Because of the improvement in medical technology, the lifetime of the contemporaries is becoming longer than it used to be. Besides, baby boomers are getting old and going to retire lately. Therefore, an aging society could be a big problem in recent years. Although the government has worked hard on social welfare to solve the problem, it seems that social welfare is not enough for elder people to take care of their retirement life nicely. That is why planning the source of income after retirement is a critical issue. Long term and stabilized characters in investment make the annuity insurance become a popular product. Responding to various kinds of demands, the insurers design different annuity insurance contracts.
Annuity products appeared in the United States in 1930. In the 30s, an annuity product was a fixed annuity, which makes fixed dollar payments to the annuitant for the term of the contract, usually until the annuitant dies. The payment of the fixed annuity was decided with assumed invested rate and actuarial survival rates. When the market rates of interest was higher than assumed invested rate of the contract, the annuity holders usually canceled their contracts and put the money into financial market for higher return. That made the insurance company’s fund outflow to other companies. To avoid that situation, the insurance company offered the advantages of withdrawing funds for free as well as flexible assumed invested rate. In the early 80s, the inflation along with the high interest rate helped to develop a variable annuity, which was focus on the return but ignored the risk of investment. A variable annuity has the property that the risk was taken by the annuity holders only. Till the late 90s, in the conditions that the interest rate went down and the performance of the stock
policyholders. For this reason, the policyholders needed another kind of the annuity product which can provide more safe and stabilized income.
With the progress of internet and the computer technology, people can receive more information in a short time in financial markets. Moreover, the stock indices were doing well in the 90s. Therefore, these are two major reasons for the development in the equity-indexed annuity (EIA) contract in the development of the computer technology and financial markets and the economic growth of the 90s. An EIA contract is still one kind of fixed annuity that earns a minimum rate of interest and offers a participation in the equity market. Comparing with other contracts linked with the stock portfolio, the EIA contracts can spread risk efficiently because the return is, instead of a single stock, based on the index-linked return.
Earlier works on valuing the EIA contracts are listed below. Tiong (2001) assumed that the underlying index follows the standard geometric Brownian motions.
He also assumed the constant interest rate for simplicity. Lin and Tan (2003) and Kijima and Wong (2007) considered a stochastic interest rate and gave a more general economic model. Under the lognormal assumption for the equity index, Hardy (2003) focused on the valuation of the ratchet EIA contracts.
The EIA contracts are relatively long dated when they compared with the other financial products. There can be many economic events or new policies over a very long period of time. Once the new information is released, the stock index usually reflects an abnormal jump. We give an example of the Dow Jones Industrial Average Index from 1999 to 2008 below.
1999/1/40.7 2001/1/2 2003/1/2 2005/1/3 2007/1/3 2008/12/31 0.8
0.9 1 1.1 1.2 1.3 1.4 1.5x 104
Time
Stock index
Figure 1 The dynamic process of the Dow Jones Industrial Average Index
1999/1/4 2001/1/2 2003/1/2 2005/1/3 2007/1/3 2008/12/31
-0.1 -0.05 -0.03 0 0.03 0.05 0.1 0.15
Time
Retrun
Figure 2 The dynamic process of the Dow Jones Industrial Average Index return
Figure 1 and 2 are the dynamics process of the Dow Jones Industrial Average Index (DJIA) and the DJIA return. In Figure 2, if we consider the ratio of price falling or rising more than 3% as a jump phenomenon, we can see that there are many jumps in 2000, 2001, 2002, 2003 and 2008. Look up the related information in these years, we find that the internet bubble occurred from 2000 to 2003; moreover there was the Financial Tsunami in 2008. Based on the jumps we found in Figure 2, we can separate
Jump Jump
Jump
Jump
the 10-year period into two states in Figure 1: One is the red-line time period which represents frequent jump phenomenon; the other is the blue-line duration which indicates few jump phenomenon.
The jump diffusion model (JDM) has been used to catch the dynamics process of the stock indices. However, as which shown in Figure 1, the arrival of any new information in the market, no matter good news or bad news, does make the stock price jump abnormally. With different frequency of the new information coming, the numbers of abnormal jumps of the stock indices is different. The JDM can not describe such phenomenon. Charles, Fuh, and Lin (2010) proposed the regime-switching jump model (RSJM) which assumes a two-state Markov modulated Poisson process (MMPP) to be the arrival process of jumps instead of a Poisson process, and pointed out that the RSJM can explain volatility clustering and long memory.
In this paper, the theoretical part is to derive valuation formulas for the EIA contract with the point-to-point design and the EIA contract with the ratchet design under the RSJM. With the 31 stock indices from 1999 to 2008, we estimate the parameters of the JDM and those of the RSJM by using the expectation-maximization (EM) algorithm. Then, we apply the supplemented EM (SEM) algorithm to calculate the asymptotic covariance matrix of the estimates. We test the significance of the RSJM and the JDM with a likelihood ratio test (LRT). After that, we discuss skewness, kurtosis, volatility clustering and volatility smile under the RSJM. Finally, we use sensitivity analysis to determine which parameter is the key driver of the price of the EIA contract under the RSJM.
The remainder of this paper is organized as follows. Section 2 reviews literatures
of the EIA contracts, the jump model and the RSJM. Section 3 illustrates the EIA contracts, the JDM and the RSJM. Section 4 derives the pricing formula of the EIA contract under the RSJM. Section 5 discusses the results of estimation, testing, moment under the RSJM; as well as describes the volatility clustering and volatility smile phenomena. Besides, a sensitivity analysis for the price of the EIA contract is provided in Section 5. Section 6 summarizes the study and gives the future work of this paper.