Transitory mortality jump modeling with renewal process and its impact on pricing of catastrophic bonds

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Abstract

A number of stochastic mortality models with transitory jump effects have been proposed for the securitization of catastrophic mortality risks. Most of the studies on catastrophic mortality risk modeling assumed that the mortality jumps occur once a year or used a Poisson process for their jump frequencies. Although the timing and the frequency of catastrophic events are unknown, the history of the events might provide information about their future occurrences. In this paper, we propose a specification of the Lee–Carter model by using the renewal process and we assume that the mean time between jump arrivals is no longer constant. Our aim is to find a more realistic mortality model by incorporating the history of catastrophic events. We illustrate the proposed model with mortality data from the US, the UK, Switzerland, France, and Italy. Our proposed model fits the historical data better than the other jump models for all countries. Furthermore, we price hypothetical mortality bonds and show that the renewal process has a significant impact on the estimated prices.

Introduction

Insurance companies and pension plans are exposed to the risk of uncertainty in future mortality. This risk may arise due to improvements in mortality or shocks such as catastrophic mortality events [1]. The latter is called catastrophic mortality risk, which is the risk that, over short periods of time, mortality rates are much higher than expected [2]. Due to a shorter lifetime of an individual or group than expected, an insurer or a pension plan may have to make sudden pay-outs to many policyholders. Hence, severe adverse financial consequences can potentially arise, such as breaches in regulatory solvency and capital requirements [3]. As a result, the management of catastrophic mortality risk is fundamental for insurance companies and pension plans.

Catastrophic events, such as infectious diseases/pandemics, natural disasters, terrorist attacks, wars, and accidents, may cause sudden increases in mortality curves, which are called mortality jumps. For instance, the Spanish flu virus killed 40–50 million people in 1918 and caused a huge jump in mortality rates. More recently, avian flu in 2006 and the Ebola virus in 2014 caused approximately 1 million deaths [4].

According to statistics from the Emergency Events Database (EM-DAT), the frequency, magnitude, and duration of natural disasters have increased since 1975. The World Disasters Report in 2016 stated that rising global temperatures caused global climate change and more natural disasters. These climate changes and natural disasters led to catastrophic events that caused many diseases and deaths in recent years. In the 1970s there were roughly 100 catastrophic events per year. This number has consistently increased more than three times in the last decade. Between 1994 and 2013, the EM-DAT recorded 6873 natural disasters that claimed 1.35 million lives on average each year. Furthermore, in 2018, there were 348 climate-related and geophysical disaster events recorded in International Disaster Database reports and 68 million people were affected around the world.

The occurrence of catastrophic events could cause a large number of deaths and hence a large number of unexpected death claims. Consequently, the financial impacts of catastrophic events on an insurer’s solvency require effective risk management to eliminate and reduce the risk [5]. In the United States, the three largest natural disasters recorded before Hurricane Katrina in 2005 caused a total insured loss of $23 billion and a few reinsurers went insolvent to pay claims [6]. Moreover, a worst pandemic could result in approximately €45 billion of additional claim expenses in Germany according to the estimations of Stracke and Heinen [7]. This amount is equivalent to 100% of the policyholder bonus reserves in the German life insurance market. Some public health experts think that a pandemic is overdue and another will inevitably occur due to the nature of inter-species transmission, intra-species variation, and altered virulence [4].

The frequency of catastrophic events and the degree to which they are accurately priced are serious concerns in managing extreme mortality risks. In recent years, catastrophic bonds have been used by insurers as a risk management tool. The first catastrophic bond was issued by Swiss Re, called Vita I, in 2003 to reduce the impact of catastrophic events. Due to the great success of that bond, many other catastrophic mortality bonds are now being issued (see [8], [9]). Several stochastic models have been developed to capture these jump effects in mortality and to value catastrophic bonds. These models differ in the type of mortality jumps and the severity of jumps. For instance, Cox et al. [10] combined a geometric Brownian motion and compound Poisson process to model age-adjusted rates. Cox et al. [10] modeled permanent mortality jumps by considering Poisson jump counts. Chen and Cox [11] used a normal distribution for jump severity, while Chen and Cummins [12] combined two types of jumps in their model. Similarly, Deng et al. [13] considered the mortality time index as a double-exponential jump process. In contrast to those studies, Liu and Li [14] investigated the age pattern of jump effects on mortality.

All of these mentioned jump models in the literature assumed that mortality jumps occur once a year, or they used a Poisson process for their jump frequencies. Due to their low probability and high impact nature, the timing and the frequency of future catastrophic events and hence mortality jumps are unpredictable [11]. On the other hand, the history of events can give information about their future occurrences. In the Poisson process, inter-arrival times between events are independent and exponentially distributed. However, the Poisson process has a limitation arising from the memoryless property of the exponential distribution. In this paper, we aim to include the history of catastrophic events. One way to incorporate the history of the events is to use duration dependence models. Instead of the constant hazard function, these models have time-varying hazard functions. This property is important for duration analysis since the hazard function is used to capture the duration dependence. The hazard function reflects the waiting times between events. For instance, an increasing hazard function represents longer waiting times between events compared to a decreasing hazard function. In these models, events are dependent in the sense that the arrival of at least one event (in contrast to none) up to time t influences the probability of a further arrival in t+Δt. There is thus a link between the counting model and timing process. This class is known as renewal processes [15].

Winkelmann [16] was the first to derive a counting process by using the renewal process with gamma distributed inter-arrival times. Many other models were derived by using different inter-arrival times afterwards. Mcshane et al. [17] used the Weibull distribution for inter-arrival times while lognormal distribution was used by Bradlow et al. [18], Everson and Bradlow [19], and Miller et al. [20] [21].

In this paper, we propose a new approach for modeling the arrivals of mortality jumps. Inter-arrival time implies the time between two jumps, and we use the renewal process for modeling. For this purpose, we detect jumps in the mortality time series and perform statistical tests for inter-arrival times of mortality jumps to show that we can use the renewal process as a counting process. After that, we use the Lee–Carter model with a jump–diffusion process to model mortality and the lognormal renewal process to model jump count probabilities. We test our model with historical data and compare the goodness of fit of the models with jump sizes and jump count processes for the US, the UK, Switzerland, Italy, and France. To the best of the authors’ knowledge, using the renewal process for jump counts is new in mortality modeling.

It is reasonable to assume that the renewal process has an impact on the pricing of catastrophic mortality bonds. To verify this impact, we use our mortality model to price a catastrophic mortality bond. In an incomplete market, the pricing problem is not explicit, although it might be met by no-arbitrage methods [2], [11], [22], [23], insurance-based methods [12], [24] or economic methods [25].

The no-arbitrage approach was often used in previous research on pricing mortality-linked securities. In the no-arbitrage condition, the market price of risks cannot uniquely be identified. As a result, an arbitrary assumption is necessary for pricing. One might also use canonical valuation to create a risk-neutral probability measure. Canonical valuation can be applied without making any arbitrary decisions [26]. For this reason, we use canonical valuation to create a risk-neutral probability measure and to obtain mortality risk premiums, which was first proposed by Stutzer [27] and then applied to the market for insurance-linked securities by Chen et al. [28], Li [22], and Li and Ng [23]. In this paper, we use the Swiss Re mortality bond as a martingale constraint. This method identifies a risk-neutral probability measure, and thus the price of the hypothetical mortality bonds can be estimated. By using a risk-neutral measure, we price the hypothetical bonds in an incomplete market.

The rest of the paper is organized as follows. Section 2 defines the renewal count process. In Section 3, mortality data are presented. Section 4 provides the specifications of the proposed model and the statistical analysis of mortality jumps. Section 5 demonstrates a numerical example of pricing mortality-linked security. Finally, Section 6 concludes the paper.

Section snippets

Renewal process

Since we want to include the history of the events in our jump frequency model, we need to use a renewal process. A renewal process is a stochastic model for events that occur randomly in time (generally called renewals or arrivals). The times between the successive arrivals are independent and identically distributed and the renewal process might be used as a foundation for building more realistic models.

A counting process {Nt}t0 is a renewal process with independent and identically

Data description

We use mortality data for the US, the UK, Switzerland, France, and Italy. The US mortality data is obtained from the National Center for Health Statistics (NCHS) for the period of 1900–2017 for all ages. The data from the UK, Switzerland, and France are obtained from the Human Mortality Database (HMD) for the period of 1922–2016 for all ages. The Italian mortality data is obtained from the HMD for the period of 1922–2014 for all ages. The data are arranged in 10-year age intervals as follows: <1

The Lee–Carter model

In the Lee–Carter model [30], mx,t denotes the central death rate of age group x in year t. The model is expressed as ln(mx,t)=ax+bxkt+ex,t,where ax is an average of lnmx,t over time t and exp(ax) represents the general shape of the mortality rates. Mortality time index kt which captures the variation of log mortality rates over time, is modulated by an age response bx that represents how slowly or rapidly mortality at each age varies when the mortality index changes [31]. ex,t is the error

Swiss Re mortality bond

We use the Swiss Re mortality bond to obtain a risk-neutral probability measure for pricing hypothetical bonds. The Swiss Re insurance company issued the first mortality risk contingent securitization in December 2003. When the bond is triggered by a catastrophic evolution of death rates of a certain population, the investors incur the loss in principal and interest. The bond provides the investor higher yield as compensation for the mortality risk taken. The bond was issued through a special

Conclusions

In this paper, we have investigated the impacts of the history of catastrophic events on mortality modeling. We use the lognormal renewal process with exponential jumps as the counting process for transitory mortality jumps. A specification of the Lee–Carter model has been proposed, which provides a better fit for different countries. We applied the proposed model to the mortality data for the US, the UK, Switzerland, France, and Italy. Our model turned out to be the best model for all

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