2.1 Assumptions

We consider an arbitrage-free and complete financial market. Moreover, we assume that there are risks that are not traded on the financial market, which we label as “actuarial” risks. These risks are assumed to be independent of the financial market. Such strong assumptions have already been imposed in some papers in the literature, such as Deelstra et al. (Reference Deelstra, Devolder, Gnameho and Hieber2020) and Belhouari et al. (Reference Belhouari, Deelstra and Devolder2025). The set of trading dates is $[0,T],$ where $T$ is the maturity. We consider a portfolio of $n$ insured individuals, all of age $x$ at time $0$ ; their remaining lifetimes are denoted by $(\tau _1, \tau _2, \ldots , \tau _n).$ We denote by $N_t$ the number of survivors up to time $t,$ formulated as $N_{t}=\sum _{j=1}^{n}\unicode {x1D7D9}_{\tau _j\gt t}$ . Furthermore, for each $t \in [0,T]$ , the remaining lifetimes of the individuals alive at time $t$ , denoted $(\tau _i)_{1 \le i \le N_t}$ , are assumed to be identically distributed. Assuming that actuarial and financial risks are independent, we consider the following filtered probability space

\begin{align*} (\Omega , \mathcal{F}, \mathbb{F}, \mathbb{P}). \end{align*}

The sample space is defined as the product $\Omega = \Omega ^a \times \Omega ^f$ , reflecting all the possible financial and mortality events. Here, $\Omega ^a$ denotes the actuarial space and $\Omega ^f$ denotes the financial space. The $\sigma$ -algebra $\mathcal{F}$ is defined as $\mathcal{F} = \mathcal{F}^a \lor \mathcal{F}^f$ , where $\mathcal{F}^a$ is the actuarial $\sigma$ -algebra and $\mathcal{F}^f$ is the financial $\sigma$ -algebra. The probability measure is $\mathbb{P} = \mathbb{P}^a \times \mathbb{P}^f$ , where $\mathbb{P}^a$ and $\mathbb{P}^f$ denote the real-world actuarial and financial probabilities, respectively. We also introduce the risk-neutral financial probability $\mathbb{Q}^{\,f}$ . The filtration $\mathbb{F} = (\mathcal{F}_t)_{t \ge 0}$ represents the information flow and is defined by $\mathcal{F}_t = \mathcal{F}_t^a \lor \mathcal{F}_t^f$ where $(\mathcal{F}_t^a)_{t \ge 0}$ is the actuarial filtration and $(\mathcal{F}_t^f)_{t \ge 0}$ is the financial filtration. The actuarial filtration $(\mathcal{F}_t^a)_{t \ge 0}$ is composed of the filtration $(\mathfrak{S}_{t})_{t \ge 0}$ generated by systematic mortality risk factors and the filtration $ (\bigvee _{i=1}^n \Im ^{i}_{t} )_{t \ge 0}$ encompassing information on unsystematic mortality risk. At any given time $t \lt T$ , for each individual $1 \le i \le n$ , we define $\Im ^i_{t} = \sigma (\unicode {x1D7D9}_{\tau _i \le u} \;:\; u \le t)$ . Thus, it follows that $\mathcal{F}^a_t = \mathfrak{S}_t \lor \left ( \bigvee _{i=1}^n \Im ^i_t \right )$ . Furthermore, we will assume that the lifetimes of the surviving insured at each instant $t$ display conditional independence, given the systematic mortality risk information at maturity $u$ .Footnote 1

For the actuarial instruments, stochastic mortality rates $(\lambda _{t})_{0\leq t \leq T}$ , adapted to the filtration $\mathfrak{S}$ will be used. Thus, the probability of an individual at time $t\le u$ to survive until maturity $u\le T$ isFootnote 2

(1) \begin{equation} _{u-t}p_{x+t}\overset {\Delta }{=}\mathbb{E}_{\mathcal{F}_t}\big[e^{-\int _{t}^u\lambda _s\textrm {d}s}\big]. \end{equation}

In addition to the actuarial risk, we include the financial risk. In the realm of financial instruments, our focus will be on a sole stochastic investment fund denoted as $(S_{t})_{0\leq t \leq T}$ , coupled with stochastic interest rates captured by $(r_{t})_{0 \leq t \leq T}.$ Footnote 3 The determination of the value for a zero-coupon bond with maturity $u\le T$ at a time $t\le u$ can be accomplished through the following equation

(2) \begin{equation} P(t,u)\overset {\Delta }{=}\mathbb{E}^{\mathbb{Q}^{\,f}}_{\mathcal{F}_t}\big[e^{-\int _{t}^ur_s\textrm {d}s}\big]. \end{equation}

2.2 Axiomatization

The examination of axioms will take place using a general class of hybrid product described by the following general set of random variables (rv.s)

(3) \begin{equation} \forall t \in [0,T],\ \ \mathcal{H}_t\overset {\Delta }{=}\left \{H \ rv/ \mathcal{F}_{t}-\mathrm{measurable} \ \& \ \mathrm{payable} \ \mathrm{at} \ t\right \}\!. \end{equation}

Furthermore, we introduce two more refined subsets, building exclusively on knowing the financial filtration or the actuarial filtration. The precise definitions of these subsets are outlined below

(4) \begin{equation} \forall t \in [0,T],\ \ \mathcal{H}_t^f\overset {\Delta }{=} \{h\ rv/ \mathcal{F}_{t}^{f}-\mathrm{measurable} \ \& \ \mathrm{payable} \ \mathrm{at} \ t \}. \end{equation}

(5) \begin{equation} \forall t \in [0,T],\ \ \mathcal{H}_t^a \overset {\Delta }{=}\left \{g\ rv/ \mathcal{F}_{t}^{a}-\mathrm{measurable} \ \& \ \mathrm{payable} \ \mathrm{at} \ t \right \}\!. \end{equation}

According to Barigou et al. (Reference Barigou, Chen and Dhaene2019), we follow some axioms that a valuation operator must verify in the following; see also Belhouari et al. (Reference Belhouari, Deelstra and Devolder2025).

Definition 1 ( $t^{u}$ -Valuation operator).Footnote ⁴ Let $t\in [0,T],$ $u \in [0,T]$ such as $t\le u$ . Let $\pi _{t}^{u}\;:\;\mathcal{H}_u\rightarrow \mathcal{H}_t,$ a function which represents the value at time $t$ of the hybrid payoff $H \in \mathcal{H}_u$ . We say that $\pi _{t}^{u}$ is a $t^u$ -valuation operator if it verifies the following axioms

• • Normalization $\pi _t^{u}(0)=0.$

• • Translation invariance $\forall x_{t},$ $\mathcal{F}_{t}$ – measurable and payable at the instant $u$ , $\forall H \in \mathcal{H}_u$

\begin{align*} \pi _{t}^{u}(H+x_{t})=\pi _{t}^{u}(H)+P(t,u)x_{t}. \end{align*}

Given that standard asset pricing in finance traditionally determines the value of a purely financial derivative $h \in \mathcal{H}_u^{f}$ by $\mathbb{E}_{\mathcal{F}^{f}_t}^{\mathbb{Q}^{\,f}}[e^{-\int _{t}^u r_s \mathrm{d}s}h]$ , one might naturally consider applying this same methodology to the broader class of hybrid life payoffs $H \in \mathcal{H}_u$ by defining the following $t^u$ -financial valuation operator $\pi _{t}^{u,f}$

(6) \begin{equation} \pi _t^{u,f}(H|\mathcal{F}_t)\overset {\Delta }{=}\mathbb{E}_{\mathcal{F}_t}^{\mathbb{P}^{a} \times \mathbb{Q}^{\,f}}\big[e^{-\int _{t}^ur_s\textrm {d}s}H\big] , \end{equation}

with $H \in \mathcal{H}_u$ and $0\le t \le u\le T.$

Definition 3 (Market-consistent property). Let $0\le t \le u\le T$ . A $t^u$ -valuation operator $\pi _t^{u}$ is market-consistent with respect to the $t^u$ -financial operator $\pi _{t}^{u,f}$ defined in ( 6 ) if and only if

\begin{align*} \forall h \in \mathcal{H}_u^f,\forall H \in \mathcal{H}_u\;:\; \pi _{t}^{u}(H+h|\mathcal{F}_t)=\pi _{t}^{u}(H|\mathcal{F}_t)+\pi _t^{u,f}(h|\mathcal{F}_t). \end{align*}

On the actuarial front, we assume that establishing actuarial consistency will draw upon two actuarial principles, namely, the standard deviation premium principle and the variance premium principle. Given the stochastic nature of interest rates in our study, we will adopt the framework proposed by Belhouari et al. (Reference Belhouari, Deelstra and Devolder2025), albeit with minor modifications.

The standard deviation premium principle is

(7) \begin{equation} \pi _{t}^{u,SD}[H|\mathcal{F}_t]=\mathbb{E }_{\mathcal{F}_t}^{\mathbb{P}^a\times \mathbb{Q}^{\,f}}\big[e^{-\int _{t}^ur_s\textrm {d}s} H\big]+ \alpha _t\sqrt {\mathbb{V}_{\mathcal{F}_t}[H]} \times \sqrt {u-t}\times P(t,u), \end{equation}

for $0\le t \le u\le T$ , $ H \in \mathcal{H}_u,$ and $\alpha _{t}\gt 0$ is the portion of the load used at time $t$ . The standard deviation term is multiplied by the factor $\sqrt {u-t}$ ; this small change is necessary to solve the dimensionality problem that arises from the different time-scales produced by the expectation and the standard deviation, see Pelsser and Ghalehjooghi (Reference Pelsser and Ghalehjooghi2016). The variance premium principle is

(8) \begin{equation} \pi _{t}^{u,V}[H|\mathcal{F}_t]=\mathbb{E }_{\mathcal{F}_t}^{\mathbb{P}^a\times \mathbb{Q}^{\,f}}\big[e^{-\int _{t}^ur_s\textrm {d}s}H\big]+ \gamma _{t}\times \mathbb{V}_{\mathcal{F}_t}[H]\times P(t,u), \end{equation}

for $0\le t \le u\le T$ , $ H \in \mathcal{H}_u.$ As the expectation is linear in monetary units while the variance is quadratic, a dimensional mismatch arises. To address this, we define $\gamma _t \gt 0$ as a loading parameter expressed in inverse monetary units. In our market-fair valuation context, detached from specific regulatory constraints, the insurer sets the parameters $\alpha _t$ and $\gamma _t$ based on its own risk appetite.

Definition 4 (Actuarial-consistent property). Let $0\le t \le u\le T$ . A $t^u$ -valuation operator $\pi _{t}^{u}$ is actuarial-consistent with respect to the $t^u$ -actuarial operator $\pi _{t}^{u,a}$ if and only if $\forall g\in \mathcal{H}_u^{a}$ ,

\begin{align*} \pi _{t}^{u}(g|\mathcal{F}_t)=\pi _t^{u,a}(g|\mathcal{F}_t), \end{align*}

with $\pi _{t}^{u,a}$ is given by ( 7 ) or ( 8 ).

Note that the operator $\pi _{t}^{u,f}$ is market-consistent but not actuarial-consistent. In contrast, the standard deviation $\pi _{t}^{u,SD}$ and variance $\pi _{t}^{u,V}$ premium principles are actuarial-consistent but do not satisfy market-consistency. Consequently, the remainder of this paper focuses on the Three-step method, which defines a fair valuation operator for assessing hybrid products.

2.3 Three-step decomposition in a dynamic setting

Within the scope of our study, we investigate the Three-step method introduced in Deelstra et al. (Reference Deelstra, Devolder, Gnameho and Hieber2020). To this end, we extend the natural dynamic approach of the Three-step decomposition. We consider the aggregate hybrid payoff for the insured entities, outlined as

(9) \begin{equation} _{t}H_{u}^{c}=h(S_{u})\sum _{j=1}^{N_{t}}g(\tau _{j}), \end{equation}

where $0\le t \le u\le T$ , $g:\;]0,\infty [\rightarrow [0,\infty [$ and $h:\;]0,\infty [\rightarrow [0,\infty [$ are two $\mathcal{F}^{a}_u$ -measurable functions and $\mathcal{F}^{f}_u$ -measurable, respectively.Footnote 5 We decompose $\frac {_{t}H_{u}^c}{N_t}\times {\unicode {x1D7D9}_{N_{t}\gt 0}}$ , with respect to the full spectrum of information, as followsFootnote 6

(10) \begin{equation} _{t}H_{u}=\frac {_{t}H_{u}^c}{N_t}\times {\unicode {x1D7D9}_{N_{t}\gt 0}}=\ _tH_u^{1}+\ _tH_u^{2} + \ _tH_u^{3}, \end{equation}

where

(11) \begin{equation} _tH_u^{1}\overset {\Delta }{=}\mathbb{E}_{\mathcal{F}_{t}\lor \mathcal{F}_{u}^f}[{}_{t}H_{u}]=\frac {\sum _{j=1}^{N_t}\mathbb{E}_{\mathcal{F}_t}[g(\tau _j)]h(S_{u})}{N_t}\times {\unicode {x1D7D9}_{N_{t}\gt 0}} . \end{equation}

(12) \begin{equation} _tH_u^{2} \overset {\Delta }{=}\ _{t}H_{u}-\mathbb{E}_{\mathcal{F}_{t}\lor \mathfrak{S}_{u} \lor \mathcal{F}_{u}^f}[{}_{t}H_{u}]=\frac {\sum _{j=1}^{N_t}(g(\tau _j)-\mathbb{E}_{\mathcal{F}_t\lor \mathfrak{S}_u}[g(\tau _j)])h(S_{u})}{N_t} \times {\unicode {x1D7D9}_{N_{t}\gt 0}}. \end{equation}

(13) \begin{align} {}_tH_u^{3} &\overset {\Delta }{=} \mathbb{E}_{\mathcal{F}_{t}\lor \mathfrak{S}_{u} \lor \mathcal{F}_{u}^f}[{}_{t}H_{u}] - \mathbb{E}_{\mathcal{F}_{t}\lor \mathcal{F}_{u}^f}[{}_{t}H_{u}]\nonumber \\ &= \frac {1}{N_t} \sum _{j=1}^{N_t} (\mathbb{E}_{\mathcal{F}_t\lor \mathfrak{S}_u}[g(\tau _j)] - \mathbb{E}_{\mathcal{F}_t}[g(\tau _j)]) h(S_{u}) \, \unicode {x1D7D9}_{N_{t}\gt 0}. \end{align}

The term $\frac {\mathbb{E}_{\mathcal{F}_t}[g(\tau _j)]}{N_t}\times {\unicode {x1D7D9}_{N_{t}\gt 0}}$ exhibits $\mathcal{F}_t$ -measurability, then $_tH_u^{1}$ is a financial claim; consequently $\pi ^{u,f}_{t}$ serves as the suitable operator for its assessment. The component (12) is diversifiable since this assertion holds

\begin{align*} &_t H_u^2 \Bigl |_{\mathcal{F}_t \lor \mathcal{F}_u^f \lor \mathfrak{S}_u} = \frac {1}{N_t} \sum _{j=1}^{N_t} ( g(\tau _j) - \mathbb{E}_{\mathcal{F}_t \lor \mathfrak{S}_u}[g(\tau _j)]) h(S_u) \Bigl |_{\mathcal{F}_t \lor \mathcal{F}_u^f \lor \mathfrak{S}_u} \\ &\xrightarrow {N_t \to \infty } \mathbb{E} \big[ ( g(\tau _1) - \mathbb{E}_{\mathcal{F}_t \lor \mathfrak{S}_u}[g(\tau _1)] \bigr ) h(S_u) \; |\; \mathcal{F}_t \lor \mathcal{F}_u^f \lor \mathfrak{S}_u \big] = h(S_u)\times 0 =0 \quad \text{a.s.} \end{align*}

Hence, we assess it by applying actuarial premium principles. Furthermore, both the standard deviation premium principle (7) and the variance principle (8) conform to pooling arguments in relation to $_{t}H_{u}^{2}$ . Indeed, they approach a zero value in an asymptotic sense.Footnote 7 Concerning the final part $_tH_u^{3}$ , it represents the systematic risk which is unaffected by both hedging and diversification. To compute its valuation, we propose an Esscher transform following Deelstra et al. (Reference Deelstra, Devolder, Gnameho and Hieber2020), detailed as follows

(14) \begin{equation} \pi _{t}^{u,n}( H|\mathcal{F}_t) \overset {\Delta }{=} \mathbb{E}_{\mathcal{F}_t}^{\mathbb{Z}_{\theta }^{a}\times \mathbb{Q}^{\,f}}\big[e^{-\int _{t}^ur_s\textrm {d}s}\times H\big], \end{equation}

$H \in \mathcal{H}_u$ , $0\le t \le u\le T$ and

(15) \begin{equation} \frac {d\mathbb{Z}_{\theta }^a}{d\mathbb{P}^a}\Big |_{\mathcal{F}_t}\overset {\Delta }{=}\frac {e^{\int _{t}^u \theta _{t,u}^{(1)} \lambda _s\textrm {d}s}}{\mathbb{E}_{\mathcal{F}_t}\big[e^{\int _{t}^u \theta _{t,u}^{(1)}\lambda _s\textrm {d}s}\big]}, \end{equation}

The explicit formula for $\theta _{t,u}^{(1)}$ depends on the specified dynamics of stochastic mortality rates and will therefore be provided later; see Subsection 3.2 for further details on this point.

Definition 9 (The dynamic Three-step method). Let $0\le t \le u\le T$ . The dynamic Three-step method is defined by

\begin{align*} \pi _t^{u,3}\big(_{t}H_{u}|\mathcal{F}_t\big)\overset {\Delta }{=}\pi _t^{u,f}\big(_tH_u^{1}|\mathcal{F}_t\big)+\pi _t^{u,a}\big(_tH_u^{2}|\mathcal{F}_t\big)+\pi _t^{u,n}\big(_tH_u^{3}|\mathcal{F}_t\big), \end{align*}

where $\pi _{t}^{u,f}$ is given by ( 6 ), $\pi _{t}^{u,a}$ is given by ( 7 ) or ( 8 ) and $\pi _{t}^{u,n}$ is given by ( 14 ).

It is easy to check that the dynamic Three-step method is fair according to our axiomatization.

3.1 Time consistency properties for financial products

Once a dynamic valuation is defined, the inquiry naturally extends to the temporal behavior of valuations across distinct time instances. The challenge lies in maintaining the hierarchical sequence of position preferences for products evaluated through dynamic valuation to safeguard rational behavior and thereby mitigate the emergence of financial arbitrage opportunities. We will first establish a mathematical formulation of this fundamental property for purely financial products, then broaden the scope to encompass hybrid life Pure Endowment products, with a particular focus on the dynamic Three-step method. Based on Acciaio and Penner (Reference Acciaio and Penner2011), we state the following definition of time-consistency.

Definition 10 (Time-consistent property for financial products). A dynamic valuation $(\pi _{t}^{u})_{0\le t \le u\le T}$ is time-consistent if and only if, $ \forall 0\le t \le u,v\le T$ , $\forall \ 0\lt s \le u-t$ , $0\lt s\le v-t$ Footnote ⁸ , $\forall$ $h \in \mathcal{H}_u^f$ , $g \in \mathcal{H}_v^f$ ,

(16) \begin{equation} \pi _{t+s}^{u}[h]\le \pi _{t+s}^{v}[g]\ \Rightarrow \pi _{t}^{u}[h]\le \pi _{t}^{v}[g]. \end{equation}

For instance, the mathematical expectation is a time-consistent dynamic because of the tower rule property. Notably, $(\pi _{t}^{u,f})_{0 \le t \le u \le T}$ is time-consistent. Alternatively, an iteratively backward scheme, closely associated to time-consistency, is frequently used in the literature.

Definition 11 (Iterated property for financial products). A dynamic valuation $(\pi _{t}^{u})_{0\le t \le u\le T}$ is iterated if and only if, $ \forall 0\le t \le u\le T$ , $\forall 0\lt s\le u-t$ , $\forall h \in \mathcal{H}_u^f$ ,

(17) \begin{equation} \pi _{t}^{t+s} \big[\pi _{t+s}^{u} [h ] \big]=\pi _{t}^{u}[h]. \end{equation}

Through Definition 11, we can progressively devise a backward-looking procedure in a discrete configuration, enabling, when it is not the case initially, the establishment of time-consistency for dynamic valuations that exhibit monotonicity.

The following subsection will examine time-consistency for hybrid-life Pure Endowment products. Fair – i.e., market-actuarial consistent – valuation operators for these products are frequently neither monotone nor time-consistent; see, for example, the operator of Møller (Reference Møller2002), the Two-step operator of Pelsser and Stadje (Reference Pelsser and Stadje2014), the Two-step operator of Barigou et al. (Reference Barigou, Chen and Dhaene2019), and the Three-step operator considered in our work. Consequently, the usual equivalence between the two versions of time-consistency (the two formulations in Definitions 10 and 11) no longer holds. Nonetheless, such counterexamples typically arise only in extreme contexts and thus have limited practical relevance. For the remainder of the paper, we will therefore adopt the iterative version of time-consistency as the reference definition. Moreover, some contributions in the financial literature consider the formulation presented in Definition 11 as the primary definition of time-consistency; see, for instance, Bion-Nadal (Reference Bion-Nadal2009).

3.2 Time-consistency in hybrid life Pure Endowment products

We investigate a hybrid life insurance product, where the longevity risk corresponds to what is commonly referred to in the literature as a Pure Endowment product. The benefit consists of generating a payment only in case of survival of the insured at a fixed maturity. No payout is made in the event of premature death. For the financial payoff, we consider a broad framework involving any measurable function $h$ of the financial investment fund. Formally, the collective payoff for a maturity $u\le T$ is

(18) \begin{equation} _{t}H_u^{c}=\left(\sum _{j=1}^{N_t}{\unicode {x1D7D9}_{\tau _{j}\gt u}}\right)\times h(S_u). \end{equation}

Under mortality risk, the number of survivors decreases over time from the initial instant to maturity. This necessitates ensuring price equilibrium across different time points, taking into account the number of survivors at each point in time. When dealing with such hybrid life Pure Endowment products, this adjustment should be integrated into the iterative time consistency property specified in Definition 11. Hence, Definition 11 extends to hybrid life Pure Endowment products as follows.

Definition 12 (Time-consistency for hybrid life Pure Endowment products). Consider the collective payoff given by ( 18 ), and let $_{t}H_{u}$ be its individual payoff. A dynamic valuation $(\pi _{t}^{u})_{0\le t \le u\le T}$ is time-consistent if and only if,

(19) \begin{equation} \pi _{t}^{t+s}\left[\frac {N_{t+s}}{N_t}\pi _{t+s}^{u} [{}_{t+s}H_{u} ]\right]=\pi _{t}^{u}[{}_{t}H_{u}]. \end{equation}

In the context of hybrid products embedding non-tradable longevity risk, time-inconsistent valuation does not give rise to exploitable arbitrage opportunities, as the difficulty of trading such risk prevents their effective implementation. Nevertheless, time consistency remains essential for insurers’ rational decision-making about the products they offer, as it ensures that choices made today remain viable over time.

3.3 The Three-step method time-consistent dynamic for hybrid life Pure Endowment products

Definition 13 (Time-consistency of the Three-step method). Consider the collective payoff given by ( 18 ). Let $_{t}H_u^{1}$ , $_{t}H_u^{2}$ , and $_{t}H_u^{3}$ be its Three-step decomposition, as defined respectively in ( 10 ), ( 11 ), and ( 12 ). Then, the dynamic Three-step method is time-consistent if and only if the equation below holds

(20) \begin{align} &\pi _{t}^{t+s,3}\left[\frac {N_{t+s}}{N_{t}}\times \left(\pi _{t+s}^{u,f}\big[{}_{t+s}H_{u}^1\big]+\pi _{t+s}^{u,a}\big[{}_{t+s}H_{u}^{2}\big]+\pi _{t+s}^{u,n}\big[{}_{t+s}H_{u}^{3}\big] \right) \right]\nonumber\\[3pt] &\quad =\pi _{t}^{u,f}\big[{}_{t}H_u^{1}\big]+\pi _{t}^{u,a}\big[{}_{t}H_u^{2}\big]+\pi _{t}^{u,n}\big[{}_{t}H_u^{3}\big],\end{align}

for all $ 0\le t \le u\le T, 0\lt s\le u-t$ .

We cannot proceed by applying the iterative time-consistency formula to each Three-step component in isolation. This comes from the fact that a unique initial decomposition at maturity $u$ cannot be preserved as time progresses. For instance, while the risk $_{t+s}H_{u}^1$ is hedgeable at time $t+s$ , the risk $\Big (\pi _{t+s}^{u,f}[{}_{t+s}H_{u}^{1}]\times \frac {N_{t+s}}{N_{t}}\Big )$ ceases to be hedgeable at time $t$ . Consequently, applying the financial operator $\pi _{t}^{t+s,f}$ to $\Big (\pi _{t+s}^{u,f}[{}_{t+s}H_{u}^{1}]\times \frac {N_{t+s}}{N_{t}}\Big )$ would be inappropriate. Likewise, there is no assurance that the risk $\Big (\pi _{t+s}^{u,a}[{}_{t+s}H_{u}^{2}]\times \frac {N_{t+s}}{N_{t}}\Big )$ will remain diversifiable at time $t$ , excluding its revaluation through a $t^{t+s}$ -actuarial operator. Viewed from time $t$ ,

(21) \begin{equation} _{t}LY_{t+s}\overset {\Delta }{=}\frac {N_{t+s}}{N_{t}}\times \big(\pi _{t+s}^{u,f}\big[{}_{t+s}H_{u}^1\big]+\pi _{t+s}^{u,a}\big[{}_{t+s}H_{u}^{2}\big]+\pi _{t+s}^{u,n}\big[{}_{t+s}H_{u}^{3}\big] \big) \end{equation}

is a payoff that matures at $t+s$ . Hence, once again, it must be decomposed into a hedgeable component, a diversifiable component, and a systematic component to apply the Three-step method and determine the valuation at time $t.$

As an introductory step, we aim to assess the time-consistency of the Three-step approach, disregarding the diversifiable risk. This results in the Three-step decomposition being simplified to a ’Two-step’ model, comprising a hedgeable part and a systematic part, as detailed below

\begin{align*} \left \{ \begin{array}{ll} _{t}H_u^{h}\overset {\Delta }{=}\mathbb{E}_{\mathcal{F}_{t}\lor \mathcal{F}_{u}^f}[{}_{t}H_u]=\mathbb{E}_{\mathcal{F}_{t}}\big[e^{-\int _{t}^u \lambda _s\textrm {d}s}\big]\times h(S_u)\times {\unicode {x1D7D9}_{N_{t}\gt 0}},\\[3pt] _{t}H_u^{sy}\overset {\Delta }{=}\ _{t}H_u-\mathbb{E}_{\mathcal{F}_{t}\lor \mathcal{F}_{u}^f}[{}_{t}H_u]=\left (\dfrac {N_u}{N_{t}}-\mathbb{E}_{\mathcal{F}_{t}}\big[e^{-\int _{t}^u \lambda _s\textrm { d}s}\big]\right )\times h(S_u)\times {\unicode {x1D7D9}_{N_{t}\gt 0}}. \end{array} \right \} \end{align*}

Therefore, $\pi _{t}^{u,3}[{}_{t}H_{u}]$ introduced in Definition 9, is reduced to

(22) \begin{equation} \pi _{t}^{u,2}[{}_{t}H_{u}]\overset {\Delta }{=}\pi _{t}^{u,f}\big[{}_{t}H_{u}^h\big]+\pi _{t}^{u,n}\big[{}_{t}H_{u}^{sy}\big]. \end{equation}

Lemma 1. The dynamic $(\pi _{t}^{u,2})_{0\le t \le u\le T}$ is time-consistent; consequently, its time-consistent valuation is given, by the one-period valuation, as follows

\begin{align*} \tilde {\pi }_{t}^{2}[{}_{t}H_T|\mathcal{F}_{t}]=\pi _{t}^{2}[{}_{t}H_T|\mathcal{F}_{t}]=\mathbb{E}_{\mathcal{F}_{t}}^{\mathbb{Q}^{\,f}}\big[e^{-\int _{t}^T r_v\textrm {d}v}h(S_T)\big]\times \mathbb{E}_{\mathcal{F}_{t}}^{\mathbb{Z}_{\theta }^{a}}\big[e^{-\int _{t}^T \lambda _v\textrm {d}v}\big], \end{align*}

with $t \in [0,T].$

Next, we include the diversifiable risk into the previous decomposition, finding the full Three-step decomposition as provided below

\begin{align*} \left \{ \begin{array}{ll} _{t}H_u^{1}=\mathbb{E}_{\mathcal{F}_{t}}\big[e^{-\int _{t}^u \lambda _s\textrm {d}s}\big]\times h(S_u)\times {\unicode {x1D7D9}_{N_{t}\gt 0}},\\[5pt] _{t}H_u^{2}=\left (\dfrac {N_u}{N_{t}}-e^{-\int _{t}^u \lambda _s\textrm {d}s}\right )\times h(S_u)\times {\unicode {x1D7D9}_{N_{t}\gt 0}},\\[5pt] _{t}H_u^{3}=\big(e^{-\int _{t}^u \lambda _s\textrm {d}s}-\mathbb{E}_{\mathcal{F}_{t}}\big[e^{-\int _{t}^u \lambda _s\textrm {d}s}\big]\big)\times h(S_u)\times {\unicode {x1D7D9}_{N_{t}\gt 0}}. \end{array} \right \} \end{align*}

Since the dynamic Three-step method is not time-consistent, we strive to iteratively construct a sequence of time-consistent valuations by applying Corollary1 in line with Definition 13.

To derive explicit time-consistent valuations based on Corollary2, it is essential to specify the dynamics of $(\lambda _{t})_{0\leq t \leq T}$ , $(S_{t})_{0\leq t \leq T}$ and $(r_{t})_{0 \leq t \leq T}.$ Under $\mathbb{P}^a$ , the stochastic mortality rate $\lambda _{t}$ follows a stochastic solution of the following equation commonly recognized as the Ornstein–Uhlenbeck intensity process (see Luciano & Vigna, Reference Luciano and Vigna2008)

\begin{align*} d\lambda _t=\mu _m\lambda _t dt+\sigma _m dW_m(t), \end{align*}

where $(W_{m}(t))_{t\ge 0}$ is a Brownian motion and $\mu _m\gt 0$ , $\sigma _m\gt 0.$ Thus, for $0\le t \le u\le T$ , $\ _{u-t}p_{x+t}=e^{-\lambda _{t}l_{t,u}+\frac {1}{2}s_{t,u}^2},$ where $l_{t,u}=\frac {e^{\mu _{m}(u-t)}-1}{\mu _{m}},$ and $s_{t,u}^2=\frac {\sigma _m^2}{2\mu _m^3}(1-e^{\mu _m(u-t)})^2+\frac {\sigma _m^2}{\mu _m^2}((u-t)-\frac {e^{\mu _m(u-t)}-1}{\mu _m}).$ The parameter $\theta _{t,u}^{(1)}$ present in (15) is adjusted in accordance with the Cost of Capital approach detailed in Belhouari et al. (Reference Belhouari, Deelstra and Devolder2025), producing $\theta _{t,u}^{(1)}=\frac {-0.16}{s_{t,u}}.$

The dynamics of the couple stochastic investment fund and stochastic interest rates will be characterized using the following Black-Scholes-Vasicek model, described under $\mathbb{P}^f$ by

\begin{align*} \left \{ \begin{array}{ll} dr_t=(b-ar_t)dt+\sigma _r\overset {\sim }{dW}_r(t),\\ dS_t=\mu _S S_tdt+S_t\sigma _S(\rho \overset {\sim }{dW}_r(t)+\sqrt {1-\rho ^2}\overset {\sim }{dW}_S(t)). \end{array} \right \}\!. \end{align*}

where $(\tilde {W}_{S}(t))_{t\ge 0}$ is a Brownian motion independent from $(\tilde {W}_{r}(t))_{t\ge 0}$ , $a$ is the constant speed of mean reversion, $b$ represents a strictly positive scalar such that the ratio $\frac {b}{a}$ is the long-term mean value and $\sigma _r$ is the constant volatility of interest rates. $\mu _S$ is the instantaneous return of $(S_t)_{t\ge 0}$ , $\sigma _S$ is the volatility of the financial investment fund, and $\rho$ defines the correlation between the financial investment fund and interest rates. Switching to the risk-neutral probability involves the application of Girsanov’s theorem, which defines the Brownian motions $(W_r(t))_{t}$ and $(W_S(t))_{t}$ in the following manner $\mathbb{P}^f$

\begin{align*} \left \{ \begin{array}{ll} W_r(t)\overset {\Delta }{=}\overset {\sim }{W}_r(t)-\nu t,\\ W_S(t)\overset {\Delta }{=}\overset {\sim }{W}_S(t)+\int _{0}^t \left(\dfrac {\mu _S-r_u}{\sqrt {1-\rho ^2}\sigma _S}+\dfrac {\nu \rho }{\sqrt {1-\rho ^2}}\right)\,du. \end{array} \right \}\!. \end{align*}

Drawing upon this, the Radon-Nikodym derivative of $\mathbb{Q}^{\,f}$ with respect to $\mathbb{P}^f$ is

\begin{align*} \frac {d\mathbb{Q}^{\,f}}{d\mathbb{P}^f}\Big |_{\mathcal{F}_t}=e^{\nu \overset {\sim }{W}_r(T)-\int _{t}^T \Big(\frac {\mu _S-r_u}{\sqrt {1-\rho ^2}\sigma _S}+\frac {\nu \rho }{\sqrt {1-\rho ^2}}\Big)\,d\overset {\sim }{W}_S(u)-\frac {1}{2}\int _{t}^T \Big(\nu ^2+\Big(\frac {\mu _S-r_u}{\sqrt {1-\rho ^2}\sigma _S}+\frac {\nu \rho }{\sqrt {1-\rho ^2}}\Big)^2\Big)\,du}. \end{align*}

Hence, the following dynamics are obtained under $\mathbb{Q}^{\,f}$

\begin{align*} \left \{ \begin{array}{ll} dr_t=(\theta -ar_t)dt+\sigma _r dW_r(t),\\ dS_t=r_tS_tdt+S_t\sigma _S(\rho dW_r(t)+\sqrt {1-\rho ^2}dW_S(t)), \end{array} \right . \end{align*}

with $\theta =b+\sigma _{r}\nu .$

3.4 The backward scheme in a dual-iteration approach

The achievement of closed forms through the recursive equation (23) with a substantial number of iterations, given the stochastic risks on both the financial and mortality fronts, as assumed in our study, poses a formidable challenge, if not an unattainable one. To simplify the problem, we consider two periods $[0,b]$ and $[b,T]$ in a constant interest rate framework,Footnote 11 and where we set $h(S_u)=1$ . In this case, we are delving into the purely actuarial, Pure Endowment product. Owing to the requirements of derivation and computation, we will use the variance premium principle, defined in (8), as the operator for evaluating the diversifiable risk component of the Three-step method. This choice enables the derivation of a closed-form solution, unlike the standard deviation premium principle. In light of the newly introduced denotations

\begin{align*} &\tilde {_{i}f_j}=e^{-i\lambda _{0}l_{b,T}e^{\mu _m b}+j\frac {(e^{\mu _{m}b}+1)}{2}\frac {l_{b,T}^{2}l_{0,b}\sigma _{m}^{2}}{2}+j\frac {\theta _{0,b}^{(1)}l_{0,b}^2l_{b,T}\sigma _{m}^2}{2}},&\\ &_{i}f^{j}_{k}=e^{-i\lambda _{0}l_{0,b}+j\frac {l_{b,T}l_{0,b}^{2}\sigma _{m}^{2}}{2}+k\frac {1}{2}s_{0,b}^{2}},& \end{align*}

we hereby present the following lemma.

Lemma 3. The Three-step valuation of the Pure Endowment product at the instant $b$ is $\tilde {\pi }_{b}^{3}[{}_{b}H_{T}|\mathcal{F}_{b}]=P(b,T)\Big (e^{-\lambda _{b}l_{b,T}+\frac {1}{2}s_{b,T}^2} e^{-\theta _{b,T}^{(1)} s_{b,T}^2}+\gamma _{b}\times \frac {e^{-\lambda _{b}l_{b,T}+\frac {1}{2}s_{b,T}^2}-e^{-2\lambda _{b}l_{b,T}+2s_{b,T}^2}}{N_b}\Big ).$ While the time-consistent valuation at the instant $0$ is

\begin{align*} \tilde {\pi }_{0}^{3}[{}_{0}H_{T}|\mathcal{F}_{0}]=\Gamma ^{(h,sy)}_{0}+\Gamma ^{(div)}_{0}, \end{align*}

where

\begin{align*} & \Gamma ^{(h,sy)}_{0}=P(0,T)\left(e^{-\lambda _{0}l_{0,T}-s_{0,T}^2(\theta _{0,T}^{(1)}-\frac {1}{2})}+\gamma _{b}\frac {e^{\frac {1}{2}s_{b,T}^2}\tilde {_{1}f_1}-e^{2s_{b,T}^2}\tilde {_{2}f_4}}{n}\right),&\\ & \Gamma ^{(div)}_{0}=\frac {1}{n}\times \gamma _{0}\times P(0,b)P(b,T)^{2} \times e^{-2s_{b,T}^{2}\big(\theta _{b,T}^{(1)}-\frac {1}{2}\big)}\big({}_{1}f^{2}_{1}- \ _{2}f^{8}_{4}\big),&\\ &\theta _{b,T}^{(1)}=\frac {-0.16}{s_{b,T}},&\\ &\theta _{0,b}^{(1)}=\frac {-0.16}{s_{0,b}}.& \end{align*}

A notable aspect is that the aggregated non-diversifiable risks, including systematic and hedgeable components of the time-consistent formula at time $0$ , incorporate loading terms arising from the valuation of the diversifiable component at time $b$ ; as shown in $\Gamma ^{(h,sy)}_{0}.$ This implies that, during the backward scheme, the valuation of the diversifiable component from the prior iteration impacts the valuation of the subsequent non-diversifiable components. Moreover, observe that, under perfect diversification of the portfolio, the diversifiable risk disappears. Correspondingly, the closed-form Three-step time-consistent price $\tilde {\pi }_{0}^{3}[{}_{0}H_{T}|\mathcal{F}_{0}]$ in Lemma3 converges to the Two-step static price, where it reduces to the best estimate, i.e., $ (P(0,T)\times e^{-\lambda _{0}l_{0,T}-s_{0,T}^2(\theta _{0,T}^{(1)}-\frac {1}{2})})$ ; thereby explicitly confirming that the diversifiable risk component operates to undermine time-consistency.

4.1 Derivation

In the preceding section, several simplistic measures were used to derive the time-consistent valuation across two periods. To comprehensively unveil the repercussions of time consistency on pricing, a more sophisticated numerical approach surpassing two iterations is imperative. This section focuses on studying the continuous-time limit of the Three-step method when integrated into the backward scheme. In order to accomplish this, and following Pelsser and Ghalehjooghi (Reference Pelsser and Ghalehjooghi2016), we will derive PDEFootnote 12 of the Three-step method time-consistent dynamic, incorporating all the stochastic processes of mortality and finance considered in our study. Considering an infinitesimally small time increment $dt$ that tends towards $0$ , we position ourselves within the $(t, t+dt)$ interval and employ

(24) \begin{equation} \tilde {\pi }_{t}^3=\pi _{t}^{t+dt,3}\left[\frac {N_{t+dt}}{N_t}\times \tilde {\pi }_{t+ dt}^3\right] \end{equation}

This time, we purposefully adopt the standard deviation premium principle to assess the actuarial component of the Three-step method. Interestingly, the variance principle enables the generation of a similar PDE, reflecting also a time-consistent price.

Theorem 1. Let $t^{(1)}$ be a strictly positive real scalar. The Three-step method time-consistent dynamic $(\tilde {\pi }_t^{3})_{t \in [0,T-t^{(1)}]}$ is a solution of the following PDE

(25) \begin{equation} r_{t} \tilde {\pi }_{t}^{3} =\Lambda ^{(1)}_{t}+\alpha _{t}\times \sqrt {\Lambda ^{(2)}_{t}}, \end{equation}

with

\begin{align*} \Lambda ^{(1)}_{t}&=\partial _{t} \tilde {\pi }^{3}_{t}+r_{t}S_{t}\partial _{S} \tilde {\pi }^{3}_{t}+(\theta -ar_{t})\partial _{r} \tilde {\pi }^{3}_{t}+ \left(\overline {\theta _{t,T}^{(1)}}+\mu _m \lambda _t\right)\partial _{\lambda } \tilde {\pi }^{3}_{t}+\frac {1}{2}\sigma _{S}^{2}S_{t}^{2}\partial _{S}^{2}\tilde {\pi }^{3}_{t}+\frac {1}{2} \sigma _{m}^{2}\partial _{\lambda }^{2}\tilde {\pi }^{3}_{t} \\&\quad +\frac {1}{2}\sigma _{r}^{2}\partial _{r}^{2}\tilde {\pi }^{3}_{t} + \rho \sigma _{r}\sigma _{S}S_{t}\partial _{rS}^{2}\tilde {\pi }^{3}_{t}+\big(\tilde {\pi }^{3*}_{t}-\tilde {\pi }_{t}^{3}\big)\lambda _{t},&\\\Lambda ^{(2)}_{t}&=\sigma _{S}^{2}S_{t}^{2}\big(\partial _{S}\tilde {\pi }^{3}_{t}\big)^2+ \sigma_{m}^{2}\big(\partial_{\lambda}\tilde{\pi}^{3}_{t}\big)^2+\sigma _{r}^{2}\big(\partial _{r}\tilde {\pi }^{3}_{t}\big)^2+2\rho \sigma _{S}\sigma _{r}S_{t}\partial _{S}\tilde {\pi }^{3}_{t}\partial _{r}\tilde {\pi }^{3}_{t}+\big(\tilde {\pi }_{t}^{3}-\tilde {\pi }_{t}^{3*}\big)^{2}\lambda _{t},&\\\overline {\theta _{t,T}^{(1)}}&=\theta _{0,T}^{(1)}\sigma _{m}^{2}\times \frac {e^{\mu _{m}(T-t)}-1}{\mu _m},&\\ \tilde {\pi }_{t}^{3*}[.]&=\tilde {\pi }_{t}^{3}[.|N_{t}-1].& \end{align*}

Represented by $\Lambda ^{(1)}_{t}$ , the first component of the PDE captures the time-consistent price of the aggregate replicable and systematic risks. The subsequent part $\Big (\alpha _{t}\times \sqrt {\Lambda ^{(2)}_{t}}\Big )$ , relates to the time-consistent value of the diversifiable risk.

Contrary to the time-consistent case, the static continuous-time valuation framework circumvents the requirement for a PDE. Indeed, a closed-form expression is readily tractable and is given by the following formula

(26) \begin{align} \pi _{t}^{3}[{}_{t}H_{T}] &=\mathbb{E}_{\mathcal{F}_{t}}^{\mathbb{Z}_{\theta }^{a}\times \mathbb{Q}^{\,f}}\big[e^{-\int _{t}^{T} r_s\textrm {d}s}\ _{t}H_{T}\big]+\alpha _{t}\times \sqrt {\mathbb{E}_{\mathcal{F}_t}[h(S_{T})^2]}\times \nonumber \\[5pt]& \sqrt {\frac { \mathbb{E}_{\mathcal{F}_t} [\mathbb{V}_{\mathcal{F}_t\lor \mathfrak{S}_{T}} [g(\tau _1) ] ]}{N_t}}\times \sqrt {T-t}\times P(t,T), \end{align}

for all $t$ $\in [0,T-t^{(1)}],$ with $t^{(1)}$ denoting a strictly positive real scalar.

Let us remark that our analysis of time-consistency shall be conducted over the interval $[0,T-t^{(1)}]$ . Consequently, this entails a static transition from time $T$ to time $T-t^{(1)}$ . Moreover, we shall exploit the Three-step closed-form solution in the static framework and adopt, as the terminal condition, the Three-step valuation at time $T-t^{(1)}$ of the payoff $_{T-t^{(1)}}H_{T}$ maturing at $T$ . The rationale behind this circumvention will be explained in detail in the subsequent subsection, which outlines the numerical resolution scheme.

4.2 Numerical scheme for the resolution of the PDE using a Pure Endowment product with terminal bonus

We will concentrate our inquiry on a hybrid life product identified as the Pure Endowment with terminal bonus. Within this setting, a financial profit-sharing element is incorporated in addition to the actuarial yield of the Pure Endowment product. In the event that the financial investment fund exceeds the capitalized premium $P$ by the insurer’s technical rate $i$ , profit-sharing between the insurer and the insured will occur, given by $h(S_T)=C \times ( 1+\beta (S_{T}-P(1+i)^T)^+ )$ , where $\beta$ represents the profit-sharing rate, and $C\gt 0$ is the guaranteed minimum capital. Therefore, the defined formulation for the hybrid collective payoff is as follows $H_{T}^{c}=N_{T}\times C \times (1+\beta (S_{T}-P(1+i)^T)^+ )$ . Working on a first-order basis, the premium $P$ is calculated through $P=(\frac {1}{1+i})^{T}\times _{T}\tilde {p}_{x}$ , where $_{T}\tilde {p}_{x}$ Footnote 13 signifies the prudent survival probability, and it is entirely allocated to the financial investment fund, resulting in $S_0=P$ . Here is the unfolding of the Three-step decomposition of the Pure Endowment product with a terminal bonus

\begin{align*} \left \{ \begin{array}{ll} _{T- t^{(1)}}H_T^{1}=\mathbb{E}_{\mathcal{F}_{T-t^{(1)}}}\big[e^{-\int _{T- t^{(1)}}^T \lambda _s\textrm {d}s}\big]\times h(S_{T})\times {\unicode {x1D7D9}_{N_{T-t^{(1)}}\gt 0}} ,\\[9pt] _{T- t^{(1)}}H_T^{2}=\Big (\frac {N_T}{N_{T- t^{(1)}}}-e^{-\int _{T- t^{(1)}}^T \lambda _s\textrm {d}s}\Big ) \times h(S_{T})\times {\unicode {x1D7D9}_{N_{T- t^{(1)}}\gt 0}} ,\\[9pt] _{T-t^{(1)}}H_T^{3}=\Big (e^{-\int _{T-t^{(1)}}^T \lambda _s\textrm {d}s}-\mathbb{E}_{\mathcal{F}_{T- t^{(1)}}}\big[e^{-\int _{T- t^{(1)}}^T \lambda _s\textrm {d}s}\big]\Big )\times h(S_{T})\times {\unicode {x1D7D9}_{N_{T- t^{(1)}}\gt 0}}, \end{array} \right \} \end{align*}

As stated in Lemma2, the dynamic Three-step method is not time-consistent because of the presence of diversifiable actuarial risks. Consequently, we will numerically address the PDE stated in Theorem1 using the finite difference method to obtain the time-consistent price. Firstly, we discretize $(t)_{t}$ , $(r_{t})_{t}$ , $(S_{t})_{t}$ , $(\lambda _{t})_{t}$ and $(N_{t})_{t}$ on the following finite grids $[0,T-t^{(1)}]$ , $[r_{\min},r_{\max}]$ , $[0,S_{\max}]$ , $[0,\lambda _{\max}]$ , and $[0,n]$ , respectively. The parameters $r_{\min}$ , $r_{\max}$ , $S_{\max}$ , and $\lambda _{\max}$ are configured to be attained with a probability that is very low.Footnote 14 We define the discretization step by $\Delta \zeta =\frac {\zeta _{\max}-\zeta _{\min}}{M_{\zeta }},$ where $\zeta \in \{t, S,r,\lambda ,N\}$ and $M_{\zeta }$ denotes a strictly positive integer that represents the number of steps in the grid for $\zeta .$ Furthermore, we denote the time-consistent option price at time $i \Delta t$ , for an underlying asset price of $j_{S} \Delta S$ , with a mortality intensity value of $j_{\lambda } \Delta \lambda$ , an interest rate value of $(r_{\min}+j_{r} \Delta r),$ and a current count of surviving policyholders $j_{N} \Delta N$ by $\tilde {\pi }_{i,j_{S},j_{r},j_{\lambda }}^{j_{N}},$ where $j_{\tilde {\zeta }} \in \{0,1,\ldots ,M_{\tilde {\zeta }}\}$ for $\tilde {\zeta }\in \{S,r,\lambda ,N\}$ and $i \in \{0,1,\ldots ,M_{t}\}.$

Given that the risk factors $\tilde {M_t}^{(1)}=\frac {N_t}{N_{t- t^{(1)}}}$ and $\tilde {M_t}^{(2)}=e^{-\int _{t- t^{(1)}}^t \lambda _s\textrm {d}s}$ are not Markovian processes,Footnote 15 we establish the terminal condition at the instant $T-t^{(1)}$ instead of at $T$ . The definitive formula below follows from the closed-form expression in (26), the valuation of the call option under stochastic interest rates, and the log-normal distribution of the investment fund under $\mathbb{P}^f$

\begin{align*} \tilde {\pi }_{T-t^{(1)}}^{3}[{}_{T-t^{(1)}}H_{T}] &=C\times \big(P\big(T-t^{(1)},T\big)+\beta \big(S_{T-t^{(1)}}\Phi \big(d_{T-t^{(1)}}\big)-P\big(T-t^{(1)},T\big)PG \\ &\quad \times \Phi \big(d_{T- t^{(1)}}-n\big(T-t^{(1)},T\big)\big)\big) \big ) \times e^{-\lambda _{T-t^{(1)}}l_{T-t^{(1)},T}+\frac {1}{2}s_{T-t^{(1)},T}^2} \\ &\quad \times e^{-\theta _{T-t^{(1)},T}^{(1)} s_{T-t^{(1)},T}^2}+C \times \alpha _{T-t^{(1)}}\sqrt {t^{(1)}}P\big(T-t^{(1)},T\big)\Big [1+2\beta \big(S_{T-t^{(1)}} \\ & \quad \times e^{\mu _{S} t^{(1)}}\Phi (\delta _{T-t^{(1)}})-PG\Phi \big(\delta _{T-t^{(1)}}-\sigma _{S}\sqrt {t^{(1)}}\big)\big)+\beta ^{2}\Big (S_{T- t^{(1)}}^{2}e^{2\big(\mu _{S}+\frac {1}{2}\sigma _{S}^2\big)t^{(1)}} \\ & \quad \times \Phi \big(\delta _{T-t^{(1)}}+\sigma _{S}\sqrt {t^{(1)}}\big) +(PG)^{2}\Phi \big(\delta _{T- t^{(1)}}-\sigma _{S}\sqrt {t^{(1)}}\big)-2PGS_{T-t^{(1)}}e^{\mu_{S} t^{(1)}} \\ & \quad \times \Phi \big(\delta _{T-t^{(1)}}\big)\Big)\Big ]^{\frac {1}{2}} \times {\unicode {x1D7D9}_{N_{T-t^{(1)}}\gt 0}} \\ & \quad \times \sqrt {\frac {1}{N_{T-t^{(1)}}}\Big(e^{-\lambda _{T-t^{(1)}}l_{T-t^{(1)},T}+\frac {1}{2}s_{T- t^{(1)},T}^2} -e^{-2\lambda _{T-t^{(1)}}l_{T-t^{(1)},T}+2s_{T-t^{(1)},T}^2}\Big)}, \end{align*}

where

\begin{align*} \delta _{T-t^{(1)}}&=\frac {\left(\mu _{S}+\frac {1}{2}\sigma _{S}^{2}\right)t^{(1)}+ln\left(\frac {S_{T-t^{(1)}}}{PG}\right)}{\sigma _{S}\sqrt {t^{(1)}}},\\ \theta _{T-t^{(1)},T}^{(1)}&=\frac {-0.16}{s_{T-t^{(1)},T}},\\ d_{T-t^{(1)}}&=\frac {\frac {1}{2}n\big(T-t^{(1)},T\big)^2+ln\left(\frac {S_{T-t^{(1)}}}{PG}\right)-ln\left(P\big(T-t^{(1)},T\big)\right)}{n\big(T-t^{(1)},T\big)},\\ n\big(T-t^{(1)},T\big)^2&=\sigma _S^2t^{(1)}+\frac {\sigma _r^2}{a^2}t^{(1)}+\frac {\sigma _r^2}{a^3}\left(\frac {-3}{2}-\frac {1}{2}e^{-2at^{(1)}}+2e^{-at^{(1)}}\right)+\frac {2\rho \sigma _S\sigma _r t^{(1)}}{a} \\ &\quad -\frac {2\rho \sigma _S\sigma _r\big(1-e^{-at^{(1)}}\big)}{a^2}. \end{align*}

The approximations for the partial derivatives existing in Theorem1 are detailed as follows

\begin{equation*} \partial _{t} \tilde {\pi }_{i,j_{S},j_{r},j_{\lambda }}^{j_{N}} = \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i,j_{S},j_{r},j_{\lambda }}^{j_{N}}}{\Delta t},\ i\lt M_{t} \end{equation*}

\begin{equation*} \partial _{S}\tilde {\pi }_{i,j_{S},j_{r},j_{\lambda }}^{j_{N}} = \begin{cases} \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}-2,j_{r},j_{\lambda }}^{j_{N}}}{2\Delta S} & \text{if } j_{S}=M_{S} \\[1em] \dfrac {\tilde {\pi }_{i+1,j_{S}+2,j_{r},j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}}}{2\Delta S} & \text{if } j_{S}=0 \\[1em] \dfrac {\tilde {\pi }_{i+1,j_{S}+1,j_{r},j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}-1,j_{r},j_{\lambda }}^{j_{N}}}{2\Delta S} & \text{else} \end{cases} \end{equation*}

\begin{equation*} \partial _{r}\tilde {\pi }_{i,j_{S},j_{r},j_{\lambda }}^{j_{N}} = \begin{cases} \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S},j_{r}-2,j_{\lambda }}^{j_{N}}}{2\Delta r} & \text{if } j_{r}=M_{r} \\[1em] \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r}+2,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}}}{2\Delta r} & \text{if } j_{r}=0 \\[1em] \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r}+1,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S},j_{r}-1,j_{\lambda }}^{j_{N}}}{2\Delta r} & \text{else} \end{cases} \end{equation*}

\begin{equation*} \partial _{\lambda }\tilde {\pi }_{i,j_{S},j_{r},j_{\lambda }}^{j_{N}} = \begin{cases} \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }-2}^{j_{N}}}{2\Delta \lambda } & \text{if } j_{\lambda }=M_{\lambda } \\[1em] \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }+2}^{j_{N}} - \tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}}}{2\Delta \lambda } & \text{if } j_{\lambda }=0 \\[1em] \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }+1}^{j_{N}} - \tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }-1}^{j_{N}}}{2\Delta \lambda } & \text{else} \end{cases} \end{equation*}

\begin{equation*} \partial _{S}^{2}\tilde {\pi }_{i,j_{S},j_{r},j_{\lambda }}^{j_{N}} = \begin{cases} \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}} -2\tilde {\pi }_{i+1,j_{S}-1,j_{r},j_{\lambda }}^{j_{N}} + \tilde {\pi }_{i+1,j_{S}-2,j_{r},j_{\lambda }}^{j_{N}}}{\Delta S^2} & \text{if } j_{S}=M_{S} \\[1em] \dfrac {\tilde {\pi }_{i+1,j_{S}+2,j_{r},j_{\lambda }}^{j_{N}} -2\tilde {\pi }_{i+1,j_{S}+1,j_{r},j_{\lambda }}^{j_{N}} + \tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}}}{\Delta S^2}, & \text{if } j_{S}=0 \\[1em] \dfrac {\tilde {\pi }_{i+1,j_{S}+1,j_{r},j_{\lambda }}^{j_{N}} -2\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}} + \tilde {\pi }_{i+1,j_{S}-1,j_{r},j_{\lambda }}^{j_{N}}}{\Delta S^2} & \text{else} \end{cases} \end{equation*}

\begin{align*} \partial _{r}^{2}\tilde {\pi }_{i,j_{S},j_{r},j_{\lambda }}^{j_{N}}=\left \{ \begin{array}{ll} \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}} -2\tilde {\pi }_{i+1,j_{S},j_{r}-1,j_{\lambda }}^{j_{N}} + \tilde {\pi }_{i+1,j_{S},j_{r}-2,j_{\lambda }}^{j_{N}}}{\Delta r^2 } \ \ \ \ \ \text{if } j_{r}=M_{r} \\[1em] \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r}+2,j_{\lambda }}^{j_{N}} -2\tilde {\pi }_{i+1,j_{S},j_{r}+1,j_{\lambda }}^{j_{N}} + \tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}}}{\Delta r^2} \ \ \ \ \ \text{if } j_{r}=0\\[1em] \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r}+1,j_{\lambda }}^{j_{N}} -2\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}} + \tilde {\pi }_{i+1,j_{S},j_{r}-1,j_{\lambda }}^{j_{N}}}{\Delta r^{2} } \ \ \ \ \text{else} \\ \end{array} \right \} \end{align*}

\begin{align*} \partial _{\lambda }^{2}\tilde {\pi }_{i,j_{S},j_{r},j_{\lambda }}^{j_{N}}=\left \{ \begin{array}{ll} \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}} -2\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }-1}^{j_{N}} + \tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }-2}^{j_{N}}}{\Delta \lambda ^2 } \ \ \ \ \ \text{if } j_{\lambda }=M_{\lambda } \\[1em] \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }+2}^{j_{N}} -2\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }+1}^{j_{N}} + \tilde {\pi }_{i+1,j_{ S},j_{r},j_{\lambda }}^{j_{N}}}{\Delta \lambda ^2} \ \ \ \ \ \text{if } j_{\lambda }=0\\[1em] \dfrac {\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }+1}^{j_{N}} -2\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}} + \tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }-1}^{j_{N}}}{\Delta \lambda ^{2} } \ \ \ \ \text{else}\\ \end{array} \right \} \end{align*}

\begin{align*} (\tilde {\pi }^{*}-\tilde {\pi })_{i,j_{S},j_{r},j_{\lambda }}^{j_{N}}=\left \{ \begin{array}{ll} 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if } j_{N}=0 \\[6pt] \left(\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}-1}-\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}}\right) \ \ \text{else} \end{array} \right \} \end{align*}

\begin{equation*} \partial _{rS}^{2}\tilde {\pi }_{i,j_{S},j_{r},j_{\lambda }}^{j_{N}} = \begin{cases} \dfrac {\substack {\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S},j_{r}-2,j_{\lambda }}^{j_{N}}\\ \qquad \;+\; \tilde {\pi }_{i+1,j_{S}-2,j_{r}-2,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}-2,j_{r},j_{\lambda }}^{j_{N}}}} {4 \Delta S \Delta r} & \text{if } j_{S}=M_{S} \ \& \ j_{r}=M_{r} \\[1.2em] \dfrac {\substack {\tilde {\pi }_{i+1,j_{S},j_{r}+1,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S},j_{r}-1,j_{\lambda }}^{j_{N}}\\\quad \;+\; \tilde {\pi }_{i+1,j_{S}-2,j_{r}-1,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}-2,j_{r}+1,j_{\lambda }}^{j_{N}}}} {4 \Delta S \Delta r} & \text{if } j_{S}=M_{S} \ \& \ j_{r}\ne M_{r},0 \\[1.2em] \dfrac {\substack {\tilde {\pi }_{i+1,j_{S}+1,j_{r},j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}+1,j_{r}-2,j_{\lambda }}^{j_{N}}\\ \quad \;+\; \tilde {\pi }_{i+1,j_{S}-1,j_{r}-2,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}-1,j_{r},j_{\lambda }}^{j_{N}}}} {4 \Delta S \Delta r} & \text{if } j_{S}\ne M_{S},0 \ \& \ j_{r}=M_{r} \\[1.2em] \dfrac {\substack {\tilde {\pi }_{i+1,j_{S}+2,j_{r},j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}+2,j_{r}-2,j_{\lambda }}^{j_{N}}\\ \quad \;+\; \tilde {\pi }_{i+1,j_{S},j_{r}-2,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}}}} {4 \Delta S \Delta r} & \text{if } j_{S}=0 \ \& \ j_{r}=M_{r} \\[1.2em] \dfrac {\substack {\tilde {\pi }_{i+1,j_{S},j_{r}+2,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}}\\ \quad \;+\; \tilde {\pi }_{i+1,j_{S}-2,j_{r},j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}-2,j_{r}+2,j_{\lambda }}^{j_{N}}}} {4 \Delta S \Delta r} & \text{if } j_{S}=M_{S} \ \& \ j_{r}=0 \\[1.2em] \dfrac {\substack {\tilde {\pi }_{i+1,j_{S}+2,j_{r}+2,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}+2,j_{r},j_{\lambda }}^{j_{N}}\\ \quad \;+\; \tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S},j_{r}+2,j_{\lambda }}^{j_{N}}}} {4 \Delta S \Delta r} & \text{if } j_{S}=0 \ \& \ j_{r}=0 \\[1.2em] \dfrac {\substack {\tilde {\pi }_{i+1,j_{S}+2,j_{r}+1,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}+2,j_{r}-1,j_{\lambda }}^{j_{N}}\\ \quad \;+\; \tilde {\pi }_{i+1,j_{S},j_{r}-1,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S},j_{r}+1,j_{\lambda }}^{j_{N}}}} {4 \Delta S \Delta r} & \text{if } j_{S}=0 \ \& \ j_{r}\ne 0,M_{r} \\[1.2em] \dfrac {\substack {\tilde {\pi }_{i+1,j_{S}+1,j_{r}+2,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}+1,j_{r},j_{\lambda }}^{j_{N}}\\ \quad \;+\; \tilde {\pi }_{i+1,j_{S}-1,j_{r},j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}-1,j_{r}+2,j_{\lambda }}^{j_{N}}}} {4 \Delta S \Delta r} & \text{if } j_{S}\ne 0,M_{S} \ \& \ j_{r}=0 \\[1.2em] \dfrac {\substack {\tilde {\pi }_{i+1,j_{S}+1,j_{r}+1,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}+1,j_{r}-1,j_{\lambda }}^{j_{N}}\\ \quad \;+\; \tilde {\pi }_{i+1,j_{S}-1,j_{r}-1,j_{\lambda }}^{j_{N}} - \tilde {\pi }_{i+1,j_{S}-1,j_{r}+1,j_{\lambda }}^{j_{N}}}} {4 \Delta S \Delta r} & \text{else} \end{cases} \end{equation*}

Excluding the boundary conditions, the backward recursive scheme assumes the ensuing structure for $i \in \{0,1,\ldots ,M_{t}-1\}$

\begin{align*} \tilde {\pi }_{i,j_{S},j_{r},j_{\lambda }}^{j_{N}} &=b_{j_{S},j_{r}}\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}}+c_{j_{S},j_{r}}^{+}\tilde {\pi }_{i+1,j_{S}+1,j_{r},j_{\lambda }}^{j_{N}}-c_{j_{S},j_{r}}^{-}\tilde {\pi }_{i+1,j_{S}-1,j_{r},j_{\lambda }}^{j_{N}}+d_{j_{r}}^{+}\tilde {\pi }_{i+1,j_{S},j_{r}+1,j_{\lambda }}^{j_{N}}\\&\quad -d_{j_{r}}^{-}\tilde {\pi }_{i+1,j_{S},j_{r}-1,j_{\lambda }}^{j_{N}}+\left(\overline {\theta _{i \Delta t,T-t^{(1)}}^{(1)}}\frac {g_{j_r}}{2\Delta \lambda }+e_{j_{r},j_{\lambda }}^{+}\right)\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }+1}^{j_{N}}- \left(\overline {\theta _{i \Delta t,T-t^{(1)}}^{(1)}}\frac {g_{j_r}}{2\Delta \lambda }+e_{j_{r},j_{\lambda }}^{-}\right) \\&\quad \times\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }-1}^{j_{N}}+ g_{j_{r}}P_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}}+g_{j_{r}}\left(\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}-1}-\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}}\right)j_{\lambda } \Delta \lambda \\&\quad +\alpha _{i \Delta t}g_{j_{r}}\sqrt {S_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}}+\left(\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}-1}-\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }}^{j_{N}}\right)^{2}j_{\lambda }\Delta \lambda }. \end{align*}

where $b_{j_{S},j_{r}}=\frac {1-\sigma _{S}^{2}j_{S}^{2}\Delta t -\frac {\sigma _{m}^2}{\Delta \lambda ^2}\Delta t-\frac {\sigma _{r}^{2}\Delta t}{\Delta r ^2}}{1+(r_{\min}+j_{r}\Delta r) \Delta t}$ , $c_{j_{S},j_{r}}^{+-}=\frac {\frac {1}{2}(r_{\min}+j_{r}\Delta r)j_{S} \Delta t+-\frac {1}{2}\sigma _{S}^2j_{S}^2\Delta t }{1+(r_{\min}+j_{r}\Delta r) \Delta t},$ $d_{j_r}^{+-}=\frac {\frac {(\theta -a(r_{\min}+j_r\Delta r)\Delta t}{2\Delta r} +-\frac {\sigma _{r}^2\Delta t}{2\Delta r ^2}}{1+(r_{\min}+j_{r}\Delta r) \Delta t}$ , $e_{j_r,j_\lambda }^{+-}=\frac {\frac {1}{2}\mu _m j_{\lambda }\Delta t+-\frac {1}{2}\frac {\sigma _{m}^2\Delta t}{\Delta \lambda ^2}}{1+(r_{\min}+j_{r}\Delta r) \Delta t},$ $g_{j_r}=\frac {\Delta t }{1+(r_{\min}+j_{r}\Delta r) \Delta t},$ $P_{i+1,j_{S},j_{r},j_{\lambda }}^{j_N}=\rho \sigma _{r}\sigma _{S}j_{S} \frac {\tilde {\pi }_{i+1,j_{S}+1,j_{r}+1,j_{\lambda }}^{j_{N}} -\tilde {\pi }_{i+1,j_{S}+1,j_{r}-1,j_{\lambda }}^{j_{N}} +\tilde {\pi }_{i+1,j_{S}-1,j_{r}-1,j_{\lambda }}^{j_{N}}-\tilde {\pi }_{i+1,j_{S}-1,j_{r}+1,j_{\lambda }}^{j_{N}} }{4\Delta r}$ and $S_{i+1,j_{S},j_{r},j_{\lambda }}^{j_N}=(\sigma _{S}j_{S})^{2}\frac {\big(\tilde {\pi }_{i+1,j_{S}+1,j_{r},j_{\lambda }}^{j_{N}}-\tilde {\pi }_{i+1,j_{S}-1,j_{r},j_{\lambda }}^{j_{N}}\big)^{2}}{4}+\sigma _{m}^{2}\frac {\big(\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }+1}^{j_{N}}-\tilde {\pi }_{i+1,j_{S},j_{r},j_{\lambda }-1}^{j_{N}}\big)^{2}}{4\Delta \lambda ^2}+\sigma _{r}^{2}\frac {\big(\tilde {\pi }_{i+1,j_{S},j_{r}+1,j_{\lambda }}^{j_{N}}-\tilde {\pi }_{i+1,j_{S},j_{r}-1,j_{\lambda }}^{j_{N}}\big)^{2}}{4\Delta r^2}+\rho \sigma _{r} \sigma _{S}j_{S}\frac {\tilde {\pi }_{i+1,j_{S},j_{r}+1,j_{\lambda }}^{j_{N}} -\tilde {\pi }_{i+1,j_{S},j_{r}-1,j_{\lambda }}^{j_{N}}}{2\Delta r}\big(\tilde {\pi }_{i+1,j_{S}+1,j_{r},j_{\lambda }}^{j_{N}} -\tilde {\pi }_{i+1,j_{S}-1,j_{r},j_{\lambda }}^{j_{N}}\big).$

It is important to note that enforcing time consistency through the backward-iterative scheme defined in (23) does not require the same risk measure operator to be applied at every iteration. Hence, the $\alpha _{i\Delta t}$ -loading parameter of this backward algorithm does not need to remain identical across periods and can be recalibrated over time between periods.

4.3 Numerical results

We consider a cohort of $n=100$ females aged $35$ contracting a Pure Endowment with a terminal bonus. Our numerical application follows the assumptions used in Belhouari et al. (Reference Belhouari, Deelstra and Devolder2025), where the parameters characterizing our financial-actuarial models can be summarized as follows (see Table 1).

Table 1.

Numerical inputs

Parameters	Numerical Values
$\alpha _t$	$15\%$
$i$	$0.9\%$
$C$	$100,000$
$T$	$10$
$P$	$0.9070291$
$r_{0}$	$1.6$ %
$a$	$40\%$
$\theta$	$0.8\%$
$\sigma _r$	$0.5\%$
$\sigma _S$	$15\%$
$\rho$	$20\%$
$\mu _{S}$	$5\%$
$\beta$	$40.19\%$
$\lambda _0$	$0.001080584$
$\mu _m$	$0.0745323698$
$\sigma _m$	$0.0002248628$

Figure 1

The Three-step method time-consistent dynamic with $M_S=35$ , $M_r=40$ , $M_{\lambda }=40$ , $M_{N}=n=100$ , and, $S_t=S_{0},$ $r_t=r_{0}$ , $\lambda _t=\lambda _{0}$ , $N_{t}=n.$ ( $t^{(1)}=1$ ).

Figure 2

The Three-step method time-consistent dynamic with $M_t=989$ , $M_S=35$ , $M_{\lambda }=40$ , $M_{N}=n=100$ , and, $S_t=S_{0},$ $r_t=r_{0}$ , $\lambda _t=\lambda _{0}$ , $N_{t}=n.$ ( $t^{(1)}=1$ ).

Figure 3

The Three-step method time-consistent dynamic with $M_t=989$ , $M_S=35$ , $M_r=40$ , $M_{N}=n=100,$ and, $S_t=S_{0},$ $r_t=r_{0}$ , $\lambda _t=\lambda _{0}$ , $N_{t}=n.$ ( $t^{(1)}=1$ ).

Figure 4

The three-step method time-consistent dynamic with $M_t=989$ , $M_{\lambda }=40,$ $M_r=40$ , $M_{N}=n=100,$ and, $S_t=S_{0},$ $r_t=r_{0}$ , $\lambda _t=\lambda _{0}$ , $N_{t}=n.$ ( $t^{(1)}=1$ ).

Figure 5

The dynamic Three-step method in the static versus time-consistent case and the best estimate under $\mathbb{Z}_{\theta }^{a}\times \mathbb{Q}^{\,f}$ . ( $t^{(1)}=1$ ).

Figure 6

The dynamic Three-step method net of the best estimate $\mathbb{E}_{\mathcal{F}_{t}}^{\mathbb{Z}_{\theta }^{a}\times \mathbb{Q}^{\,f}}[e^{-\int _{t}^T r_s\textrm {d}s}\ _{t}H_{T}]$ in the static versus time-consistent cases. ( $t^{(1)}=1$ ).

Prior to elucidating the numerical results, we must preliminarily verify the convergence of the finite difference method. Since the discretization step $\Delta N$ remains invariably equal to $1$ ,Footnote 16 it obviates the need for any calibration of the number of steps concerning the number of survivors. As depicted in Figure 1, the convergence of time is reached after $989$ iterations, implying that approximately one iteration occurs every three days and one tier. Figures 2–4 respectively investigate the convergence of interest rates, mortality rates, and the financial investment fund. The stability of the numerical values is deemed sufficient for $M_S=35$ , $M_r=40$ and $M_{\lambda }=40$ . Figure 5 contrasts the time-consistent values of the Three-step method with its static counterparts and the best estimate calculated under $\mathbb{Z}_{\theta }^{a}\times \mathbb{Q}^{\,f}$ . The dynamic Three-step method follows a monotonic trend analogous to that of the best estimate, reflecting a price upward over time. Furthermore, it demonstrates that the static values of the dynamic Three-step method exhibit a modest increase, ultimately resulting in a $3.7\%$ cumulative growth by time $0$ . By including the diversifiable part, the dynamic Three-step method loses its time-consistent property, which it previously maintained with a valuation formula equivalent to $\mathbb{E}_{\mathcal{F}_{t}}^{\mathbb{Z}_{\theta }^{a}\times \mathbb{Q}^{\,f}}[e^{-\int _{t}^T r_s\textrm {d}s}\ _{t}H_{T}]$ , as referenced in Lemma1. Therefore, to better understand the effect of time-consistency, it is important to compare the time-consistent Three-step prices, net of this best estimate, with the one-period Three-step prices, net again of this best estimate.Footnote 17 As we can see in Figure 6, the static values undergo a substantial increase, ending in an $185\%$ increase by time $0$ . To address this issue, we follow Devolder and Lebègue (Reference Devolder and Lebègue2016) and adjust the $\alpha _{i \Delta t}$ safety loads portion applied in the backward recursive scheme. In our study, we propose a linear decrease of the safety loads portion over the entire time frame, within the backward algorithm,Footnote 18 as detailed below

\begin{align*} \alpha _{i\Delta t} = \left \{ \begin{array}{ll} \dfrac {0.15}{989}\, i. & 0 \le i \le 989 \end{array} \right \}\!. \end{align*}

As illustrated in Figure 6, the linear recalibration of the safety loading portion successfully reduces the percentage increase induced by time consistency at time $0$ to a value below $100\%$ , namely $23\%$ . Our decision to implement a backward-decreasing safety loading portion is primarily motivated by the objective of reducing the time-consistent price, which would otherwise have been prohibitively high. Moreover, this choice of a monotonically decreasing safety loading portion is also supported in the actuarial literature, see, for example, Hürlimann (Reference Hürlimann2010) and Devolder and Lebègue (Reference Devolder and Lebègue2016), where in the context of dynamic capital requirement calculation, the analog of the safety loading portion corresponds to the confidence level applied to Value-at-Risk. With this portion loading recalibration approach, we achieve a good compromise between a time-consistent evaluation methodology and a value that is not overly costly relative to the benchmark static value. This therefore proves that it is possible to switch to a time-consistent valuation without generating an unreasonable risk margin.