Application of Compound Poisson Process in Pricing Catastrophe Bonds: A Systematic Literature Review

Abstract

The compound Poisson process (CPP) is often used in catastrophe risk modeling, for example, aggregate loss risk modeling. Hence, CPP can be involved in pricing catastrophe bonds (CAT bonds) because it requires a catastrophe risk modeling method. However, studies of how the application of CPP in pricing CAT bonds is still scarce. Therefore, this study aims to conduct a systematic literature review (SLR) on how CPP is used in pricing CAT bonds. The SLR consists of three stages: the literature selection, bibliometric analysis, and gap analysis. At the literature selection stage, the 30 articles regarding the application of CPP in pricing CAT bonds are obtained. Then, the conceptual and nonconceptual structures of the articles are mapped at the bibliometric analysis stage. Finally, in the gap analysis stage, the application of CPP in pricing CAT bonds from the previous studies is analyzed, and new research opportunities are studied. This research can be a reference for researchers regarding the application of CPP in pricing CAT bonds and can motivate them to design more beneficial ways of pricing CAT bonds with CPP in the future.

1. Introduction

The frequency of catastrophic events has experienced an increasing trend in almost all countries worldwide in the last five decades. This trend occurs for various catastrophes, such as wildfires, droughts, earthquakes, extreme weather, floods, landslides, storms, and volcanoes. This increasing trend for each catastrophe in the last five decades can be seen in Figure 1 (source: https://www.emdat.be) (accessed on 15 May 2020). Figure 1 shows that the occurrence frequency of each catastrophe each year is around the increasing trend line.

The increasing trend in catastrophic world events certainly causes a rising trend in economic losses. The World Meteorological Organization states that the economic loss from weather and climate catastrophes in 2010 alone was seven times that of 1970 [1], let alone the total economic loss for all types of catastrophes. Furthermore, this increasing trend is expected to continue in the future in line with the increasingly significant weather and climate changes [2].

The high economic losses due to catastrophe certainly burden the country in its efforts to overcome them. Sometimes, the quantity of catastrophe relief funds that have been prepared is not commensurate with the economic losses incurred. Finally, the budget and social assistance became the mainstay [3]. Therefore, contingency costs to overcome catastrophe must be sought as soon as possible [4].

One alternative way to obtain contingency costs in strengthening the preparation of funds for catastrophe prevention is through insurance. Of course, the mechanism used is not an ordinary insurance mechanism because the financial situation of both the insurer and the reinsurer will be depressed if they bear the risk of significant catastrophe losses alone [5]. Therefore, a unique insurance mechanism used is needed. In the last three decades, efforts have been made to design unique mechanisms for insurers and reinsurers. This effort is finally obtained by linking the risk of state economic losses due to a catastrophe with financial security in the capital market [6]. In other words, insurers and reinsurers involve capital market investors to jointly bear the risk of economic losses due to a catastrophe in a country.

Catastrophe bonds (CAT bonds) are one of the most successful securities in linking the risk of state economic losses due to a catastrophe with financial securities in the capital market [7]. CAT bonds can provide a hefty contingency cost to countries. In 2006, Mexico used an earthquake catastrophe insurance mechanism through a CAT bond [8]. It was followed by other Caribbean countries, which used the exact mechanism to provide contingency costs in response to hurricane catastrophes and earthquakes in 2014 [9]. Lastly, Colombia, Peru, Chile, and Mexico use CAT bonds to finance the risk of earthquake losses.

Although several countries have used CAT bonds, their fair pricing mechanism, which is a fundamental stage, must be studied further. The use of CAT bonds is still very new, thus, there is little research on its pricing, and it can continue to be developed. CAT bond pricing is undoubtedly different from traditional bond pricing [10] because the factors involved vary. In addition to involving financial factors, CAT bonds also consider other factors, namely the catastrophe risk factor. The catastrophe risk factors include catastrophe intensity, single catastrophe loss or magnitude (in an earthquake catastrophe), and spatial elements.

In CAT bond pricing, catastrophe risk needs to be modeled first. So far, one of the popular methods used to model catastrophe risk is the compound Poisson process (CPP). CPP can be used to determine aggregate losses or magnitude. In other words, CPP can integrate the model of catastrophe intensity and loss or magnitude factors. Two types of CPP can be used, homogeneous and nonhomogeneous CPP. Homogeneous CPP was used when catastrophe intensity was assumed as being constant, whereas nonhomogeneous CPP was used when catastrophe intensity was assumed as being nonconstant [11]. Therefore, CPP can measure catastrophe risk in the CAT bond pricing process.

The SLR on the CAT bond that has been carried out previously is briefly described in this paragraph. An overview of the CAT bond terminology was first presented by Canabarro et al. [12]. Their explanation shows the difference between traditional and CAT bonds based on their risk and return. They also describe a simple model used in CAT bond pricing. Then, Linnerooth-Bayer and Amendola [13] explain the form of loss-sharing of state losses after the catastrophe, and they also describe what financial instruments can be used for this loss-sharing. The results of their study show that the CAT bond is the most efficient instrument for the country to restore its economy. Lastly, Skees et al. [14] explained the importance of transferring state catastrophe risk to the capital market to help fund catastrophe countermeasures for low-income countries. They also describe several countries that have succeeded in providing funds for catastrophe prevention through CAT bonds.

Based on this introduction, this study aims to conduct a systematic literature review (SLR) on how to apply CPP in CAT bond pricing. The differences, which are the novelty of our study, are as follows:

(a)

This study discusses a technical problem in CPP application in pricing CAT bonds;

(b)

This study analyzes the conceptual and nonconceptual structures of the literature;

(c)

This study discusses the gaps in previous studies.

The SLR consists of three stages: the literature selection, bibliometric analysis, and gap analysis. At the literature selection stage, this is conducted using the Scopus and Science Direct databases. Then, in the bibliometric analysis, we present the mapping of conceptual and nonconceptual structures, such as mapping the number of literary publications each year, mapping the number of literature citations, mapping literature publishers, and mapping the literature keyword network. To facilitate the process of bibliometric analysis, we use the help of the R software via the Biblioshiny Web Interface and VosViewer. Finally, a gap analysis is carried out by examining the following matters first:

(a)

Financial and catastrophic factors involved;

(b)

The method used to measure risks of financial and catastrophe;

(c)

The method used to estimate CAT bond price.

After (a) to (c) is assessed, the gap is determined based on the involvement of factors and methods that have not been carried out in previous studies. This research is expected to provide knowledge for researchers about the application of CPP in CAT bond pricing, and this research is also expected to motivate other researchers to design more beneficial methods of CAT bond pricing with CPP in the future.

2. Materials and Methods

2.1. Literature Collection

The literature is searched on two reliable database search engines, Scopus [15] and Science Direct. The literature collection is conducted based on the following criteria:

(a)

The type of literature is the final journal article;

(b)

The title, abstract, or keyword of the literature contains the words (“catastrophe bond” OR “CAT bond” OR “catastrophic bond”) AND (“price” OR “pricing”). We specifically do not use keywords related to CPP. It is conducted so that the collected literature becomes more general and more numerous;

(c)

The literature is written in English;

(d)

The literature is published in peer-reviewed international journals;

(e)

The literature aims to design a way for pricing CAT bonds using CPP.

2.2. Methods of Literature Selection

In this study, the literature’s fulfillment of criteria (e) is conducted manually (not using a search engine). It is carried out to avoid not detecting the literature by the machine even though the literature is under the topic of this research. The manual selection stage carried out refers to Firdaniza et al. [16] as follows:

(a)

Removing duplicates and the unavailable literature from each database;

(b)

Reading the abstract of articles;

(c)

The reading of articles one-by-one in its entirety.

2.3. A Brief Explanation of Bibliometric Analysis

Bibliometric analysis is a series of descriptions and investigations from a list of the literature that becomes a reference in the writing of new literature. The benefit of the bibliometric analysis is that the conceptual and nonconceptual structures of the literature list can be studied clearly and systematically [17]. The conceptual structure includes the mapping of words contained in the title, abstract, and keywords. In contrast, the nonconceptual structure provides the mapping of the most relevant researchers on a topic and the mapping of publishers and journals from the most relevant literature. This analysis can help researchers determine collaborators in their research, and it can permit researchers to choose the best journal and publisher for their literature.

This study uses bibliometric analysis to analyze the conceptual and nonconceptual structures of the collected literature. To simplify the bibliometric analysis process, we use the software VosViewer and R. The package used in the R software is “bibliometrix”, and the command used is “biblioshiny()” [18].

3. Results

3.1. Literature Selection

Our literature collection was conducted on 15 May 2022. The number of literature that meets criteria (a) to (d), listed in Section 2.1, is 81 from the Scopus database and 18 from the Science Direct database. In other words, we attained 99 articles from both databases. The summary can be seen in

Table 1. The Number of Literature Collected from Scopus and Science Direct Databases.

Literature Database	The Number of Literature
Scopus	81
Science Direct	18

.

After the literature from each database is obtained, the next step is the fulfillment of criteria (e) manually. The stages of the manual literature selection are as follows:

(a)

The first stage is removing duplicates and the unavailable literature from each database. After each piece of literature is checked, all the literature is available, and the number of duplicate literature found is 18. We deleted these 18 pieces of literature, leaving 81 pieces of literature selected for the next stage.

(b)

The second stage is the advanced literature selection stage through reading the abstract. In the abstract section, we look at the purpose of the literature. The literature that aims to design ways to price CAT bonds is chosen. After each literature abstract is read, 49 pieces of literature are obtained and then selected for the next stage.

(c)

The third stage is the final literature selection stage by reading it one-by-one in its entirety. The selected article is an article on CAT bond pricing that applies CPP. The result is a total of the 30 articles obtained. These articles are reviewed later.

We provide a file in the form of “.bib”, which contains a database of the 30 articles studied at the following link: bit.ly/CATBondLiteratureDatabase. A manual literature selection process summary can be seen visually in Figure 2.

3.2. Bibliometric Analysis

The 30 articles obtained in the final selection stage were published in different years. Details regarding the publication year of the 30 articles are presented in Figure 3.

Figure 3 shows that the article on how to price CAT bonds using CPP was first published in 2002, while the most recently was published in 2022. The most articles published in one year was five, which was in 2017. Then, there was no publication on pricing CAT bonds in 2005, 2006, 2007, 2011, 2012, and 2016. Then, the number of articles published each year tends to increase. It can be seen from the trend line, which has a positive gradient. The indication is that using CAT bonds to provide contingency costs to overcome country catastrophe can become increasingly popular yearly.

When researchers write articles about pricing CAT bonds with CPP, information about the most impactful articles on the topic is required. Therefore, we analyzed the articles with the most impact on pricing CAT bonds with CPP in 30 articles based on the number of citations. The top 10 articles with the most citations are presented visually in Figure 4.

Figure 4 shows that the article by Lee and Yu [19] is the most impactful article on pricing CAT bonds with CPP based on the number of citations. The article has the most citations, 95. Then, there is information that 8 out of 10 articles with the most impact based on the number of citations were published after 2010. It indicates no significant relationship between the year of publication and the number of citations. Instead of having a relationship with the year of publication, the number of citations may be influenced by other factors, such as article quality, the novelty of the designed model, and article accessibility. More detailed information about the top 10 articles with the most citations can be seen in

Table 2. Brief Overview of the Top 10 Articles with the Most Citations.

Author(s)	Title	Keywords	Indexed Database	The Number of Citations
Lee and Yu [19]	Pricing Default-Risky CAT Bonds with Moral Hazard and Basis Risk	-	Scopus	95
Nowak and Romaniuk [20]	Pricing and Simulations of Catastrophe Bonds	Catastrophe Bonds, Monte Carlo Simulations, Risk, Stochastic Processes	Scopus and Science Direct	39
Vaugirard [21]	Pricing Catastrophe Bonds by an Arbitrage Approach	Catastrophe Bonds, Incomplete Market, Jump-diffusion Process, Monte Carlo Simulations, Path-dependent Digital Options	Scopus	38
Egami and Young [22]	Indifference Prices of Structured Catastrophe (CAT) Bonds	Catastrophe (CAT) Bond, Exponential Utility, Indifference Price, Jump-diffusion, Reinsurance Strategy, Structured Derivative Security	Scopus and Science Direct	35
Jarrow [23]	A Simple Robust Model for CAT Bond Valuation	CAT Bond, Catastrophe Events, Reduced Form Model, Reinsurance	Scopus and Science Direct	28
Ma and Ma [24]	Pricing Catastrophe Risk Bonds: A Mixed Approximation Method	Catastrophe Risk Bonds, Compound Nonhomogeneous Poisson Process, Mixed Approximation Method, PCS Loss, Stochastic Interest Rates	Scopus	28
Schmidt [25]	Catastrophe Insurance Modeled by Shot Noise Processes	CAT Bonds, Catastrophe Derivatives, Marked Point Process, Minimum Distance Estimation, Self-Exciting Processes, Shot Noise Processes, Tail Dependence	Scopus	15
Nowak and Romaniuk [26]	Catastrophe Bond Pricing for The Two-Factor Vasicek Interest Rate Model with Automatized Fuzzy Decision Making	Automated Decision Making, Catastrophe Bonds, Fuzzy Numbers, Monte Carlo Simulations, Stochastics Processes, Vasicek Model	Scopus	14
Lai et al. [27]	The Valuation of Catastrophe Bonds with Exposure to Currency Exchange Risk	3D Brownian Motion, Brownian Bridge, CAT Bond Valuation, Catastrophic and Currency Exchange Risk, Importance Sampling, Jump-Diffusion Process	Scopus and Science Direct	10
Nowak and Romaniuk [28]	Valuing Catastrophe Bonds Involving Correlation and CIR Interest Rate Model	Asset Pricing, Catastrophe Bonds, CIR Model, Monte Carlo Simulations, Stochastic Models	Scopus	10

.

Table 2 shows that if the articles are reviewed from the keywords used, all articles use the keywords “catastrophe bonds” or “CAT bonds”. In addition, almost fifty per cent of the 10 articles use the keyword “Monte Carlo simulation”. Finally, if the articles are reviewed from their indexing point of view, almost half of the 10 articles presented are indexed in the Scopus and Science Direct databases, while the others are only in Scopus.

In designing articles regarding the pricing of CAT bonds with CPP, one researcher can collaborate with other researchers. Determination of this collaboration requires measurement first. Therefore, we listed the top 10 researchers on this topic by the number of articles they have written as references for the relevant authors to collaborate with. The top 10 authors of articles on determining the price of CAT bonds by applying the most CPP are presented visually in Figure 5.

Figure 5 shows that Romaniuk is the most relevant author of a study on how to price CAT bonds with CPP based on the articles that have been published. He published 4 articles, the highest among all authors. Nowak is in second place as the author of an article on determining CAT bond price using CPP, with 3 articles written.

Researchers need to consider the journal that published the articles. Therefore, we sorted the journals of the 30 articles reviewed by publication quantity. The visualization of the 10 most relevant journals that publish articles on determining the price of CAT bonds using CPP is presented in Figure 6.

Figure 6 shows that Insurance: Mathematics and Economics is the journal with the most significant number of articles publishing articles about how to price CAT bonds using CPP, namely 3 articles. Then, Risks, Discrete Dynamics in Nature and Society, and Astin Bulletin each have 2 published articles. Furthermore, the remainder is a 1 article publication. Next, in more detail about the publishers of the 30 articles, we present a visualization of the top 10 relevant publishers publishing articles on how to price CAT bonds with CPP in Figure 7. Figure 7 shows that Elsevier publisher published the most articles on how to price CAT bonds with 10 articles. Then, Taylor and Francis came in second with 5 articles, and so on.

Next is the analysis of topics that are often discussed between articles. The measure used is the number of similar words that often appear in articles’ titles, abstracts, or keywords. We visually present the number of similar words in the titles, abstracts, or keywords of articles and their relationship in Figure 8. Visualizations are made using the VosViewer software. Only words with multiple occurrences are considered to simplify it.

The circle size of each word represents how often the word is discussed in the 30 articles. The larger the word circle, the more often the word is discussed, and vice versa. Then, the connector lines between word circles represent the presence or absence of linkages between words in the 30 articles. The more connector lines that fit into a word circle, the more connections between the words in the circle and other words. Then, the color of the word circle represents the cluster. Word circles of the same color indicate that the word circles are in the same cluster. Finally, the distance between the word circles represents the strength of the linkage between the word circles. The closer the distance between the word circles, the stronger the relationship between the words. Figure 8 shows three-word circle clusters colored red, green, and blue. Then, Figure 8 also shows that the circle containing the word “catastrophe bonds” has the largest size. The word “catastrophe bonds” is often discussed in the 30 articles. The second-largest word circle is the “costs” word circle. The circle of the words “costs” is also related to the circle of the words “catastrophe bonds”. It can be seen from the presence of a connector line connecting the two-word circles. It indicates that the topic of discussion that is often discussed with “catastrophe bonds” is the issue of costs. These costs can be in the form of economic losses due to the destruction of essential country infrastructure. Then, Figure 8 also shows the existence of word circles that refer to the approach used in determining the price of CAT bonds, namely the word circles “stochastic processes”, “stochastic models”, and “random processes”. These three-word circles indicate that instead of using a deterministic approach, the widely used approach is a probabilistic approach through a stochastic process. In addition, there is also a circle of words “jump-diffusion process”. The jump-diffusion process is a form of development of CPP, which is also one of the methods in the stochastic process approach. Furthermore, in Figure 8, there are also word circles that refer to the factors involved in determining the price of CAT bonds, namely “stochastic interest rate”, “catastrophic event”, and “losses”. Finally, there is also the existence of word circles that refer to the model simulation methods in Figure 8, namely “monte carlo methods”, “monte-carlo simulations”, “monte carlo simulations”, and “computer simulations”. It indicates that not all CAT bond pricing models from each article have a closed-form solution, hence, these simulation methods determine the solution.

3.3. SLR Results

In this section, we analyze the extent to which the application of CPP in the pricing of CAT bonds has been carried out so far. The analysis includes how CPP is applied in pricing CAT bonds, what financial factors are involved in pricing CAT bonds, and what methods are used to estimate the price of CAT bonds.

3.3.1. The Application of CPP in the Pricing of CAT Bonds

In general, of the 30 articles reviewed, CPP was applied to design the first time the claim-triggering event of CAT bonds occurred. The claim-triggering event occurs when the attachment point of the specified claims index is exceeded within the life span of CAT bonds. The part of the claim index most used is the indemnity index, an index of actual losses experienced when a catastrophe occurs, or an index of losses based on the property claim service.

Two types of CPP used in the 30 articles described the claim-triggering event of CAT bonds for the first time. The first is a homogeneous CPP. In the homogeneous CPP, the catastrophe intensity is assumed to be constant. The first time claim-triggering event of CAT bonds designed with homogeneous CPP is as follows:

$$\tau =inf\left\{t:L_t>\mu \right\},$$

(1)

where $\tau$ is the first time the claim-triggering event of the CAT bond occurred, $\mu$ represents the attachment point, and $L_t$ is the homogeneous CPP representing the aggregate of catastrophic losses until time $t$. In more detail, $L_t$ is expressed as follows:

$$L_t={\displaystyle \sum }_{i=1}^{N_t}{X}_i,$$

(2)

where $N_t$ is a homogeneous Poisson process with an intensity of $\lambda >0$ representing the frequency of catastrophes that occur until time $t$, and $\left\{X_i,i=1,2,\dots ,N_t\right\}$ is a sequence of random variables representing the $i$-th single catastrophe loss.

The next type of CPP is a nonhomogeneous CPP. In the nonhomogeneous CPP, the catastrophe intensity is assumed to be not constant. In other words, the $N_t$ involved in Equation (2) is represented as a nonhomogeneous Poisson process with an intensity of ${\lambda }_t>0$. There are several assumptions applied to the CPP used in pricing CAT bonds in the 30 articles, which are as follows:

(a)

Losses for each catastrophe are assumed to be independent and identically distributed. In other words, the losses of one catastrophe with those of another do not affect each other, and the losses of each catastrophe follow the same probability distribution.

(b)

The number of catastrophe events is independent of catastrophe losses. In other words, the number of catastrophe events does not affect the losses experienced.

Assumptions (a) and (b) may be inappropriate. It is because every assumption of freedom does not always occur. However, it made the CAT bond pricing process simpler to design and simulate. Homogeneous and nonhomogeneous CPP users from the 30 articles are presented in

Table 3. Homogeneous and Nonhomogeneous CPP Users.

CPP Type	Frequency of Article	The Articles
Homogeneous	17	[3,9,19,20,21,22,25,27,28,29,30,31,32,33,34,35,36]
Nonhomogeneous	13	[23,24,26,37,38,39,40,41,42,43,44,45,46]

.

Table 3 shows that homogeneous CPP is more widely used in pricing CAT bonds than nonhomogeneous CPP. The catastrophe intensity is not always constant in every unit of time, but homogeneous CPP is the most widely used.

In addition to the single claim-triggering index, some articles in the 30 articles apply CPP in pricing CAT bonds with a multiple claim trigger index. These articles were written by Chao and Zou [3] and Ibrahim et al. [46]. Both use the loss index and the death index. In these two articles, CPP is applied to the design of one of the claim-triggering event times and the two claim-triggering event times. The time when one of the claim-triggering events of CAT bonds occurred designed with CPP is as follows:

$${\tau }_{min}=\min \left\{{\tau }_L,{\tau }_D\right\},$$

(3)

where ${\tau }_L$ represents the claim-triggering event of the loss index of CAT bonds that occurred for the first time, and ${\tau }_D$ represents the claim-triggering event of the death index of CAT bonds that occurred for the first time. In more detail, ${\tau }_L$ and ${\tau }_D$ are, respectively, expressed as follows:

$${\tau }_L=\mathit{inf}\left\{t:L_t>{\mu }_L\right\}$$

(4)

and

$${\tau }_D=\mathit{inf}\left\{t:D_t>{\mu }_D\right\},$$

(5)

where $L_t$ represents the CPP which represents the aggregate of catastrophe losses, $D_t$ represents the CPP which represents the aggregate of deaths, ${\mu }_L$ represents the attachment point of the aggregate of catastrophe losses, and ${\mu }_D$ represents the attachment point of the aggregate of catastrophe deaths. Meanwhile, the two claim-triggering events of CAT bonds that occurred for the first time designed with CPP are as follows:

$${\tau }_{max}=\max \left\{{\tau }_L,{\tau }_D\right\}.$$

(6)

As a result of using the multiple claim-triggering index, additional assumptions are imposed on the CAT bonds pricing model. These assumptions are as follows:

(a)

The number of deaths for each catastrophe is assumed to be independent and identically distributed. It means that the number of deaths of one catastrophe with those of another does not affect each other, and the number of deaths for each catastrophe follows an equal probability distribution.

(b)

The number of catastrophe events is independent of the number of catastrophe deaths. It means that the number of catastrophic events does not affect the number of deaths and vice versa.

(c)

Catastrophic losses are independent of the number of catastrophe deaths. It means that catastrophe losses do not affect the number of deaths and vice versa.

3.3.2. Analysis of Financial Factors Involved in Pricing CAT Bonds

This section analyzes financial factors and assumptions that apply to it in modeling CAT bond pricing in the 30 articles studied. Similar to the standard bond pricing, the CAT bond pricing of the 30 articles obtained involves the interest rate factor. The interest rate factor determines the present value of the coupon and principal payments. Then, there are two assumptions used to involve this interest rate factor. The two assumptions used are where some use constant and nonconstant interest rates. The number of articles that apply the assumption of constant and nonconstant interest rates is presented in

Table 4. The Frequency of Articles Applying Constant and Nonconstant Interest Rate Assumptions.

Interest Rate Assumptions	Frequency of Article	The Articles
Constant	12	[9,22,25,29,30,31,32,33,34,37,38,39]
Nonconstant	18	[3,19,20,21,23,24,26,27,28,35,36,40,41,42,43,44,45,46]

.

Table 4 shows that of the 30 articles, 12 articles apply a constant interest rate assumption, while the other 18 articles apply the assumption of a nonconstant interest rate. It indicates that the assumption of a nonconstant interest rate is predominantly more widely used. It is also appropriate where interest rates fluctuate in each period. In addition to involving the interest rate factor, there is 1 article involving other financial factors, namely the basic risk and credit risk factors. These factors are included in Lee and Yu’s [19] article.

Of the 18 articles that use nonconstant interest rates, we analyze the methods used to model them. After our analysis, there are 6 methods used. These methods are the Cox–Ingersoll–Ross (CIR) model, the Vasicek model, the Hull–White model, the CIR and Vasicek models, the robust approach, and the autoregressive integrated moving average (ARIMA) model. The number of articles using these methods is presented in

Table 5. The Frequency of Use of the Interest Rate Models.

Interest Rate Model	Frequency of Article	The Articles
CIR	8	[3,19,24,28,41,43,44,45]
Vasicek	5	[21,26,27,35,36]
Robust Approach	2	[23,40]
CIR and Vasicek	1	[20]
Hull–White	1	[42]
ARIMA	1	[46]

.

Table 5 shows that almost half of the 18 articles that use a nonconstant interest rate design their interest rates using the CIR model. Then, another 5 articles used the Vasicek model. The robust approach is used in 2 articles, and the other methods are used in 1 article each. The use of the CIR model is commonly used because, in this model, the interest rate is guaranteed not to have a negative value. It is under the actual situation wherein the interest rate is not negative in almost every country worldwide. Then, to model another financial factor, the credit risk factor, Lee and Yu [19] adopted the Duan, Moreau, and Sealey [47] model.

3.3.3. The Analysis of the CAT Bond Price Estimation Methods

Models can have closed or non-closed solutions. If the model has a closed-form solution, the solution can be determined by ordinary arithmetic operations. If the model does not have a closed-form solution, the solution is difficult or even impossible to decide on by common arithmetic operations. In this section, we analyze the model solutions from the 30 articles. The number and percentage of articles that have and do not have closed-form solutions are presented in Figure 9.

Figure 9 shows that of the 30 models in the 30 articles, 25 models (83%) do not have a closed-form solution, and 5 other models (17%) have a closed-form solution. Thus, of the 30 existing models, these models generally do not have a closed-shaped solution. For models that have a closed-form solution, the authors are Jarrow [23], Sun et al. [40], Georgiopoulos [32], Tang and Yuan [36], and Deng et al. [9]. They do not require an alternative method to obtain the solution because this can be determined by common arithmetic operations, whereas models that do not have a closed-form solution must use an alternative method to determine the solution. Therefore, we analyzed the alternative methods used from the 25 articles with these non-closed solutions. After the analysis is carried out, to determine the closed-form solution of the model used in 25 articles, there are 7 methods used. The methods used are the Monte Carlo method, the Quasi-Monte Carlo method, the mixed distribution approach method, the Wang-Double-Factor model, the numerical integral method, the combination mixed distribution approach method and Nuel recursive method, and a combination of the Monte Carlo method and the stochastic iteration method. The number and percentage of articles using these methods are presented in Figure 10.

Figure 10 shows that the Monte Carlo method is the most widely used method to obtain a non-closed solution from the model. The reason for the many uses of the Monte Carlo method is that the Monte Carlo method is very intuitive. That is, this method is easy to reason with logic; hence, why it is widely used. For the other methods, each is used uniquely. The combination of the Monte Carlo method and the stochastic iteration equation was used by Romaniuk [29], and the Quasi-Monte Carlo method was used by Albrecher et al. [37]. Haslip and Kaishev [30] used the integral numerical method, and Ma and Ma [24] used a mixed approximation method. The Wang-Double-Factor model method was used by Chen et al. [31], and the combination of a mixed approximation method and the Nuel recursive method was used by Ibrahim et al. [46].

4. Discussions

In this section, we discuss the results obtained from Section 3. The discussion includes presenting fascinating facts and the presentation of gaps from the 30 articles that discuss how to apply CPP to pricing CAT bonds.

4.1. The Facts of Bibliometric Analysis and SLR Results

If the 30 articles are reviewed from the number of citations provided in Section 3.2, the articles published after 2010 are cited more frequently; they even fill many places in the top 10 articles with the greatest number of citations. However, the year of publication does not significantly affect the number of citations. That is, other factors influence it. Other factors may be the quality of the article, the suitability of the model with the actual situation, and the ease of obtaining articles.

The next fact is that the number of articles on CPP application pricing CAT bonds tends to increase yearly. The popularity of CAT bonds as a provider of catastrophe contingency costs is rising, so knowledge about how to price them is increasingly needed. If we look at the current condition where the frequency of catastrophes continues, research on how to price CAT bonds with CPP will increase in the future.

Based on the analysis presented in Table 3 in Section 3.3.1, the use of homogeneous CPP is more than nonhomogeneous CPP, even though the nonhomogeneous CPP is better in describing the aggregate of losses and deaths because it is more in line with the actual situation. The possible reason for this is for the sake of model simplicity. An example is what was conducted by Lee and Yu [19]. In addition, another reason is a necessity due to the use of an assumption or other method. An example is Chao and Zou [3], who used homogeneous CPP due to copulas in designing random vectors of loss risk and fatality risk, which require identical multivariate distributions.

Based on the analysis of the number of claim-triggering events used in the 30 articles studied, there are exciting things where the use of multiple claim-triggering events on CAT bonds has begun to grow in the last four years. Chao and Zou [3] first used this type of claim-triggering event. It was then continued by Ibrahim et al. [46]. It could indicate that the need for CAT bonds with multiple claim-triggering event types may increase. It can undoubtedly happen along with the increasing frequency of catastrophes in the last five decades, such as the data and facts described in Section 1.

4.2. Gap Analysis

First, we analyze the general model gap. Almost all of the existing CAT bond pricing models with CPP are not specific in referring to certain types of catastrophes. The catastrophes that these models refer to are catastrophes in general. Thus, developing CAT bond price modeling with CPP can refer to inevitable catastrophes, such as CAT bonds for earthquakes, volcanoes, or floods.

Next, we analyze the model gaps based on the factors involved. For CAT bonds that use a single claim-triggering event, if each model is considered from the catastrophe severity factor, almost all use catastrophic losses, even though other severity factors can be used. Examples are the maximum earthquake magnitude, the maximum area flooded, and the maximum radius of the area affected by volcanic eruptions. Therefore, this may be developed. For CAT bonds involving multiple claim-triggering events, almost all combinations of catastrophe severity factors used are catastrophic losses and number of deaths, although other combinations of severity factors can also be used. For example, the combinations of loss and magnitude of earthquakes for earthquake CAT bonds can be considered. Therefore, this can also be a research opportunity in the future. Then, if each model is viewed considering the financial factors involved, no model consists of the inflation rate factor even though this factor significantly affects the present value of money time-by-time. Therefore, the involvement of these factors can be a development.

The following model gap analysis is carried out based on the methods used in modeling financial and catastrophe factors. The methods used to model financial factors are fair, with many models being used. Meanwhile, the methods used to model the catastrophe factor are still slightly varied. Because the severity factor that almost all models use is catastrophe loss, practically all models use a random addition process:

$$L_t={\displaystyle \sum }_{i=1}^{N_t}{X}_i,$$

(7)

where $L_t$, $t=0, 1, 2,\dots ,$ represents the aggregate loss at time $t$, $X_i$, $i=1, 2,\dots ,N_t$ represents the $i$-th catastrophe loss, and $N_t$ represents the Poisson process with a catastrophe intensity of $\lambda >0$. A random maximum value process can be applied as an alternative to modeling other catastrophe severity factors, such as maximum earthquake magnitude:

$$M_t=\max \left\{Y_i,i=1, 2, \dots , N_t\right\},$$

(8)

where $M_t$,$t=0, 1, 2,\dots$ is the maximum earthquake magnitude until time t, and $Y_i$, $i=0, 1, 2,\dots ,N_t$ represents the magnitude of the $i$-th earthquake. The involvement of this method can be a new opportunity in the future.

5. Conclusions

This study presents a systematic literature review on the application of CPP in pricing CAT bonds. In collecting articles from the Scopus and Science Direct databases, we obtained 99 articles on how to apply CPP in pricing CAT bonds. After the manual selection process, namely duplication, abstraction, and full paper selection, the 30 articles on CPP application in pricing CAT bonds are collected. Then, the articles are bibliometrically analyzed using R software and VosViewer. The bibliometric analysis results are an overview of the 30 articles on how to apply CPP in pricing CAT bonds. One of the results is that articles on how to apply CPP in pricing CAT bonds tend to increase from 2002 to 2022. It indicates that research on this topic is becomes more popular from time-to-time as the demand for CAT bonds increases due to the tendency of the increasing frequency of world catastrophes.

Models from the 30 articles are analyzed as well. This analysis includes how CPP is applied in pricing CAT bonds, what financial factors are involved in pricing CAT bonds, and what methods are used to estimate the price of CAT bonds.

In general, of the 30 articles reviewed, CPP was applied to design the first time the claim-triggering event of CAT bonds occurred. Two types of CPP were used in the 30 articles to describe the claim-triggering event of CAT bonds for the first time, namely homogeneous CPP and nonhomogeneous CPP. In more detail, all models predominantly involve a single claim-triggering event. Only two articles involve multiple claim-triggering events. CAT bonds with a single claim-triggering event used catastrophe losses to measure the severity of the catastrophe, whereas for CAT bonds with multiple claim-triggering events, all used catastrophe losses and the number of deaths to measure the severity of the catastrophe.

We also analyze the financial factor involved in the models. The involved fundamental financial factor is the interest rate. It calculates the present value of the coupon and principal payments. Two assumptions are used to model the interest rates, namely constant interest rates and nonconstant interest rates. In addition, another financial factor is only addressed by Lee and Yu [19], namely the credit and the basis risk. The methods used to model nonconstant interest rates are very diverse, for example, the CIR model, the Vasicek model, the Hull–White model, the robust approach, and the ARIMA model. The credit and the basis risk are modeled by Lee and Yu [19], referred to as the model of Duan, Moreau, and Sealey [47].

In terms of the form of the model solution, they have or do not have a closed-form solution. For a model with a closed-form solution, the solution can be determined by ordinary arithmetic operations, while for a model with no closed-form solution, the solution is determined by the simulation method. The Monte Carlo simulation method is widely used to determine the closed solution.

Finally, the model gaps of the 30 articles are also analyzed. In general, all articles are not referring to a specific type of catastrophe. Therefore, designing a CAT bond pricing model with CPP for certain types of catastrophes can be an opportunity for future research. Then, because the catastrophe referred to is a catastrophe in general, almost all measures of catastrophe severity used in the model are catastrophic losses. If the CAT bond price model designed refers to a specific type of catastrophe, for example, an earthquake, the catastrophe severity measure that can be used is the maximum earthquake magnitude. Moreover, there is no model for determining the price of CAT bonds with CPP that involves other financial factors, namely the inflation factor. Therefore, this is also an opportunity for new research in the future. Finally, all methods used to model catastrophe risk factors were CPP for random summation. It can be varied by using a random maximum value, i.e., the maximum value of some $N$ random variables, where $N$ represents the random variable of the number of catastrophes. It is hoped that this research can be used as a reference for developing research on how to price CAT bonds with CPP and can motivate other researchers to design more beneficial methods of CAT bond pricing in the future.