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.
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 [
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 [
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 [
The inapplicability of the assumption of financial data following the log normality distribution in option pricing was early proposed by Mandelbrot [
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. [
The remainder of this paper is structured as follows. In Section
We first introduce the definition of MEGB2 distribution.
Set
Meanwhile, if parameter
Given
If
Based on the distribution of MGB2, MEGB2 distribution is defined as follows. Just as the relationship between the Lognormal and Normal distribution, define
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
Based on the study of Yang et al. [
For pricing the guaranteed unitized participating life insurance, we start with a basic endowment insurance contract
Firstly, we consider a basic endowment insurance contract
Next, in addition to the features of the basic contract
The expected value of the benefits to the insured will be increased by annual dividend. Set annul benefits for the insured as
Equation (
According to Bacinello [
In the payment form of single premium, (
Then,
Denoting
Insurance policy reserves will be invested in the financial market, taken as a special portfolio of assets. First introduce the following notation:
Combining and solving (
then,
In order to calculate the annual equilibrium premium of contract
At time
At time
Then, the expression of
Considering the annual dividend, the actual benefit payable of contract
Assume that the different assets obey the MEGB2 joint distribution. Then, combined with the formulation of single premium about
The difference
Finally, we consider the framework on the basis of contract
The insured will select an optimal admissible exercise strategy according to some information at time
The single premium
We denote by
Assume that the insured paid the single premium
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
Additionally, the expectation of
In order to get the expected value
From (
According to the research findings of Yang et al. [
The density function of
Then, we derive
The derivation procedure is detailed in Appendix
In order to price the guaranteed unitized participating life insurance, we need to get the expected assets value
In order to price the contract
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
The descriptive statistics of stock index.
logarithmic yield of CSI 300 | ||||
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Min. | Max. | Median | Mean | Deviation |
-0.09154 | 0.06499 | 0.000422 | 0.000251 | 0.015931 |
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logarithmic yield of CSI Smallcap 500 | ||||
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Min. | Max. | Median | Mean | Deviation |
-0.08926 | 0.06393 | 0.001878 | 0.000382 | 0.018479 |
Using the parameters we estimated above, the single premium of contract from
Then, according to the single premium of contract from
As comparison, using the initial parameters we set above, when the return rates of two different indexes follow the Multivariate Normal distribution with parameters
Then, according to the single premium of contract from
Figure
Histograms and scatter plot of CSI 300 and CSI Smallcap 500.
We use the EGB2 distribution as the marginal distribution to fit the data. The parameter estimation results and the fitting effect are shown in Table
Parameter estimation results of EGB2 distribution.
Stock index | | | | |
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CSI 300 | 0.0005088 | -0.0001914 | 0.01827691 | 0.0177666 |
CSI Smallcap 500 | 0.0016684 | 0.00181833 | 0.1486097 | 0.1641946 |
Fitting effect of EGB2 distribution as marginal distribution.
The p.d.f and c.d.f of the MEGB2 copula with EGB2 marginals.
Density and cumulative distribution contour plots of distribution having MEGB2 copula.
The results in Table
The premiums with respect to the various parameters.
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Then, we compare the values of embedded annual dividend option and surrender option under two different assumptions of the distribution of the return rate. The results are shown in Figures
Parameter sensitivity analysis of annual dividend option between MEGB2 and Multivariate Normal asset returns.
Parameter sensitivity analysis of surrender option between MEGB2 and Multivariate Normal asset returns.
We can see that in most cases the values of annual dividend option and surrender option under the MEGB2 distribution of the return rate is higher than the result under the Multivariate Normal distribution. With the increase of age
In this paper, some put options are embedded in this type of insurance contract. These options may cause huge downside risks for the insurer. Therefore, managing loss risks of the underlying assets and designing a risk hedging strategy are quite necessary. In this section, we study the optimal hedging decision for this kind of guaranteed participating life insurance product when it implements the dynamic hedging strategy. The problem of hedging in insurance markets has been extensively studied, such as Carr and Picron [
In this strategy, we use the hedging portfolio to hedge the possible loss of our proposed insurance contracts. In this case, we trade index futures and riskless assets at the end of each
Using a loss function of Least-Squares method, this optimization problem of Delta dynamic hedging strategy with respect to the hedging amounts (
The optimal holdings are then obtained by solving a least square problem. Then, we use the distribution of profit and loss (P&L) and some standard risk measures to evaluate the hedge effects. When the returns of the underlying assets follow MEGB2 distribution, Figures
The descriptive statistics of P&L.
Return distribution | Mean | Deviation | Median | 95%-VaR | 99%-VaR |
---|---|---|---|---|---|
MEGB2 | -0.00018 | 0.0207 | -0.0023 | 0.0366 | 0.0601 |
Multivariate Normal | -0.00027 | 0.0194 | -0.0018 | 0.0345 | 0.0552 |
Hedging amounts (
Distribution of P&L on MEGB2 returns.
In this paper, we assume that the returns of the underlying assets follow MEGB2 distribution. However, in financial market, the assumption that returns from Normal distribution is more generally applied and have been widely used in pricing of all kinds of life insurance products. So, in order to investigate the robustness of this hedging strategy in a more general condition, we use the same hedging strategy to manage the risk when the returns of the underlying assets follow a Multivariate Normal distribution. Then, we compared the hedging results of MEGB2 distribution with the results of Multivariate Normal distribution. Figures
Hedging amounts (
Distribution of P&L on MNormal returns.
According to the characteristics of guaranteed unitized participating life insurance, considering the death rate, surrender, and minimum guaranteed yield rate dividend policy, and assuming that the return rates follow the MEGB2 distribution, we establish the valuation model of guaranteed unitized participating life insurance based on the fair value measurement. In this article, a multifactor fair valuation pricing model of guaranteed unitized participating life insurance contract is built in four steps. By the valuation model, the guaranteed unitized participating life insurance can be divided into four parts: the basic endowment insurance contract, contract with annual dividend option, contract with terminal dividend option, and contract with surrender option. Finally, we give the single premium of the guaranteed unitized participating life insurance under different influence factors.
From the results of study on pricing of guaranteed unitized participating life insurance embedded surrender options, when the return rates follow the MEGB2 distribution assumption, the value of surrender option is sensitive to the pricing parameters including the age, minimum guaranteed yield rate, the annual dividend ratio, and the risk-free interest rate. Compared with the pricing results that the return rates
Finally, due to insurer facing the infinite risks on falling in underlying asset prices, we designed an optimal dynamic hedging strategy for this insurance contract. By hedging the same underlying stock index futures, we can calculate optimal hedging amounts. The results of P&L illustrate that this kind of dynamic hedging strategy is functional and effective. By comparing the results of underlying assets following Multivariate Normal distribution, our strategy is useful and can be applied in different loss situations when appropriate pricing of contract is established.
The probability density function (p.d.f) of the GB2 distribution:
Set
Meanwhile, suppose that
Then the unconditional p.d.f. of
Given
If
The joint conditional cumulative distribution function of
Then, the unconditional cumulative distribution function of
Additionally, let
The
We can get a conclusion that
Combined with (
In order to get the expected value
From (
Then, the conditional p.d.f of
Assuming
In addition, combining the relation between GB2 distribution and standard Beta distribution by (
Setting
Combined with (
Furthermore, the unconditional cumulative distribution function of
Then the probability density function of
The MEGB2 copula function is constructed as follows:
Inputting the cumulative distribution function
As can be seen,
When
We have the p.d.f of the EGB2 distribution:
Then, we derive
Since
then,
That is,
In order to price the guaranteed unitized participating life insurance under the MEGB2 distribution, in addition to calculating
By (
Therefore
In Section
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China (Grant nos. 71371021, 71571007, 71333014, and 71873012). Particularly, the authors thank Professor Debbie Dupuis and seminar participants’ comments at the 10th International Conference of the ERCIM WG on Computational and Methodological Statistics.