2.1 The financial market

We consider a risky asset $\mathcal{S}$ whose time- $t$ price, $S_t$ , is described under the objective probability measure $\mathbb{P}$ by the following dynamics:

(2.1) \begin{equation} \mathrm{d}S_t=\mu _t S_t \mathrm{d}t + \sigma _t S_t \mathrm{d}W_t. \end{equation}

The stochastic differential equation above suggests that the price of the risky asset follows a geometric Brownian motion, where the time-varying parameters are the drift $\mu _t$ and the relative volatility $\sigma _t$ .

We then characterize the bond market and assume deterministic dynamics of the term structure of interest rates, implying that the time- $t$ instantaneous forward rate for a future date $\tau \gt t$ , $f_{t,\tau }$ , coincides with the actual time- $\tau$ instantaneous interest rate, $r_\tau$ . Consequently, denoting with $B\left (t,\tau \right )$ the time- $t$ price of a default-free zero-coupon bond maturing at time $\tau \gt t$ , the following relation holds:

(2.2) \begin{equation} r_\tau = f_{t,\tau } = -\dfrac { \partial \ln \left (B \left ( t,\tau \right ) \right ) } {\partial \tau }. \end{equation}

Different zero-coupon bonds $\mathcal{B}_i$ ( $i\in \left \{1,\ldots ,N_{\mathcal{B}}\right \}$ ) are traded on the financial market, with maturities given by $T_{\mathcal{B}_i}$ and time- $t$ prices denoted by $B_{i,t}=B(t,T_{\mathcal{B}_i} )$ .

Finally, we assume that European put options $\mathcal{P}_i$ ( $i\in \left \{1,\ldots ,N_{\mathcal{P}}\right \}$ ) written on the underlying risky asset $\mathcal{S}$ are traded on the market. The strike prices and maturities of the put options are $K_{\mathcal{P}_i}$ and $T_{\mathcal{P}_i}$ , respectively, while the time- $t$ prices of the options are denoted by $P_{i,t}=P(t,T_{\mathcal{P}_i},K_{\mathcal{P}_i})$ .

In the following section, we derive a strategy replicating the financial claim and provide additional details on the maturities of the bonds, as well as on the maturities and strike prices of the put options, that are needed to construct the replicating strategy. The time- $t$ trading strategy ( $t\in \left \{0,\ldots ,n-1\right \}$ ) is defined as a multi-dimensional process $\left \{\boldsymbol{\theta }_t\right \}_{t\in \left \{0,\ldots ,n-1\right \}}$ predictable with respect to the filtration $\mathbb{F}^W$ , i.e., $\boldsymbol{\theta }_t$ is $\mathcal{F}_{t}^W$ -measurable. More precisely, $\boldsymbol{\theta }_t=\left (\theta ^{\mathcal{S}}_t,\theta ^{\mathcal{B}_1}_t,\ldots ,\theta ^{\mathcal{B}_{N_{\mathcal{B}}}}_t,\theta ^{\mathcal{P}_1}_t,\ldots ,\theta ^{\mathcal{P}_{N_{\mathcal{P}}}}_t\right )$ , where $\theta ^{\mathcal{S}}_t$ is the number of units invested in the stock $\mathcal{S}$ at time $t$ , i.e., for the time interval $\left (t,t+1\right ]$ , while $\theta ^{\mathcal{B}_i}_t$ and $\theta ^{\mathcal{P}_i}_t$ are the analogous for bonds and put options.

We consider a continuous-time financial market on the time horizon $[0,n]$ and work under the objective probability measure $\mathbb{P}$ , where the source of uncertainty is the Brownian motion $W$ . Let $\lambda =\left (\lambda _t\right )_{t\in [0,n]}$ be an $\mathbb{F}^W$ -adapted market price of risk process.Footnote 1 Define the stochastic discount factor (density process) $\xi ^\lambda =\left (\xi _t^\lambda \right )_{t\in [0,n]}$ by

(2.3) \begin{equation} \xi _t^\lambda =\exp \left \{-\int _{0}^{t}\lambda _s\, \mathrm{d}W_s-\frac {1}{2}\int _{0}^{t}\lambda _s^{2}\, \mathrm{d}s\right \}. \end{equation}

Under the above integrability conditions, $\xi ^\lambda$ is a strictly positive $\mathbb{P}$ -martingale. Hence, following Munk (Reference Munk2013), we define an equivalent probability measure $\mathbb{Q}$ on $(\Omega ,\mathcal{F}_n)$ via the Radon–Nikodym derivative $\frac {\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}\Big |_{\mathcal{F}_n} \;:\!=\; \xi _n^\lambda ,$ so that $\xi _t^\lambda =\mathbb{E}^{\mathbb{P}}\left [\left .\frac {\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}\right |\mathcal{F}_t^W\right ]$ for all $t\in [0,n]$ .

In our framework, the absence of arbitrage and the completeness of the market are equivalent to the existence of a unique equivalent martingale measure $\mathbb{Q}$ . Under $\mathbb{Q}$ , the discounted price processes are $\mathbb{F}^W$ -martingales and, in particular, for $t\in \{1,\ldots ,n\}$ ,

(2.4) \begin{align} & S_{t-1} = \mathbb{E}^{\mathbb{Q}}_{t-1} \left [e^{-\int _{t-1}^{t}r_s\,\mathrm{d}s}\,S_t \right ], \end{align}

(2.5) \begin{align} & B_{i,t-1} = \mathbb{E}^{\mathbb{Q}}_{t-1} \left [e^{-\int _{t-1}^{t}r_s\,\mathrm{d}s}\,B_{i,t} \right ] \quad \forall i\in \left \{1,\ldots ,N_{\mathcal{B}}\right \}, \end{align}

(2.6) \begin{align} & P_{i,t-1} = \mathbb{E}^{\mathbb{Q}}_{t-1} \left [e^{-\int _{t-1}^{t}r_s\,\mathrm{d}s}\,P_{i,t} \right ] \quad \forall i\in \left \{1,\ldots ,N_{\mathcal{P}}\right \}. \end{align}

Note that we use the notation $\mathbb{E}^{\mathbb{Q}}_t\left [\cdot \right ]=\mathbb{E}^{\mathbb{Q}}\big[\cdot \big |\mathcal{F}_{t}^W\big ]$ . For a proof of this equivalence, we refer to Delbaen and Schachermayer (Reference Delbaen and Schachermayer2006).

2.2 The cohort of policyholders

We assume that, at time $t=0$ , a cohort of $\ell _0\in \mathbb{N}_{\neq 0}$ policyholders with identical characteristics (same entry age $x$ , policy term, premium payment pattern, etc.) enters the insurer’s portfolio. The insurer sells equity-linked endowment policies with a Cliquet guarantee to this cohort. Hence, risks are homogeneous across policyholders, except for the sum insured: each policyholder $k\in \{1,\ldots ,\ell _0\}$ subscribes to the same contract structure with an individual sum insured $c_{k,0}$ . We assume that lapses (surrenders) are not allowed over the contract term, so that the only cause of contract termination before maturity is death.

Let $T_{k,x}$ denote the (unconditional) future lifetime in years of policyholder $k$ measured from time $t=0$ . Working in discrete time (annual periods), we define the residual lifetime at time $t$ by the convention

(2.7) \begin{equation} T_{k,x+t} \;:\!=\; \max [T_{k,x}-t,0], \quad t=0,1,\ldots , \quad k\in \{1,\ldots ,\ell _0\}. \end{equation}

In particular, $T_{k,x+t}=0$ when death occurs at or before time $t$ . If $\{T_{k,x}\gt t\}$ , death occurs after time $t$ and $T_{k,x+t}=T_{k,x}-t$ represents the remaining lifetime measured from time $t$ .

Consistent with the benefit structure of an endowment contract, we introduce the following indicator:

(2.8) \begin{equation} \mathbb{I}_{k,t}\;:\!=\; \begin{cases} \unicode {x1D7D9}_{\{T_{k,x+t}\in (0,1]\}}, & t=0,\ldots ,n-2,\\ \unicode {x1D7D9}_{\{T_{k,x+t}\gt 0\}}, & t=n-1. \end{cases} \qquad k\in \{1,\ldots ,\ell _0\}. \end{equation}

For $t=0,\ldots ,n-2$ , the event $\{\mathbb{I}_{k,t}=1\}$ means that policyholder $k$ dies during the period $(t,t+1]$ , thereby representing the occurrence of the insured event that triggers a death benefit payment at time $t+1$ . Accordingly, the sequence $\{\mathbb{I}_{k,t}\}_{t=0}^{n-2}$ describes whether the policy is in force up to each time $t$ . The final line in (2.8) is a contractual convention: at maturity (time $n$ ), the endowment benefit is payable only if the contract is still in force at time $n-1$ , which is equivalent to $\{T_{k,x+t}\gt 0\}$ . This convention allows the use of a single indicator process $\{\mathbb{I}_{k,t}\}_{t=0}^{n-1}$ to model both the in-force status over time and the endowment payoff at maturity.

For each $t=0,\ldots ,n-2$ , the indicator $\mathbb{I}_{k,t}$ is observed at time $t+1$ and is therefore $\mathcal{F}_{t+1}^{\mathbb{I}}$ -measurable (and hence also $\mathcal{F}_{t+1}$ -measurable). Moreover, under Assumption 2.1, $\mathbb{I}_{k,t}$ is independent of $\mathcal{F}_t^W$ .

Assumption 2.3 (Conditional i.i.d. mortality (among survivors)). For each $t=0,\ldots ,n-2$ , let $Q_{x+t}$ denote the stochastic one-year death probability at age $x+t$ in the standard actuarial sense, i.e., conditional on being alive at time $t$ .

Define the (random) set of policyholders alive at time $t$ by

(2.9) \begin{equation} \mathcal L_t \;:\!=\; \{k\in \{1,\ldots ,\ell _0\}:\ T_{k,x}\gt t\}, \qquad \ell _t \;:\!=\; |\mathcal L_t|, \end{equation}

where $\ell _t$ is the number of policyholders alive at time $t$ .

Conditional on $Q_{x+t}$ , the random variables $\{\mathbb I_{k,t}\}_{k\in \mathcal L_t}$ are independent and identically distributed. Hence,

(2.10) \begin{equation} \mathbb{I}_{k,t}\,\big |\,\left (Q_{x+t},\,k\in \mathcal L_t\right )\sim \mathrm{Ber}(Q_{x+t}). \end{equation}

Moreover, for any distinct $h,k$ ,

(2.11) \begin{equation} \mathbb{P}(\mathbb{I}_{h,t}=1,\mathbb{I}_{k,t}=1\,|\,Q_{x+t},\,h\in \mathcal L_t,\,k\in \mathcal L_t) = Q_{x+t}^{2}. \end{equation}

Finally, we define the best-estimate (one-year) death probability at time $t$ as

(2.12) \begin{equation} q_{x+t}\;:\!=\; \mathbb{E}^{\mathbb{P}}\left [Q_{x+t}\,\big |\,\mathcal F_t\right ], \end{equation}

i.e., the conditional expectation given the information available up to time $t$ .

At a generic time $t\in \{1,\ldots ,n-1\}$ , the sum insured in force for policyholder $k$ is given by

(2.13) \begin{equation} C_{k,t}\;:\!=\; c_{k,0}\prod _{s=0}^{t-1}\left (1-\mathbb{I}_{k,s}\right ), \qquad k\in \{1,\ldots ,\ell _0\}, \end{equation}

which equals $c_{k,0}$ if no death has occurred up to time $t$ and $0$ otherwise. This representation is convenient at the portfolio level, as it allows for modeling the in-force status of contracts and to aggregate mortality-driven cash flows across policyholders.

3.1 Premiums and benefits

We first define the $\mathbb{F}$ -adapted vector of the cashflows, generically indicated with $\textbf {X}$ :

(3.1) \begin{equation} \textbf {X}_{(0)}=(\textit {X}_0,\textit {X}_1,\ldots ,\textit {X}_n). \end{equation}

Considering two vectors of random variables, $\textbf {X}$ and $\textbf {Y}$ , both of dimension $n\times 1$ , we assume

(3.2) \begin{align} \mathbb{E}\left [\sum _{t=0}^{n}X^2_t\right ]&\lt \infty , \end{align}

(3.3) \begin{align} \mathbb{E}\left [\sum _{t=0}^{n}X_t\cdot Y_t\right ]&\lt \infty . \end{align}

Consequently, $\textbf {X}\in L^2_{n+1}\left (\mathbb{P},\mathbb{F}\right )$ , where $L^2_{n+1}\left (\mathbb{P},\mathbb{F}\right )$ is a $(n+1)$ -dimensional Hilbert space. For a generic time $t$ , we define inflows and outflows, representing cashflows as follows:

(3.4) \begin{equation} X_\tau \;:\!=\; \begin{cases} -X_\tau ^{in} & \text{if $\tau =t$,} \\[5pt] X_\tau ^{out}-X_\tau ^{in} & \text{if $t\lt \tau \lt n$,}\\[5pt] X_\tau ^{out} & \text{if $\tau =n$}, \end{cases} \end{equation}

where

(3.5) \begin{align} X_\tau ^{in} & \;:\!=\; \sum _{k=1}^{\ell _0}C_{k,\tau }\cdot \pi \cdot U_{\tau ,\tau }^{in}, \end{align}

(3.6) \begin{align} X_\tau ^{out} & \;:\!=\;\sum _{k=1}^{\ell _0}C_{k,\tau -1}\cdot \mathbb{I}_{k,\tau -1}\cdot U_{\tau ,\tau }^{out}. \end{align}

In formula (3.4), the generic time- $\tau$ cashflow, $X_\tau$ , is given by the sum of two components: the inflow $X_\tau ^{in}$ and the outflow $X_\tau ^{out}$ . According to Assumption 3.1, the former are paid at the beginning of each time period and the latter at the end.

The inflow received from the $k$ -th policyholder is the product of three terms: the sum insured at time $\tau$ , $C_{k,\tau }$ , the regular premium rate, $\pi$ , and the payout of a unit zero-coupon bond maturing at $\tau$ , $U_{\tau ,\tau }^{in}$ . Although the last factor is identically equal to one and could be formally omitted, its role will be clear when the hedging portfolio and the mathematical reserve are determined.

Regarding the outflows, the benefit of the $k$ -th policyholder is the product of the sum insured at time $\tau -1$ , $C_{k,\tau -1}$ , a binary variable equal to $1$ if the beneficiary becomes eligible for the benefit at time $t$ , $\mathbb{I}_{k,\tau -1}$ , and the (unitary) revaluation of the sum assured, $U_{\tau ,\tau }^{out}$ . In accordance with the definition of a Cliquet guarantee (see Hipp, Reference Hipp1996; Bacinello, Reference Bacinello2001), such revaluation is defined as:

(3.7) \begin{equation} U_{\tau ,\tau }^{\textit {out}}\;:\!=\;\prod _{s=1}^{\tau } (1+\max [R_s,g] ), \end{equation}

where $R_s=\frac {S_s-S_{s-1}}{S_{s-1}}$ is the arithmetic return of the stock between time $s-1$ and time $s$ , while $g$ is the guaranteed per-period return.

The quantity $U^{out}_{\tau ,\tau }$ is the time- $\tau$ payoff factor per unit of sum insured generated by the Cliquet crediting rule. It represents the value at time $\tau$ of one monetary unit that is revalued each year by the credited gross return $1+\max [R_s,g]$ . Accordingly, $U^{out}_{\tau ,\tau }$ is an $\mathcal{F}^W_\tau$ -measurable random variable (i.e., a contingent claim), whereas $U^{out}_{t,\tau }$ denotes its time- $t$ market-consistent value obtained via the replicating portfolio constructed below.

3.2 Replication and pricing of the contingent claim

Given $\tau \in [0,\ldots ,n]$ , the quantities $\left (U_{t,\tau }^{in}\right )_{t=0,\ldots ,n}$ and $\left (U_{t,\tau }^{out}\right )_{t=0,\ldots ,n}$ are $\mathbb{F}^W$ -adapted stochastic processes representing the time- $t$ values of the financial portfolios $\mathcal{U}^{in}_{t,\tau }$ and $\mathcal{U}^{out}_{t,\tau }$ , respectively used at time $t$ to replicate the time- $\tau$ quantities $U_{\tau ,\tau }^{in}$ and $U_{\tau ,\tau }^{out}$ . While replicating the inflows is straightforward through zero-coupon bonds paying off future premiums, replicating the outflows is more challenging, due to the stochastic and nonlinear nature of the benefit in equation (3.7). The following theorem provides a replicating strategy and the price of the contingent claim.

Theorem 1. The claim $U_{\tau ,\tau }^{\textit {out}}$ is $t$ -hedgeable because there exists a time- $t$ self-financing strategy $\boldsymbol{\theta }_{t,\tau }\in \Theta _{t,\tau }$ , with the following positions in the asset $\mathcal{S}$ and a European put option $\mathcal{P}_t$ with a strike price $S_t(1+g)$ expiring at time $t+1$ :

(3.8) \begin{align} \theta _{t,\tau }^{\mathcal{S}} & =\frac {U_{t,\tau }^{\textit {out}}}{S_t+P_t}, \end{align}

(3.9) \begin{align} \theta _{t,\tau }^{\mathcal{P}_t} & =\frac {U_{t,\tau }^{\textit {out}}}{S_t+P_t}. \end{align}

$U_{t,\tau }^{\textit {out}}$ is the time- $t$ price of the claim,

(3.10) \begin{equation} U_{t,\tau }^{\textit {out}}=U_{t,t}^{\textit {out}}\prod _{h=t}^{\tau -1}(1+p_h), \end{equation}

where $P_t=S_t p_t$ is the no-arbitrage price of the put option $\mathcal{P}_t$ and $p_{t}$ represents the Black-Scholes price of a European put option on an underlying asset with current price equal to one, strike price equal to $1+g$ , time-to-maturity equal to the time period $\Delta t$ (equal to one under yearly frequency), interest rate $r_{t,t+1}=\frac {1}{\Delta t}\int _{t}^{t+1}r_{s}\mathrm{d}s$ , and volatility $\sigma _{t,t+1}=\sqrt {\frac {1}{\Delta t}\int _{t}^{t+1}\sigma _{s}^{2}\mathrm{d}s}$ :

(3.11) \begin{align} p_{t} &=(1+g) e^{-r_{t,t+1}\Delta t}\mathcal{N}\left (-\frac {\log \frac {1}{1+g}+\left (r_{t,t+1}-\frac {\sigma _{t,t+1}^{2}}{2}\right )\Delta t}{\sigma _{t,t+1}\sqrt {\Delta t}}\right )\nonumber \\ &\quad -\mathcal{N}\left (-\frac {\log \frac {1}{1+g}+\left (r_{t,t+1}+\frac {\sigma _{t,t+1}^{2}}{2}\right )\Delta t}{\sigma _{t,t+1}\sqrt {\Delta t}}\right ), \end{align}

where $\mathcal{N}\left (\cdot \right )$ is the cumulative distribution function of a standard normal random variable.

The self-financing strategy described in Theorem 1 is dynamic, as the position in the risky asset, $\theta _{t,\tau }^{\mathcal{S}}$ , is adjusted at the beginning of each time period, while the remaining fraction of the portfolio value is invested in a European put option expiring at the end of the relevant time period. The strategy is such that the portfolio is invested in an equal number, $\frac {U_{t,\tau }^{\textit {out}}}{S_t+P_t}$ , of units of the stock and the put option. The put options have a strictly positive payoff when the stock return is lower than $g$ , so that the minimum guaranteed return is delivered.

The time- $t$ value of the claim maturing at a future date $\tau$ equals the revaluation of the sum insured up to the current time, $U_{t,t}^{\textit {out}} = \prod _{s=1}^{t}(1+\max [R_s,g])$ , increased by the cumulative multiplicative effect of the put option prices normalized by the asset value, $p_h=\frac {P_h}{S_h}$ , applied at each subsequent time step. This additional component ensures that the portfolio value is sufficient to purchase the put options required to guarantee the minimum return in all remaining periods.

3.3 The mathematical reserve and the valuation portfolio approach

Article 76 of Directive 2009/138/EC (European Parliament and Council, 2021) requires that the insurance and reinsurance undertaking hold technical provisions measured at fair value, i.e., “the current amount insurance and reinsurance undertakings would have to pay if they were to transfer their insurance and reinsurance obligations immediately to another insurance or reinsurance undertaking”.

Moreover, the Solvency II Delegated Regulation (see European Parliament and Council, 2014) requires that, with reference to non-hedgeable liabilities where the assessment necessitates the separate quantification of the BE and the RM, this second component should not be included in the calculation of the SCR. We therefore present a market-consistent approach for the calculation of the BE; this approach is coherent with Clemente et al. (Reference Clemente, Della Corte, Savelli and Zappa2025), Wüthrich et al. (Reference Wüthrich, Bühlmann and Furrer2010) and Wüthrich and Merz (Reference Wüthrich and Merz2013).

The purpose of this section is to show the construction of a VaPo and determine its value. A VaPo is a collection of financial instruments designed to replicate a given series of future cashflows. We will outline the steps to build a VaPo, ensuring the final result aligns with the cohort approach. Indicating with $Q\left (\textbf {X}\right )$ a valuation functional, the mapping $\textbf {X} \mapsto Q\left (\textbf {X}\right )$ assigns a monetary value, i.e., the market-consistent price, to each cashflow $\textbf {X}\in L^2_{n+1}(\mathbb{P},\mathbb{F})$ .Footnote 2 We define the mathematical reserve $R_t$ at time $t$ as:

(3.12) \begin{equation} R_t\;:\!=\; Q_t(\textbf {X}_{(t)}) . \end{equation}

1. Step 1: Choice of financial instruments. The first step involves selecting the financial basis: referring to Theorem 1, we focus on two groups of financial instruments, i.e., those used to replicate the outflows and inflows, respectively. We denote with $\mathcal{U}^{out}_{t,\tau }$ the portfolio of financial instruments used at time $t$ to replicate the outflow $X_\tau ^{out}$ , which consists in a linear combination of financial instruments available on the market $\mathcal{M}$ , with weights $\boldsymbol{\theta }_{t,\tau }$ . Hence,

   (3.13) \begin{equation} \mathcal{U}^{out}_{t,\tau } = \theta ^{\mathcal{S}}_{t,\tau }\cdot \mathcal{S}+\theta ^{\mathcal{P}_t}_{t,\tau }\cdot \mathcal{P}_t . \end{equation}
   Focusing on the inflows, the portfolio replicating a unitary cashflow at a future time $\tau$ , i.e., the term $U_{\tau ,\tau }^{in}$ in equation (3.5), is given by a zero-coupon bond paying one dollar at time $\tau$ :
   (3.14) \begin{equation} \mathcal{U}^{in}_{t,\tau } = \mathcal{B}_{\tau }. \end{equation}

2. Step 2: Determine the number of shares of the hedging portfolio. After defining the financial instruments required to replicate outflows and premiums, we determine the BE of the positions in these instruments that the insurance company needs to take in order to replicate the future cashflows $\textbf {X}_{(t)}$ . Starting from period $t$ , future outflows begin at time $t+1$ and require long positions in the VaPo, while the premiums are received from time $t$ and correspond to short positions:

   (3.15) \begin{equation} \begin{aligned} \textbf {X}_{(t)} & \mapsto VaPo(\textbf {X}_{(t)}) \\ VaPo(\textbf {X}_{(t)}) & = \sum _{k=1}^{\ell _0} \sum _{s=0}^{n-t-1}C_{k,t}\cdot \mathbb{E}_t^{\mathbb{P}}\left [\left (\prod _{h=0}^{s-1}(1-\mathbb{I}_{k,t+h})\right )\cdot \mathbb{I}_{k,t+s}\right ]\cdot \mathcal{U}_{t,t+s+1}^{out} \\ & \quad -\sum _{k=1}^{\ell _0} \sum _{s=0}^{n-t-1}C_{k,t}\cdot \pi \cdot \mathbb{E}_t^{\mathbb{P}}\left [\left (\prod _{h=0}^{s-1}(1-\mathbb{I}_{k,t+h})\right )\right ]\cdot \mathcal{U}_{t,t+s}^{in}. \end{aligned} \end{equation}

3. Step 3: Apply an accounting principle to the VaPo in order to obtain a monetary value for the portfolio. Equation (3.15) provides the positions in a set of financial instruments allowing for hedging outflows and inflows. In this final step, we calculate the BE by assigning a monetary value to each financial instrument to obtain a valuation of the VaPo.

   (3.16) \begin{equation} \begin{aligned} VaPo(\textbf {X}_{(t)}) & \mapsto \upsilon _t\left (VaPo(\textbf {X}_{(t)})\right )=Q_t(\textbf {X}_{(t)})= R_{t} \\ R_{t} & = \sum _{k=1}^{\ell _0} \sum _{s=0}^{n-t-1}C_{k,t}\cdot \mathbb{E}_t^{\mathbb{P}}\left [\left (\prod _{h=0}^{s-1} (1-\mathbb{I}_{k,t+h} )\right )\cdot \mathbb{I}_{k,t+s}\right ]\cdot U_{t,t+s+1}^{out} \\ & \quad -\sum _{k=1}^{\ell _0} \sum _{s=0}^{n-t-1}C_{k,t}\cdot \pi \cdot \mathbb{E}_t^{\mathbb{P}}\left [\left (\prod _{h=0}^{s-1}(1-\mathbb{I}_{k,t+h} )\right )\right ]\cdot U_{t,t+s}^{in} . \end{aligned} \end{equation}
   The accounting principle $\left (\upsilon _t\right )_{t\in \left [0,n\right ]}$ assigns a monetary value to the financial instruments and their linear combinations. Among all possible accounting principles, $\upsilon$ represents the fair value, as required by Solvency II. Notably, each financial instrument is valued at the market price under the assumption of no arbitrage.

4.1 Characteristics of demographic – idiosyncratic risk

The following theorem provides a compact formula for the CDR due to the idiosyncratic component.

Theorem 2. The idiosyncratic component of the Claims Development Result for a cohort of $\ell _0$ policyholders can be calculated as:

(4.7) \begin{equation} CDR^{Idios}_{t+1}=\left (\sum _{k=1}^{\ell _0}C_{k,t}\cdot \big(\mathbb{E}_t^{\mathbb{P}}[\mathbb{I}_{k,t}]-\mathbb{I}_{k,t}\big)\right )\cdot \eta _{t+1}, \end{equation}

where $\eta _{t+1}$ is the stochastic Sum-at-Risk (SAR) rate, defined as:

(4.8) \begin{equation} \eta _{t+1}= U_{t+1,t+1}^{out}-\beta _{t+1}, \end{equation}

where $\beta _{t+1}$ is the BE rate at time $t+1$ based on the demographic basis defined at time $t$ , defined in formula ( 4.4 ).

The idiosyncratic component of the CDR is given by the product of two factors. The first is the most significant from a demographic standpoint and depends on the difference $\mathbb{E}_t^{\mathbb{P}}\left [\mathbb{I}_{k,t}\right ]-\mathbb{I}_{k,t}$ , taking negative values when actual mortality is higher than expected mortality between $t$ and $t+1$ , and positive values when it is lower.Footnote 4 The sums insured across the entire cohort of $\ell _0$ policyholders, $C_{k,t}$ , act as a scaling factor in this term.

The second term, $\eta _{t+1}$ , represents the SAR rate of the contract. For traditional policies with fixed (non-revalued) benefits, the outflow settled at time $t+1$ upon death is deterministic and equal to one per unit of sum insured, i.e. $U^{out}_{t+1,t+1}=1$ . In that setting, the best-estimate rate typically satisfies $0\lt \beta _{t+1}\lt 1$ , so that the SAR rate is positive.

In our framework, instead, $\eta _{t+1}=U^{out}_{t+1,t+1}-\beta _{t+1}$ is random at time $t$ , since both $U^{out}_{t+1,t+1}$ and $\beta _{t+1}$ depend on market outcomes between $t$ and $t+1$ , while the demographic assumptions underlying $\beta _{t+1}$ are fixed at time $t$ . Economically, $U^{out}_{t+1,t+1}$ is the amount required to settle the policy immediately at time $t+1$ upon death, whereas $\beta _{t+1}$ is the market-consistent continuation value of the contract at time $t+1$ . Hence, the sign of $\eta _{t+1}$ determines whether idiosyncratic deviations generate losses through excess deaths or through excess survivals: if $\eta _{t+1}\gt 0$ , an excess of deaths requires additional resources to settle the claim at $t+1$ , while if $\eta _{t+1}\lt 0$ the opposite mechanism prevails and excess survivals are costly.

The possibility of a negative SAR rate is specific to equity-linked contracts with path-dependent guarantees. Indeed, equation (3.10) implies that, for any $t+1\lt \tau \le n$ ,

(4.9) \begin{equation} {U^{out}_{t+1,\tau }=U^{out}_{t+1,t+1}\prod _{h=t+1}^{\tau -1}(1+p_h)\ \ge \ U^{out}_{t+1,t+1},} \end{equation}

with strict inequality whenever $p_h\gt 0$ on a set of positive probability. Therefore, the longer the policy remains in force, the greater the value of the future benefit that must be hedged, since in each remaining period the insurer must be able to finance the purchase of the put option required to enforce the guaranteed rate. When $\eta _{t+1}\lt 0$ (i.e. $\beta _{t+1}\gt U^{out}_{t+1,t+1}$ ), the immediate settlement value upon death is smaller than the continuation value. Intuitively, an early death shortens the period over which the guarantee must be provided and releases part of the replicating resources, whereas an unexpectedly large number of surviving policyholders extends the hedging horizon and may render the financial positions established at time $t$ insufficient. In this case, additional options and stock positions are required to preserve the guarantee over the remaining periods, so that idiosyncratic losses are driven by deviations towards higher-than-expected survivorship.

Applying the Tower Property to equation (4.5) implies that the expected value of the idiosyncratic component of the CDR is zero, i.e.,

(4.10) \begin{equation} \mathbb{E}^{\mathbb{P}}_t\big[CDR^{Idios}_{t+1}\big]=0. \end{equation}

In line with intuition, there is no expected accidental profit or loss, as, on average, accidental mortality aligns with expected mortality (see also Clemente et al., Reference Clemente, Della Corte, Savelli and Zappa2025 and Wüthrich & Merz, Reference Wüthrich and Merz2013).

The conditional variance of $CDR^{Idios}_{t+1}$ is given by the following formula (proved in Appendix D):

(4.11) \begin{equation} \mathrm{Var}_t\big [CDR_{t+1}^{Idios}\big ] = \big ( \ell _t\,\bar {C}_t^{2}\,q_{x+t}(1-q_{x+t} ) +\big (\ell _t^{2}\, \big(\bar {C}_t^{1} \big)^{2}-\ell _t\,\bar {C}_t^{2}\big)\sigma ^{2}_{Q_{x+t}} \big)\, \mathbb{E}_t^{\mathbb{P}}\left [\eta _{t+1}^{2}\right ], \end{equation}

where $\bar {C}_t^j$ is the raw moment of order $j$ of the vector of the sums insured and $\sigma ^2_{Q_{x+t}}$ is the variance of the parameter of the mixed Bernoulli random variable, $Q_{x+t}$ . Specifically, we observe that the idiosyncratic volatility consists of two components: the first is fully diversifiable by increasing the cohort size, while the second arises from the uncertainty in the BE of the Bernoulli parameter. Importantly, equation (4.11) relies on the assumption that, given the mortality parameter $Q_{x+t}$ , the deaths of individual policyholders are conditionally independent and identically distributed. A natural extension would be to relax this assumption using a frailty model (Olivieri and Pitacco, Reference Olivieri and Pitacco2016), which introduces a latent random effect capturing unobserved heterogeneity and dependence between policyholders. This would allow the model to account for correlated mortality shocks.

Under Solvency II regulation, the Standard Formula is the primary method for calculating the SCR. For Life Underwriting Risk, the SCR related to mortality and longevity is determined as the difference in Basic Own Funds between the base scenario and stressed scenarios, where mortality rates are increased by 15% and decreased by 20%, respectively. This calculation must be consistent with a VaR measure, based on a one-year time horizon and a 99.5% confidence level.

Our objective is to use the properties of the previously defined CDR to explicitly quantify the SCR and to propose an alternative methodology. Specifically, the idiosyncratic component of the SCR, denoted $SCR^{Idios}$ , is calculated as the negative 0.5% quantile of the distribution of $CDR_{t+1}^{Idios}$ .

To derive the distribution of $CDR_{t+1}^{Idios}$ and estimate the corresponding $SCR^{Idios}$ , we employ the procedure outlined in Algorithm 1 (Appendix E). The algorithm includes the generation of mortality scenarios, the simulation of stochastic cashflows, and the calculation of the capital requirement based on extreme quantiles.

4.2 Demographic – trend risk

As shown in formula (4.6), the CDR for trend risk depends on changes in BEs resulting from updates to demographic assumptions, i.e., from $\mathcal{F}_t^{\mathbb{I}}$ to $\mathcal{F}_{t+1}^{\mathbb{I}}$ . In practical terms, this requires updating the parameters of the forecasting model used for longevity and recalculating the BEs accordingly.

In the present framework, systematic mortality (trend) risk is assumed to be fully unhedgeable. Accordingly, the SCR computed in Section 5 is obtained under the modeling assumption that no mortality-linked financial instruments are used for risk mitigation. The reported capital absorption, therefore, reflects exposure to systematic mortality shocks in an unhedged setting. This modeling choice is intended to isolate the impact of demographic trend risk on the liability side and to ensure comparability with the standard Solvency II benchmark, under which no explicit hedge of systematic mortality risk is typically assumed.

At the same time, the literature documents the existence of mortality-linked securities, including longevity swaps, survivor bonds, and $q$ -forwards, which can, in principle, hedge part of the systematic longevity exposure, subject to liquidity and basis-risk considerations. For instance, zero-coupon longevity bonds linked to the survival index of a reference population provide traded exposure to aggregate mortality improvements and may embed a longevity risk premium (see Agarwal et al., Reference Agarwal, Ewald and Wang2025). From a market-consistent perspective, the availability of such instruments would enlarge the reference market and modify the replicating strategy: liability values could be represented by self-financing portfolios composed of standard financial assets and mortality-linked securities. In that case, the resulting SCR would capture only the residual (unhedged) component of systematic mortality risk, potentially compounded by longevity basis risk arising from an imperfect match between the insured portfolio and the reference population underlying the traded index.

In the following, we adapt the algorithm introduced in Clemente et al. (Reference Clemente, Della Corte, Savelli and Zappa2025) to facilitate the calculation of the SCR for trend demographic risk. Before outlining the details of the algorithm, we provide a qualitative overview of the methodology.

We begin at time $t$ by fitting a chosen stochastic mortality projection model – such as the LC model (Lee, Reference Lee2000), a cohort extension (Renshaw and Haberman, Reference Renshaw and Haberman2006), or a two-factor framework (Cairns et al., Reference Cairns, Blake and Dowd2006) – to the current training data $DB_t$ , thus establishing baseline forecasts for future mortality rates. To capture uncertainty around those forecasts, we then perform a bootstrap procedure over $M$ iterations (following Brouhns et al., Reference Brouhns, Denuit and Vermunt2002), drawing alternative parameter sets and generating a distribution of one-year mortality rates.

For each bootstrap draw, we simulate individual survival outcomes, treating each policyholder’s death as a Bernoulli trial with the drawn one-year mortality probability, and thereby obtain a stochastic profile of remaining sums insured at time $t+1$ . We then merge these simulated outcomes back into $DB_t$ to form an enriched dataset $DB^m_{t+1}$ for each scenario $m$ , and refit our mortality model on that dataset. This step naturally combines trend-risk uncertainty with idiosyncratic fluctuations, since each refitted model incorporates both long-term trend and idiosyncratic random variation (Zhu & Bauer, 2022).

Next, we simulate the evolution of our asset portfolio under the objective probability measure $\mathbb{P}$ and recompute optimal investment strategies for all future dates. Combining these updated financial valuations with the scenario-specific mortality exposures, we calculate, scenario by scenario, the CDR driven by trend risk. Finally, we examine the distribution of these simulated CDRs and take its 0.5% quantile (i.e., the 99.5% worst-case outcome) as the SCR for trend risk.

Algorithm 2, outlined in Appendix E, describes the procedure to obtain the distribution of $CDR_{t+1}^{Trend}$ and compute the associated $SCR^{Trend}$ . Subsequently, Algorithm 3 describes how to determine the overall $CDR$ distribution, as defined in formula (4.2), which combines idiosyncratic and trend risks, along with the calculation of the corresponding $SCR$ for total demographic risk.

5.1 Simulation parameters

Table 1 presents the baseline parameter values and main simulation assumptions. Our analysis focuses on a cohort consisting of 10,000 policyholders at the time of policy inception (i.e., at time 0). This cohort is selected assuming homogeneous risks within the group. Each policyholder is assigned an individual sum insured, drawn once from a log-normal distribution with a mean of 100,000 and a standard deviation equal to twice the mean. The vector of individual sums insured, $\mathbf{c}_0$ , is then kept constant in all simulations. A highly skewed distribution of sums insured allows the analysis to account for the impact of high-net-worth individuals, whose deaths can drive substantial losses, especially in cohorts of moderate size.

Table 1.

Input characteristics of the numerical analysis

At inception, all policyholders are aged 50. In the following subsection, we will also examine scenarios involving younger (age 30) and older (age 70) cohorts. The equity-linked endowment policies under consideration have a 10-year term, which is relatively short. We will also consider longer-term contracts to illustrate how, in particular, trend risk increases with policy duration.

The valuation is conducted at time $t=1$ , considering the market conditions at the end of year 2024. Consequently, at the time of valuation, the contract has a remaining duration of nine years. By simulating $CDR_{t+1}$ , we assess the one-year risk over the interval between $t=1$ and $t+1=2$ . To evaluate the SCR, we then focus on the losses in the worst-case scenario at a 99.5% confidence level. The valuation time $t$ , and thus the remaining duration of the contract, will be varied in subsequent analyses.

In the baseline scenario, the insurer considers mortality probabilities derived from the LC model (see Lee, Reference Lee2000; Brouhns et al., Reference Brouhns, Denuit and Vermunt2002) to be realistic. However, for pricing purposes, aiming to ensure a positive expected profit, the insurer adopts a demographic table based on second-order mortality rates increased by 20%.

The financial guarantee embedded in the policy is set at 1%, consistent with values typically applied in the insurance market and lower than the prevailing risk-free spot rate at the time of valuation. Indeed, the EIOPA one-year spot rate at the end of the year 2024 is approximately 2%. Consequently, assuming a volatility of the risky asset $\sigma =15\%$ and a market price of risk $\lambda =0.3$ , in line with the historical values of the main market indices, the risky asset is projected to yield an expected return of 6.5%. A sensitivity analysis using alternative values of $\sigma$ will be conducted at the end of the section.

The results are obtained using the R software environment R Core Team (2023), based on 10 million Monte Carlo simulations. The mortality model and the bootstrap procedure are implemented with the help of the StMoMo package (Villegas et al., Reference Villegas, Kaishev and Millossovich2018). Due to the relatively slow execution of the bootstrap estimation, primarily caused by the computational intensity of the functions provided by StMoMo, we verify that a moderate number of bootstrap iterations is sufficient to yield stable outcomes. In particular, 1,000 bootstrap simulations are adequate to achieve convergence in the distribution of $Q_{x+t}$ . As a result, we limit the number of iterations for the parametric bootstrap to 1,000 and subsequently resample the resulting parameters 10 million times. We maintain the full set of 10 million simulations for the generation of Bernoulli random variables, the modeling of financial components, and the trend risk refitting. A large number of simulations is required to obtain accurate results, especially for the estimation of extreme quantiles. The numerical stability of the Monte Carlo procedure is addressed in Appendix F through a convergence analysis of the resulting SCR estimates, which justifies the number of simulations adopted.

Figure 1

Simulated distributions of the Claims Development Result (CDR) for idiosyncratic, trend, and total demographic risks based on 10 million simulations.

5.2 Baseline analysis

Figure 1 displays the distribution of the CDR for idiosyncratic and trend risks, as well as the total CDR, while Table 2 reports the summary statistics of the distributions, together with the corresponding SCR. All distributions are centered around zero; however, the standard deviation associated with idiosyncratic risk is slightly higher than that arising from trend risk. The relative importance of the two components strongly depends on the duration of the contract, as discussed in the sensitivity analysis that follows.

Table 2.

Characteristics of the simulated distributions of Claims Development Result (CDR) and corresponding Solvency Capital Requirement (SCR) for idiosyncratic, trend, and total demographic risks based on 10 million simulations

Note: Mean, standard deviation, and SCR are in thousands of euros.

Regarding idiosyncratic risk, the overall variance is mainly influenced by the variability of the sum insured paid in the event of death during the year. This variance is attributable to two distinct sources: a diversifiable component due to process variance, and a systematic component stemming from uncertainty in parameter estimation (addressed here via a bootstrap approach). Our analysis indicates that the coefficient of variation (CV) of $Q_{x+t}$ is approximately 5%, implying that the diversifiable component accounts for 98.7% of the total variance of the CDR attributable to idiosyncratic risk.

In contrast, the main source of variability in trend risk emerges from the ongoing revision of second-order demographic assumptions during the year. Specifically, the second-order mortality rates at time $t+1$ are re-estimated by refitting the mortality model conditional on the updated filtration available at that time. The expected values of future mortality rates remain centered around the second-order rates used at time $t$ . We observe that the coefficients of variation of future annual mortality rates range from 5% to 7.5%, and tend to increase over time. Consequently, the SCR for trend risk is highly sensitive to the remaining duration of the contract.

Finally, the benefits of aggregation become apparent when the total demographic risk is assessed. Our analysis shows that the procedure, which implicitly links idiosyncratic and trend risks, results in a correlation of 0.11 between $CDR^{Idios}_{t+1}$ and $CDR^{Trend}_{t+1}$ . A positive correlation can be attributed to the shared volatility in mortality between times $t$ and $t+1$ : by comparing formula (4.5) with formula (4.6), it can be observed that, when the policy features a negative SAR rate (in this simulation, $\mathbb{E}^{\mathbb{P}}_t\left [\eta _{t+1}\right ]=-0.66$ ), an increase in longevity leads to greater idiosyncratic risk and, consequently, a higher number of BE liabilities that must be reserved for the policyholders still alive. The negative SAR rate can be easily explained by Theorem 1: since hedging a payoff that occurs further in the future requires larger positions in the risky asset and the put option, the idiosyncratic risk for a cohort of equity-linked endowment policies becomes a survival risk, in which a policyholder living longer than expected leaves the company without sufficient holdings in the risky asset and put option to fully hedge the position. However, the correlation remains low because the trend component is influenced by several other factors, primarily the revision of technical bases. Consequently, when the aggregate demographic risk is considered, a diversification benefit of 22% emerges in the total SCR relative to the sum of the SCRs attributable to idiosyncratic and trend risks.

5.3 Sensitivity analysis

In the following, we assess the impact of key model parameters on the SCR. Figure 2 shows the impact of the mortality models used for the estimation of the death rate by comparing the SCRs obtained using the LC model (Lee, Reference Lee2000; Brouhns et al., Reference Brouhns, Denuit and Vermunt2002), the original version of the Cairns–Blake–Dowd (CBD) model (Cairns et al., Reference Cairns, Blake and Dowd2006), and the Renshaw-Haberman (RH) model (Renshaw & Haberman, Reference Renshaw and Haberman2006). We observe that the proposed approach captures parameter uncertainty within the idiosyncratic component and the model’s sensitivity to new observations during the refitting procedure. The selection of the mortality model significantly affects the SCR for the three risks considered. For idiosyncratic risk, the LC model results in the highest SCR, indicating greater sensitivity to individual mortality variations compared to the CBD and RH models. Although the LC and RH models provide similar expected annual mortality rates, the relative volatility of $Q_{x+t}$ based on bootstrap is 5.1% and 4.3%, respectively. Regarding the CBD model, we observe both a lower expected annual mortality rate and a relative volatility close to that of the RH model, leading to the lowest SCR for idiosyncratic risk.

Figure 2

Solvency Capital Requirement (SCR) for idiosyncratic, trend, and total demographic risks based on alternative mortality models: Lee-Carter (LC), Cairns–Blake–Dowd (CBD), and Renshaw–Haberman (RH).

The difference between the LC model and the other models becomes more pronounced when trend risk is considered, indicating that the CBD and RH models are less sensitive to trend risk. When extending the time horizon to the remaining coverage period of the contract, the relative volatility of mortality rates increases significantly under the LC model. Specifically, the CV rises from 5.0% to 7.3% for the LC model, while it increases from approximately 3.5% to 5.3% for the other models. The slight difference between the RH and CBD models is due to a more conservative estimation of the second-order probability in the RH model.

Overall, the choice of mortality model has a substantial effect on SCR outcomes. The LC model consistently produces higher capital requirements across all risk components, offering a more conservative estimate of demographic risk. This highlights the importance of model selection in accurately assessing and managing demographic risks in actuarial practice.

Figure 3

Solvency Capital Requirement (SCR) for idiosyncratic, trend, and total demographic risks based on alternative values of $t$ (valuation time) and $x$ (age).

Figure 4

Solvency Capital Requirement (SCR) for idiosyncratic, trend, and total demographic risks based on alternative values of $n$ (duration of the contract) and $\sigma$ (volatility of the risky asset).

In Figure 3, we show the effect of varying cohorts, defined by different ages at inception $x$ and different valuation times $t$ . As expected, a higher age at inception implies a higher capital requirement due to increased expected mortality rates and greater volatility. The increase is noticeably higher for trend risk, which reflects the variability in the refitted mortality rates throughout the remaining coverage period of the contract. At time 1, the idiosyncratic SCR increases by a factor of 13 when the policyholder’s age rises from 30 to 70. For trend risk, the increase is nearly twice as large.

When varying the valuation time and focusing on idiosyncratic risk, the standard deviation of sums insured in the event of death is influenced by several opposing factors. On one hand, volatility rises with $t$ , due to the increasing average annual death probability over time. On the other hand, the number of surviving policyholders and the expected sums insured, as well as the average value of the SAR rate, decrease. As a result, the idiosyncratic SCR does not necessarily follow a monotonic trend over time, although a reduction in the amount is observed as the contract nears its end. In contrast, the SCR for trend risk declines markedly as the contract approaches expiry, primarily due to the diminishing impact of rate revisions on BE calculations over a shorter remaining horizon. In the final year, when the event becomes certain, trend risk effectively vanishes. This downward trend is particularly pronounced for cohorts of older policyholders.

Finally, in Figure 4, we consider different contract durations $n$ and alternative return volatilities of the risky asset $\sigma$ . A longer contract duration does not affect the next year’s volatility of mortality rates; therefore, in formula (4.11), the only element influenced by the duration is the SAR rate. Due to the extended horizon, a noticeable increase in the variability of the expected SAR rate is observed, leading to a higher capital requirement for idiosyncratic risk. For trend risk, the longer time horizon results in increased volatility of both the SAR rate and mortality rates, which together contribute to a greater increase in the SCR. Specifically, as the duration increases from 10 to 30 years, the SCR rises by a factor of 50.

A crucial parameter is the Geometric Brownian Motion volatility $\sigma$ , which appears in the dynamics of the price of the risky asset in (2.1). As expected, higher volatility in the financial model implies a greater capital requirement. This effect is more pronounced for trend risk. Notably, compared to the reference scenario, the combination of a longer duration and a higher $\sigma$ results in the SCR increasing by a factor of 10 for idiosyncratic risk and by a factor of 86 for trend risk. The overall SCR obtained is equal to 10.81% of the sum insured at the beginning of the year, whereas in the reference scenario, this ratio is 0.18%.

In Figure 5, we evaluate the effects of the relative variability of the insured sums and of the minimum guaranteed rate on the SCR for both idiosyncratic and trend risks. The variability of the insured sums plays a crucial role in the variability of demographic risk, particularly for idiosyncratic risk. Specifically, in the base case where the minimum guaranteed rate is $g = 1\%$ , an increase in the CV of the insured sums from 0 to 2 results in a 42% increase in capital requirements for idiosyncratic risk, whereas the increase is approximately 8.2% for trend risk. Conversely, when focusing on the minimum guaranteed rate and holding the relative variability of the insured sums fixed at 1, raising the rate from 0% to 2% leads to an SCR increase of 36% for idiosyncratic risk and 13% for trend risk.

Overall, both the variability of the insured sums and the minimum guaranteed rate have a greater impact on idiosyncratic risk than on trend risk. However, the CV is more influential for idiosyncratic risk, while the minimum guaranteed rate exerts a stronger effect on trend risk.

Figure 5

Solvency Capital Requirement (SCR) for idiosyncratic and trend demographic risks based on alternative values of minimum guaranteed rate ( $g$ ) and coefficient of variation of the insured sums (CV).