Discrete Dynamics in Nature and Society

Volume 2019, Article ID 9439786, 16 pages

https://doi.org/10.1155/2019/9439786

## Valuation of Guaranteed Unitized Participating Life Insurance under MEGB2 Distribution

^{1}School of Economics & Management, Beihang University, Beijing, 100083, China^{2}Department of Statistics, University of Wisconsin, Madison, WI 53706, USA

Correspondence should be addressed to Junzhang Hao; nc.ude.aaub@zjoah

Received 6 September 2018; Revised 16 December 2018; Accepted 16 January 2019; Published 6 February 2019

Academic Editor: Francisco R. Villatoro

Copyright © 2019 Haitao Zheng et al. 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

Crisis events have significantly changed the view that extreme events in financial markets have negligible probability. Especially in the life insurance market, the price of guaranteed participating life insurance contract will be affected by a change in asset volatility which leads to the fluctuations in embedded option value. Considering the correlation of different asset prices, MEGB2 (multivariate exponential generalized beta of the second kind) distribution is proposed to price guaranteed participating life insurance contract which can effectively describe the dependence structure of assets under some extreme risks. Assuming the returns of two different assets follow the MEGB2 distribution, a multifactor fair valuation pricing model of insurance contract is split into four components: the basic contract, the annual dividend option, the terminal dividend option, and the surrender option. This paper studies the effect of death rate, minimum guaranteed yield rate, annual dividend ratio, terminal dividend ratio, and surrender on the embedded option values and calculates the single premium of the insurance contract under different influence factors. The Least-Squares Monte Carlo simulation method is used to simulate the pricing model. This article makes a comparison in the sensitivity of the pricing parameters under the MEGB2 distribution and Multivariate Normal distribution asset returns. Finally, an optimal hedging strategy is designed to cover the possible risks of the underlying assets, which can effectively hedge the risks of portfolio.

#### 1. Introduction

Today participating life insurance products are quite popular in insurance market all over the world. In these policies the insured not only gets the guaranteed annual minimum benefit but also receives proceeds from an investment portfolio. Accurate pricing of life insurance participating policies can be traced back to Wilkie [1] through the fair valuation of the embedded options. He initiated the use of modern option pricing theory to study the embedded options in bonuses on participating life-insurance policies. Besides embedding a term life insurance and a minimum guaranteed interest rate, these policies include bonus participation rules and surrender regarding the insurer’s benefits. Grosen and Jrgensen [2] utilized a barrier option framework to found the influence of regulatory intervention rules for reducing the insolvency risk of the policies. Concerned with the study of a unit-linked policy with interest rate guarantees, Shen and Xu [3] studied the valuation of unit-linked policies with or without surrender options. Steffensen [4] provided a general framework about the analysis of surrender and paid-up options and the actuarial approach comparison done by Linnemann [5, 6]. Gerber, Shiu, and Yang [7] calculated the value of equity-linked death benefits in various participating life insurance products. Bacinello [8] supposed the pricing of participating life insurance policies with surrender options using a recursive binomial tree approach. Bernard et al. [9] used techniques which were developed in Carr et al. [10] to achieve the result of the optimal surrender strategy for a variable annuity contract embedded with minimum guaranteed accumulation benefits.

The pricing of participating life insurance products with guarantees has been largely studied under the assumptions of the Black-Scholes option pricing model by Tiong [11], Milevsky and Posner [12], Lin et al. [13], Ng et al. [14], Gatzert and Holzmüller [15], and Fan [16]. However, when the historical prices of the underlying assets are analyzed and used as the basis for setting the pricing assumptions with respect to the risk model, it is possible to recognize that asset returns rarely meet the standard Brownian motion. Since the last financial crisis, the pricing modeling assumptions which are frequently used in the field of option market and risk management in the financial industry have gradually come into focus. Crisis events such as the collapse of Lehman Brothers have radically changed the view that extreme events in financial markets have negligible probability. Over the last decade, the life insurance industry’s situation has deteriorated due to substantial changes in both the economic and the regulatory environment. However, changing market conditions, for instance, a change in asset volatility, may affect these models quite differently. There has been a growing pragmatic and theoretical shift in interest from the modeling of “normal” asset market conditions to the shape and fatness of the tails of the distributions of asset returns which characterize statistical models for extreme events. The Black-Scholes model cannot explicitly account for the negative skewness and the excess kurtosis of asset returns. Typically, in periods when the left skewness of asset prices increases, the Black-Scholes model will overprice out-of-the-money call options and underprice in-the-money call options relative to when there is greater symmetry in the distribution function. Therefore, when we are pricing different forms of investment guarantees typically embedded in participating life insurance products, the brusque variations of the financial market require considering the unpredictable variations of the risk model hypotheses.

The modeling of financial prices is a key assumption to conjecture in the theoretical analysis of participating life insurance products and particularly the dynamics of the value of the referenced financial portfolio or market index; also the embedded options in contracts are very sensitive to the tails of the underlying distribution. It is important to identify the distribution of stock returns over expansion-recession cycles and the occurrence of catastrophic events when the pricing of both the participation and the surrender options are affected by the value of the reference portfolio. Therefore, asymmetries and leptokurtic have to be taken into account. The research has developed models with regime-switching schemes [17] to cope with these stylized facts, or with method of Lévy process, such as Cont [18], Ballotta [19], Coleman et al. [20], Siu et al. [21], Fard and Siu [22], Yu [23], Huang and Yu [24], Lin et al. [25], Orozco-Garcia and Schmeiser [26], and Kélani and Quittard-Pinon [27]. However, Melick and Thomas [28] mentioned that it was more natural to begin with an assumption about the future distribution of the underlying asset, rather than the particular stochastic process by which it evolved and to use option prices to directly recover the parameters of that distribution. Therefore, extreme value theory is a kind of robust framework to analyze the tail behavior of distributions during the financial crisis, which has been applied extensively in various fields especially in the insurance industry, in case of insufficient solvency problem for insurance companies.

The inapplicability of the assumption of financial data following the log normality distribution in option pricing was early proposed by Mandelbrot [29]. The systematic study of extreme value theory for risk management and financial models was done by Embrechts et al. [30–32] and Mc Neil [33]. Aparicio and Hodges [34] and Corrado [35] studied the option pricing based on generalized Beta functions of the second kind and generalized Lambda distribution respectively. Rocco [36] summarized some new developments of extreme value theory for finance. Yang et al. [37] investigated the applications in modeling multivariate long-tailed data by a generalized beta copula. Markose and Alentorn [38] argued that there was difference in pricing options used Generalized Extreme Value (GEV) distribution and Normal distribution for asset returns. The results presented that how the option price was highly sensitive to the changes in the tail shape, which was distinct to its sensitivity to the variance of the return distribution. Cui and Ma [39] studied the pricing synthetic CDO with MGB2 distribution. Ma et al. [40] evaluated the default risk of bond portfolios with extreme value theory. Zheng et al. [41] made a valuation of guaranteed participating life insurance under GEV distribution.

Existing literatures rarely considered the pricing for participating life insurance products with applications in modeling multivariate shape and heavy tailed asset returns. With fluctuations in the financial markets, does the price of participating life insurance products change sharply under the changing financial market? This will be a problem worth studying. Therefore, the framework of Yang et al. [37] is used in our article; we introduce MEGB2 distribution to fit the distributions of asset returns in order to highlight the characteristic features of heavy tail, the skew, and the dependence structures in asset returns. Then we make a pricing for participating life insurance products embedded the surrender option and analyze the sensitivity of parameters. Finally, we study the differences of pricing of participating life insurance products between MEGB2 joint distribution and Multivariate Normal distribution for asset returns.

The remainder of this paper is structured as follows. In Section 2 we briefly illustrate the MEGB2 asset returns and introduce our valuation framework about four contracts with different embedded options and the single premium of every contract is explicitly written out. We make an empirical study to analyze the influence of different initial parameters on single premium in Section 3. As a comparison, we discuss the parameters sensitivity under the assumption of Multivariate Normal distribution for asset returns. Then, a comparison is made about the difference between the two pricing results under two different multivariate distributions. Moreover, an optimal hedging strategy for this type of insurance product is also designed. Finally our study has been concluded in Section 4. The Appendix is derivation of MEGB2 copula and derivation of assets expected value under MEGB2 distribution.

#### 2. Valuation Model under MEGB2 Asset Returns

##### 2.1. MEGB2 Distribution

We first introduce the definition of MEGB2 distribution.

Set as a real* n*-dimensional random vector on . Each given* θ* follows a generalized gamma distribution with the probability density function (p.d.f.):

Meanwhile, if parameter* θ* is a random variable which follows an inverse gamma distribution with parameters (

*q*,1), i.e.,

*InvGa (*

*θ*~*q*,1), then the unconditional p.d.f. of

*isand then*

*θ**X*=(,…,) is said to follow multivariate GB2 distribution (MGB2) and the marginal coordinate follows GB2(, , ,

*q*) with density:

Given* θ*, the conditional cumulative distribution function of iswhere , is the Gamma function.

If , we can get . From the above relation, the cumulative distribution function of can be derived:where is a cumulative distribution function of the standard Beta distribution with the parameters (,* q*) and the function expression is

Based on the distribution of MGB2, MEGB2 distribution is defined as follows. Just as the relationship between the Lognormal and Normal distribution, define* G*=*LnX* as a real* n*-dimensional random vector on . If follows MGB2 distribution, then the variable is said to be distributed as a multivariate exponential generalized beta of the second kind (MEGB2).

Although the MEGB2 distribution is a useful tool to deal with the complex multivariate long-tailed data, there are still some constraints faced by the MEGB2 distribution. Firstly, the MEGB2 distribution requires the univariate marginal distributions belonging to the same family. Secondly, the analytical expression of MEGB2 distribution becomes increasingly complicated when dimension rises. Thirdly, all margins of the MEGB2 distribution must have a common parameter* q*, while other parameters may vary across margins. For these reasons, we now introduce the corresponding MEGB2 copula.

Based on the study of Yang et al. [37], we construct the* n*-dimensional MEGB2 copula which is defined bywhere . It is easier to acquire the* n*-dimensional MEGB2 distribution by using the MEGB2 copula function. Details of the derivation are given in Appendix A.

##### 2.2. The Structure of the Contract

For pricing the guaranteed unitized participating life insurance, we start with a basic endowment insurance contract including an initial guaranteed death benefit. According to the features of different options, three options are included into the basic contract. On the basis of the basic endowment insurance contract, we define a model for a contract added an annual dividend option to ; a contract added a terminal dividend option to ; a contract added a surrender option to . Finally, the fair valuation of guaranteed unitized participating life insurance contract can be obtained.

###### 2.2.1. Basic Contract

Firstly, we consider a basic endowment insurance contract with periodic premium payments. For the contract , if they die within the term of the contract, the insured will receive the death benefits from the insurer. Meanwhile, the insured will obtain the expected premium payments in the case of survival until maturity. It is assumed that the death benefits are equal to the survival payments. According to the actuarial mathematics theory for an endowment policy, if the age of the insured is* x*, single premium can be calculated as follows:where denotes death or survival payments and* r* is used as the annual compounded interest rate for discounting future benefits and premiums. represents the probability that the insured dies between* t*-1 and* t*^{th} year of contract and _{t}*p*_{x} is the probability for an year old policyholder surviving for the next years. The accounts will receive the annual premium at each policy time. The single premium can be used to calculate the value of annual dividend option.

###### 2.2.2. Contract with Annual Dividend Option

Next, in addition to the features of the basic contract , we include an annual dividend option and call this contract . The single premium is set as .

The expected value of the benefits to the insured will be increased by annual dividend. Set annul benefits for the insured as ; then =+annual dividends. Since is the benefit payable at time* t* (*t*=2,,*T*), we need to consider the insurance policy reserve of this contract at the end of each policy year in order to calculate . We denote by _{t}*V* the insurance policy reserve at time* t* (*t*=1,,*T*-1) and _{t}*V* can be calculated according to the actuarial equivalence principle for the endowment policy:

Equation (9) is just a theoretical formula because is unknown among the equation. Hence, _{t}*V* cannot be calculated directly. In order to price the contract with an annual dividend option, assuming that the added value of insurance policy reserve at the end of the policy year is determined by a certain percentage of the reserve at the beginning of the policy year. Then

According to Bacinello [8], (Δ*V*)_{t} and (Δ*C*)_{t} exist in the following relationship:where (Δ*P*)_{t} is the increase value of the premium in the case of the annual premium.

In the payment form of single premium, (11) should be written as

Then,

Denoting as the ratio of change annual payment, we have

Insurance policy reserves will be invested in the financial market, taken as a special portfolio of assets. First introduce the following notation: denotes the special portfolio and represents the rate of return on the reference portfolio at the* t*^{th} year of contract. Annual benefits will be allocated to the insured at a certain percentage which is denoted as* η*. Then insurance policy reserves will be increased . According to the provisions of the participating life insurance, fund growth rate cannot be less than the minimum guaranteed yield rate . We can obtain some relationships written as follows:

Combining and solving (9) and (10), we can deduce the expression of annual benefits

then,

In order to calculate the annual equilibrium premium of contract , supposing financial market risks and mortality risks are independent, we assume that the risks of insurance companies are neutral. On the basis of these two assumptions, the specific calculation is implemented in two steps. Firstly, the expected value of the benefit is discounted to the beginning of the insurance period and denote as discounted present value. Considering the expected value of the death risk, by *π*() we can calculate the single premium .

At time* t*=1, one gets

At time* t*>1,where , the minimum guaranteed yield rate is , and annual dividend ratio is denoted as* η*.

Then, the expression of is given by

###### 2.2.3. Contract with Terminal Dividend Option

Considering the annual dividend, the actual benefit payable of contract at time is given bywhere the value of asset is denoted as at the expired date of contract. When is greater than _{T}*V*, the insured will receive the final dividend which is a certain proportion *β* of surpluses. The terminal dividend of the insured is expressed as *β*( - )^{+}.

Assume that the different assets obey the MEGB2 joint distribution. Then, combined with the formulation of single premium about , the single premium of can be written aswhere .

The difference - = [* π*()-

*()]*

*π*_{T}

*p*

_{x}in (21) is the value of terminal dividend option under the MEGB2 asset returns model.

###### 2.2.4. Contract with Surrender Option

Finally, we consider the framework on the basis of contract and include a surrender option named contract , which the policyholder can exercise annually at time* t*=1, 2,…,*T* until maturity. When exercising the surrender option at time* τ*, the policyholder can obtain the insurance policy reserve at time of termination of the contract . The contract payoff at time of surrender

*t*=

*can be written out as follows:*

*τ*The insured will select an optimal admissible exercise strategy according to some information at time* τ*, which includes the assets value, reserve, and risk-free interest rate. For the insured, the optimal surrender exercise time is the moment that can obtain the maximum expected value of cash flow. The expected total discounted payoff of cash flow acquired by the policyholder after exercising the surrender option is given by

The single premium of contract with surrender option is denoted by

We denote by all the values at stopping time from 0 to* T*-1. minus equals the value of surrender option.

##### 2.3. Assets Value and Return on Assets Model

Assume that the insured paid the single premium within the expiration date of the policy. The insurance company will use the single premium as the initial capital of the insured to set up the capital account, and the account funds are invested in the specified portfolio for the return on investment. Suppose that an insurance company invests funds in risky assets and uses to represent the value of the assets at the end of time* t*. can be expressed aswhere is annual weighted rate of return on investment of the asset in the interval . represents the investment proportions of assets in the interval , which satisfied . can be determined by mean-variance model of Markowitz. is the return rate of the assets in the interval and assume obeys the MEGB2 distribution.

Because the annual dividend and terminal dividend are related to the changes of annual payments, we need to investigate the changes of the assets value. The insured have their own capital accounts for their insurance contracts. The accounts will receive the annual premium at each policy time and form a new initial value of portfolio combined with the existing portfolio. Assume that investment strategy does not change due to the inflow of funds and the annual return on asset follows the same distribution, which is = , = . Let denote the value of assets at time before the inflow of annual premium. Then, combined with (25), can be expressed aswhere

Additionally, the expectation of is given by

In order to get the expected value of , the distribution of should to be investigated.

From (4), we can calculatewhere .

According to the research findings of Yang et al. [37], the MEGB2 Copula function is constructed as follows:

The density function of iswhere .

Then, we derive* E*():

The derivation procedure is detailed in Appendix B.

#### 3. Data Analysis and Empirical Results

##### 3.1. Model Solution

In order to price the guaranteed unitized participating life insurance, we need to get the expected assets value* E*() at the end of each policy year. According to (6), the calculation of* E*() relates to the distribution of the return on assets and single premium . We suppose that the return on assets follows the MEGB2 distribution. Then the parameters of marginal distributions ((; , , ,* q*)) and the parameters (, ,* q*) of copula function should be estimated. The expectation of annual return rate can be simulated after having estimated the corresponding parameters. Actually,* E*() is used to calculate the terminal dividend, which existed at contract and , whereas other contracts can be calculated only by the annual return rate of the assets value. Hence, when pricing contract , we use the single premium of contract to simulate the expected value of assets in each policy year. Similarly, pricing contract , we use the single premium of contract . The contract uses the single premium of contract to simulate the expected value of assets.

In order to price the contract with three options mentioned above, we should simulate from to . Firstly, according to the actuarial principle, we calculate the single premium of contract . Secondly, we assume return on assets following the MEGB2 distribution and then calculate the single premium of contract . Thirdly, based on the single premium of contract , we calculate the single premium of contract . Finally, we can calculate the single premium of contract by the Least-Squares Monte Carlo simulation (LSM) method. The difference of single premium between contracts is the value of embedded option. Specifically, the value of annual dividend is equal to - ; the value of terminal dividend is equal to - ; the value of surrender dividend is equal to - .

##### 3.2. Parameter Setting and Premium Results

In order to empirical analysis, we need to decide the parameters involved in the previous sections firstly. The mortality used in this study was derived from the experience life table of China Life Insurance (2010-2013). The experience life table includes four groups of mortality for Chinese people. Because the impact of pricing for different population mortality is not the focus of this article, we use the nonpension male mortality as the initial parameter. Then the initial age of the insured is set as 40 years old. The basic payment of guaranteed unitized participating life insurance is set as 10000 yuan and the insurance period is 10 years. Considering that the scheduled interest rate of China Life Insurance Policy was failed to 2.5% in 1999 and has maintained this level so far, therefore, the minimum guaranteed yield rate is set as 2.5%. The risk-free interest rate is set as the current one-year fixed deposit rate 1.5%. Ratios of annual dividend and final dividend, according to the provisions of the CIRC (China Insurance Regulatory Commission), are set as 75% and 50%, respectively. In this study, we select the CSI 300 stock index and CSI Smallcap 500 index as the two underlying assets in the portfolio and use the logarithmic yield of these two indexes from June 2012 to June 2017 to estimate the parameters of marginal EGB2 distribution and MEGB2 copula function and the parameters of MEGB2 copula are =1.12, =0.057, and* q*=0.028. The descriptive statistics results of stock index are shown in Table 1. The weights of the two underlying assets in the portfolio are =0.33, =0.67, respectively, which are calculated by mean-variance model.