3.1. Dynamic programming approach

In order to solve (14), we implement a dynamic programming algorithm for discrete stochastic control problems (see, e.g., Bertsekas, Reference Bertsekas2005 and Seierstad, Reference Seierstad2009). For each $j=0,1,\ldots , n+1,$ and each value of policy account $W_j$ , benefit base $A_j$ and health status $z_j$ , we denote by $V_j(W_j, A_j, z_j)$ the contract value at time $j$ (before the periodic withdrawal) and by $v_j(W_j, A_j, z_j)$ the contract value at the same time if the policyholder is alive. Clearly $V_j(W_j, A_j, z_j)=\mathbf{1}_{\{\tau \gt j\}}v_j(W_j, A_j, z_j)$ and $V_{0}=V_0(P,P,1)=v_0(P,P,1)$ . Since the algorithm proceeds backward, we take $n+1$ as the starting point and define the following terminal condition:

(15) \begin{equation} v_{n+1}(W_{n+1},A_{n+1},z_{n+1})\equiv 0. \end{equation}

Note that the specific value of $v_{n+1}$ is irrelevant since it is multiplied by the survival indicator $\mathbf{1}_{\{\tau \gt n+1\}}$ , which is almost surely zero, to determine the contract value $V_{n+1}$ .

Then, to proceed backward, we define the Bellman recursive equation of the problem as follows:

(16) \begin{align} &v_j(W_j, A_j, z_j)=\sup _{y_j \in Y_j}\Big ( \mathbf{1}_{\{\lambda \gt j\}}\ C_j^{(h)}(y_j, A_j)\,+\, \mathbf{1}_{\{\lambda \leq \, j\}}\ C_j^{(i)}(y_j, A_j) \\&\quad +\mathbb{E}^Q\big [\mathbf{1}_{\{\tau \leq \, j+1\}}w(W_j,R_{j+1},y_j)\text{e}^{-r}|W_j,A_j,z_j,\tau \gt j\big ] \\&\, \,\,\,\,\,+\,\mathbb{E}^Q\big [ \mathbf{1}_{\{\tau \gt j+1\}}v_{j+1}\big(w(W_j,R_{j+1},y_j),a_{j+1}^{(h)}(W_j,A_j,y_j)\,\mathbf{1}_{\{\lambda \gt j\}}+a_{j+1}^{(i)}(W_j,A_j,y_j)\,\mathbf{1}_{\{\lambda \leq \, j\}}, z_{j+1}\big)\text{e}^{-r}\nonumber\\ &\qquad \qquad |W_j,A_j,z_j,\tau \gt j\big ]\Big ),\qquad j=n,n-1,\dots ,1. \end{align}

In the first expectation of (16), using the stochastic independence between financial and demographic factors, the definition of the policy account value in (8), and the martingale property $\mathbb{E}^Q[(1+R_{j+1})e^{-r}]=1$ , we obtain

(17) \begin{equation} \mathbb{E}^Q\big [\mathbf{1}_{\{\tau \leq \, j+1\}}w(W_j,R_{j+1},y_j)\text{e}^{-r}|W_j,A_j,z_j,\tau \gt j\big ]=\max \{W_j - y_j,\,0\}(1-\varphi )Q(\tau \leq \, j+1\, |\, z_j, \tau \gt j). \end{equation}

In the second expectation, we can factorize the conditional $Q$ -expectation inside, and after applying the tower property, we obtain

(18) \begin{equation} \begin{aligned} &\mathbb{E}^Q\big [ \mathbf{1}_{\{\tau \gt j+1\}}v_{j+1}\big(w(W_j,R_{j+1},y_j),a_{j+1}^{(h)}(W_j,A_j,y_j)\,\mathbf{1}_{\{\lambda \gt j\}}+a_{j+1}^{(i)}(W_j,A_j,y_j)\,\mathbf{1}_{\{\lambda \leq \, j\}}, z_{j+1}\big )\text{e}^{-r}\\[5pt] &\qquad \qquad |W_j,A_j,z_j,\tau \gt j\big ]\\[5pt] &= \mathbb{E}^Q\big[v_{j+1}\big(w(W_j,R_{j+1},y_j),a_{j+1}^{(h)}(W_j,A_j,y_j)\,\mathbf{1}_{\{\lambda \gt j\}}+a_{j+1}^{(i)}(W_j,A_j,y_j)\,\mathbf{1}_{\{\lambda \leq \, j\}}, 1\big)\text{e}^{-r}\\[5pt] &\qquad \qquad |W_j,A_j,z_j\big]\,Q(\tau \gt j+1,\, z_{j+1}=1\, |\, z_j, \tau \gt j)\\[5pt] &+ \mathbb{E}^Q\big [v_{j+1}\big(w(W_j,R_{j+1},y_j),a_{j+1}^{(h)}(W_j,A_j,y_j)\,\mathbf{1}_{\{\lambda \gt j\}}+a_{j+1}^{(i)}(W_j,A_j,y_j)\,\mathbf{1}_{\{\lambda \leq \, j\}}, 0\big)\text{e}^{-r}\\[5pt] &\qquad \qquad |W_j,A_j,z_j\big ]\,Q(\tau \gt j+1,\, z_{j+1}=0\, |\, z_j, \tau \gt j). \end{aligned} \end{equation}

Finally, the initial contract value is:

(19) \begin{equation} \begin{aligned} v_0(P,P,1)=q^{h}_{0}P(1-\varphi )&+\mathbb{E}^Q\big [v_1(P(1+R_1)(1-\varphi ),P,1)\,e^{-r}\big ]p_0^{h,h} \\&+\mathbb{E}^Q\big [v_1(P(1+R_1)(1-\varphi ),P,0)\,e^{-r}\big ]p_0^{h,i}, \end{aligned} \end{equation}

where $q^{h}_{0}$ , $p_0^{h,h}$ and $p_0^{h,i}$ are defined as in Section 3.

3.2. Bang-bang analysis

The recursive equation (16) requires, at each time step $j=n,n-1,\dots 1$ , to solve a real-valued optimization problem where the domain $Y_j^{(m)}$ (4) of the single policy $y_j$ depends both on the health status of the policyholder and the contract phase (accumulation or income phase). Even if withdrawals can take place only once a year, for, e.g., a 50-year-old policyholder this would imply around 65 time steps. Moreover, at each time step the problem must be solved for every possible triplet of state variables $(W_j,A_j,z_j)$ and therefore the computational effort could be substantial. A property that would drastically reduce this effort is the bang-bang condition, which states that the set of the optimal withdrawals consists of a finite number of choices. In what follows, we prove this condition without assuming convexity or monotonicity of the contract value function. In particular, in the accumulation phase if the policyholder is healthy, the set of optimal withdrawals consists of three choices only: maximum allowable purchase, zero withdrawal, complete surrender, i.e., the maximization can be restricted to the subset $\{-\pi A_j, 0,W_j\}$ of $Y_j^{(h)}$ . If instead she is disabled, the set of optimal withdrawals consists of four choices only: maximum allowable purchase, zero withdrawal, withdrawal at the contractual amount, complete surrender, namely the subset $\{-\pi A_j, 0, g^{(i)} A_j, W_j\}$ of $Y_j^{(i)}$ . In the income phase, the set of optimal withdrawals again consists of three choices only: zero withdrawal, withdrawal at the contractual amount, complete surrender, and hence the maximization can be restricted to the subset $\{ 0, g^{(m)} A_j, W_j\}$ , $m=h, i$ .

The economic intuition behind this condition can be explained as follows according to the contract structure. In the accumulation phase, for the healthy policyholder could be convenient to purchase $y_j=-\pi A_j$ to maximize the benefit base (and thus future guaranteed withdrawals during the income phase), especially if her expected residual lifetime is sufficiently long. However, it is not advantageous to purchase an amount $y_j$ with $-\pi A_j\lt y_j\lt 0$ since the benefit base increase is not maximized. Also, the policyholder loses the advantage of investing her money outside the policy, potentially in a risk-free asset yielding more than the policy account (due to the insurance fee). Hence, the optimal strategy would be no-withdrawal ( $y_j=0$ ). It is possible to withdraw an amount $y_i\gt 0$ but it is subject to the withdrawal penalty $k_j y_j$ (see footnote 2) as well as to a second penalization due to the reduction of the benefit base according to the pro-rata rule, since the policyholder loses the roll-up incentive. This suggests that withdrawing the entire account value $W_j$ , i.e., surrendering the contract, may be more convenient.

For the disabled policyholder in the accumulation phase, the same arguments apply to $y_j=-\pi A_j$ and $y_j=0$ . If $0\lt y_j\lt g^{(i)}A_j$ , the policyholder loses the roll-up incentive without taking full advantage of the guarantee: if she does not need the entire amount $g^{(i)}A_j$ and prefers to benefit from the investment of what exceeds her needs, it may be preferable to invest this money in a risk-free asset, which, as just observed, yields more than the policy account value. On the other hand, if $W_j\gt g^{(i)}A_j$ , it is possible to withdraw an amount $y_j\gt g^{(i)}A_j$ , but in this case the withdrawal penalty $k_j(y_j-g^{(i)}A_j)$ is applied. Moreover, if $g^{(i)}A_j\lt y_j\lt W_j$ , there is a second penalization due to the pro-rata adjustment rule. Therefore, it would be more convenient to withdraw the entire account value $W_j$ .

In the income phase, we come to similar conclusions as above and thus the set of optimal withdrawals for the healthy policyholder is $\{ 0, g^{(h)} A_j, W_j\}$ , while for the disabled policyholder is $\{ 0, g^{(i)} A_j, W_j\}$ .

The following result states that this intuition is right.

3.3. Algorithm

We outline the algorithm employed to value the GLWB variable annuity with the LTC option. The execution of the algorithm requires, first, a discretization over the state variables in order to select grid points; secondly, the interpolation of the value function $v$ , for each withdrawal date, over such a grid in order to compute the expectations in (18) and (19).

Step 0. For each $j = 1, \dots ,n+1$ discretize the state space $[0, \infty )$ for $W_j$ and $(0, \infty )$ for $A_j$ into a set of equidistant points to generate a regular grid:

\begin{equation*} \begin{aligned} &\mathbb{W} = \{w_0, \dots , w_L\},\quad \;\;0 = w_0 \lt w_1 \lt \dots \lt w_L,\\ &\mathbb{A}\; = \{a_0, \dots , a_K\},\quad \;\;\;0 \lt a_0 \lt a_1 \lt \dots \lt a_K, \end{aligned} \end{equation*}

where $\Delta _W$ and $\Delta _A$ denote the spacing between consecutive points in $\mathbb{W}$ and $\mathbb{A}$ , while $L$ and $K$ are the total number of grid points, respectively.

Step 1. Start at $n+1$ by setting $v_{n+1}(w_{\ell }, a_k, z_{n+1}) = 0$ for each $(w_{\ell }, a_k, z_{n+1}) \in \mathbb{W} \times \mathbb{A}\times \{0, 1\}$ .

Step 2. Proceed backwards: for $j = n,n-1,\dots ,1$ :

(I) Bilinearly interpolate $\left (w_{\ell },a_k, v_{j+1}(w_{\ell },a_k,1)\right )$ and $\left (w_{\ell },a_k, v_{j+1}(w_{\ell },a_k,0)\right )$ for $\ell =0,1,\dots ,L$ and $k=0,1,\dots ,K$ , to construct the functions $\, \tilde{\kern-2pt v}_{j+1}(w, a, 1)$ and $\, \tilde{\kern-2pt v}_{j+1}(w, a, 0)$ for $w\geq 0$ , $a\gt 0$ ;

(II) for each $(w_{\ell },a_k) \in \mathbb{W} \times \mathbb{A}$ and considering the definition of $g_j^{(h)}$ given in (1), compute

\begin{equation*} \begin{aligned} v_j(w_{\ell },a_k,1)= &\sup _{y\in Y_j^{(h)}}\Bigg \{y-k_j^{(h)}\max \{y-g_j^{(h)}a_k,0\}+q_j^h\max \{w_{\ell }-y,0\}(1-\varphi )\\ &+ e^{-r}\bigg [{\left (\int _0^\infty \, \tilde{\kern-2pt v}_{j+1}(\, \tilde{\kern-2pt w}\text{e}^{\delta },\, \tilde{\kern-2pt a}^{(h)},1)\,f_1(\delta )\, \text{d}\delta \right )}p_j^{h,h}\\ &+ \left (\int _0^\infty \, \tilde{\kern-2pt v}_{j+1}(\, \tilde{\kern-2pt w}\text{e}^{\delta },\, \tilde{\kern-2pt a}^{(h)},0)\,f_1(\delta )\, \text{d}\delta \right )p_j^{h,i}\bigg ] \Bigg \},\\ v_j(w_{\ell },a_k,0)= &\sup _{y\in Y_j^{(i)}}\Bigg \{y-k_j^{(i)}\max \{y-g^{(i)}a_k,0\}+q_j^i\max \{w_{\ell }-y,0\}(1-\varphi )\\ &+ e^{-r}\left (\int _0^\infty \, \tilde{\kern-2pt v}_{j+1}(\, \tilde{\kern-2pt w}\text{e}^{\delta },\, \tilde{\kern-2pt a}^{(i)},0)\,f_1(\delta )\, \text{d}\delta \right )p_j^{i,i} \Bigg \}, \end{aligned} \end{equation*}

where $\, \tilde{\kern-2pt w}=\text{e}^{r+d}\max \{w_{\ell }-y,0\}(1-\varphi )$ with $d$ the drift adjustment (12), $f_1$ is the density of the Lévy process given by Equation (13) and $\, \tilde{\kern-2pt a}^{(m)}=a_{j+1}^{(m)}(w_{\ell }, a_k, y)$ , $m=h, i$ .

Step 3. The value of the contract at inception is

\begin{equation*} \begin{aligned} v_0(P,P,1) &= q_0^h P(1-\varphi )+e^{-r}\Bigg [{\left ( \int _0^\infty \, \tilde{\kern-2pt v}_1\left (P\text{e}^{r+d+\delta }(1-\varphi ),P,1\right )\,f_1(\delta )\, \text{d}\delta \right )}p_0^{h,h}\\ &+\left (\int _0^\infty \, \tilde{\kern-2pt v}_1\left (P\text{e}^{r+d+\delta }(1-\varphi ),P,0\right )\,f_1(\delta )\, \text{d}\delta \right )p_0^{h,i}\Bigg ]. \end{aligned} \end{equation*}

Note that the density $f_1$ can be calculated through inversion of the characteristic function (11) (see Bailey & Swarztrauber, Reference Bailey and Swarztrauber1994) and the constant $d$ via numerical integration. Note that a regular grid is preferred for bilinear interpolation because it ensures fast, stable, and accurate computations. The uniform spacing allows for direct indexing, efficient memory access, and guarantees that interpolation is performed over well-defined rectangular cells. This structure minimizes computational overhead and avoids geometric ambiguities.

4.1. Transition and death probabilities

To determine the transition and death probabilities, we consider the study of Baione et al. (Reference Baione, Conforti, Levantesi, Menzietti and Tripodi2016) concerning the estimation of the biometric model for the LTC insurance. More in detail, the study exploits health-related data provided by the Italian National Institute of Social Security (INPS) to fit a three-state (healthy, disabled, dead), continuous time Markov model. The dataset refers to the period $2002-2012$ , covers ages 20–95, and includes the following detailed information, classified by gender, age, and calendar year:

• • Number of disabled,

• • Deaths among disabled,

• • Inception on disabled state.

For each calendar year, the estimated transition and death probabilities are graduated aiming to extrapolate probabilities for the age range 96–120. Finally, the authors apply the Lee-Carter model with the extension of Brouhns et al. (Reference Brouhns, Denuit and Vermunt2002) to obtain, over the calendar years $2013-2043$ , projected mortality tables for healthy and disabled individuals, as well as the table of transition probabilities from healthy to disabled status.

For the purposes of the present work, we adopt the projected transition and death probabilities stemming from the study of Baione et al. (Reference Baione, Conforti, Levantesi, Menzietti and Tripodi2016). As a result, in our model we take $\omega =120$ , and we consider the cohort of the Italian population aged $x =50$ in $2013$ , following that $n=70$ .

4.2. Financial model

As already mentioned in Section 3, for the reference price we assume an exponential Lévy process. Within this class, we adopt the Variance-Gamma (VG) specification. This choice is motivated by the need to illustrate the valuation framework under a return process that departs from the standard geometric Brownian motion (GBM) while remaining analytically tractable. In particular, the VG model allows for non-Gaussian return distributions while admitting a closed-form characteristic function, which makes it convenient for numerical valuation and simulation of insurance contracts with embedded guarantees. Since the focus of the paper is on the valuation structure of the contract rather than on the empirical modeling of financial returns, this specification should mainly be interpreted as illustrative. The VG model is defined through the characteristic function

(20) \begin{equation} \Phi _t(u)= \exp \left (-\frac {t}{v}\ln \left (1-iu\mu v +\frac {1}{2}\sigma ^2 u^2 v\right )\right ), \end{equation}

with $\mu \in \mathbb{R}$ , $\sigma ,\, v\gt 0$ . The process is a pure-jump Lévy process with infinite activity and paths of finite variation, first introduced by Madan and Seneta (Reference Madan and Seneta1990) and further developed in Madan et al. (Reference Madan, Carr and Chang1998). It allows to control skewness and kurtosis of the return distribution and to correct some biases in option pricing implied by the Black and Scholes model (Black & Scholes, Reference Black and Scholes1973). The parameters of the process used are the same as in Kirkby and Nguyen (Reference Kirkby and Nguyen2021), namely $\sigma = 0.2$ , $v = 0.85$ , and $\mu =0$ , which were calibrated using S&P 500 market data. Accordingly, for the interest rate $r$ , unless otherwise specified, we set a baseline level of $3.5\%$ , consistent with the term structure of risk-free interest rates in the U.S. market for very long maturities.Footnote ⁴ This ensures consistency with the market used for the calibration of the VG process and provides a representative benchmark for long-term risk-free rates.

4.3. Contract parameters

The contract parameters in the basic case are summarized in Table 1.

Table 1.

Parameters of GLWB annuity with an accumulation phase and LTC option

Table 1. Long description

The table presents contract parameters for a guaranteed lifetime withdrawal benefit annuity with an accumulation phase and long-term care option. It has six rows and three columns. The columns are labeled ‘Contract parameter’, ‘Notation’, and ‘Value’. The rows are as follows: Row 1: Contract parameter, Bonus rate; Notation, b subscript j superscript h equals b subscript i superscript h; Value, 5 percent. Row 2: Contract parameter, Penalty rate; Notation, k subscript j superscript h equals k subscript i superscript h; Value, 3 percent. Row 3: Contract parameter, Withdrawal rate (healthy); Notation, g subscript j superscript h; Value, 3 percent. Row 4: Contract parameter, Withdrawal rate (disabled); Notation, g subscript j superscript r equals 2 times g subscript j superscript h; Value, 6 percent. Row 5: Contract parameter, Fee rate; Notation, phi; Value, 2.32 percent. Row 6: Contract parameter, Withdrawal dates; Notation, empty; Value, yearly.

The choice of the withdrawal rate $g^{(h)}=3\%$ is motivated by the fact that, with a (constant) interest rate of $3.5\%$ and the survival probabilities of healthy people, the (actuarial) value of a whole-life annuity for a healthy man until death is $19.93$ , so that its reciprocal, which would be the conversion rate in a standard lifelong annuity, is $5.02\%$ . While the (actuarial) value of a whole-life annuity for a healthy female until death is $20.97$ , so that its reciprocal is $4.77\%$ . However, buying such annuities, while being protected against the longevity risk (as in the case of the GLWB rider), completely loses the property of the investment fund. Then, if the policyholder dies during the first years of the contract, his/her heirs do not receive anything while, in our case, the whole account value remains to them. Moreover, within the annuities mentioned above, surrender is never admitted in order to avoid adverse selection. That is the reason why $g^{(h)}$ that we take as base case must be lower than the conversion rate of both annuities. To support the healthcare costs of a disabled policyholder, we assume to double her withdrawal rate. As far as the roll-up rate is concerned, it was chosen high enough in order to disincentive withdrawals regardless of health status, while concerning the level of the fee, our calculations show that, with the basic parameters here considered, the fair fee rate for a male should be equal to $1.94\%$ and the one for a female should be equal to $2.7\%$ . We have chosen a base level for $\varphi =2.32\%$ as the average of these two values. This implies that the contract is overpriced (i.e., $V_0\lt P$ ) for male policyholders while it is underpriced (i.e., $V_0\gt P$ ) for female policyholders.

4.4. Numerical accuracy analysis

To assess the numerical accuracy of our algorithm and validate the results presented below, we conduct the following analysis using the transition and death probabilities for females as the baseline. Numerical calculations require imposing upper bounds on $W_j$ and $A_j$ that are sufficiently large to ensure a negligible probability of exceeding them. To select appropriate bounds, we compute the initial contract value for various upper bounds, denoted as $a_K$ and $w_L$ , as shown in Table 2. Grid parameters are uniformly spaced with $W_j$ at unit intervals (2 $W_0$ -7 $W_0$ ) and $A_j$ at 0.5-unit intervals (1.5 $A_1$ -4 $A_1$ ).

Table 2.

Initial values under different upper bounds for $A_j$ and $W_j$

Table 2. Long description

A table comparing initial values under different upper bounds for alpha and W_L. The table has 7 columns and 6 rows, including headers. Column headers are alpha vs W_L, 2W_0, 3W_0, 4W_0, 5W_0, 6W_0, and 7W_0. Row labels are 1.5 alpha 1, 2 alpha 1, 2.5 alpha 1, 3 alpha 1, 3.5 alpha 1, and 4 alpha 1. Row 1: 1.5 alpha 1, 100.8753, 100.5665, 100.4946, 100.4855, 100.4791, 100.4758. Row 2: 2 alpha 1, 101.6528, 101.3424, 101.2621, 101.2504, 101.2431, 101.2381. Row 3: 2.5 alpha 1, 101.7505, 101.4904, 101.4107, 101.4028, 101.3950, 101.3907. Row 4: 3 alpha 1, 101.7508, 101.5092, 101.4297, 101.4208, 101.4141, 101.4098. Row 5: 3.5 alpha 1, 101.7508, 101.5114, 101.4319, 101.4229, 101.4163, 101.4120. Row 6: 4 alpha 1, 101.7508, 101.5115, 101.4321, 101.4231, 101.4165, 101.4122.

With $a_K$ fixed, the initial contract value decreases as $w_L$ increases. Conversely, for fixed $w_L$ , the contract value increases with $a_K$ . When both parameters increase simultaneously, the diagonal entries in Table 2 show that the contract value increases until $a_K=3A_1$ and $w_L=5W_0$ , then begins to decline. This pattern indicates that the benefit base $A_j$ reaches its upper bound before $3.5A_1$ , so that further increases in $W_j$ start to dominate and reduce the contract value. To prevent this reduction, we therefore set $w_L=6W_0$ and $a_K=3.5A_1$ as upper bounds, where both $W_0$ and $A_1$ equal $P=100$ .

Given the chosen upper bounds, we explore the convergence of initial contract values for different grid spacings $\Delta _W$ and $\Delta _A$ ; results are shown in Table 3. As both parameters decrease, the initial values and their volatility decrease accordingly. These values consistently exceed $P=100$ because, given the average fair fee (as explained in Section 4.3), the contract is underpriced for female policyholders. When both $\Delta _W$ and $\Delta _A$ fall below 2.5, value variations become marginal, while computational time increases substantially (Table 4). Therefore, we set both $\Delta _W$ and $\Delta _A$ to 2.5, yielding $L=w_L/\Delta _W=240$ and $K=a_K/\Delta _A=140$ grid points for subsequent numerical examples.

Table 3.

Initial values under different spacing $\Delta _A$ and $\Delta _W$

Table 3. Long description

A table with five columns and five rows, showing initial contract values for different grid spacings and parameters. The columns are labeled with different values of ΔW (10, 7.5, 5, 2.5, 1), and the rows are labeled with different values of ΔA (10, 7.5, 5, 2.5, 1). Each cell in the table contains a numerical value representing the initial contract value for the corresponding grid spacing and parameter. The values decrease as both ΔW and ΔA decrease. Notable trends include the marginal variations in values when both ΔW and ΔA fall below 2.5.

Table 4.

Mean computational time (in milliseconds), averaged over 10 independent iterations, required for the calculation of the initial values reported in Table 3 for different $\Delta _A$ and $\Delta _W$

Table 4. Long description

A table comparing mean computational time in milliseconds for different grid spacings and values. The table has 5 rows and 5 columns. Column headers are Time in ms, ΔW = 7.5, ΔW = 5, ΔW = 2.5, and ΔW = 1. Row labels are ΔA = 10, ΔA = 7.5, ΔA = 5, ΔA = 2.5, and ΔA = 1. Row 1: 2362, 3138, 4699, 9361, 23424. Row 2: 3097, 4097, 6132, 12229, 30591. Row 3: 4667, 6203, 9250, 18478, 46217. Row 4: 9259, 12317, 18420, 36782, 92405. Row 5: 23147, 30773, 46023, 92258, 257137.

4.5. Sensitivity analysis with respect to the contractual parameters and interest rates

In this subsection, we focus on the sensitivity of the initial value of the GLWB variable annuity with accumulation phase to relevant contractual parameters and interest rate. In particular, we analyze the product both with and without the LTC option for male and female policyholders. Note that in the case without the LTC option, it is sufficient to set the same withdrawal rate. Figure 1 shows the variation in the initial values of the two types of annuity held by male and female policyholders as the roll-up rate changes. Note that for low levels of roll-up rate there is a slight level of growth in the initial values of the two annuities, while once the roll-up rate exceeds a certain threshold, the initial values exhibit a significant increase, particularly in the case of the annuity with the LTC option. In addition, the initial value of annuities held by female policyholders exhibits greater sensitivity to variations in the roll-up rate and can generate higher benefits from an increase in the roll-up rate compared to those held by male policyholders. This phenomenon can be attributed to the fact that women are both more likely to remain healthy and have a longer expected remaining life span than men. This effect can be quantified using the available transition probabilities, from which we obtain that the probability of remaining healthy up to age 120 is $0.31\%$ for females and $0.0024\%$ for males, corresponding to a gap of $0.3076\%$ . Consistently, over the considered roll-up range, the initial contract value including the LTC option increases by $15.27\%$ for males and $24.48\%$ for females. This indicates that females exhibit a relative sensitivity to roll-up variations that is approximately $60\%$ higher than that of males. In Figure 2, we analyze the initial annuity values for male and female policyholders across both annuity types, assessing their sensitivity to changes in the guaranteed withdrawal rate. The latter varies from $0$ to $7\%$ but, as noted in Section 4.3, $g$ -levels close to $5.02\%$ for healthy male policyholders and to $4.77\%$ for healthy female policyholders would be completely impractical, as they produce extremely high contract values. As expected, the initial contract values of the two types of annuities strongly increase with $g$ . In particular, for $g\leq 2\%$ such values have an almost constant trend and it is more convenient to surrender the contracts than to keep them. Conversely, with higher withdrawal rates, the initial contract values increase and decrease the incentive for policyholders to leave the contract so as to benefit from high guaranteed withdrawals. Moreover, similar to the roll-up rate, the initial value of annuities held by female policyholders is more sensitive to changes in the withdrawal rate than those held by male policyholders. Figure 3 highlights the influence of the insurance fee rate on the initial values of the two types of annuity maintaining fixed all the remaining basic parameters. From this figure, one can capture at first glance the decrease of the initial values of the two annuities with respect to $\varphi$ as well as the differences among the contract values, which are higher for female policyholders both with and without LTC option. Finally, note that the fair fee rate $\varphi ^*$ is given by the abscissa of the intersection between the contract value graphs and the horizontal line at level $P = 100$ : in the absence of LTC option, the fair fee rate is $\varphi ^* = 1.71\%$ for male policyholders and $\varphi ^* = 2.28\%$ for female policyholders. When the LTC option is included, the fair fee rate increases to $\varphi ^* = 1.94\%$ for males and $\varphi ^* = 2.7\%$ for females. In Figure 4, we illustrate how the initial contract values of the two types of annuity vary with respect to the interest rate $r$ . Note that $r$ is a key parameter which has a very strong effect on the contract values. As expected, we observe a general decreasing of the contract values in both annuities, but with greater impact in GLWB annuities that include the LTC option.

Figure 1.

Initial contract value without and with the LTC option for different roll-up rates.

Figure 2.

Initial contract value without and with the LTC option for different withdrawal rates.

Figure 3.

Initial contract value without and with the LTC option for different fee rates.

Figure 4.

Initial contract value without and with the LTC option for different interest rates.

4.6. Optimal withdrawal strategies for different contractual parameters

This subsection explores the optimal withdrawal behavior of male and female policyholders holding GLWB variable annuities with accumulation phase and LTC option. As already noted in Section 3, a withdrawal strategy that maximizes the solution to the optimization problem (14) does not necessarily correspond to the optimization of the policyholder’s preferences. Rather, it represents the adverse withdrawal strategy from the insurer’s viewpoint, thus serving as an intermediate step in the pricing process. Nevertheless, understanding the policyholders’ implied optimal withdrawal behavior is crucial for providing valuable insights to practitioners and researchers. That is why we show the optimal withdrawal strategies of policyholders at different points in time. More in detail, in the following graphs, we highlight the optimal withdrawal at time $j$ , i.e., $y_j$ , as a function of both the account value $W_j$ ( $y$ -axis) and the benefit base $A_j$ ( $x$ -axis) for the basic set of contractual parameters, investigating the impact of gender and assuming $z_j=1$ , that is, the policyholder is healthy. Subsequently, we analyze the impact of variations in the bonus rate and the guaranteed withdrawal rate on withdrawal behavior. The associated graphical results are omitted here for brevity and reported in Section 2 of the Supplementary Material. The case $z_j=0$ is shown in Section 3 of the Supplementary Material for the basic contract parameters listed in Table 1. Our analysis shows that disabled policyholders will always choose guaranteed withdrawal or surrender as their optimal withdrawal strategy, regardless of whether they are in the accumulation or income phase. For this reason, we consider it more interesting to investigate the case in which the insured is healthy at the various maturities considered.

Figure 5.

Optimal withdrawal strategies, in the base case, of a healthy male policyholder (upper) and female policyholder (lower) for $j=1, 5, 15$ . The value $1$ denote the additional purchase, $2$ the no-withdrawal strategy, and $4$ the surrender.

Figure 5 Long description

Panel A: A contour plot with j = 1. The x-axis is labeled A with a range from 0 to 350. The y-axis is labeled W with a range from 0 to 600. The plot shows a gradient of colors from yellow to blue, indicating different levels of density. Panel B: A contour plot with j = 5. The x-axis is labeled A with a range from 0 to 350. The y-axis is labeled W with a range from 0 to 600. The plot shows a similar gradient of colors from yellow to blue, indicating different levels of density. Panel C: A contour plot with j = 15. The x-axis is labeled A with a range from 0 to 350. The y-axis is labeled W with a range from 0 to 600. The plot shows a similar gradient of colors from yellow to blue, indicating different levels of density. Panel D: A contour plot with j = 1. The x-axis is labeled A with a range from 0 to 350. The y-axis is labeled W with a range from 0 to 600. The plot shows a gradient of colors from yellow to blue, with a distinct dark blue region near the lower left corner. Panel E: A contour plot with j = 5. The x-axis is labeled A with a range from 0 to 350. The y-axis is labeled W with a range from 0 to 600. The plot shows a similar gradient of colors from yellow to blue, with a distinct dark blue region near the lower left corner. Panel F: A contour plot with j = 15. The x-axis is labeled A with a range from 0 to 350. The y-axis is labeled W with a range from 0 to 600. The plot shows a similar gradient of colors from yellow to blue, with a distinct dark blue region near the lower left corner.

The results concerning the basic set of contractual parameters are reported in Figures 5 and 6.

Figure 5 shows that when $W_j\geq 100$ and the benefit base is relatively low, surrender is optimal for both male (upper panels) and female (lower panels) policyholders: an elevated policy account value enables immediate and substantial liquidity upon surrender, whereas a lower benefit base reduces the attractiveness of the long-term guarantee provided by the contract, incentivising the policyholder to exit the contract immediately. Conversely, when $W_j\lt 100$ and for high values of $A_j$ , it is always convenient to no-withdraw both for males and females. Interestingly, if $40\leq W_j\leq 180$ and $40\leq A_j\leq 160$ at $j = 1$ , females optimally choose to make additional purchases, whereas males never do so. This difference arises because, under the chosen basic parameters, male policyholders pay an insurance fee above their actuarially fair one, making additional purchases economically disadvantageous due to the disproportionately higher charges on these incremental amounts. Conversely, female policyholders pay a fee below their actuarially fair one, rendering additional purchases financially attractive in the early years of the contract. Moreover, at $j = 5$ , the region associated with optimal additional purchases completely vanishes for female policyholders as well. A possible explanation for this phenomenon is that over time, the advantage from the roll-up rate becomes relatively less attractive compared to the prevailing interest rate environment. Additionally, the fees charged on the growing account value (inclusive of these purchases) might progressively erode their perceived net benefit, further discouraging additional payments. Thus, the combination of diminishing roll-up attractiveness and ongoing fee charges jointly explains why the economic incentive for females to make additional purchases ultimately vanishes by year five. Finally, when analyzing the optimal withdrawal strategy over time, we further notice that as time progresses, the surrender region expands more significantly for males than for females. This occurs because females have longer life expectancies and thus stand to gain greater cumulative benefits from continuing the contract into future periods. Let us now focus on the Figure 6. For high values of the personal account, it is convenient to surrender the contract, since in these cases, the GLWB guarantee is deeply out of the money and the probability of this rider being triggered before the policyholder’s death is extremely low. On the other hand, when $W_j$ approaches $180$ and the benefit base exceeds $200$ for males or $160$ for females, the most convenient choice, at least for $j=16, 20$ , is not to withdraw anything in order to take advantage of the roll-up feature by increasing the benefit base. This implies a higher LTC payment in case of disability at the subsequent anniversaries and/or a higher withdrawal of the guaranteed amount. For relatively low levels of the account value and for benefit base values greater or equal to $100$ , it is always optimal to withdraw the guaranteed amount $g A_j$ , allowing the policyholder to secure a stable income even when the account value is low. As time goes on, the surrender zone expands to the detriment of the no-withdrawal zone that shrinks until disappearing completely at $j=30$ . To complete the analysis of the base case, we notice that, as expected, the guaranteed amount withdrawal zone increases as the policyholders grow old.

Figure 6.

Optimal withdrawal strategies, in the base case, of a healthy male policyholder (upper) and female policyholder (lower) for $j=16, 20, 30$ . The value $2$ denotes the no-withdrawal strategy, $3$ the guaranteed withdrawal, and $4$ the surrender.

We now focus on the effect of the roll-up rate on withdrawal behavior. In particular, when $b=4.25\%$ (i.e., $-15\%$ with respect to the base case), during the accumulation phase, we observe a complete absence of the purchase zone already at $j=1$ for both male and female policyholders. Moreover, there is a slight reduction of the no-withdrawal zone, resulting in an expanded surrender zone. During the income phase, we note that for male policyholders, the no-withdrawal zone completely disappears (with respect to the base case) in favor of the guaranteed amount withdrawal zone, while the surrender zone remains (almost) the same as in the base case. Also, for female policyholders there is a reduction of the no-withdrawal region, particularly pronounced in the transition between $j=16$ and $j=20$ . Such a sharp decrease, driven by the lower roll-up rate making no-withdrawal less attractive over time, leads to an enlargement of the guaranteed amount withdrawal region, while the surrender region again remains largely unchanged.

When $b = 5.75\%$ (i.e., $+15\%$ with respect to the base case), during the accumulation phase we observe a general reduction of the surrender zone in favor of both purchase and no-withdrawal zones. In particular, at $j=1$ , one can see a major use of the purchase strategy by females, partially absorbing the no-withdrawal area, as well as the use of such a strategy by males in contrast to the base case. Interestingly, for this higher roll-up rate, the economic incentive to make additional purchases remains at $j = 5$ for female policyholders, in contrast to the base case where this incentive disappears entirely by year five. This clearly illustrates the critical impact of the roll-up rate in maintaining policyholder incentives to invest additional funds over time. Conversely, even with the higher roll-up rate, the purchase region disappears for males by year five, reflecting their higher than-actuarially-fair fee and thus a weaker incentive for continued purchases. Finally, at $j=15$ , only the surrender and non-withdrawal zones coexist, with the former region expanding at the expense of the latter. During the income phase, we note that the surrender zone reduces a bit from the base case, while, as expected, the no-withdrawal zone expands, absorbing part of the guaranteed-withdrawal area both for males and females at $j=16$ and $j=20$ . The surrender zone at $j=30$ remains (almost) the same as in the base scenario.

We now illustrate the impact of variations in the withdrawal rate $g$ on the optimal withdrawal strategy. In particular, reducing the withdrawal rate to $g=2.55\%$ (i.e., $-15\%$ with respect to the base case) leads, during the accumulation phase, to an increase of the surrender on the no-withdrawal zone both for male and female policyholders and, for the latter, also the total lack of the purchase area. That increase, also visible during the income phase, spills over more or less evenly on both the no-withdrawal zone and the guaranteed-withdrawal zone when $j=16$ and $j=20$ , while it is to the disadvantage of the guaranteed-withdrawal zone when $j=30$ . Symmetrically, when $g=3.45\%$ (i.e., $+15\%$ with respect to the base case), we highlight a decrease of the surrender zone to the advantage of the purchase zone for males at $j=1$ and for females at both $j=1$ and $j=5$ . Furthermore, at $j = 5$ , male policyholders experience an increase in the no-withdrawal region (with respect to the base case), a trend also observable for both genders at $j = 15$ . The reduction of the surrender zone, still observable during the income phase, leads to more or less similar growth of both the non-withdrawal area and the guaranteed withdrawal area when $j=16$ and $j=20$ , while it implies an increase in the guaranteed withdrawal area alone when $j=30$ . Note that this increase/decrease effect on surrender is not solely attributable to variation in the contractual parameter $g$ but also to the fact that for low $g$ ’s the contract is overpriced not only for males, since we are maintaining fixed the fee rate at the level of $2.32\%$ , but also for females. In contrast, for high $g$ ’s the contract is underpriced.

4.7. Sensitivity analysis with respect to the financial model

In this section, we examine both the sensitivity of the initial value of the GLWB variable annuity with accumulation phase and LTC option, and the optimal withdrawal strategies for a healthy policyholder (previously derived under baseline contractual parameters) to alternative specifications of the underlying asset dynamics. In addition to the VG process, we also consider three alternative exponential Lévy processes, the GBM, the Merton’s Jump Diffusion (MJD) and the Carr, Geman, Madan, Yor (CGMY), with their respective parameters as given in Kirkby and Nguyen (Reference Kirkby and Nguyen2021). Their characteristic functions are respectively given by

1. 1. Geometric Brownian Motion (GBM)

   \begin{equation*}\Phi _t(u)=\exp \left (iurt-\frac {1}{2}\,\sigma ^2u^2t\right)\!, \qquad \sigma \gt 0. \end{equation*}

2. 2. Merton’s Jump Diffusion (MJD)

   \begin{equation*}\Phi _t(u)=\exp \left (iurt-\frac {1}{2}\,\sigma ^2u^2t+\lambda _{J}t\left (e^{iu\mu _{J}-\frac {1}{2}\sigma _{J}^2 u^2}-1\right )\right)\!,\end{equation*}
   with $\mu _J \in \mathbb{R}$ , $\sigma ,\, \sigma _{J},\, \lambda _J\gt 0.$

3. 3. Carr, Geman, Madan, Yor (CGMY)

   \begin{equation*}\Phi _t(u) = \exp \left ( C \, \Gamma (\!-{\mathcal Y})t \left [(M - iu)^{\mathcal Y} - M^{\mathcal Y} + (G + iu)^{\mathcal Y} - G^{\mathcal Y}\right ]\right) \!,\end{equation*}
   with $G,\,M\geq 0$ , $C \gt 0$ , ${\mathcal Y} \lt 2$ and $\Gamma$ is the gamma function.

In Table 5, we report the parameters used for the four different processes and the corresponding results in terms of initial contract value for male and female policyholders under the basic set of contractual parameters. As we can see, the ordering of the contract values across the different asset models can be explained by the different degree of return dispersion and downside risk implied by the calibrated processes. Since the variable annuity embeds convex payoff components (withdrawal guarantees and surrender options), its value increases with the probability of large drawdowns of the fund value. Among the considered models, the Variance Gamma process generates the highest level of variability and tail risk, thus leading to the largest contract value. The Merton jump–diffusion model, which incorporates negative jumps, produces a lower but still relatively high valuation. The GBM, driven only by continuous diffusion, implies a smaller probability of abrupt drawdowns and therefore a lower contract value. Finally, under the adopted calibration, the CGMY process exhibits comparatively milder downside risk, resulting in the lowest valuation. This naturally leads to the observed ordering $V_0^{CGMY}\lt V_0^{GBM}\lt V_0^{MJD}\lt V_0^{VG}$ , where the meaning of the symbols is clear from the context, for both male and female policyholders.

Table 5.

Contract value for different asset price models

Table 5. Long description

A table comparing contract values for different asset price models for male and female policyholders. The table has 10 rows and 5 columns. The columns are labeled Model, CGMY, GBM, MJD, and VG. The rows are labeled Male and Female, with additional rows for specific parameters under each model. For male policyholders, the parameters are as follows: CGMY: C = 0.02, G = 5, M = 15, Y = 1.2, V0 = 94.85; GBM: σ = 0.15, V0 = 94.97; MJD: σ = 0.12, λJ = 0.4, μJ = -0.12, V0 = 95.45; VG: σ = 0.2, μ = 0, v = 0.85, V0 = 98.66. For female policyholders, the parameters are as follows: CGMY: V0 = 94.87; GBM: V0 = 95.38; MJD: V0 = 96.18; VG: V0 = 101.35.

Table 6 reports the percentage changes in the use of optimal withdrawal strategies during the accumulation phase under alternative asset price dynamics (CGMY, GBM, and MJD), relative to the VG model, based on the parameter sets in Table 5. As we can observe moving away from the VG model leads to a reduction in the use of the no-withdrawal strategy and, symmetrically, to an increase in surrender decisions. This pattern is consistent with the ordering of contract values: since VG yields the highest valuation, it implies a larger continuation value, making it optimal to keep the contract in force and defer withdrawals. Under alternative models associated with lower contract values, the continuation value decreases, and policyholders optimally anticipate surrender. A similar argument applies to additional purchases, which are more attractive for female policyholders when the continuation value is higher.

Table 6.

Percentage changes in the use of optimal strategies during the accumulation phase under different asset pricing models (CGMY, GBM, and MJD), reported separately for male and female individuals. Each cell reports the optimal controls with associated percentage deviations in parentheses, relative to the baseline variance gamma process

Table 6. Long description

A table comparing percentage changes in optimal withdrawal strategies during the accumulation phase under different asset pricing models (CGMY, GBM, and MJD) for male and female individuals. The table has 15 rows and 6 columns. Column headers are labeled as CGMY, GBM, and MJD for both male and female individuals. Each cell reports the optimal controls with associated percentage deviations in parentheses, relative to the baseline variance gamma process. Row labels are numbered from 1 to 15. Row 1: Male: CGMY -πA(0.00%), GBM -πA(0.00%), MJD -πA(0.00%); Female: CGMY -πA(-2.15%), GBM -πA(-2.15%), MJD -πA(-2.15%). Row 2: Male: CGMY 0(-13.12%), GBM 0(-9.22%), MJD 0(-7.60%); Female: CGMY 0(-12.58%), GBM 0(-8.14%), MJD 0(-6.29%). Row 3: Male: CGMY Wj(13.12%), GBM Wj(9.22%), MJD Wj(7.60%); Female: CGMY Wj(14.73%), GBM Wj(10.30%), MJD Wj(8.44%). Row 5: Male: CGMY -πA(0.00%), GBM -πA(0.00%), MJD -πA(0.00%); Female: CGMY -πA(0.00%), GBM -πA(0.00%), MJD -πA(0.00%). Row 6: Male: CGMY 0(-11.83%), GBM 0(-8.30%), MJD 0(-6.82%); Female: CGMY 0(-13.25%), GBM 0(-9.25%), MJD 0(-7.57%). Row 7: Male: CGMY Wj(11.83%), GBM Wj(8.30%), MJD Wj(6.82%); Female: CGMY Wj(13.25%), GBM Wj(9.25%), MJD Wj(7.57%). Row 15: Male: CGMY -πA(0.00%), GBM -πA(0.00%), MJD -πA(0.00%); Female: CGMY -πA(0.00%), GBM -πA(0.00%), MJD -πA(0.00%). Row 16: Male: CGMY 0(-7.97%), GBM 0(-5.62%), MJD 0(-4.55%); Female: CGMY 0(-9.26%), GBM 0(-6.60%), MJD 0(-5.35%). Row 17: Male: CGMY Wj(7.97%), GBM Wj(5.62%), MJD Wj(4.55%); Female: CGMY Wj(9.26%), GBM Wj(6.60%), MJD Wj(5.35%).

Table 7, which refers to the income phase, shows a related but more nuanced pattern. While the reduction in the no-withdrawal strategy across models persists, the sensitivity of the continuation region to the underlying asset dynamics is reduced, as decisions are increasingly driven by the withdrawal at the contractual amount. In particular, surrender becomes more likely as one moves away from the VG benchmark, while the use of guaranteed withdrawals increases as one approaches it, consistent with the higher value of embedded guarantees. Overall, the results confirm that different asset dynamics affect not only the contract value but also the structure of optimal withdrawal decisions in a coherent way.

Table 7.

Percentage changes in the use of optimal strategies during the income phase under different asset pricing models (CGMY, GBM, and MJD), reported separately for male and female individuals. Each cell reports the optimal controls with associated percentage deviations, relative to the baseline VG process. As we are considering only healthy policyholders, to lighten the notation within the table, we will use $g$ instead of $g^{(h)}$

Table 7. Long description

The table presents percentage changes in the use of optimal strategies during the income phase under different asset pricing models (CGMY, GBM, and MJD) for male and female individuals. The table is divided into two main sections: one for males and one for females, each with data for different time points (16, 20, and 30). Each section contains three columns for the asset pricing models and rows for different strategies (0, gA, and W) with associated percentage deviations relative to the baseline VG process. For males at time point 16, the values are: CGMY: 0 (-12.01%), gA (4.47%), W (7.54%); GBM: 0 (-12.01%), gA (6.68%), W (5.33%); MJD: 0 (-11.97%), gA (7.66%), W (4.31%). For females at time point 16, the values are: CGMY: 0 (-16.74%), gA (7.90%), W (8.84%); GBM: 0 (-12.76%), gA (6.40%), W (6.36%); MJD: 0 (-9.84%), gA (4.69%), W (5.15%). Similar data is presented for time points 20 and 30 for both males and females. The table highlights how different asset dynamics affect the optimal withdrawal decisions.

4.8. Study of contract fee across different parameters

In this subsection, we analyze the impact of variations in the bonus rate and guaranteed withdrawal rate on the fair fee for our GLWB variable annuity with accumulation phase and LTC option. As shown in Tables 8 and 9, the fair fee is significantly more responsive to changes in the withdrawal rate than to variations in the roll-up rate. Moreover, these tables illustrate that fair fees for female policyholders are generally higher than those for male policyholders. Consequently, when employing a homogeneous fee structure for both genders, female policyholders may effectively contribute less than their fair share of the total expenses compared to male policyholders.

Table 8.

Fair fee for different roll-up rates $b$

Table 8. Long description

A table comparing fair fees for male and female policyholders under different roll-up rates. The table has 2 rows and 5 columns. The columns are labeled with different roll-up rates: b equals 0.0425 minus 15 percent, b equals 0.0475 minus 5 percent, b equals 0.05, b equals 0.0525 plus 5 percent, and b equals 0.0575 plus 15 percent. The rows are labeled Male and Female. Row 1: Male, 0.0141 minus 27.32 percent, 0.0174 minus 10.31 percent, 0.0194, 0.0216 plus 11.34 percent, 0.0268 plus 38.14 percent. Row 2: Female, 0.0195 minus 27.77 percent, 0.0242 minus 10.37 percent, 0.027, 0.0301 plus 11.48 percent, 0.0379 plus 40.37 percent.

Table 9.

Fair fee for different withdrawal rates $g$

Table 9. Long description

A table comparing fair fees for male and female policyholders under different withdrawal rates. The table has 2 rows and 5 columns. Column headers are Case, Male, and Female. Row labels are g = 0.0255 (- 15 percent), g = 0.0285 (- 5 percent), g = 0.03, g = 0.0315 (+ 5 percent), g = 0.0345 (+ 15 percent). Row 1: Male, 0.0121 (- 37.62 percent); Female, 0.0168 (- 37.77 percent). Row 2: Male, 0.0167 (- 13.91 percent); Female, 0.0232 (- 14.07 percent). Row 3: Male, 0.0194; Female, 0.027. Row 4: Male, 0.0224 (+ 15.46 percent); Female, 0.0313 (+ 15.92 percent). Row 5: Male, 0.0294 (+ 51.54 percent); Female, 0.0418 (+ 54.81 percent).