Discrete Dynamics in Nature and Society

Volume 2018, Article ID 5068480, 9 pages

https://doi.org/10.1155/2018/5068480

## Multiple-Event Catastrophe Bond Pricing Based on CIR-Copula-POT Model

^{1}School of Economics and Management, Fuzhou University, Fuzhou 350116, China^{2}Institute of Investment and Risk Management, Fuzhou University, Fuzhou 350116, China

Correspondence should be addressed to Wen Chao; moc.361@4102newoahc

Received 19 December 2017; Accepted 5 April 2018; Published 25 June 2018

Academic Editor: Alicia Cordero

Copyright © 2018 Wen Chao and Huiwen Zou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Catastrophe events are attracting increased attention because of their devastating consequences. Aimed at the nonlinear dependency and tail characteristics of different triggered indexes of multiple-event catastrophe bonds, this paper applies Copula function and the extreme value theory to multiple-event catastrophe bond pricing. At the same time, floating coupon and principal payoff structures are adopted instead of fixed coupon and principal payoff structures, to reduce moral hazard and improve bond attractiveness. Furthermore, we develop a CIR-Copula-POT bond pricing model with CIR stochastic rate and estimate flood multiple-event triggered catastrophe bond price using Monte Carlo simulation method. Finally, we implement the sensitivity analysis to show how catastrophe intensity, maturity date, and the dependence affect the prices of catastrophe bonds.

#### 1. Introduction

Different kinds of natural disasters occur frequently in the world over the past several years. These low-frequency and high-losses catastrophic events have a serious influence on peoples life and the stability society. Traditionally, when the catastrophic events occur, the national finance and social aid would be used to compensate the catastrophe losses. When faced with the natural disaster losses, the insurance companies themselves cannot satisfy the demand of catastrophe risk due to the large financial pressure and the restrictions of business ability. In recent years, there appear some kinds of important insurance-linked securities (ILSs) in the international catastrophe insurance market. And catastrophe (CAT) bonds are one of the most prominent ILSs, which transfer the consequence of CAT financial risks from issuers to investors. CAT bonds not only improve the risk bearing capacity of insurance companies, but also bring more investment choices to capital market. Reasonable pricing is the critical point in the CAT bonds issuing and trading.

Recently, some research efforts have been devoted to the catastrophe bonds pricing. Lee and Yu [1] developed a contingent-claim model to price CAT bonds with consideration of default risk, basis risk, and moral hazard. Egami and Young [2] presented a method for pricing structured CAT bonds based on utility indifference pricing. Härdle and Cabrera [3] calibrated the government parameter index-triggered CAT bonds with Mexico earthquake data; they proved that mixing reinsurance with CAT bond can reduce exposure risk and default risk. Z. G. Ma and C. Q. Ma [4] proposed a mixed approximation method to find the numerical solution for the price of catastrophe risk bonds. Nowak and Romaniuk [5] implemented the CAT bond pricing model described by the two-factor Vasicek model. Moreover, they proposed an automated approach for decision-making in fuzzy environment with relevant examples presenting this method.

While the catastrophe risks have obvious thick tail features, it is more reasonable to use extreme value theory (EVT) to characterize the tail characteristic of catastrophic losses distribution. Zimbids et al. [6] studied the Greece earthquakes data using advanced techniques from the extreme value theory. Moreover, they evaluated the CAT bond price using Monte Carlo simulation techniques and stochastic iterative equations. Shao et al. [7] applied equilibrium pricing theory and EVT to construct a multiple-variable CAT bond for California earthquakes. Ma et al. [8] employed a doubly stochastic Poisson and Peak over Threshold (POT) to price zero-coupon catastrophe bonds.

Since one-triggering-event CAT bond is difficult to meet the diverse needs of investors, the multiple-event CAT bond starts to rise because of its advantages. Recently, several studies have mainly focused on multiple-event CAT bonds. For example, Woo [9] addressed multiple-event risk securitization as a good way of transferring terrorism risk to capital markets. Reshetar [10] developed a framework for pricing of a multiple-event coupon paying CAT bond. It should be noted that they do not take fully the fat tail features of catastrophic risks into account. Considering that, this paper tries to combine Copula function and EVT to model a multiple-event CAT bond. The details are evaluating the marginal distributions of flood catastrophe losses and deaths via EVT. Furthermore, this paper employs the Copula function to model a joint distribution of losses and deaths. To make the price have a more applied value, we describe the spot interest rate by the CIR stochastic interest model.

The remainder of the paper is organized as follows: Section 2 briefly describes the framework of CAT claim model and stochastic interest model. Section 3 presents an empirical analysis. Section 4 is devoted to Monte Carlo simulation and sensitivity analysis. Finally, Section 5 offers conclusions for this research.

#### 2. Valuation Framework

##### 2.1. Pricing Model for the CAT Bond

Throughout this paper, we use the following assumptions:(i) is a Poisson process with the intensity ;(ii) and denote the economy losses and the number of deaths, respectively;(iii) and are two sequences of independent and identically distribution random variables;(iv), and are mutually independent.

We consider a coupon paying CAT bond, namely, paying a certain percentage coupons to investors at the end of the year and returning a certain percentage of principal at maturity date. In this case of catastrophe event, both coupon and principal are at risk. We choose the catastrophe losses and death tolls as trigger indicators. When one of the indicators is triggered, the current and future coupons are paid in proportion to cumulative catastrophe losses. And the principal is also paid in proportion to the cumulative losses only when both indicators are triggered simultaneously. The structures of payoff are given by the following:

the coupon paying framework:

the principal paying framework:where and stand for the attachment point for property losses and deaths, respectively, is coupon value, is bond maturity date, and denotes the percentage of principal paying under different cases. Moreover, in order to avoid the denominator meaninglessness (the number of deaths equal to zero), we replace with in (1).

Compared with the previous researches about multiple-event CAT bond, we adopt floating coupon and principal payment structures to replace the fixed coupon and principal payment structures. This will prevent insurance companies from increasing catastrophe losses on purpose after catastrophe events, resulting in moral hazard.

Let denote the price of CAT bond; then can be calculated by the following formula:where is the price at the time of a risk-free interest rate.

##### 2.2. Interest Rate Model

Cox, Ingersoll, and Ross (CIR) [11] model can not only describe the mean-reverting characteristic of interest rate, but also guarantee nonnegative interest rate, which is just the characteristic in real interest rate markets. Therefore, in this paper, the spot interest rate is assumed to follow CIR model. The interest rate model can be expressed as where is the speed of mean-reverting, is an mean of interest rate in the long run, is the volatility of the interest rate, and is a standard Brownian process. In the CIR model, the risk-neutral pricing of zero-coupon bond is given by following equation: where

##### 2.3. Extreme Value Theory and Modeling

Extreme value theory provides two methods to portray the extreme value behavior of observations, namely, Block Maxima Method (BMM) model and Peak Over Threshold model. However, BMM model is only interested in the behavior of the sample maximum, which could cause vast valid data missing. To take advantage of data information, the POT model (see [12]), which considers all observations exceeding a certain threshold value, shall be introduced to discuss the data behavior.

Suppose a random variable with the distribution function and a high threshold ; can be viewed as the statistical extremes; then the excess distribution function of iswhich implies that

According to the theorem of PBdH (1975), as is large enough, the excess distribution in (10) can be approximated by the generalized Pareto distribution (GPD); that is, where and are, respectively, the shape parameter and the scale parameter.

Then substituting (11) into (10), we obtain

#### 3. Empirical Analysis

##### 3.1. Description of Data

Our data consists of flood events that are recorded in Global Archive of Large Flood Event, provided by Dartmouth College since 1985. And our study mainly considers the losses value exceeding 100 thousand dollars. Thus, the total of 827 pairs of observations for losses and deaths are picked out. Directly analyzing the data, we will find that the fitting performance of data is not very well. Aimed at improving fitting accuracy, the data is adjusted to logarithm method to eliminate the magnitude difference.

Before applying EVT, heavy-tailed characteristic of flood data should be discussed. In general, there are two methods to judge the heavy-tailed behavior: numerical method and the exponential quantile-quantile (Q-Q) plot. Here the two methods shall be used for analysis. Figure 1 obviously shows the tail of exponential QQ plots appears in a convex shape. Besides, the kurtosis values are bigger than three. Therefore, the graphs strongly suggest that the hypotheses that the data follows GPD is acceptable.