Valuation of Variable Long-term Care Annuities with Guaranteed Lifetime Withdrawal and Limited Hospitalization Coverage Bene ts and Two Investment Funds

Population aging on the one hand and increasing of expenditures for medical services via technology development on the other hand have created some problems for the insurance industry and have converted it into one of the most risky areas. Thus, it is obvious that designing the innovative product is one of the most basic needs in the field of health insurance. This study proposes a new variable annuity product that includes the benefits of guaranteed lifelong withdrawal benefit option, long-term care coverage and limited hospitalization coverage. This innovative product has been evaluated under two investment funds. A Monte Carlo algorithm has been employed to calculate the fair value of the product and numerical study has been conducted to illustrate the capability of this product. The innovative product may provide a more comprehensive solution for the aging problem of the population that is challenging in the current societies.


Introduction
Greater awareness of longevity risk and improvements in medical technology are two important reasons for the popularity of health care insurance products and the ever-increasing demand for such products. This study proposes a new variable annuity product that includes the benefits of guaranteed lifelong withdrawal benefit option, long-term care coverage and limited hospitalization coverage.
The life care annuity (LCA) which is a combination of a long-term care (LTC) and lifetime annuity (LA) reduces the effect of strict underwriting process and adverse selection (Murtaugh, Spillman and Warshawsky, 2001;Webb, 2009;Brown and Warshawsky, 2013). Murtaugh et al. (2001) showed that the LCA reduces the cost of the LTC policy in some situations; therefore, more people can have access to it. They proved that the combination of a LTC and a LA can become more attractive through a positive correlation between mortality and disability. Webb (2009) maintained that the LCA products provide a more economical product than buying the LTC insurance and an annuity contract separately whenever there exists some management fees. In 2013, Brown  (2018) for the first time. Due to the complex calculation, they proposed a Monte Carlo algorithm that utilizes the variance reduction technique (Glasserman, 2004;Asmussen and Glynn, 2007).
Modeling the transition probability of the policyholder's health state is one of the most important steps to evaluate the above-mentioned products. Many authors have modeled their product by means of a Multi-state model based on the Markovian framework. For instance, Pitacco (1995Pitacco ( , 2013Pitacco ( , 2014 has offered a Multi-state model as a powerful tool for interpreting disability and the LTC insurances. Albarran et al. (2005) presented a new estimation method for a multiple state model and applied it to the disability insurance in Spain's aging population. Pritchard (2006) parameterized a multiple state model for the LTC and disability insurance contracts using the interval-censored longitudinal data. Also, he estimated costs of the LTC insurance contract for This study is organized as follows. Section 2 explains product details. Theoretical foundation of the product reviews in Section 3. Data explanation and numerical studies are addressed in Sections 4 and 5, respectively. Finally, Section 6 provides some brief conclusions and suggestions.

Product specifications
The health state of policyholders and their corresponding transition probability play fundamental roles in evaluation of an insurance health product. For simplicity, a seven-state continuous-time Markov process has been considered for classification of the individual's health status based on disability in performing instrumental activities of daily living (IADLs, including light housework, laundry, grocery shopping, meal preparation, getting around outside, money management, using the telephone) and activities of daily living (ADL, including eating (level 1), bathing (level 2), dressing (level 3), moving around (level 4), doing personal hygiene (level 5) and going to the toilet (level 6)), see Pritchard (2006), Haberman and Pitacco, (1998) for more details.
State of each policyholder begins from state 1 (healthy ) and moves to state 2 (ADL level=1), state 3 (ADL level=1 and 2), state 4 (ADL level=3 and 4), state 5 (ADL level=5 and 6), state 6 (stay at a hospital) and finally state 7 (death). Therefore, if M x (t) represents the state of the policyholder at time t when s/he has bought the policy at age x, then the 7 × 7 transition probability matrix is stands for the transition probability that the policyholder moves to state j from state i after a time t. This study follows Pritchard's (2006) recommendation for transition rates.  A 2 ) HB x t−1 , denotes the insured's hospitalization costs up to L days during the whole contract's period in the previous year.
A 3 ) G t denotes yearly random withdrawal benefits in year t up to βw x 0 ), β ∈ (0, 1). Note that the policyholder determines the value of yearly random withdrawal benefits.
Furthermore, the insurer withdraws 1)) and M , which denote the guarantee fee and fixed management fee, at the beginning of year t, respectively. Note that the invested amount, in the two investment funds, grows based on the constant interest rate i and the return rate R t and reaches W t at time t.
Hereafter, it is assumed assume that account value which was invested in the risky fund RF (t) at time t follows the Geometric Brownian Motion (GBM), therefore its return rate R t holds where µ and σ 2 are the expected drift rate and the volatility of the process, respectively, and B t denotes the standard Brownian motion process, see Shreve (2004) and Oksendal (2013) for more details.

Theoretical representation of the product
As discussed in the previous section, the insurer has three obligations: medical costs SB x t , hospitalization costs HB x t and withdrawal benefits for its policyholder. This study assumes that medical costs of a policyholder who has bought the contract at age x are non-zero only whenever the policyholder loses 3 or more ADLs. In the other word, SB x t that is only paid to a policyholder who moves to state 4 or 5 during the period [t − 1, t) and is 0 otherwise, can be modeled by where c and π are given positive constant and fixed inflation rate, respectively.
When a policyholder stays at the hospital during the period [t − 1, t) and moves to state 6, the hospitalization costs HB x t are paid to her/him. This study assumes that the hospitalization benefits are limited and have been provided up to L days from the beginning of the contract; therefore, such reimbursement depends on two factors: 1) total number of hospitalization days for the policyholder at age x + t who has bought the contract at age x (denoted by T t x ), and 2) the number of years during which the policyholder, who has bought the contract at age x, has received the hospitalization benefits (denoted by D x ).
It is obvious that modeling of the hospitalization days T t x and the hospitalization benefits D x are the first steps for determining the hospitalization costs.
Suppose the counting process N t x represents the number of hospitalization visits at the t th policy year. Moreover, suppose that the policyholder in her/his i th visit at the t th policy year has stayed M x t,i days in the hospital. Therefore, the hospitalization days for the policyholder at the policy year t are For simplicity hereafter, it is assumed that the two counting processes N t x and M t,i x are two independent Poisson processes with rates λ N and λ M , respectively.
Considering that y shows the average age at which the total expected hospitalization days reaches L for the first time, the random number of years that the policyholder can be benefited from the hospitalization coverage is determined as follows: Hereafter, it is assumed that the daily payment for hospitalization at time t is modeled by where φ and b are a given inflation protection rate for medical costs and a given positive constant, respectively. Then HB t x can be formulated by where I A (t) is the indicator function. Given that a limitation has been taken into account for the number of hospitalization days for the product specified in the section 2, two different products can be designed considering this feature. Therefore, based on the expiration condition, the following two slightly different products can be considered. The following represents a mathematical model for product 1.
Product 1. Suppose that a policyholder, at age x, pays w x 0 and buys an insurance contract with assumptions from A 1 to A 4 that have been presented in the section 2. Additionally: (1) at time t, the insurer invests the δ percent of the investable amount in a risky fund with the return rate R t and the rest (i.e., the 1 − δ percent) in a risk-free fund with the constant interest rate i and (2) the policy will be expired either when the policyholder dies (state 7) or his/her money reaches zero.
Then, W t , holds where W − t and W + t denote the account value at year t before and after withdrawals, respectively.
And the random positive integer number K x stands for death time of a policyholder who has bought the product at age x.
In order to find the fair value of this product, the expected present value of cash flows should be calculated. Therefore, the discounted cash flows (DCF) method at time t is first used by a money market account γ(t). Due to the risk-neutrality of γ(t), the dynamics of γ(t) can be given by the dγ(t) γ(t) = rdt and its solution is γ(t) = γ(0)e rt , where r is a given constant rate. Without loss of generality, suppose that γ(0) = 1 and therefore γ(t) = e rt . Based on this observation, the fair value of Product 1 can be expressed as where E() denotes the expected value. Now, the product 1 is exposed to an extra expiration condition and the product 2 is introduced. The mathematical model for such product is represented below.
Product 2. Suppose that a policyholder, at age x, pays w x 0 and buys an insurance contract with assumptions from A 1 to A 4 that have been presented in the section 2. Additionally: (1) at time t, the insurer invests δ percent of the investable amount in a risky fund with the return rate R t , and the rest (i.e., 1 − δ percent) in a risk-free fund with the constant interest rate i, and (2) the policy will be expired either when the policyholder dies (state 7), or when s/he reaches the maximum hospitalization days L at year D x or her/his invested amount reaches zero.
Then, W t holds where Y x = min{D x , K x } and W − t , W + t and K x have already been defined. Similar to the Product 1, the fair value of Product 2 can be calculated via the following equation where E() denotes the expected value.

Data Explanation
Since solving two Equation (8) and Equation (10)  The interest rate is equal to i = 5% and δ = 0.7. Figure 2 shows return rates of the two investment funds that are assumed in this study.
The R software (version 3.6.3) and packages markovchain, actuar, mgcv, car and MASS have been employed for the simulation study. The R codes are provided on request.
Output: Fair value Simulate a seven-states Markov process for the policyholder based on a given transition matrix in Equation (1); 7 Simulate the number of days that the policyholder may be hospitalized (state 6) without any constraint; 8 Simulate a sample path from GBM (µ, σ); then, determine the return rate R t ;

Numerical Results
Now using the assumptions in section 4, fair value of Product 1 and product 2 are calculated. Table   1 and Figure 3 represent fair value of the two products under the above assumptions.
Since increasing the policyholder's entrance age increases the medical costs, it is expected that the fair value of both products will be enhanced as the age is increased. In addition, since Product 1 provides benefits for a longer time, therefore, its fair value is higher than the Product 2. Table 1 and Figure 3 verifiy this expectation and show that as the entrance age increases, the fair value of  Figure 3: Fair values of two products for different entrance ages.
both products is increased and the product 2 is less expensive than the product 1. Also, Figure 3 shows the pattern of fair value for both products as a function of the policyholder's age.
Both products have L day hospitalization benefits. Certainly, such hospitalization day is effective on the fair value of both products. To illustrate such effect, consider a policyholder who bought the products at age 60 and suppose that the total hospitalization days L can be varied between 20 to 200 days. Figure 4 illustrates the effect of L on fair values of the two products. As it is expected, the length of hospitalization days L is effective on the fair value of both products directly. Furthermore, when L is increased, fair value of these products has a similar behavior. This observation can be due to that fact that the account value is relatively small for large L and the difference between fair values of the two products is very small.   The random "withdrawal benefit" rate G t is another factor that may influence the fair value of the products. This study assumes that the policyholder withdraws the random amount G t that has been selected from the interval (0, βw x 0 ) at the beginning of year t. Since increasing the "withdrawal benefit" rate G t increases the benefits that have been received by the policyholder directly, it is expected that when β is increased, the fair value of both products will be increased too. Suppose a policyholder who buys the products at age 60 with the limitation L = 10 days. Table 2 shows the effect of the coefficient β in "withdrawal benefit" on the fair value of products and verifies the above anticipation.
As mentioned earlier, δ% represents the investment rate in a arbitrage free investment fund and (1 − δ)% represents the investment rate in a risk-free investment fund with the interest rate i. It is obvious that the fair value of both products is influenced by the value of δ. Figure 2 shows that the return rate under the GBM process is greater than the risk-free investment fund. Therefore, it is expected that fair value of both products will be increased when δ is increased. Table 3 shows the effect of δ on the fair value of our products.
The current study assumes that (1 − δ)% of the account value is invested with a fixed interest rate.  Thus, it is obvious that the constant interest rate i effective on the fair value of the products too.
In Table 4, the effect of i on fair value of these two products has been analyzed. In this table, i varies from 3% to 18% and policyholder's entrance age is 60. Table 4 illustrates that as the interest rate i is increased, the fair value of both products will be increased too.
Up to now, the inflation rate for hospitalization cost over time has been assumed equal to zero (e.g. φ = 0). Now, the effect of the two inflation rates π and φ on fair value of both products is analyzed.
Consider a policyholder who buys the products at age 60 with the limitation L = 10 days. Table   5 shows such effect. As expected, the fair value of products has directly been influenced by two inflation rates π and φ. This finding can be interpreted by the fact that increasing such inflation rates will directly increase the benefits that have been paid to the policyholder and consequently fair value of both products is increased.

Conclusion and suggestions
The present study proposed and evaluated two new products, i.e. the variable annuity products The next study has assumed that the age of the patient and the time he/she refers to receive medical services are effective on the average health costs and has suggested an stochastic model for the expected health expenses. Hence, given this model, health expenses can be estimated more accurately and health insurance products will be designed more precisely.

Compliance with Ethical Standards
Funding: There is no financial support for the study.
Authors declare that they have no conflict of interest.
Ethical approval: This study does not contain any studies with human participants or animals performed by any of the authors.

Acknowledgements
We would like to thank referees whose helpful comments led to substantial improvements in this paper.