1 Report

Overall Assessment

I appreciate the manuscript, and the open-source R software package made available by the author. Yet, I think clear documentation is also important. Although DeepTriangle can be a potentially useful tool for the actuarial community, I think the manuscript omits a lot of details, and hence would need a major revision before it could be accepted into Risks. I explain why in the following comments.

Specific Comments

Equation (1) What function did you use for g?

Equation (7) If the goal is predicting the ultimate claims, why is OS needed? Define OS more clearly.

First line after equation (8): What does it mean to divide a vector by NPE?

Paragraph after (9): Why does joint modeling of P and OS improve the prediction for P? The author claims that it helps, but does not explain why in detail.

Section 3.3: The model architecture is described very briefly. I do no think this provides enough details for someone to reproduce the work.

It is not clear what loss function is used for the training. It seems like equation (18) is the loss function, but it is not clear.

The author is not clear why a GRU is used, and why it is applied I-1 times.

The author is not clear about what exactly a "company code" is, and how features are extracted from this input to form the embedding. The author simply states that companies similar to each other are mapped to similar embeddings. Please explain how the embeddings are obtained.

The dataset used for the analysis should be explained in more detail. More summary statistics should be included.

Equation (15) pops out of nowhere. It is not clear where exactly the Relu activation function is used. I assume it is used for the final output P and OS. If this is true, the author should make this more clear.

The results should also be presented more clearly, in order for a reader to be convinced that the approach has an advantage.

I also think the author should explain why Keras and Tensorflow is used, instead of other neural network platforms. It is ok to use commercial packages for research as long as the research question is clear. Yet, perhaps the prediction advantage in this paper is coming from Keras and Tensorflow. In order to show that the specific framework described in the paper is contributing to the improvement, I think the author should compare apples to apples: One should compare a Tensorflow implemented neural network, to another Tensorflow implemented neural network.

Conclusion

In conclusion, I think the author can include a lot more details than the current manuscript does. Upon improvement in the aspects mentioned above, I think the article could be a good contribution to Risks.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

1. The literature review could be more informative and the contribution of each reference could be explained in more detail to add value for the reader. Moreover, there could be a discussion beyond neural networks applied to claims reserving, especially with regard to other claims reserving methods where iterative updates of the model parameters are performed. Some more recent references could be added and discussed, with an emphasis on publications in Risks, e.g.
Martinek (2019): Analysis of Stochastic Reserving Models By Means of NAIC Claims Data, Risks 7(2)
Peremans, Van Aelst & Verdonck (2018): A Robust General Multivariate Chain Ladder Method, Risks 6(4)
Chukhrova & Johannssen (2017): State Space Models and the Kalman-Filter in Stochastic Claims Reserving: Forecasting, Filtering and Smoothing, Risks 5(2)
2. There is an outline of the paper missing.
3. I miss motivation for the denomination "DeepTriangle". It sounds nice, but there should be a concise motivation for this denomination.
4. Some notations seem inappropriate, e.g. L45: x_n_x (using another index would be better); L49: no need for brackets; L50: where begins the summation and where does it end?; L52: "the functions g^[l]...", this is only one function, please add in brackets l = 1,...,L; please use common notations, i.e. X_{i,j} instead of P_{ij} and so on; L79: add the abbreviation UL in brackets & substitute the element-sign by an equal-sign; do not use sometimes i,j and sometimes ij in the index, be consistent throughout the paper; moreover, you sometimes use [...] and sometimes <...> for superscripts without a specific reason, please use a consistent notation.
5. There are a few typos: L25: utilizes -> utilize, L26: embeds -> embed, L44: sentence seems incomplete, L53: delete the comma after W^[l]

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Summary

I think the paper can be accepted into Risks for publication.

Details

I would like to thank the author for submitting a revised version of the paper. The paper has been improved, and I appreciate the responses to some of the concerns that I had in my previous review.

The author has made available the source code for DeepTriangle, which I believe is a valuable resource for the research community. For this reason, I decide that enough detail is provided. I think the paper can be accepted into Risks for publication.

This work is a valuable example of utilizing Keras in the R computing environment, within the loss reserving context. I think more such examples are needed for the research community based on R.

Reviewer 2 Report

Thank you for considering my remarks!