An Alternative Source of Funding to Mitigate Flood Losses through Bonds: A Model for Pricing Flood Bonds in Indonesian Territory

Abstract

Indonesia suffers significant economic losses from floods, and state budget allocations are often inadequate. Flood bonds provide an alternative funding source, but the pricing framework is complex due to simultaneous flood and financial risk considerations. Therefore, this study aims to model flood bond prices as an alternative flood funding in Indonesia. The model is formulated using the risk-neutral-pricing measure with the stochastic assumption of the force of interest. The claim trigger is represented as maximum rainfall, which is modeled as a continuous-stochastic process with a discrete-time index. Given the varying patterns of rainy and dry seasons, we assume both durations are dynamic. Then, we provide the approximate model solution for the government to estimate bond prices quickly. This estimation shows that the bond’s trigger point is proportional to the bond prices. Additionally, bond prices are proportional to the dry season duration and inversely proportional to the rainy season duration. We also show that using a stochastic force of interest yields significant differences from a constant one except for the constant as data average. This study can help the Indonesian government price flood bonds and provide more tools for related meteorological and climatological institutions to calculate the probability of future maximum rainfall.

1. Introduction

Indonesia has a high rainfall danger with an annual average cumulative rainfall of 1500 to 4000 mm/year [1]. High rainfall generally occurs during the rainy season, which occurs from October to March on average. The geographical location at the equator, precisely between two continents (Asia and Australia) and two oceans (Indian and Pacific), causes sunlight throughout the year, high evaporation of seawater, and the rapid formation of rain clouds. Monsoon winds and diverse mountain topography also increase rainfall intensity through orographic processes.

The high rainfall danger in Indonesia makes this country vulnerable to weather catastrophes, especially floods. Based on data from The International Disaster Database (accessed on 2 February 2024 at https://public.emdat.be/) as visually presented in Figure 1, there are, on average, eight flood catastrophes yearly. This average is the second highest worldwide after China. The highest frequency of floods occurred in 2020, with 25 annual catastrophes, while the lowest was in 2004, with one flood catastrophe. According to Figure 1, the frequency of flood catastrophes tends to increase each year, as indicated by the positive slope of the linear trend line in Figure 1. Meteorologically, this increasing trend is due to the intensification of the La Niña phenomenon, which increases rainfall, especially during the rainy season. Increased tropical storm activity in the western Pacific Ocean also brings extreme rainfall [2]. Climatologically, global climate change has caused an increase in sea surface temperature by 0.8 °C since 1901. It increases evaporation and atmospheric moisture, resulting in higher and more intense rainfall in Indonesia [3].

Unfortunately, the increasing tendency of flood frequency in Indonesia is still slightly mitigated. Hence, the increasing frequency tendency positively correlates with the losses experienced [4], where one flood disaster causes average economic losses of 50.7 million dollars. In 2020 alone, economic losses due to flooding in Indonesia reached more than 2.1 billion USD or around 31.5 trillion IDR (exchange rate 1 USD = 15,000 IDR) (accessed on 2 February 2024 on National Disaster Management Agency of the Republic of Indonesia, https://data.bnpb.go.id/). These economic losses include damage to infrastructure, damage to homes, decreased agricultural productivity, and significant disruption to business activities. According to a report of The World Bank Group & Asian Development Bank [5], the intensity and frequency of extreme rainfall in Indonesia will increase further in the future because of climate change. It could cause annual economic losses to increase to 4.3 billion USD by 2030.

To fund flood mitigation and recovery, the Indonesian government has generally relied on two sources of funds: the state budget (central or regional) and social assistance (from the private sector, international institutions, communities, and other donors). These funds are usually used for infrastructure projects such as building embankments, normalizing rivers, drainage systems, and providing catastrophe management tools. However, this funding is often insufficient to overcome all significant flooding problems because the amount is limited [6,7]. Therefore, new sources of funding to deal with floods are needed.

Efforts to increase sources of funds to mitigate flood catastrophes in Indonesia have been started through insurance. Based on data from the International Disaster Database (Accessed on 18 July 2024 on https://public.emdat.be/), since 2000, one flood catastrophe has received insurance benefits of 7.2 million dollars on average. However, this benefit’s average amount is only 14.27 percent of the actual losses on average, so it is not significant enough to cover it. For example, in the 2013 Jakarta Flood, insurance benefits only covered 3.33 percent of actual losses, while in the 2014 Tangerang Flood, insurance benefits only covered 9.99 percent of actual losses. These data can also be seen in Figure 2. Therefore, the insurance mechanism still needs to be effective in funding flood mitigation in Indonesia, and other sources of funds are needed.

The World Bank and several other countries have tried to identify alternative sources of funding for catastrophe management (including floods) [8,9]. As a result of this identification, insurance-linked securities (ILSs) became a new mechanism [10,11]. Through this mechanism, the community is invited to play a role in flood funding as ILS investors. Communities as investors not only pursue financial profits but also support social goals to increase preparedness and resilience to catastrophes. Given the high risk of flooding, they will be given significant compensation from this investment in return. Among existing securities, bonds are the most successful securities in ILS. Bonds have a clear interest (coupon) distribution schedule, fast liquidity, and moderate financial risk. Investors and insureds prefer these advantages in this scheme. These are, after this, referred to as flood bonds. Many countries have issued flood bonds. Under historical data on flood bond issuance taken on 1 March 2024, from ARTEMIS (https://www.artemis.bm/deal-directory), the UK used catastrophe bonds worth 150 million USD to transfer flood risks in April 2007. Then, in May 2018, Japan used catastrophe bonds worth 320 million USD in funding measures to mitigate the consequences of typhoons, floods, and earthquakes. The United States used 300 million USD from flood bonds in April 2019.

Based on the previous paragraph, flood bonds can be used as an alternative source of funds for flood management in Indonesia. However, to issue it, one thing that must be conducted is to study the pricing framework in Indonesia. The flood bond pricing framework is complex to formulate because prices are simultaneously influenced by financial and flood risk variables, e.g., interest rates, rainfall, duration of the rainy season, and duration of the dry season. Therefore, this research aims to model flood bond prices in Indonesia. Considering that the rainy season and dry season patterns in Indonesia are not always constant, we consider it in the designed model. To model the trigger event, we use the maximum rainfall trigger index to quantify it. This index has the advantage that the measurement is quickly carried out, objective, and transparently based on the flood catastrophe that occurred. These advantages minimize administrative costs in assessing claims and the potential for moral hazard in claims. We model this index as a continuous stochastic process with a discrete-time index. Then, we model flood bond prices using a risk-neutral-pricing measure assuming a stochastic force of interest. After the modeling was conducted, we applied the model to rainfall and interest rate data in Indonesia. Finally, we also analyze the influence of rainfall and financial risks on flood bond prices, e.g., trigger point of rainfall, bond term, force of interest, and time to buy the bond. This study can increase the tools available for calculating the probability of maximum rainfall in the future for meteorological or climatological institutions in this country. Apart from that, this study can certainly be used by flood bond issuers in Indonesia in determining flood bond prices that better reflect the real situation based on the inconstant duration of rainy and dry seasons and the stochastic force of interest.

2. Related Research

Before reviewing articles on flood bond pricing modeling, we describe some research on catastrophe bond pricing modeling of general catastrophes. This research became popular in the 2000s. Cox and Pedersen [12] used equilibrium pricing theory to develop a method for pricing catastrophe bonds based on a model of interest rate structure and catastrophe risk probability. This model has been used as a basis for modeling in subsequent research. Lee and Yu [13] developed a catastrophe bond pricing model considering moral hazards, basis risks, and stochastic interest rates. Albrecher et al. [14] examined quasi-Monte Carlo methods and variance reduction algorithms for pricing catastrophe bonds involving integrals with discontinuous and infinite-dimensional integrands. Zimbidis et al. [15] modeled earthquake catastrophe bond prices using an extreme value theory (EVT) approach and presented numerical results using Monte Carlo simulations and stochastic iterative equations. Jarrow [16] provided a simple solution for a catastrophe bond (CAT) pricing model using a model for the evolution of the Libor term structure of interest rates.

Research on modeling catastrophe bond prices has become increasingly popular since 2011. Nowak and Romaniuk [17] developed the model of Cox and Pedersen [12] by describing interest rate dynamics using the Cox–Ingersoll–Ross and Hull–White models. Ma and Ma [18] applied the approximation method introduced by Chaubey et al. [19] to determine the solution of the model by Lee and Yu [13]. Liu et al. [20] used the Jarrow and Turnbull model and EVT methods to model catastrophe bond prices with credit risk involved. Ma et al. [21] modeled catastrophe bond prices using a doubly stochastic Poisson process with intensity function proposed by Black et al. [22]. Georgiopoulos [23] proposed a stochastic optimization model for pricing catastrophe bonds. Giuricich and Burnecki [24] modeled catastrophe bond prices using a left-truncated heavy-tailed distribution of a nonhomogeneous compound Poisson process. Deng et al. [25] modeled catastrophe bond prices for drought catastrophe financing using a homogeneous compound Poisson process. Then, Ibrahim et al. [26] modeled catastrophe bond prices with an aggregate loss trigger index and a number of fatalities using a nonhomogeneous compound Poisson process. Sukono et al. [27] developed the model of Ibrahim et al. [26] involving the inflation rate and the correlation rate between trigger indices. Both are modeled using vector autoregression and copula models. Tang et al. [28] developed the model of Ibrahim et al. [26] by modeling the trigger indices using the peaks-over-threshold (POT) method. Then, Manathunga and Deng [29] modeled pandemic bond prices using the stochastic logistic growth model.

After providing a brief previous research overview on modeling catastrophe bond prices in general, we review research with the main topic of modeling flood bond prices until 4 June 2024. After searching in the Scopus, Springer, and Taylor databases, four previous studies discussed this topic. Chen et al. [30] modeled flood bond prices with a single-period maturity. They used the trigger indemnity (actual loss) index to determine the trigger event. They designed the index using a homogeneous compound Poisson process. The model they studied did not have a closed-form solution, so the solution was approximated by the Wang-double-factor transformation method. Once the model was designed, they applied it using data on extreme flood losses in China from 1961 to 2009. In the application, they found that the trigger point of flood bonds was proportional to their price.

Chao and Zou [31] modeled multi-period flood bond prices using parametric trigger and indemnity indices. The parametric and indemnity trigger indices were represented by the number of fatalities and aggregate losses from extreme flooding, respectively. They designed the marginal distributions of both indices using a combination of a homogeneous compound Poisson process and peaks-over threshold method. Then, the joint distribution of the two indices was modeled with an Archimedean copula. To model flood bond prices, they used a risk-neutral-pricing measure and a stochastic interest rate based on the Cox–Ingersoll–Ross (CIR) model. The model they designed did not have a closed-form solution, so the solution was approached using the Monte Carlo method. After designing the model, they applied it to flood loss and fatality data provided by Dartmouth College since 1981. In the application, they found that flood intensity, period, and correlation between indices had an inverse relationship with flood bond prices.

Li et al. [32] designed a flood bond pricing model with a trigger index for the number of dangerous floods that occur. Dangerous flood was defined as flooding accompanied by rainfall that exceeds a specified threshold value. The number of these dangerous floods was modeled with compound distributions. This compound distribution consisted of the dangerous flood frequency distribution represented by the Poisson distribution and the rainfall distribution represented by the generalized Pareto distribution (GPD). The flood bond pricing model was modeled using a risk-neutral-pricing measure and stochastic interest rates based on the Vasicek model. The bond price model did not have a closed-form solution, so the solution was approximated by the Wang transformation method. The model was then applied to flood catastrophe data in China from 2004 to 2019, which was accessed in the Ministry of Water Resources of the People’s Republic of China.

Ibrahim et al. [33] designed a flood bond pricing model with a closed-form solution. A closed-form solution was obtained because they assumed the interest rates were constant. The model was designed with an indemnity trigger index. The index was modeled using a homogeneous compound Poisson process. To estimate the solution of the model, they applied the Normal, Normal-Power 2, Gamma, and Gamma Inverse-Gaussian approximation methods described by Chaubey et al. [19] and Reijnen et al. [34]. After the model was designed, they applied it to flood data in Indonesia from 2010 to 2021, which was accessed at the National Catastrophe Management Agency of the Republic of Indonesia. On that application, they found that the loss an investor might experience on a flood bond was half the possible gain.

We identify the gaps based on a review of the research carried out. Then, we made these gaps novelties for our research. These novelties are as follows:

a.

This study uses a parametric trigger index represented by the maximum rainfall in each dekadal (10 days) each year.

b.

This study considers the duration of the rainy and dry seasons each year, which are assumed to be not constant. This inconstancy is adjusted to the actual conditions regarding the varying patterns of the country’s rainy and dry seasons.

c.

This study examines the effect of buying time on flood bond prices so that investors can use it as a guide about when it is the right time to buy bonds based on the claim probability and price they expect.

3. Flood Bond: Simply and Brief Explanation

Flood bonds, one type of catastrophe bond, are financial securities linked to flood risk belonging to a sponsor (government or other institution, insurer, or reinsurer). This linking is conducted by sponsors by sharing some or all the flood risk they bear with investors in the capital market. These linking aims to obtain a broader coverage capacity to face the risk of enormous flood losses. The enormous risk of major flood losses provides significant compensation to investors of these securities in return. If the claim trigger event for a flood bond does not occur within its life span, the investor receives in full of all periodic coupons and the redemption value. However, they must be prepared to risk losing some or all of it if the triggering event occurs [35].

In simple terms, flood bonds have three main entities: sponsors, special purpose vehicles (SPVs), and investors. The sponsor is the insured (government or another institution, insurer, or reinsurer) of the flood bond. They share flood risks with investors in the capital market through bonds, form special purpose vehicles (SPVs) that will issue bonds, and pay premiums to the SPV. Once an SPV is established and issued bonds, they collect bond funds from investors and premiums from sponsors. These funds are then managed by investing them in short-term safe securities, e.g., treasury bonds. The investment results will later be paid as coupons to investors.

In flood bonds, the trigger event is measured by various indices. Several indexes are commonly used, namely the indemnity index, parametric index, model index, and industry index. The indemnity index is measured by actual losses from flooding experienced by sponsors. Then, the parametric index is measured through specific parameters related to flood catastrophes, e.g., maximum rainfall and the number of fatalities. Meanwhile, the model index is measured based on risk analysis results from risk modeling agencies, e.g., Risk Management Solution (RMS) and Applied Insurance Research (AIR) Worldwide. Finally, the industry index is measured by the losses incurred by the industry, e.g., the issuer. Combinations between trigger indices are also possible. This study uses a parametric index, maximum rainfall, as the trigger index of flood bonds. This index has the advantage that it measures quickly, objectively, and transparently based on the flood that occurred [36,37,38]. These advantages minimize administrative costs in assessing claims and the potential for moral hazard in claims.

4. The Model

4.1. Assumptions

The following are the assumptions used in modeling the flood bond prices in this study:

a.

The term of a flood bond is $T$ years, where $T$ is a positive integer.

b.

All variables are modeled in $(\mathrm{\Omega },\mathcal{F},{\mathcal{F}}_k,\mathrm{P})$, where $\mathrm{\Omega }$ is the set of states of the world, $\mathcal{F}$ is the sigma-algebra on $\mathrm{\Omega }$, ${\mathcal{F}}_k\subset \mathcal{F}$ is increasing filtration with $k\in \left[0,T\right]$, and $\mathrm{P}$ is a probability measure (also called the probability set function) on $\mathcal{F}$. Suppose that $\mathfrak{A}$ is a random variable, and $\mathcal{A}\in \mathcal{F}$. Then, $\mathrm{P}\left(\mathfrak{A}\in \mathcal{A}\right)=\mathrm{P}\left[\left(\omega \in \mathrm{\Omega }|\mathfrak{A}\left(\omega \right)\in \mathcal{A}\right)\right]$.

c.

The face value of the flood bond is $K$, where this value is paid at the end of year $T$.

d.

Coupons from flood bonds are paid annually in the amount of $C$ at the end of each year $t$ with $t=1, 2,\dots , T.$

e.

One month is divided into three dekadals: the 1st to the 10th is called the first dekadal, the 11th to the 20th is called the second dekadal, and the 21st to the end of the month is called the third dekadal. In other words, there are 36 dekadals in one year.

f.

Every year begins with the first rainy season. Then, the season continues with the dry season. Finally, the year ends with the second rainy season. It is adjusted to the duration of rainy and dry seasons in Indonesia in reality [39]. For other countries, it can be reformulated.

g.

The duration of rainy and dry seasons is discrete with dekadal units. The start of the first rainy season in year $t$ is the 1st dekadal. Then, the start of the dry season in year $t$ is the $d_t$-th dekadal, and the start of the second rainy season in year $t$ is the $r_t$-th dekadal. In other words, the duration of the first rainy season is $\left\{\mathrm{1,2},\dots ,d_{t-1}\right\}$, the duration of the dry season is $\left\{d_t,d_{t+1},\dots ,r_{t-1}\right\}$, and the duration of the second rainy season is $\left\{r_t,r_{t+1},\dots ,36\right\}$.

h.

The start of the dry season $d_t$ is determined as the 1st dekadal with cumulative rainfall of fewer than fifty millimeters and is followed by the next two dekadals, each of which is less than fifty millimeters of cumulative precipitation. Then, the start of the second rainy season $r_t$ is determined as the 1st dekadal with cumulative rainfall of more than or equal to fifty millimeters and is followed by the next two dekadals, each of which is more than or equal to fifty millimeters of cumulative precipitation.

i.

Maximum rainfall at each dekadal does not affect each other. In reality, this assumption may not hold. However, assumptions are made to facilitate the modeling process, likewise with the use of assumptions on the following random variable.

j.

Maximum rainfall values over the $m$-th dekadal in the rainy season in a year, denoted by $R_m$ with $m=1, 2,\dots ,d_t-1,r_t,r_t+1,\dots , 36$, are assumed to have identical probability distribution characteristics. In other words, $\mathrm{P}\left(R_m\le \mathcal{u}\right)=\mathrm{P}\left(R\le \mathcal{u}\right)=F_R\left(\mathcal{u}\right),$ for all $m=1, 2,\dots ,d_t-1,r_t,r_t+1,\dots , 36$. Then, maximum rainfall values over the $n$-th dekadal in the dry season of a year, denoted by $D_n$ with $n=d_t, d_t+1,\dots ,r_t-1$, are assumed to have identical probability distribution characteristics. In other words, $\mathrm{P}\left(D_n\le \mathcal{u}\right)=\mathrm{P}\left(D\le \mathcal{u}\right)=F_D\left(\mathcal{u}\right),$ for all $n=d_t, d_t+1,\dots ,r_t-1$.

k.

The annual force of interest $I_t$ for all $t=1, 2, \dots , T$ in one year is assumed to not affect the annual force of interest in other years and have the same distribution characteristics. In other words, $\mathrm{P}\left(I_t\le \mathcal{i}\right)=\mathrm{P}\left(I\le \mathcal{i}\right)=F_I\left(\mathcal{i}\right)$, for all $t=1, 2, \dots , T$.

4.2. Maximum Rainfall Trigger Index Model

Maximum rainfall from the $c_1$-th dekadal to $c_2$-th dekadal in year $t$ with $1\le c_1\le c_2\le 36$ and $t=1,2,\dots ,T$ is designed as follows:

a.

When $c_1$ and $c_2$ are in the first rainy season interval, the value is equal to the maximum values of $R_{c_1}$, $R_{c_1+1}$, $\dots$, and $R_{c_2}$.

b.

When $c_1$ is in the first rainy season interval, and $c_2$ is in the dry season interval, the value is equal to the maximum value of $R_{c_1}$, $R_{c_1+1}$, $\dots$, $R_{d_t-1}$, $D_{d_t}$, $D_{d_t+1}$, $\dots$, and $D_{c_2}$.

c.

When $c_1$ is in the first rainy season interval, and $c_2$ is in the second rainy season interval, the value is equal to the maximum value of $R_{c_1}$, $R_{c_1+1}$, $\dots$, $R_{d_t-1}$, $D_{d_t}$, $D_{d_t+1}$, $\dots ,D_{r_t-1}$, $R_{r_t}$, $R_{r_t+1}$, $\dots$, and $R_{c_2}$.

d.

When $c_1$ and $c_2$ are in the dry season interval, the value is equal to the maximum values of $D_{c_1}$, $D_{c_1+1}$, $\dots$, and $D_{c_2}$.

e.

When $c_1$ is in the dry season interval, and $c_2$ is in the second rainy season interval, the value is equal to the maximum value of $D_{c_1}$, $D_{c_1+1}$, $\dots$, $D_{r_t-1}$, $R_{r_t}$, $R_{r_t+1}$, $\dots$, and $R_{c_2}$.

f.

When $c_1$ and $c_2$ are in the second rainy season interval, the value is equal to the maximum values of $R_{c_1}$, $R_{c_1+1}$, $\dots$, and $R_{c_2}$.

Mathematically, we denote it by ${\mathrm{M}}_{c_1,c_2,t}$ formulated as follows:

$${\mathrm{M}}_{c_1,c_2,t}=\left\{\begin{array}{cc}\max \left\{R_{c_1},\dots ,R_{c_2}\right\}& :c_1=1,2,\dots d_t-1,c_2=1,2,\dots ,d_t-1\hfill \\ \max \left\{R_{c_1},\dots ,R_{d_t-1},D_{d_t},\dots ,D_{c_2}\right\}& :c_1=1,2,\dots d_t-1,c_2=d_t,d_t+1,\dots ,r_t-1\hfill \\ \max \left\{R_{c_1},\dots ,R_{d_t-1},D_{d_t},\dots ,D_{r_t-1},R_{r_t},\dots ,R_{c_2}\right\}& :c_1=1,2,\dots d_t-1,c_2=r_t,r_t+1,\dots ,36\hfill \\ \max \left\{D_{c_1},\dots ,D_{c_2}\right\}& :c_1=d_t,d_t+1,\dots ,r_t-1,c_2=d_t,d_t+1,\dots ,r_t-1\hfill \\ \max \left\{D_{c_1},\dots ,D_{r_t-1},R_{r_t},\dots ,R_{c_2}\right\}& :c_1=d_t,d_t+1,\dots ,r_t-1,c_2=r_t,r_t+1,\dots ,36\hfill \\ \max \left\{R_{c_1},\dots ,R_{c_2}\right\}& :c_1=r_t,r_t+1,\dots ,36,c_2=r_t,r_t+1,\dots ,36\hfill \end{array}\right.$$

After ${\mathrm{M}}_{c_1,c_2,t}$ is defined, we then formulate its cumulative distribution function as follows:

$$\mathrm{P}\left({\mathrm{M}}_{c_1,c_2,t}\le \mathcal{u}\right)=\left\{\begin{array}{cc}\mathrm{P}\left(\max \left\{R_{c_1},\dots ,R_{c_2}\right\}\le \mathcal{u}\right)& :c_1=1,2,\dots d_t-1,c_2=1,2,\dots ,d_t-1\hfill \\ \mathrm{P}\left(\max \left\{R_{c_1},\dots ,R_{d_t-1},D_{d_t},\dots ,D_{c_2}\right\}\le \mathcal{u}\right)& :c_1=1,2,\dots d_t-1,c_2=d_t,d_t+1,\dots ,r_t-1\hfill \\ \mathrm{P}\left(\max \left\{R_{c_1},\dots ,R_{d_t-1},D_{d_t},\dots ,D_{r_t-1},R_{r_t},\dots ,R_{c_2}\right\}\le \mathcal{u}\right)& :c_1=1,2,\dots d_t-1,c_2=r_t,r_t+1,\dots ,36\hfill \\ \mathrm{P}\left(\max \left\{D_{c_1},\dots ,D_{c_2}\right\}\le \mathcal{u}\right)& :c_1=d_t,d_t+1,\dots ,r_t-1,c_2=d_t,d_t+1,\dots ,r_t-1\hfill \\ \mathrm{P}\left(\max \left\{D_{c_1},\dots ,D_{r_t-1},R_{r_t},\dots ,R_{c_2}\right\}\le \mathcal{u}\right)& :c_1=d_t,d_t+1,\dots ,r_t-1,c_2=r_t,r_t+1,\dots ,36\hfill \\ \mathrm{P}\left(\max \left\{R_{c_1},\dots ,R_{c_2}\right\}\le \mathcal{u}\right)& :c_1=r_t,r_t+1,\dots ,36,c_2=r_t,r_t+1,\dots ,36\hfill \end{array}\right..$$

(1)

Note that for each $m=1, 2,\dots ,d_t-1,r_t,r_t+1,\dots ,36$ and $n=d_t, d_t+1,\dots ,r_t-1$, it holds as follows [40]:

$$\max \left\{R_m,D_n\right\}\le \mathcal{u}\equiv \left\{R_m\le \mathcal{u},D_n\le \mathcal{u}\right\}.$$

(2)

Then, because the maximum rainfall values in each dekadal do not affect each other, the following equation holds:

$$\mathrm{P}\left(R_m\le \mathcal{u},D_n\le \mathcal{u}\right)=\mathrm{P}\left(R_m\le \mathcal{u}\right)\mathrm{P}\left(D_n\le \mathcal{u}\right),$$

(3)

for each $m=1, 2,\dots ,d_t-1,r_t,r_t+1,\dots ,36$ and $n=d_t, d_t+1,\dots ,r_t-1$. Based on Equations (2) and (3), the CDF of ${\mathrm{M}}_{c_1,c_2,t}$ in Equation (1) can be rewritten as follows:

$$\mathrm{P}\left({\mathrm{M}}_{c_1,c_2,t}\le \mathcal{u}\right)=\left\{\begin{array}{cc}\mathrm{P}\left(R_{c_1}\le \mathcal{u}\right)\dots \mathrm{P}\left(R_{c_2}\le \mathcal{u}\right)& :c_1=1, 2,\dots d_t-1, c_2=1, 2,\dots ,d_t-1 \hfill \\ \mathrm{P}\left(R_{c_1}\le \mathcal{u}\right)\dots \mathrm{P}\left(R_{d_t-1}\le \mathcal{u}\right)\mathrm{P}\left(D_{d_t}\le \mathcal{u}\right)\dots \mathrm{P}\left(D_{c_2}\le \mathcal{u}\right)& :c_1=1, 2,\dots d_t-1, c_2=d_t, d_t+1,\dots ,r_t-1 \hfill \\ \mathrm{P}\left(R_{c_1}\le \mathcal{u}\right)\dots \mathrm{P}\left(R_{d_t-1}\le \mathcal{u}\right)\mathrm{P}\left(D_{d_t}\le \mathcal{u}\right)\dots \mathrm{P}\left(D_{r_t-1}\le \mathcal{u}\right)\mathrm{P}\left(R_{r_t}\le \mathcal{u}\right)\dots \mathrm{P}\left(R_{c_2}\le \mathcal{u}\right)& :c_1=1, 2,\dots d_t-1, c_2=r_t, r_t+1,\dots ,36 \hfill \\ \mathrm{P}\left(D_{c_1}\le \mathcal{u}\right)\dots \mathrm{P}\left(D_{c_2}\le \mathcal{u}\right)& :c_1=d_t, d_t+1,\dots ,r_t-1, c_2=d_t, d_t+1,\dots ,r_t-1\hfill \\ \mathrm{P}\left(D_{c_1}\le \mathcal{u}\right)\dots \mathrm{P}\left(D_{r_t-1}\le \mathcal{u}\right)\mathrm{P}\left(R_{r_t}\le \mathcal{u}\right)\dots \mathrm{P}\left(R_{c_2}\le \mathcal{u}\right)& :c_1=d_t, d_t+1,\dots ,r_t-1, c_2=r_t, r_t+1,\dots ,36 \hfill \\ \mathrm{P}\left(R_{c_1}\le \mathcal{u}\right)\dots \mathrm{P}\left(R_{c_2}\le \mathcal{u}\right)& :c_1=r_t, r_t+1,\dots ,36, c_2=r_t, r_t+1,\dots ,36 \hfill \end{array}\right..$$

(4)

Then, recall that for every $m$, $R_m$ has an identical distribution, and for every $n$, $D_n$ also has an identical distribution. These are as mentioned in point j in Section 4.1. Therefore, Equation (4) can be rewritten as follows:

$$\mathrm{P}\left({\mathrm{M}}_{c_1,c_2,t}\le \mathcal{u}\right)=\left\{\begin{array}{cc}{F}_R^{c_2-c_1+1}\left(\mathcal{u}\right)& :c_1=1,2,\dots d_t-1,c_2=1,2,\dots ,d_t-1\hfill \\ F_R^{d_t-c_1}\left(\mathcal{u}\right)F_D^{c_2-d_t+1}\left(\mathcal{u}\right)& :c_1=1,2,\dots d_t-1,c_2=d_t,d_t+1,\dots ,r_t-1\hfill \\ F_R^{d_t-c_1+c_2-r_t+1}\left(\mathcal{u}\right)F_D^{r_t-d_t}\left(\mathcal{u}\right)& :c_1=1,2,\dots d_t-1,c_2=r_t,r_t+1,\dots ,36\hfill \\ F_D^{c_2-c_1+1}\left(\mathcal{u}\right)& :c_1=d_t,d_t+1,\dots ,r_t-1,c_2=d_t,d_t+1,\dots ,r_t-1\hfill \\ F_D^{r_t-c_1}\left(\mathcal{u}\right)F_R^{c_2-r_t+1}\left(\mathcal{u}\right)& :c_1=d_t,d_t+1,\dots ,r_t-1,c_2=r_t,r_t+1,\dots ,36\hfill \\ F_R^{c_2-c_1+1}\left(\mathcal{u}\right)& :c_1=r_t,r_t+1,\dots ,36,c_2=r_t,r_t+1,\dots ,36\hfill \end{array}\right.$$

(5)

To determine the survival distribution function (SDF) of ${\mathrm{M}}_{c_1,c_2,t}$, we can use Equation (5) to obtain the following equation:

$$\mathrm{P}\left({\mathrm{M}}_{c_1,c_2,t}>\mathcal{u}\right)=1-\mathrm{P}\left({\mathrm{M}}_{c_1,c_2,t}\le \mathcal{u}\right).$$

Then, we formulate the maximum rainfall from the $c_1$-th dekadal in year $t_1$ to the $c_2$-th dekadal in year $t_2$ with $1\le c_1,c_2\le 36$ and $1\le t_1<t_2\le T$ as follows:

$${\mathfrak{M}}_{c_1,t_1,c_2,t_2}=\max \left\{{\mathrm{M}}_{c_1,36,t_1},{\mathrm{M}}_{\mathrm{1,36},t_1+1},\dots ,{\mathrm{M}}_{1,c_2,t_2}\right\}.$$

Conveniently, the CDF of ${\mathfrak{M}}_{c_1,t_1,c_2,t_2}$ can be formulated as follows:

$$\begin{gathered} \mathrm{P}\left({\mathfrak{M}}_{c_1,t_1,c_2,t_2}\le \mathcal{u}\right)=\mathrm{P}\left(\max \left\{{\mathrm{M}}_{c_1,36,t_1},{\mathrm{M}}_{\mathrm{1,36},t_1+1},\dots ,{\mathrm{M}}_{1,c_2,t_2}\right\}\le \mathcal{u}\right), \\ =\mathrm{P}\left({\mathrm{M}}_{c_1,36,t_1}\le \mathcal{u},{\mathrm{M}}_{\mathrm{1,36},t_1+1}\le \mathcal{u},\dots ,{\mathrm{M}}_{1,c_2,t_2}\le \mathcal{u}\right), \\ =\mathrm{P}\left({\mathrm{M}}_{c_1,36,t_1}\le \mathcal{u}\right)\mathrm{P}\left({\mathrm{M}}_{\mathrm{1,36},t_1+1}\le \mathcal{u}\right)\dots \mathrm{P}\left({\mathrm{M}}_{1,c_2,t_2}\le \mathcal{u}\right). \end{gathered}$$

(6)

In other words, the CDF of ${\mathfrak{M}}_{c_1,t_1,c_2,t_2}$ is determined based on the CDF of ${\mathrm{M}}_{c_1,c_2,t}$ in Equation (5). To determine the SDF of ${\mathfrak{M}}_{c_1,t_1,c_2,t_2}$, we can use Equation (6) to obtain the following equation:

$$\mathrm{P}\left({\mathfrak{M}}_{c_1,t_1,c_2,t_2}>\mathcal{u}\right)=1-\mathrm{P}\left({\mathfrak{M}}_{c_1,t_1,c_2,t_2}\le \mathcal{u}\right).$$

(7)

4.3. Flood Bond Price Model

Suppose that a flood bond with a term of $T$ years is issued at the beginning of the $c_1$-th dekadal in the year $t_1$. In this study, annual coupons are paid to investors at the end of each year. In other words, the $t$-th annual coupon is paid at every $\mathcal{c}\left(c_1\right)$-th dekadal in year $\mathcal{t}\left(c_1,t\right)$, where $\mathcal{c}\left(c_1\right)$ and $\mathcal{t}\left(c_1,t\right)$ are functions, each expressed as follows:

$$\mathcal{c}\left(c_1\right)=\left\{\begin{array}{cc}36& :c_1=1\hfill \\ c_1-1& :c_1=2,3,\dots ,36\hfill \end{array}\right.,$$

and

$$\mathcal{t}\left(c_1,t\right)=\left\{\begin{array}{cc}{t}_1+t-1& :c_1=1,t=1,2,\dots ,T\hfill \\ t_1+t& :c_1=2,3,\dots ,36,t=1,2,\dots ,T\hfill \end{array}\right..$$

Then, the redemption value of the flood bonds is paid to investors at the end of the year T. In other words, the redemption value is paid on the $\mathcal{c}\left(c_1\right)$-th dekadal in year $\mathcal{t}\left(c_1,T\right)$. The annual coupon payment scheme and redemption value are intact if the maximum rainfall up to the payment date does not exceed the trigger point and in proportion if the maximum rainfall up to the payment date exceeds the trigger point. Mathematically, the annual coupon payment scheme and redemption value are denoted as $P_t$ and are formulated as follows:

$$\begin{gathered} P_t=\left\{\begin{array}{cc}C{1}_{\left\{{\mathfrak{M}}_{c_1,t_1,\mathcal{c}\left(c_1\right),\mathcal{t}\left(c_1,t\right)}\le \omega \right\}}(\mathcal{u})+{\lambda }_{1}C{1}_{\left\{{\mathfrak{M}}_{c_1,t_1,\mathcal{c}\left(c_1\right),\mathcal{t}\left(c_1,t\right)}>\omega \right\}}(\mathcal{u})& :t=1,2,\dots ,T-1\hfill \\ (C+K)1_{\left\{{\mathfrak{M}}_{c_1,t_1,\mathcal{c}\left(c_1\right),\mathcal{t}\left(c_1,t\right)}\le \omega \right\}}(\mathcal{u})+({\lambda }_{1}C+{\lambda }_{2}K)1_{\left\{{\mathfrak{M}}_{c_1,t_1,\mathcal{c}\left(c_1\right),\mathcal{t}\left(c_1,t\right)}>\omega \right\}}(\mathcal{u})& :t=T\hfill \end{array}\right., \\ =\left\{\begin{array}{cc}C{1}_{\left\{{\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}\le \omega \right\}}(\mathcal{u})+{\lambda }_{1}C{1}_{\left\{{\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}>\omega \right\}}(\mathcal{u})& :c_1=1,t=1,2,\dots ,T-1\hfill \\ C{1}_{\left\{{\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}\le \omega \right\}}(\mathcal{u})+{\lambda }_{1}C{1}_{\left\{{\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}>\omega \right\}}(\mathcal{u})& :c_1=2,3,\dots ,36,t=1,2,\dots ,T-1\hfill \\ (C+K)1_{\left\{{\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}\le \omega \right\}}(\mathcal{u})+({\lambda }_{1}C+{\lambda }_{2}K)1_{\left\{{\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}>\omega \right\}}(\mathcal{u})& :c_1=1,t=T\hfill \\ (C+K)1_{\left\{{\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}\le \omega \right\}}(\mathcal{u})+({\lambda }_{1}C+{\lambda }_{2}K)1_{\left\{{\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}>\omega \right\}}(\mathcal{u})& :c_1=2,3,\dots ,36,t=T\hfill \end{array}\right.. \end{gathered}$$

(8)

where $\omega$ is the trigger point, and ${\lambda }_1$ and ${\lambda }_2$ are, respectively, the proportion of annual coupon payment and redemption value when the maximum rainfall trigger point until the payment date is exceeded,

$$1_{\left\{{\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}\le \omega \right\}}\left(\mathcal{u}\right)=\left\{\begin{array}{cc}1& :\mathcal{u}\in \left\{{\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}\le \omega \right\}\\ 0& :\mathcal{u}\notin \left\{{\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}>\omega \right\}\end{array}\right.,$$

$$1_{\left\{{\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}\le \omega \right\}}\left(\mathcal{u}\right)=\left\{\begin{array}{cc}1& :\mathcal{u}\in \left\{{\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}\le \omega \right\}\\ 0& :\mathcal{u}\notin \left\{{\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}>\omega \right\}\end{array}\right.,$$

$$1_{\left\{{\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}>\omega \right\}}\left(\mathcal{u}\right)=\left\{\begin{array}{cc}1& :\mathcal{u}\in \left\{{\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}>\omega \right\}\\ 0& :\mathcal{u}\notin \left\{{\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}\le \omega \right\}\end{array}\right.,$$

and

$$1_{\left\{{\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}>\omega \right\}}\left(\mathcal{u}\right)=\left\{\begin{array}{cc}1& :\mathcal{u}\in \left\{{\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}>\omega \right\}\\ 0& :\mathcal{u}\notin \left\{{\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}\le \omega \right\}\end{array}\right..$$

To determine the price of flood bonds, we use the summation of the present value of the expected annual coupon and the expected redemption value based on the risk-neutral pricing measure $Q$. For example, now is the beginning of the $c_1$-th dekadal. The present value of a payment of one currency unit in the next $t$ years ahead is formulated as follows:

$$B\left(c_1,t\right)={\mathbb{E}}_Q\left(\left.e^{-\sum _{s=1}^t{I}_s\left(1-{\displaystyle \frac{c_1}{36}}\right)+I_{s+1}\left({\displaystyle \frac{c_1}{36}}\right)}\right|{\mathcal{F}}_{{\displaystyle \frac{c_1}{36}}}\right),$$

(9)

where ${\mathbb{E}}_Q$ is the statistical expectation under $Q$ (risk-neutral-pricing measure). In Equation (9), the form $\left(1-{\displaystyle \frac{c_1}{36}}\right)$ indicates the time of compounding one currency unit with the force of interest $I_s$ in year $s$. Then, the form ${\displaystyle \frac{c_1}{36}}$ indicates the time of compounding one currency unit with the force of interest $I_{s+1}$ in year $s+1$. Based on the present value concept in Equation (9), the flood bond price is designed as the sum of the present values of expected $P_t$ with $t=\mathrm{1,2},\dots ,T$. Mathematically, it is formulated as follows:

$$V_{c_1,t_1,T}=\sum _{t=1}^T{\mathbb{E}}_Q\left(\left.P_t{e}^{-\sum _{s=1}^t{I}_s\left(1-{\displaystyle \frac{c_1}{36}}\right)+I_{s+1}\left({\displaystyle \frac{c_1}{36}}\right)}\right|{\mathcal{F}}_{{\displaystyle \frac{c_1}{36}}}\right).$$

(10)

Based on Cox and Pedersen [12], after converting the physical probability measure $\mathrm{P}$ to the risk-neutralized measure $Q$, it is assumed that the original distribution characteristics of the maximum rainfall process remain intact. Events dependent on rainfall risk variables and events dependent on financial variables have no effect on one another under the risk-neutralized measure $Q$ (for more detail, see Cox and Pedersen [12]). Consequently, Equation (10) can be rewritten as follows:

$$V_{c_1,t_1,T}=\sum _{t=1}^T{\mathbb{E}}_{\mathrm{P}}\left(\left.P_t\right|{\mathcal{F}}_t\right){\mathbb{E}}_Q\left(\left.e^{-\sum _{s=1}^t{I}_s\left(1-{\displaystyle \frac{c_1}{36}}\right)+I_{s+1}\left({\displaystyle \frac{c_1}{36}}\right)}\right|{\mathcal{F}}_{{\displaystyle \frac{c_1}{36}}}\right)=\sum _{t=1}^T{\mathbb{E}}_{\mathrm{P}}\left(\left.P_t\right|{\mathcal{F}}_{{\displaystyle \frac{c_1}{36}}}\right)B\left(c_1,t\right),$$

(11)

where $B\left(c_1,t\right)$ is determined using Equation (9), and ${\mathbb{E}}_{\mathrm{P}}\left(\left.P_t\right|{\mathcal{F}}_t\right)$ is determined based on Equations (6) and (7) as follows:

$${\mathbb{E}}_{\mathrm{P}}\left(\left.P_t\right|{\mathcal{F}}_t\right)=\left\{\begin{array}{cc}C\mathrm{P}\left({\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}\le \omega \right)+{\lambda }_{1}C\mathrm{P}\left({\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}>\omega \right)& :c_1=1,t=1,2,\dots ,T-1\hfill \\ C\mathrm{P}({\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}\le \omega )+{\lambda }_{1}C\mathrm{P}\left({\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}>\omega \right)& :c_1=2,3,\dots ,36,t=1,2,\dots ,T-1\hfill \\ (C+K)\mathrm{P}\left({\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}\le \omega \right)+({\lambda }_{1}C+{\lambda }_{2}K)\mathrm{P}\left({\mathfrak{M}}_{c_1,t_1,36,t_1+t-1}>\omega \right)& :c_1=1,t=T\hfill \\ (C+K)\mathrm{P}({\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}\le \omega )+({\lambda }_{1}C+{\lambda }_{2}K)\mathrm{P}\left({\mathfrak{M}}_{c_1,t_1,c_1-1,t_1+t}>\omega \right)& :c_1=2,3,\dots ,36,t=T\hfill \end{array}\right..$$

(12)

5. Numerical Simulation

5.1. Data Description

The primary data used in this study are daily rainfall data in Bandung Regency, West Java Province, Indonesia, from 1 January 1982 to 31 December 2023. These data were obtained on 1 March 2024 openly at the Prediction of Worldwide Energy Resource (POWER) provided by the National Aeronautics and Space Administration (NASA) of the United States of America at the following link: https://power.larc.nasa.gov/data-access-viewer/. From these data, we processed it to produce the following data:

a.

Data on cumulative rainfall per dekadal in Bandung Regency, West Java Province, Indonesia, from the 1st dekadal in 1982 to the last dekadal in 2023. The data size is 1.512 and has units of mm/dekadal.

b.

Data for the start of the dry and the second rainy seasons from 1982 to 2023. Data have dekadal units. Both data have the same size, namely 42. The dry season starts in the 16th dekadal every year on average. Then, the earliest dry season occurs in the 8th dekadal and the latest occurs in the 22th dekadal. The second rainy season, on average, starts in the 30th dekadal every year. Then, the earliest the second rainy season occurs is in the 23th dekadal and the latest is in the 35th dekadal. A visualization of both data is given in Figure 3.

c.

Maximum rainfall data per dekadal in the rainy and dry seasons from the 1st dekadal in 1982 to the last dekadal in 2023. The data sizes for the rainy and dry seasons are 915 and 597, respectively. Both data are in units of mm/day. The average maximum rainfall in the dry season is 8.26 mm/day. Then, the maximum rainfall in the dry season is at least 0 mm/day and at most 64.68 mm/day. Meanwhile, the average maximum rainfall in the rainy season is 26.94 mm/day. Then, the maximum rainfall in the rainy season is at least 1.20 mm/day and at most 226.27 mm/day. To determine the shape of the distribution, we analyzed the skewness and kurtosis of the two datasets. The data skewness in the dry and rainy seasons is 1.81 and 2.48, respectively. Both skewness values are positive, which indicates that both data distributions are skewed to the left. Then, the data kurtosis in the dry and rainy seasons is 6.17 and 35.92, respectively. Both kurtosis values are more than three, which indicates that both data distributions are sharper than the normal distribution (leptokurtic). From the results of the skewness and kurtosis analysis of the two data, the conclusion is that the distribution form of the two data is a heavy-right-tail distribution. Visually, this shape can be seen in the histogram of the two data in Figure 3.

Apart from that, the second primary data are data on the annual effective interest rate in Indonesia from 1991 to 2023. These data are then transformed into a force of interest with the equation $I_t=\ln (1+I_t^{\prime })$, where $I_t$ is the force of annual interest and $I_t^{\prime }$ is the annual effective interest rate. The data size is 33 and was openly obtained on 1 March 2024 from Bank Indonesia at the following link: https://www.bi.go.id/id/statistik/indikator/BI-Rate.aspx. The average force of interest in each year is 0.092329. Then, the force of interest has an average deviation of 0.053137 from the average. The skewness and kurtosis of the data are 2.247412 and 7.234458, respectively. In other words, the data distribution is skewed to the left and is sharper than a normal (leptokurtic) curve. We can also say that the data distribution is a heavy-right-tail distribution. A histogram of the data is given in Figure 3.

5.2. Fitting Rainfall and Force of Interest Data Distribution

The distribution analysis results in Section 5.1 show that the distributions of maximum rainfall data per dekadal in the rainy and dry seasons are heavy-right-tailed. The distribution of force of interest data also has the same form. Therefore, this section will match the distribution of these data with a theoretical distribution with heavy-right-tailed characteristics. Many theoretical distributions have it, e.g., gamma, beta, Burr, Pareto, Dagum, and inverse Gaussian distributions. Before matching, the parameters of the theoretical distributions are estimated first. Several methods can be used in this assessment, e.g., the maximum likelihood (ML) method. Then, to check the match between the distribution of the three data and the theoretical one, several tests can be used, e.g., Kolmogorov–Smirnov (KS) and Anderson–Darling (AD) tests. The KS test is nonparametric and does not depend on certain distribution assumptions. The AD test is a modification of the KS test, which emphasizes the tail of the distribution. The theoretical distribution that most closely fits the empirical distribution has the lowest KS and AD statistical values, denoted by $D_{\alpha }$ and $A_{\alpha }$, where α represents the significance level. The KS and AD statistical values are determined as follows [41]:

$$D=\sup \left\{\left|F_{\mathfrak{n}}\left(x\right)-F\left(x\right)\right|:x\in X\right\}$$

and

$$A=-{\displaystyle \frac{1}{\mathfrak{n}}}\sum _{\mathcal{n}=1}^{\mathfrak{n}}\left(2\mathcal{n}-1\right)\left\{\log F\left(x_{\mathcal{n}}\right)+\log \left[1-F\left(x_{\mathfrak{n}-\mathcal{n}+1}\right)\right]\right\}-\mathfrak{n},$$

where $\mathfrak{n}$ represents the data size, $F\left(x\right)$ is the theoretical distribution and $F_{\mathfrak{n}}\left(x\right)$ is the empirical distribution.

In this study, we estimated the parameters of theoretical distributions and fitted them to three data distributions using the EasyFit software version 5.5. In more detail, parameter estimation was carried out using the ML method, while fitness tests were carried out using KS and AD tests. In summary, using the software, we found the following:

a.

The Dagum distribution ($\alpha =4.2444,\beta =18.175,\kappa =0.15306$) is the most fit compared to other theoretical distributions to describe the distribution of maximum rainfall data in the dry season. In more detail, $\alpha$ and $\kappa$ represent shape parameters, while $\beta$ is a scale parameter. The statistical values of the KS and AD tests are $D=0.02213$ and $A=0.2513$, respectively. The critical values with significance level $\alpha =0.05$ from the KS and AD tests are $D_{0.05}=0.05558$ and $A_{0.05}=2.5018$, respectively. Therefore, this distribution was chosen to represent the distribution of maximum rainfall data in the dry season. Mathematically, this is written as follows:

$$F_D\left(\mathcal{u}\right)={\left[1+{\left({\displaystyle \frac{\mathcal{u}}{\beta }}\right)}^{-\alpha }\right]}^{-\kappa }={\left[1+{\left({\displaystyle \frac{\mathcal{u}}{18.175}}\right)}^{-4.2444}\right]}^{-0.15306}.$$

b.

The Burr distribution $(\alpha =2.8735,\beta =32.627,\kappa =1.9002)$ is the most fit compared to other theoretical distributions to describe the distribution of maximum rainfall data in the rainy season. In more detail, $\alpha$ and $\kappa$ represent shape parameters, while $\beta$ is a scale parameter. The statistical values of the KS and AD tests are $D=0.02114$ and $A=0.38158$, respectively. The critical values with significance level $\alpha =0.05$ from the KS and AD tests are $D_{0.05}=0.04489$ and $A_{0.05}=2.5018$, respectively. Therefore, this distribution was chosen to represent the distribution of maximum rainfall data in the rainy season. Mathematically, this is written as follows:

$$F_R\left(\mathcal{u}\right)=1-{\left[1+{\left({\displaystyle \frac{\mathcal{u}}{\beta }}\right)}^{\alpha }\right]}^{-\kappa }=1-{\left[1+{\left({\displaystyle \frac{\mathcal{u}}{32.627}}\right)}^{2.8735}\right]}^{-1.9002}.$$

c.

The Inverse-Gaussian distribution $(\lambda =0.27875,\mu =0.09233)$ is the more fit compared to other theoretical distributions to describe the distribution of the annual force of interest data, where $\lambda$ and $\mu$ are the shape and the mean parameters, respectively. The statistical values of the KS and AD tests are $D=0.0798$ and $A=0.36667$, respectively. The critical values with significance level $\alpha =0.05$ from the KS and AD tests are $D_{0.05}=0.2342$ and $A_{0.05}=2.5018$, respectively. Therefore, this distribution was chosen to represent the annual force of interest data distribution. Mathematically, this is written as follows:

$$\begin{gathered} F_I\left(\mathcal{i}\right)=\mathrm{\Phi }\left[\sqrt{{\displaystyle \frac{\lambda }{\mathcal{i}}}}\left({\displaystyle \frac{\mathcal{i}}{\mu }}-1\right)\right]+e^{{\displaystyle \frac{2\lambda }{\mu }}}\mathrm{\Phi }\left[-\sqrt{{\displaystyle \frac{\lambda }{\mathcal{i}}}}\left({\displaystyle \frac{\mathcal{i}}{\mu }}+1\right)\right], \\ =\mathrm{\Phi }\left[\sqrt{{\displaystyle \frac{0.27875}{\mathcal{i}}}}\left({\displaystyle \frac{\mathcal{i}}{0.09233}}-1\right)\right]+e^{{\displaystyle \frac{2\left(0.27875\right)}{0.09233}}}\mathrm{\Phi }\left[-\sqrt{{\displaystyle \frac{0.27875}{\mathcal{i}}}}\left({\displaystyle \frac{\mathcal{i}}{0.09233}}+1\right)\right], \end{gathered}$$

where $\mathrm{\Phi }$ represents the cumulative distribution function of the standard normal random variable.

5.3. Forecasting Duration Ranges of Rainy and Dry Seasons

This section contains the forecasting stages for the start of the dry and the second rainy seasons in the following years. Many forecasting models can be used. In this research, we use a simple moving-average time series model. The selection of this model is based on data that do not have extreme fluctuations (stable) [42]. This stability can be seen from the visualization of both data in Figure 3. This stability can also be shown visually through the Box and Whisker diagram of the two data in Figure 4, which shows no outliers.

Mathematically, the simple moving-average models of order $\mathcal{p}$ for forecasting the start of the dry and the second rainy seasons are formulated as follows [42]:

$$d_{\mathcal{t}+1}={\displaystyle \frac{1}{\mathcal{p}}}\sum _{j=1}^{\mathcal{p}}{d}_{\mathcal{t}-j+1}+e_{\mathcal{t}}$$

and

$$r_{\mathcal{t}+1}={\displaystyle \frac{1}{\mathcal{p}}}\sum _{j=1}^{\mathcal{p}}{r}_{\mathcal{t}-j+1}+{\mathcal{e}}_{\mathcal{t}},$$

where $\mathcal{t}=\mathcal{p}+1,\mathcal{p}+2,\dots ,T$, $T$ represents data size, $e_{\mathcal{t}}$ and ${\mathcal{e}}_{\mathcal{t}}$ represents error. To determine the order of $\mathcal{p}$, we conduct experiments for $\mathcal{p}=1, 2, 3, 4,$ and $5$. The one selected is the one with the best accuracy. In this study, accuracy is measured by mean-absolute-percentage error (MAPE), which is calculated using the following equation:

$${\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}}_{\mathcal{p}}={\displaystyle \frac{1}{T-\mathcal{p}}}\sum _{\mathcal{t}=\mathcal{p}+1}^T\left|{\displaystyle \frac{r_{\mathcal{t}}-{\widehat{r}}_{\mathcal{t}}}{r_{\mathcal{t}}}}\right|\times 100\%$$

or

$${\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}}_{\mathcal{p}}={\displaystyle \frac{1}{T-\mathcal{p}}}\sum _{\mathcal{t}=\mathcal{p}+1}^T\left|{\displaystyle \frac{d_{\mathcal{t}}-{\widehat{d}}_{\mathcal{t}}}{d_{\mathcal{t}}}}\right|\times 100\%,$$

where ${\widehat{r}}_{\mathcal{t}}$ and ${\widehat{d}}_{\mathcal{t}}$ are the forecasts of $r_{\mathcal{t}}$ and $d_{\mathcal{t}}$, respectively. In summary, we find that the MAPE of the single moving-average model of orders 1, 2, 3, 4, and 5 of r_t is 25%, 21.7%, 20%, 18%, and 18.3%, respectively. Then, the MAPE of the single moving average model of order 1, 2, 3, 4, and 5 of $d_{\mathcal{t}}$ is 12.13%, 10.4%, 9.31%, 9.33%, and 9.4%, respectively. Based on these MAPE values, $r_{\mathcal{t}}$ and $d_{\mathcal{t}}$ are modeled with a single moving average of order 4 and 3, respectively.

5.4. Estimating Flood Bond Price

This section contains estimates of flood bond prices. In this study, the flood bond pricing model in Equation (11) does not have a closed-form solution. In more detail, this form results from the absence of a solution to $B\left(c_1,t\right)$ in the equation. Therefore, we determine the solution numerically using the Monte Carlo method. Because only the form $B\left(c_1,t\right)$ does not have a closed-form solution, computing the numerical solution of Equation (11) can be conducted quickly. Numerically, the solution of the flood bond price model in Equation (11) can be approximated by the following equation:

$${\widehat{V}}_{c_1,t_1,T}={\displaystyle \frac{1}{\mathcal{W}}}\sum _{\mathcal{w}=1}^{\mathcal{W}}\sum _{t=1}^T{\mathbb{E}}_{\mathrm{P}}\left(\left.P_t\right|{\mathcal{F}}_{{\displaystyle \frac{c_1}{36}}}\right)e^{-\sum _{s=1}^t{\mathcal{i}}_{s,\mathcal{w}}\left(1-{\displaystyle \frac{c_1}{36}}\right)+{\mathcal{i}}_{s+1,\mathcal{w}}\left({\displaystyle \frac{c_1}{36}}\right)},$$

(13)

where $\mathcal{W}$ represents the number of simulations, and ${\mathcal{i}}_{s,\mathcal{w}}$ is the random value of $I_s$ generated in the $\mathcal{w}$-th simulation with $s=\mathrm{1,2},\dots ,t$ and $\mathcal{w}=1, 2,\dots ,\mathcal{W}$. To start estimation, the values of the variables in Equation (13) are determined first. We assume that flood bonds are issued at the beginning of the $c_1=1$st dekadal in year $t_1=2023$ and have a term of $T=3$ years. Then, the annual coupon and redemption value are assumed to be $C=0.1$ USD and $K=1$ USD, respectively. Then, the trigger point for the maximum rainfall index is assumed to be $\omega =100$ mm/day. When the trigger point is exceeded, the annual coupon and redemption value are paid in the proportions ${\lambda }_1=0$ and ${\lambda }_2=0.5$, respectively. By conducting $\mathcal{W}=\mathrm{500,000}$ simulations, we visually provide all the estimation results in the histogram in Figure 5. The description of all estimation results is as follows:

a.

The average of the estimation results is 0.9339297. In other words, the estimated price of flood bonds in Bandung Regency is ${\widehat{V}}_{\mathrm{1,2023,3}}=0.9339297$ USD.

b.

The standard deviation of the estimation results is 0.07329027. In other words, the average deviation of flood bond price estimates from the mean is 0.07329027 USD.

c.

The range of the estimated results is 0.7066545. This value is so large. Hence, many simulations are needed to obtain the no bias estimation. It is why we conducted 500,000 simulations.

6. Discussion

This section contains an analysis of the effect of rainfall and financial risks on flood bond prices. The rainfall and financial risks analyzed include trigger points, term, stochastic and constant forces of interest, and time to buy bonds. The first is analyzing the effect of trigger points on flood bond prices. To simplify the analysis, we assume that all variables except the trigger point are the same as those used in Section 5.4. Meanwhile, we assume the trigger point is a set $\mathcal{U}=\{50, 75, 100, 125, 150, 175, 200\}$ with units of mm/day. For each member of $\mathcal{U}$, we determine the price of the flood bond corresponding to it. Briefly, we visually provide flood bond prices for each U member in Figure 6.

Figure 6 illustrates that trigger points positively correlate with flood bond prices. The higher the trigger point, the higher the price of flood bonds, and the lower the trigger point, the lower the price of flood bonds. This result is validated logically, where if the trigger point is high, the chance of claiming from flood bonds will be low. This results in high investor demand for these bonds. Hence, the price becomes expensive. Then, if the trigger point is low, the chance of a claim from flood bonds will be high. This results in low investor demand for these bonds. Hence, the price becomes cheap. The positive correlation results between trigger points and flood bond prices obtained in this study align with the results conducted by Chao and Zou [31] and Ibrahim et al. [33].

Next is an analysis of the effect of the stochastic and constant force of interest on flood bond prices. We assume all variable values used are the same as in Section 5.4 except for the term. We assume the term is a set $\mathcal{T}=\{1, 2, 3, 4, 5, 6\}$ with units of year(s). For each member of $\mathcal{T}$, we determine the price of a flood bond assuming a stochastic and constant force of interest. After that, we determine the differences. In more detail, the constant force of interest is assumed to be 0.09233, which is the average of the data from 1990 to 2023. Briefly, we visually provide the results of the analysis in Figure 7.

Figure 7 illustrates that the estimated price of flood bonds, assuming a stochastic and constant force of interest, significantly differs if the estimated constant value is far from its annual average. However, the choice of a constant force of interest in Equation (13) of its annual average is very reliable to use because the difference is not significant, although this is not analytical proof. Next is an analysis of the effect of the term of a flood bond on its price. This analysis can be carried out through Figure 7, where it appears that the term of the flood bond has a negative correlation with its price. The longer the term of a flood bond, the lower the price, and the shorter the term of a flood bond, the higher the price. It is validated logically in the same way as traditional bonds [43], where if the term of the flood bond is long, the chance of a claim occurring will be high. This results in low investor demand for these bonds. Hence, the price becomes cheap. Then, if the term of the flood bond is short, the chance of a claim occurring will be low. This results in high investor demand for these bonds. Hence, the price becomes expensive.

Next is an analysis of the effect of the buying time of flood bonds on its price. To analyze it, we assume that the variable values in the model are the same as in Section 5.4 except for $c_1$. We let $c_1$ be the set ${\mathcal{C}}_1=\left\{1, 2,\dots , 36\right\}$. For each $c_1$ in ${\mathcal{C}}_1$, we calculate the price of the flood bond corresponding to it. The flood bond price calculation used two force of interest assumptions: stochastic and constant. In summary, the analysis results are given visually in Figure 8.

Figure 8 illustrates that flood bonds tend to have the same price at the 1st to 16th dekadal, increase at the 17th dekadal, tend to be the same again until the 28th dekadal, increase at the 29th dekadal, and tend to be the same again until the final dekadal. After this was analyzed further, we found the cause. There is a difference in the duration of the dry and rainy seasons at each buying time for the flood bonds. Flood bonds with the more extended dry season duration have the highest prices and vice versa. This cause can be visually seen in Figure 8. The more extended duration of the dry season is related to the buying time of flood bonds in the 29th to 36th dekadals. As a result, bond prices in this dekadal range are the highest compared to other dekadals. Then, the shortest duration of the dry season is related to the buying time of flood bonds in the 1st to 16th dekadals. As a result, flood bond prices in this dekadal range have the lowest prices compared to other dekadals. Rationally, it is validated because the longer the dry season duration, the lower the chance of high rainfall. Then, the lower the chance, the higher the demand for flood bonds. Hence, the price becomes high. On the contrary, when the duration of the dry season is short, the same logic applies. Then, flood bonds with a long rainy season duration have low bond prices, and vice versa. This can be visually seen in Figure 8. The longest duration of the rainy season is owned by the buying time of flood bonds in the 1st to 16th dekadals. As a result, flood bond prices in this dekadal range have the lowest prices compared to other dekadals. Then, the shortest duration of the rainy season is owned by the buying time of flood bonds in the 29th to 36th dekadals. As a result, flood bond prices in this dekadal range are the highest compared to other dekadals. Rationally, it is validated because the longer the duration of the rainy season, the higher the chance of heavy rainfall. Then, the higher the chance, the lower the demand for flood bonds. As a result, the price is low. On the contrary, when the duration of the rainy season is short, the same logic applies.

7. Conclusions

This study aims to model flood bond prices in Indonesia, which assumes that the durations of the rainy and dry seasons are not constant. This inconstancy is adjusted to the actual conditions regarding the varying patterns of the country’s rainy and dry seasons. To model the trigger event, we use the maximum rainfall trigger index to quantify it. This index has the advantage that it measures quickly, objectively, and transparently based on the flood catastrophe that occurred. These advantages minimize administrative costs in assessing claims and the potential for moral hazard in claims. We model this index as a continuous stochastic process with a discrete-time index. Then, we model flood bond prices using a risk-neutral-pricing measure assuming a stochastic force of interest.

After the model was designed, the numerical simulation was conducted using rainfall and interest rate data in Indonesia. The simulation shows that the maximum rainfall trigger point is proportional to the price of flood bonds. In other words, the higher the trigger point, the more expensive the flood bond will be, and vice versa. It is validated logically, where if the trigger point is high, the chance of claiming from flood bonds will be low. This results in high investor demand for these bonds. Hence, the price becomes expensive. This trigger point sensitivity to flood bond prices aligns with the results conducted by Chao and Zou [31] and Ibrahim et al. [33]. Then, using a stochastic force of interest provides a significant difference compared to using a constant force of interest if the estimated constant value is far from the annual average of the data. Suppose a constant force of interest is still to be used. In that case, the annual average is better used for results closer to estimates with a stochastic force of interest, even though this is not analytically evident. Then, the duration of the rainy season is inversely proportional to the price of flood bonds, while the duration of the dry season is proportional to the price of flood bonds. In other words, the longer the dry season, the lower the chance of heavy rainfall. Then, the lower the chance, the higher the demand for flood bonds. Hence, the price becomes high. On the contrary, the longer the rainy season, the higher the chance of heavy rainfall. Then, the higher the chance, the lower the demand for flood bonds. Hence, the price becomes low.

This study can increase the tools available for calculating the probability of maximum rainfall in the future for related meteorological or climatological institutions in this country. Apart from that, flood bond issuers in Indonesia can undoubtedly use this study to determine flood bond prices that better reflect the actual situation in the form of inconstant duration of rainy and dry seasons and stochastic force of interest. The designed model has shortcomings that can be used in future research. The designed model does not consider the intensity of flood catastrophes. This consideration can better describe the flood conditions in Indonesia. Then, our model still needs to consider the inflation rate. By involving it, investors’ return estimates will be more accurate because they have considered reducing factors.