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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.

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 [

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. [

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 [

The remainder of the paper is organized as follows: Section

Throughout this paper, we use the following assumptions:

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:

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

Cox, Ingersoll, and Ross (CIR) [

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 [

Suppose a random variable

According to the theorem of PBdH (1975), as

Then substituting (

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

Exponential QQ plot of economic losses (a), exponential QQ plot of deaths (b).

In POT model, if the losses after logarithm excess threshold

Since the logarithm of death number cannot be taken when its value is zero, we should take two steps to fit as follows. Firstly, for the deaths number exceeding one, we can also use the above-mentioned methods to give the following condition distribution of catastrophe death:

One of the main challenges in POT model is the selection of an optimal threshold for fitting the model. In practical application, mean excess function plot is usually used to set the threshold. Generally, the threshold is valid if the mean excess plot becomes roughly linear which starts from certain threshold level. For more details, see Embrechts et al. [

Figures

Mean excess plot of losses.

Mean excess plot of deaths.

Figure of calibration scale parameters and scale parameters of losses.

Figure of calibration scale parameters and scale parameters of deaths.

After setting the threshold values, the maximum likelihood method will be used to estimate the other parameters. Estimates are given in Table

POT parameters estimates.

Parameters | | | | | | |
---|---|---|---|---|---|---|

Economic losses | 0.0376 | 1.0428 | 20.5 | 95 | 827 | – |

Deaths | 0.1747 | 0.8693 | 5.8 | 62 | 786 | 0.0496 |

We use the diagnostic plots to check whether the GPD fits the data. Figures

Diagnostic plots of losses.

Diagnostic plots of deaths.

Copula function is called joint function, which connects joint distribution function with marginal distribution function. It usually studies the nonlinear relationship among variables. And Copula function mainly includes Elliptic Copula family and Archimedean Copula family. The later was used more often because the Elliptic Copula cannot describe the asymmetric relation of variables. Therefore, in this paper, three common Archimedean Copula functions, namely, Gumbel, Clayton, and Frank Copula, will be adopted to undertake related studies.

First of all, we plug the estimates of parameters shown in Table

Parameter estimates of Copula function and K-S test results.

Copula function | Clayton Copula | Gumbel Copula | Frank Copula |
---|---|---|---|

| 0.4578 | 1.3401 | 2.3764 |

K-S value | 0.0221 | 0.1494 | 0.5933 |

| 0.8292 | 0.0000 | 0.0000 |

Q-Q plot of Gumbel Copula (a), Clayton Copula (b), and Frank Copula (c).

In this section, we will concentrate on valuing the price of CAT bond. Before the estimation, some related parameter values need to be set. According to the relative data of the real insurance market, the basic parameter values are given in Table

Basic parameter values.

Text interpretation | Symbol | Value |
---|---|---|

Initial interest rate value | | 0.02 |

Speed of mean-reverting | | 0.004 |

Long-run interest rate mean | | 0.02 |

Volatility of the interest rate | | 0.006 |

Poisson process intensity | | 3 |

Face value | | 100 USD |

Coupon value | | 3 USD |

Maturity time (year) | | 3 |

Attachment point of losses | | 19.5471 USD |

Attachment point of deaths | | 4.7005 USD |

Recalling that the explicit solutions of (

To determine the price, we implement

The bond prices decrease as the intensity gets stronger; the discount values of coupons also show the inverse relationship with intensity from Table

The sensitivity of intensity to cat bond when

Intensity | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Coupon discount | 6.8436 | 5.4078 | 4.2841 | 3.3853 | 2.6808 |

Principal discount | 77.9042 | 64.2392 | 53.1307 | 43.8916 | 36.4381 |

Bond prices | 84.3878 | 69.6470 | 57.4148 | 47.2769 | 39.1189 |

Future principal | 82.6978 | 68.1919 | 56.4000 | 46.5924 | 38.6802 |

As can be seen from Table

The sensitivity of maturity to cat bond when

Maturity | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Coupon discount | 1.4484 | 2.8774 | 4.2791 | 5.6532 | 6.9923 |

Principal discount | 80.7633 | 65.4306 | 53.1625 | 43.0465 | 34.8572 |

Bond prices | 82.2117 | 68.3100 | 57.4417 | 48.6997 | 41.8495 |

Future principal | 82.3924 | 68.0926 | 56.4337 | 46.6070 | 38.4900 |

As shown in Tables

As we know, the relationship between Kendall rank correlation coefficient

Effect of dependence

| 0.0578 | 0.4578 | 0.8578 | 1.2578 | 1.6578 |
---|---|---|---|---|---|

| 0.0281 | 0.1862 | 0.3001 | 0.3861 | 0.4532 |

Principal discount | 0.0646 | 0.0809 | 0.0941 | 0.1056 | 0.1154 |

Trigger frequency of two indicators triggered | 82.2117 | 68.3100 | 57.4417 | 48.6997 | 41.8495 |

Bond prices | 64.0219 | 57.3949 | 52.5902 | 48.5170 | 45.7122 |

Catastrophe bonds triggered by multiple-event have a lower catastrophic risk than the one triggering event, while they have higher yields and more market potential than ordinary bonds. This paper designs a multiple-event triggering pricing model and combines the Copula function and the POT method in extreme value theory to study the pricing of catastrophe bonds. The pricing model not only retains the advantages of the previous multiple-event trigger model, but also has the characteristics of low risk and high return. Some improvements have been made, floating coupon and principal payment structures as replacements for the original fixed coupon and principal payment structures, so as to reduce moral hazard and improve bond attractiveness. In addition, we implement Monte Carlo simulation to price CAT bonds using Global Archive of Large Flood Event, provided by Dartmouth College since 1985. Finally, the sensitivity analysis about the parameters of pricing model is also conducted. The empirical studies reveal that the price has the inverse relationship with arrival intensity, maturity time, and Copula dependence coefficient. The effect of intensity on the price of CAT bond is more pronounced than the effect of maturity time.

In recent years, the frequency and loss affected by natural disasters in the world have been constantly expanding, which poses a more severe challenge to the traditional catastrophe insurance market. Faced with low-loss frequency and high-loss severity catastrophe risk, the capacity of insurance companies for catastrophe is very limited, making catastrophe risk not yet effectively dispersed in the insurance market. Catastrophe bonds and other derivatives emerged, connecting the insurance market with the capital market and well transferring the catastrophe risk to the capital market. The model put forward in this paper enriches existing research on catastrophe bond pricing, especially in multiple-event catastrophe bond pricing. Furthermore, our research not only provides theoretical guidance for insurers to price the multiple-event catastrophe bond, but also provides a low-risk investment products to investors, enriching their investment portfolios.

The authors declare that they have no conflicts of interest.