Valuation of Equity-Linked Death Benefits on Two Lives with Dependence

Round 1

Reviewer 1 Report

The paper's topic and conducted research are very important and justified to be presented in a high-quality Journal. The subject is very important for the literature. The paper is informative, but some issues need to be addressed carefully. My decision is a minor revision, with some amendments. Please see my comments and suggestions below.

Comment 1. The abstract could be improved.

Comment 2. The authors should add theoretical contributions and managerial implications in the section discussions or conclusions. Also, widening the limitations of this research, because they are not only those mentioned by the authors.

Comment 3. In section discussion, the authors could compare each result analysis with previous literature. For example, point out the similarities and differences between this study and previous literature. In doing so, it demonstrates how this study fills the gaps in the previous literature.

Comment 4. The number of references is not adequate.

Good luck for your work!

Author Response

 The authors would like to thank the reviewer for his/her valuable comments to improve the final draft of the paper.

Comment 1) The abstract could be improved.
Answer
The paper has been revised to address this comment

Comment 2) The authors should add theoretical contributions and managerial implications in the section discussions or conclusions. Also, widening the limitations of this research, because they are not only those mentioned by the authors.
Answer
Currently, this is done in the discussion and conclusion sections. By considering a
distribution that belongs to the Phase-type family, we generalize Gerber et al (2012, 2013)'s work and analyse how option prices will change when considering the lifetime of the married couple.

Comme 3) In section discussion, the authors could compare each result analysis with previous literature. For example, point out the similarities and differences between this study and previous literature. In doing so, it demonstrates how this study fills the gaps in the previous literature
Answer
Currently, this is done in the discussion and conclusion sections. To the best of the author's knowledge, Gerber et al (2012, 2013)'s work is closest to ours. Their works, however, do not consider the couple lifespan time in the analysis and therefore do not analyse the impact of one person's death on the option price. As there is evidence that the death of one person in a married couple affects the death of the other, we not only fill this gap but also incorporate the dependence into the lifetime distribution of the couple.

Comment 4) The number of references is not adequate.
Answer
We checked and corrected the typo.

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper covers a very interesting and dificult topic, it follows an usual pattern, every section having good and enough information. The methods are adequate, the discussions is correct and the conclusion is also sufficient. The quality of the English seems to me very good - even if it deserves a careful analysis to detect any eventual insuficiency, punctuation change or other change.

Author Response

 The authors would like to thank the reviewer for his/her valuable comments to improve the final draft of the paper.

Comment

The paper covers a very interesting and dificult topic, it follows an usual pattern, every section
having good and enough information. The methods are adequate, the discussions is correct and
the conclusion is also sufficient. The quality of the English seems to me very good - even if it deserves a careful analysis to detect any eventual insuficiency, punctuation change or other change

Answer:

Now that the paper has been revised, the language quality has been improved.

Author Response File: Author Response.pdf

Reviewer 3 Report

Review on “Valuation of equity-linked death benefits on two lives with dependence”

Summary:

This article investigates the valuation problem of equity-linked death benefit policies for a family, exactly for a wife and a husband. It generally follows the Gerber et al. (2012) and contributes to the academia by extending the study from an individual to a married couple. The concept of joint life of last survival in pensions is well adopted and a FGM copula is used to describe the dependency of remaining lifetimes within a married couple. As for modelling the remaining lifetime for each member,  a special case that a unimodal weighted exponential distribution is utilized for better illustrating lifetime data than the linear combination of exponentials. The highlight is that closed and analytical valuation expressions are found for various life-contingent option-like payoffs.

Overall, this paper deals with an interesting problem, and exposition is well structured. That lots of math is required is also taken into consideration. I recommend publishing this article after careful revisions.

Major:

1. Please explain the meaning of (-1) in equation (6) and (9). If it just returns the negative value, then it would be better to remove it to display the discounted densities in their original forms.

2. It seems that only out-of-money options are considered. Is it okay to provide in-the-money counterparts, by applying the put-call parity?

3. In the numerical example, I doubt that mu takes -0.05, a negative value. As mu stands for the expected return rate, it shall be a positive number such that mu-1/2sigma^2>0 as well.

Minor:

1.  The reference on page 2, Shevchenko and Xiaolin (2016) is not correct because Xiaolin is the given name, not the surname. The correct one is Shevchenko and Luo (2016).

2. In the second line following the subsection 2.1 Multiple-life insurance model, the verb “allow” should be replaced by “allows”.

3. On page 5, please adjust the long math expressions in Definition 1 and 2 to make the article well formatted. Besides, please introduce the MWE distributions. What does MWE stand for? What’s the difference of a WE and MWE distribution?

4. On page 5, please check Definition 1 and 2. For now, they seem the same to me.

5. Please go through the manuscript again carefully and correct grammar mistakes. I’ve found some; see the middle of page 7, the sentence should be “Let us recall…”. In the next paragraph, the sentence should be “Let us prove…”. 

6. In the Appendix part, the numbering is reversed. Please correct that.

Author Response

 The authors would like to thank the reviewer for his/her valuable comments to improve the
final draft of the paper.
Major Comment
Comment 1) Please explain the meaning of (-1) in equation (6) and (9). If it just returns the
negative value, then it would be better to remove it to display the discounted densities in their
original forms
Answer
The (-1) come from the integration of ???(?),?(?) (?, ?) and ???(?̅),?(?̅) (?, ?) with respect to ?.
We just write it in a nice format to avoid having the (-) in the other side.
Comment 2) It seems that only out-of-money options are considered. Is it okay to provide inthe-money counterparts, by applying the put-call parity.
Answer
In-the-money options can be priced using put-call parity. The result was not presented in the
paper because of this reason (see Remark 3).
Comment 3) In the numerical example, I doubt that mu takes -0.05, a negative value. As mu
stands for the expected return rate, it shall be a positive number such that mu-1/2sigma^2>0
as well
Answer:
First, we would like to thank the reviewer for his comment. We correct it in the updated
version where we choose ? = ? - 0.5 ∗ ?2. This is a special case where our valuation
coincides with the risk neutral pricing as highlight in Remark 1 in the present paper and in
Gerber et al. (2012) in their numerical illustration. (See Tables [1-3])
Minor Comment
Comment 1) The reference on page 2, Shevchenko and Xiaolin (2016) is not correct because
Xiaolin is the given name, not the surname. The correct one is Shevchenko and Luo (2016).
Answer
This typo and their similarities were corrected in the updated version
Comment2) In the second line following the subsection 2.1 Multiple-life insurance model, the
verb “allow” should be replaced by “allows”
Comment 5) Please go through the manuscript again carefully and correct grammar
mistakes. I’ve found some; see the middle of page 7, the sentence should be “Let us
recall…”. In the next paragraph, the sentence should be “Let us prove…”
Answer
The paper was once again edited to account for those typo and mistake.
Comment 3) On page 5, please adjust the long math expressions in Definition 1 and 2 to
make the article well formatted. Besides, please introduce the MWE distributions. What does
MWE stand for? What’s the difference of a WE and MWE distribution?
Answer
This paragraph was now well formatted.
There is only one definition (Definition 1) in this paragraph, the second definition is a
mistake (a duplicate form of Definition 1). We hence delete Definition 2.
Comment 4) On page 5, please check Definition 1 and 2. For now, they seem the same to me.
Answer
The two definitions are identical because Definition 2 duplicates Definition 1. In the new
version, it has been deleted

Author Response File: Author Response.pdf