2.1. Hedgeable claims

We assume that there is an arbitrage-free financial market with $n^f$ traded payoffs with maturity T. The traded assets can be bought and sold by all market participants at any quantity. The bank account at time T is denoted by $Y_0$ . Since we assume a deterministic rate r, we have that, starting from an initial investment of 1 at time $t=0$ , the bank account is given by $Y_0=\text{e}^{rT}$ . The payoff of asset i at time T is denoted by $Y_i$ . We use the random vector $\boldsymbol{{Y}}=\left(Y_0, Y_1, Y_2, \ldots, Y_{n^{f}}\right)$ to denote the bank account and the $n^f$ traded assets. The unique time-0 price $y_i$ of the payoff $Y_i$ can be observed in the market. We assume that $y_0=1$ . A financial derivative S is a function of the traded assets, that is, $S=h\left(\boldsymbol{{Y}}\right)\!,$ for some Borel-measurable function h. We denote the set of all financial derivatives by $\mathcal{C}^{\boldsymbol{{Y}}}$ . Throughout the paper we assume that all functions we encounter are Borel measurable.

A claim $S^{h}\in \mathcal{C} $ is a hedgeable claim if we have a real vector $\boldsymbol{\nu}\in \mathbb{R}^{n^f+1}$ such that $S^h = \boldsymbol{\nu} \cdot \boldsymbol{{Y}}, $ almost surely. The set of hedgeable claims is denoted by $\mathcal{C}^h.$ The vector $\boldsymbol{\nu}$ is called the hedge of the claim $S^h.$ We assume that the tradeable assets are non-redundant, that is, we cannot use a subset of the traded assets to replicate the payoff of another traded asset. Assume $ \boldsymbol{\theta}\in \mathbb{R}^{n^f+1}$ . Then $\boldsymbol{\theta} \cdot \boldsymbol{{Y}}=0$ implies $\boldsymbol{\theta} = \left( 0,0,\ldots, 0\right)$ . Since we assume the traded assets to be non-redundant, the hedge $\boldsymbol{\nu}$ is unique. Of course, a hedgeable claim is a financial derivative, that is, we have $\mathcal{C}^h \subseteq \mathcal{C}^{\boldsymbol{{Y}}}$ . In case both sets coincide, the market is complete.

A hedgeable claim can be managed by buying at time $t=0$ the hedge $\boldsymbol{\nu}$ at the market price $\boldsymbol{\nu}\cdot \boldsymbol{{y}}$ . Since the hedge will replicate the payoff of the liability, the residual liability, that is, the liability after taking into account the payout of the hedge, is zero. In order to avoid arbitrage opportunities, the price of a hedgeable claim should be equal to the unique price of the hedge. We assume there is a risk neutral probability measure. Note that in our setting, the existence of a risk neutral probability measure is equivalent to the absence of arbitrage. $\mathbb{Q}$ which is equivalent to the real-world probability measure $\mathbb{P}$ . The time-0 prices can be expressed as $y_i=\text{e}^{-rT}\mathbb{E}_{\mathbb{Q}}\left[Y_i \right],\ \text{for}\ i=0,1,2,\ldots,n^{f}.$ The price of a hedgeable claim should be determined using a discounted risk neutral expectation. The financial market, however, is assumed to be incomplete, which implies there are different risk neutral measures and there may be financial derivatives which cannot be perfectly replicated.

2.2. Diversifiable claims

The claim $S^d \in \mathcal{C}$ is said to be a diversifiable claim if there exist i.i.d. risks $V_1, V_2,\ldots, V_{n} \in \mathcal{C}$ such that

(2.1) \begin{equation} S^d = \frac{1}{n}\sum_{i=1}^{n} V_i,\end{equation}

where $n>1$ . A diversifiable claim is a combination of independent claims $V_i$ in (2.1) which are attributed to individual policies. An insurer holding a diversifiable claim insures a finite number of individual and independent policies. However, we assume the insurer can always decide to increase the number of policies in the portfolio if necessary, that is, we assume there exist more copies of $V_i$ than the insurer currently holds in its portfolio. Indeed, in order for a claim to be diversifiable, the insurer holding the claim should be able to diversify, that is, increase the number of policies (n) in its portfolio. The claim $S^d$ is not perfectly diversified since only a finite number of policies are included. Since we assume that $\mathbb{E}[V_1]$ is finite which implies that the weak law of large numbers can be applied if we allow n to grow. Longevity risk, for example, impacts the future lifetime of a large group of policyholders simultaneously in the same direction. Although the longevity risk of different groups may be independent (e.g., longevity in the United States and China), one cannot find a large number of independent copies which means an insurer may not find enough independent copies to appropriately diversify the risk.

We can determine a value and the corresponding risk management strategy for a diversifiable claim. A diversifiable claim could be a linear combination of n independent hedgeable claims $V_i$ . In this case, the diversifiable claim is also hedgeable and one can construct a hedging strategy. The value of the diversifiable claim then corresponds with the price to set up the hedging strategy, that is, pricing is based on risk neutral valuation. Assume now the diversifiable claim is not hedgeable. The law of large numbers states that if n tends to infinity, the claim $S^d$ converges to $\mathbb{E}[V_1]$ , where this expectation is taken under the real-world distribution $\mathbb{P}$ . In reality, an insurer can never hold a portfolio with infinitely many contracts. Therefore, the realization of $S^d$ fluctuates around the expectation $\mathbb{E}[V_1]$ . For a sufficiently large n, the central limit theorem shows that the distribution of $S^d$ is approximately normal distributed, that is

(2.2) \begin{equation} S^d\approx\mathbb{E}\left[S^d\right] + \sqrt{\text{Var}\left[S^d\right]} \Phi^{-1}(U), \end{equation}

where $\Phi$ is the cdf of a standard normal distribution and U is a uniform distribution.

The value of a non-hedgeable diversifiable claim $S^d$ should correspond with the amount of money required to be able to pay the claim with a sufficiently large likelihood. A possible value for the diversifiable claim can be determined using the following valuation:

(2.3) \begin{equation} \rho^{a}\left[S^d \right] = \text{e}^{-rT}\left(\mathbb{E}\left[S^d \right] + \lambda\left[S^d \right]\right), \end{equation}

where $\lambda\left[S^d \right] \in \mathbb{R}$ is a risk margin capturing deviations of $S^d$ from the average claim amount. The capital $ \rho^{a}\left[S^d \right]$ is invested in the risk-free bank account until the maturity T of the claim. Taking into account (2.2), the probability of insolvency is expressed as follows:

(2.4) \begin{equation} \mathbb{P}\left[S^d>\text{e}^{rT} \rho^{a}\left[S^d \right]\right] \approx 1-\Phi\left(\frac{\lambda\left[S^d \right]\sqrt{n}}{\sqrt{\text{Var}[V_1]}} \right). \end{equation}

We conclude that if we have a non-hedgeable diversifiable claim and a valuation with a given risk margin determined by $\lambda$ , we can reduce the insolvency probability by increasing the size of the portfolio. For a given insolvency probability, the risk margin is a decreasing function of the portfolio size.

A valuation $\rho^a$ which has the form of (2.3) and where $\lambda\left[S^d \right]$ is $\mathbb{P}$ -law invariant is called an actuarial valuation. Note that a valuation $\rho^a$ is $\mathbb{P}-$ law invariant if $S_1\overset{d}{=}S_2$ implies $\rho^a\left[ S_1\right]=\rho^a\left[ S_2\right]$ , where ‘ $\overset{d}{=}$ ’ stands for equality in distribution under the probability measure $\mathbb{P}.$ The set with all actuarial valuations is denoted by $\mathcal{A}$ . The risk margin $\lambda$ can be determined using the standard deviation of the claim, which results in the standard deviation principle. The value of a claim $S\in\mathcal{C}$ valuated using the standard deviation principle is then given by

(2.5) \begin{equation} \rho^a[S] = \mathbb{E}[S] + \beta \times \sqrt{\text{Var}[S]}, \end{equation}

where $\beta>0$ is called the safety loading. A $\mathbb{P}-$ law invariant valuation $\rho^a$ can always be written as $\rho^a\left[S \right]= \text{e}^{-rT}\left(\mathbb{E}\left[S \right]+ \left(\text{e}^{rT}\rho^a\left[S\right] - \mathbb{E}\left[ S\right]\right)\right),$ which shows we can write the valuation $\rho^a\left[S \right]$ as in (2.3). Therefore, the class of actuarial valuations can equivalently be defined as the class of all $\mathbb{P}-$ law invariant valuations.

Figure 1. The distribution of the claim S given by $S=\frac{1}{n^a}\sum_{i=1}^{n^a}X_i$ , where $p_0=0.75, p_1=0.85$ and $n^a=1000$ . The solid vertical line corresponds with the value of the claim when a standard deviation principle with $\beta=1$ is used. The dashed lines represent the value of the conditional claims $S_0$ and $S_1$ .

If we condition on the longevity index Z, the claim is diversifiable. Indeed, consider the random variable $S_k= S\ |Z=k,\ \text{where} \ k=0,1. $ Then $S_k$ is a diversifiable claim. Moreover, if we use a standard deviation principle $\rho^a$ to valuate the claim, we find that $\rho^a\left[ S_k\right]\rightarrow \mathbb{E}[S_k]$ if $n^a\rightarrow +\infty.$ The two dashed lines in Figure 1 correspond with the value of the claims $S_k$ , when a standard deviation principle is used. In this paper, we propose a new valuation that takes into account that valuating the claim S using solely a valuation based on diversification is not appropriate, but will exploit the fact that the conditional claims $S_0$ and $S_1$ are diversifiable.

2.3. Hybrid claims

In this paper, we consider an insurance company holding a portfolio with liabilities. The portfolio consists of a total of $n^{a}$ policyholders. We assume the random variable $X_i$ is modeling policyholder specific risks. For example, $X_i$ can be a random variable taking the value 1 if the policyholder survives until time T and 0 otherwise. We assume that claims depending on the insurer’s policyholder specific risks are not traded in the financial market and are therefore not perfectly replicable by the traded assets $\boldsymbol{{Y}}$ . However, the random variables $X_i$ , $i=1,2\ldots,n^{a}$ can be dependent on the financial market, meaning that they may be partially hedgeable.

We assume the portfolio of the insurer is also exposed to a number of non-traded systematic risks. We denote these non-traded systematic risks by the random vector $\boldsymbol{{Z}}=\left(Z_1, Z_2, \ldots, Z_{n^{s}}\right)$ . We assume that the random variables $X_i$ denoting the policyholder specific risks are conditionally independent:

(2.6) \begin{equation}\mathbb{P}\left[ X_1\leq x_1, X_2\leq x_2, \ldots, X_{n^a}\leq x_{n^a}|\ \boldsymbol{{Y}}=\boldsymbol{{y}}, \boldsymbol{{Z}}=\boldsymbol{{z}}\right]=\prod_{i=1}^{n^a}\mathbb{P}\left[X_i\leq x_i|\ \boldsymbol{{Y}}=\boldsymbol{{y}}, \boldsymbol{{Z}}=\boldsymbol{{z}} \right].\end{equation}

We thank the anonymous referee to point out that this condition is a simplifying assumption. The complexity of some practical situations may distort the conditional independence.

Consider a claim S which can be expressed as $S=h\left(\boldsymbol{{Z}}\right)$ for some function h. This claim is called a systematic claim, and we denote the set with all such claims by $\mathcal{C}^{\boldsymbol{{Z}}}$ . The set with all claims containing financial and systematic information is denoted by $\mathcal{C}^{\boldsymbol{{Y}},\boldsymbol{{Z}}}$ , that is, if $S\in \mathcal{C}^{\boldsymbol{{Y}},\boldsymbol{{Z}}},$ then $S=h\left(\boldsymbol{{Y}},\boldsymbol{{Z}} \right)$ for some function h.

Definition 2.1. A claim S is said to be a pure actuarial claim if S is independent of the financial and non-traded systematic risks. We denote by $\mathcal{C}^{\perp,\boldsymbol{{Y}}, \boldsymbol{{Z}}}$ the set of all pure actuarial claims

\begin{align*} \mathcal{C}^{\perp,\boldsymbol{{Y}}, \boldsymbol{{Z}}}= \left\{S\in \mathcal{C}\ \text{and}\ S\ \text{is independent of}\ \boldsymbol{{Y}}\ \text{and}\ \boldsymbol{{Z}} \right\}. \end{align*}

We will investigate a portfolio of policies, where the payoff for policyholder i at maturity T is given by $S^f\times g\left(X_i, \boldsymbol{{Z}} \right).$ The claim $S^f$ is a financial derivative and g is a known function. Note that the financial derivative $S^f$ is not necessarily hedgeable, that is, it may be that $S^f\notin \mathcal{C}^h$ .

The per-policy liability to the insurance company is then given by

(2.7) \begin{equation}S=S^f\times \frac{\sum_{i=1}^{n^{a}}g\left( X_i,\boldsymbol{{Z}}\right)}{n^{a}}.\end{equation}

The structure (2.7) together with Condition (2.6) allows to decompose the hybrid claim in three distinctive parts which individually require a different type of valuation. The aim of this paper is to propose a valuation that shows how to combine these different type of valuations.

In this paper, we assume that hedgeable claims are valuated using risk neutral valuation and pure actuarial claims are valuated with a given actuarial valuation; see (2.3). The idea of the 3-step hedge-based valuation is that we have an actuarial valuation at hand that we want to use for valuating the diversifiable part of the claim. Consider a claim S of the form (2.7) and define the conditional random variable $S_{\boldsymbol{{y}}, \boldsymbol{{z}}}$ which is given by $S_{\boldsymbol{{y}}, \boldsymbol{{z}}}=\left(S|\ \boldsymbol{{Y}}=\boldsymbol{{y}},\ \boldsymbol{{Z}}=\boldsymbol{{z}}\right)\!.$ Then we find from Condition (2.6) that $S_{\boldsymbol{{y}}, \boldsymbol{{z}}}$ is a diversifiable claim (see (2.1)) and therefore can be valuated using an actuarial valuation. Moreover, the random vector $\boldsymbol{{Z}}$ can be interpreted as the non-traded systematic effects. The valuation proposed in this paper then assumes that the systematic effects should be valuated with a different valuation than the diversifiable part. We assume that we know how to deal with diversifiable claims through a particular choice of an actuarial valuation $\rho^a$ . Since we only have a claim that is diversifiable given the outcomes of the traded and non-traded systematic effects, we can only determine the value of the claim in each systematic scenario. Otherwise stated, once we know $\boldsymbol{{y}}$ and $\boldsymbol{{z}}$ , we can valuate this claim $S_{\boldsymbol{{y}}, \boldsymbol{{z}}}$ using the actuarial valuation $\rho^a$ . Since the outcomes of the systematic scenario are random, we will determine a value $\rho^a\left[ S_{\boldsymbol{{y}}, \boldsymbol{{z}}}\right] $ for each possible realization of $\boldsymbol{{Y}}$ and $\boldsymbol{{Z}}$ . In the last step of our valuation, we apply a separate valuation that takes into account the uncertainty about the systematic scenario, that is, the values for $\boldsymbol{{y}}$ and $\boldsymbol{{z}}$ , that will eventually unfold.

Note that not all claims in $\mathcal{C}$ are product claims of the form (2.7). For such a general hybrid claim, however, one may not be able to define a diversifiable part which should be valuated with an actuarial valuation. Indeed, diversifiability of a claim implies the insurer pools many policyholders with the same policy and is able to increase the size of the portfolio. If the hybrid claim does not contain a diversifiable part, that is, in case we do not have the part $\frac{\sum_{i=1}^{n^{a}}g\left( X_i,\boldsymbol{{Z}}\right)}{n^{a}}$ together with the Condition (2.6), it may not be intuitively clear why one has to combine different valuation principles for the residual part of the claim. An example of a claim that does not have the structure as in (2.7) are option-type payoffs, for example, $S= \left(S^f\times \frac{\sum_{i=1}^{n^{a}}g\left( X_i,\boldsymbol{{Z}}\right)}{n^{a}}-K\right)_+$ , where $(x)_+=\max(x,0)$ . Another example is the claim $S=S^f\times \prod_{i=1}^n X_i$ . If the $X_i$ takes value 1 in case policyholder i survives and 0 otherwise, the claim S only pays the amount $S^f$ if everyone survives. However, these examples are rather theoretical and by restricting the set of claims to product claims of the form (2.7), we cover most realistic situations.

By considering claims S which can be expressed as in (2.7), we assume that each of the $n^a$ policyholders receives the same financial benefit $S^f$ . If an insurer sells different types of contracts, we have that the financial benefit depends on the policyholder i, that is, we have to write $S_i^f$ . In this case, we first subdivide the contracts in portfolios where each policy receives the same financial benefit and secondly, we valuate each portfolio separately. In Bacinello et al. (Reference Bacinello, Millossovich, Olivieri and Pitacco2011), the authors also consider product claims and describe different choices for $S^f$ , encompassing the most common minimum death and living guarantees. Their valuation then assumes a perfectly diversified portfolio, that is, there is no risk margin in the actuarial valuation and the financial market is independent of the mortality risk, that is, $\boldsymbol{{Y}}$ is independent of $\boldsymbol{{Z}}$ and $\boldsymbol{{X}}$ ; see also Brennan and Schwartz (Reference Brennan and Schwartz1976b), Deelstra et al. (Reference Deelstra, Devolder, Gnameho and Hieber2020) for the one-period case and Ballotta et al. (Reference Ballotta, Eberlein, Schmidt and Zeineddine2021), Delong et al. (Reference Delong, Dhaene and Barigou2019b) for the multiperiod case. Product claims with dependent financial and non-financial risks are considered in Deelstra et al. (Reference Deelstra, Grasselli and Van Weverberg2016), Salahnejhad and Pelsser (Reference Salahnejhad and Pelsser2020), Barigou et al. (Reference Barigou, Linders and Yang2022).

We write g instead of $g(X_i, \boldsymbol{{Z}})$ if no confusion is possible. The claim S is called a hybrid claim, since it is a combination of different type of risks. The aim of this paper is to determine a valuation framework to valuate the hybrid claim S.

3.1. Definition

A hybrid claim depends on the information about the traded assets $\boldsymbol{{Y}}$ . It is then reasonable to impose that a valuation of a hybrid claim takes into account the market prices $\boldsymbol{{y}}$ of the traded assets. The valuation of the hybrid claim should be ‘consistent’ with the available market information. Such valuations are called market-consistent valuations. However, employing the market information may not be appropriate for a pure actuarial claim. Therefore, we also require that valuations are model consistent. Combining market consistency with model consistency leads to the class of fair valuations; see also Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017).

If we decompose a hybrid claim in a hedgeable part $\boldsymbol{\nu}\cdot \boldsymbol{{Y}}$ and a residual part S, then any market-consistent value is the sum of the value of the hedgeable and residual part. One can buy the hedgeable claim $\boldsymbol{\nu}\cdot \boldsymbol{{Y}}$ at the price $\boldsymbol{\nu}\cdot \boldsymbol{{y}}$ and therefore the value of the hedgeable part is unambiguously determined by its hedging cost. The residual part S, on the contrary, has to be valuated using the valuation $\rho$ . If the value of a hybrid claim is determined with a market-consistent valuation, this value is consistent with the prices of the available traded financial assets in that it takes into account the market price to buy the hedge.

The set of all model-consistent valuations is denoted by $\mathcal{M}$ . Expression (3.2) states that a model-consistent valuation employs a given actuarial valuation $\pi$ , which is usually different from $\rho$ , for pricing a pure actuarial claim. Note that the set of actuarial valuations is a subset of the model-consistent valuations, that is, $\mathcal{A} \subseteq \mathcal{M}.$ For general hybrid claims, the valuation $\rho$ will be a combination of the hedging cost $\boldsymbol{\nu}\cdot \boldsymbol{{y}}$ and a model-consistent valuation.

In Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017), the authors consider a slightly different definition of a model-consistent valuation. Indeed, in their setting, a valuation is model consistent if an actuarial valuation is used for any claim which is independent of the financial assets. Hence, claims which depend on non-traded systematic information, but are independent of the financial market are also valuated using an actuarial valuation. Note, however, that using a less restrictive definition for a model-consistent, and therefore also a fair valuation does not change the results of Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017) which we will need in this paper.

We have to define the notion of a hedger. These concepts were first defined in Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017) and we refer to this paper for a detailed discussion and more properties on hedgers and hedge-based valuations. A hedger is a function $\boldsymbol{\theta}\;:\;\mathcal{C}\rightarrow \mathbb{R}^{n^f+1}$ which is normalized, that is, $\boldsymbol{\theta}_0=\left(0,0,\ldots, 0 \right)$ and translation invariant, that is, $\boldsymbol{\theta}_{S+a} = \boldsymbol{\theta}_S+\left( \text{e}^{-rT}a, 0,0,\ldots, 0\right)$ , for $S\in \mathcal{C}$ and $a\in \mathbb{R}.$ The hedging strategy $\boldsymbol{\theta}_S=\left(\theta_S^{(0)}, \theta_S^{(1)}, \ldots, \theta_S^{\left(n^f\right)} \right)$ is also called the hedge for the claim S. The component $\theta_S^{(i)}$ of the hedge determines the amount of units to be bought of the traded asset $Y_i$ . It is reasonable to assume that the hedge of a hedgeable claim $\boldsymbol{\nu}\cdot \boldsymbol{{Y}}$ corresponds with $\boldsymbol{\nu}$ . Consider the set $\mathcal{C}^{\perp, \boldsymbol{{Y}}, \boldsymbol{{Z}}}$ containing the claims that are independent of the traded assets $\boldsymbol{{Y}}$ and the non-traded systematic risks $\boldsymbol{{Z}}$ . It is also reasonable to assume that the hedge for a claim in $\mathcal{C}^{\perp, \boldsymbol{{Y}}, \boldsymbol{{Z}}}$ only consists of a position in the risk-free bank account. Therefore, we assume that all hedgers we encounter in this paper are fair hedgers, as defined in Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017).

The idea of market-consistent, model-consistent and fair hedgers was first introduced in Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017). The following theorem was proven in Theorem 3 of Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017) and characterizes the class of valuations that are both market and model consistent.

The valuation defined in (3.3) is called a hedge-based valuation and was introduced in Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017) in a one-period setting. In Barigou and Dhaene (Reference Barigou and Dhaene2019), Barigou et al. (Reference Barigou, Chen and Dhaene2019) and Chen et al. (Reference Chen, Chen, Dhaene and Yang2021), the authors consider the hedge-based valuations in a multi-period setting, whereas continuous-time versions of the hedge-based valuations are considered in Delong et al. (Reference Delong, Dhaene and Barigou2019a,b).

A hedged-based valuation employs a fair hedger $\boldsymbol{\theta}$ to determine the hedgeable part of a hybrid claim. This hedgeable part can be priced using the available market information. The remaining, residual, part of the claim is then priced using an appropriate model-consistent valuation. We say that $S-\boldsymbol{\theta}_S\cdot \boldsymbol{{Y}}$ is the residual part of the hybrid claim S.

4.1. Definition of the 3-step hedge-based valuation

One may argue that, when using the hedge-based valuation, one can use an actuarial valuation to valuate the residual part. The actuarial valuation should then be determined such that it corresponds with setting up a sufficiently large capital buffer as the risk management strategy to cope with the residual part. For example, one may use a standard deviation principle with a sufficiently large safety loading, or the VaR/TVaR with a confidence level close to 1. An example of such a valuation is explored in Example 2. Theorem 3.1, however, states that there exist fair valuations which employ a model-consistent, but not an actuarial, valuation to valuate the residual part. In this paper, we will consider these fair valuations. A first example of such a valuation is introduced in Example 3.

Example 2 (Capital for the residual part). By decomposing a claim S in a hedgeable part and a residual part using a hedger $\boldsymbol{\theta}$ , we mainly have to focus on the valuation of the residual part. Indeed, the value of the hedgeable part is uniquely determined by its hedging cost. One possibility to valuate the residual part is to determine a sufficiently large capital buffer. For example, one could use the following valuation:

\begin{align*} \rho[S] =\boldsymbol{\theta}_S\cdot \boldsymbol{{y}} + \pi\left[ S - \boldsymbol{\theta}_S\cdot \boldsymbol{{Y}}\right], \end{align*}

where $\pi$ is an actuarial valuation. The amount $\pi\left[ S - \boldsymbol{\theta}_S\cdot \boldsymbol{{Y}}\right]$ is a capital buffer which is invested in the risk-free asset until the maturity T of the claim. For example, one can use the VaR to determine the capital buffer.

Example 3 (The quantile hedging valuation of Barigou et al. (Reference Barigou, Bignozzi and Tsanakas2021)). In Barigou et al. (Reference Barigou, Bignozzi and Tsanakas2021), the authors start from the hedge-based valuation, but propose to use a quantile hedging approach for the residual part of the claim. Assume that $i\in(0,1)$ is the cost-of-capital rate. The quantile hedging valuation is defined as follows:

(4.1) \begin{equation}\rho\left[S\right] = \boldsymbol{\theta}_S\cdot \boldsymbol{{y}} + i\times \boldsymbol{\eta}^{(l)}_{S-\boldsymbol{\theta}_S\cdot \boldsymbol{{Y}} }\cdot \boldsymbol{{y}},\end{equation}

where $\boldsymbol{\theta}$ is a fair hedger and $ \boldsymbol{\eta}^{(l)}$ is a hedger defined as follows:

(4.2) \begin{equation} \boldsymbol{\eta}^{(l)}_S = \arg \min_{\boldsymbol{\beta}\in \mathbb{R}^{n^f+1}}\mathbb{E}\left[l\left( S-\boldsymbol{\theta}_S\cdot \boldsymbol{{Y}} - \boldsymbol{\beta}\cdot \boldsymbol{{Y}}\right) \right],\end{equation}

and $l\;:\; \mathbb{R}\rightarrow [0,+\infty)$ is a convex function satisfying $l(x)=0 \leftrightarrow x=0$ . The residual part is valuated using a model-consistent valuation which is not necessarily actuarial. Instead of using a capital buffer for coping with the residual part of the hybrid claim, the underlying risk management strategy is buying an appropriate investment portfolio. Note that the hedger $\boldsymbol{\eta}^l$ is a fair hedger.

The difference between Examples 2 and 3 lies in the valuation of the residual part. Example 2 uses a ‘passive’ risk management strategy where the risks of the residual part are covered by a capital buffer. Example 3, on the other hand, uses an ‘active’ risk management strategy which invests in an appropriate investment portfolio. In this section, we will build a valuation framework that incorporates both approaches.

We can now define the class of $\left(\boldsymbol{{Y}},\boldsymbol{{Z}}\right)-$ conditional valuations, which are valuations that map a hybrid claim into the set $\mathcal{C}^{\boldsymbol{{Y}}, \boldsymbol{{Z}}}$ .

Note that we only require conditionally translation invariance for hedgeable claims. In the remainder of the paper, we will write ‘conditional valuation’ instead of ‘ $\left(\boldsymbol{{Y}}, \boldsymbol{{Z}}\right)-$ conditional valuation’ if no confusion is possible.

Assume that we have an actuarial valuation $\rho^a \in \mathcal{A}$ . In case a claim is independent of the financial and the non-traded systematic risks, we require that the claim is valuated using $\rho^a$ . Indeed, since we consider claims of the form (2.7), independence of the financial market and the systematic risks implies that the claim is diversifiable, which justifies the use of an actuarial valuation. If a conditional valuation is consistent with this requirement, we say that the valuation is conditional model-consistent.

Definition 4.2. The conditional valuation $\pi\left[\cdot|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right]\;:\; \mathcal{C}\rightarrow \mathcal{C}^{\boldsymbol{{Y}}, \boldsymbol{{Z}}}$ is a conditional model-consistent valuation if pure actuarial claims are valuated with an actuarial valuation:

(4.3) \begin{equation}\text{e}^{-rT}\pi\left[S|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right]=\rho^a\left[S\right],\ \text{for}\ S\in\mathcal{C}^{\perp,\boldsymbol{{Y}}, \boldsymbol{{Z}}},\end{equation}

where $\rho^a\in \mathcal{A}$ is a model-consistent valuation.

An example of a conditional model-consistent valuation is the conditional standard deviation principle, which is defined as follows:

(4.4) \begin{equation}\rho^a\left[S|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right] = \mathbb{E}\left[ S|\ \boldsymbol{{Y}}, \boldsymbol{{Z}}\right] + \beta \sqrt{\text{Var}\left[ S|\ \boldsymbol{{Y}}, \boldsymbol{{Z}}\right] }.\end{equation}

We can now define the 3-step hedge-based valuation. In a first step, an appropriate hedging strategy $\boldsymbol{\theta}_S$ is used to offset a hedgeable part of a hybrid claim S. In the second valuation step, the residual claim $S-\boldsymbol{\theta}_S\cdot \boldsymbol{{Y}}$ is transformed by using a conditional model-consistent valuation. Indeed, since we consider claims of the form (2.7), conditioning on $\boldsymbol{{Y}}$ and $ \boldsymbol{{Z}}$ will transform the residual claim in a diversifiable claim for which the actuarial valuation $\rho^a$ can be used. The last, and third step valuates the remaining claim using a model-consistent valuation $\rho^s$ .

Definition 4.3. (3-step hedge-based valuation). A valuation $\rho$ is said to be a 3-step hedge-based valuation if for any claim $S\in \mathcal{C}$ , it can be expressed as follows:

(4.5) \begin{equation} \rho[S]=\boldsymbol{\theta}_S\cdot \boldsymbol{{y}} + \rho^{s}\left[ \rho^a\left[S-\boldsymbol{\theta}_S\cdot \boldsymbol{{Y}}|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right] \right], \end{equation}

where $\boldsymbol{\theta}$ is a fair hedger, $\rho^{s}$ is a model-consistent valuation principle and $\rho^a$ is a conditional model-consistent valuation principle.

The first term in (4.5) corresponds with using risk neutral valuation for the hedgeable part of a claim. The conditional valuation corresponds with using a valuation which is based on diversification. There are different choices for the valuation $\rho^s$ , depending on the risk management strategy one wants to implement. In the following example, we discuss a possible choice for the systematic valuation $\rho^s.$

Example 4 (Linear systematic valuation $\boldsymbol\rho^{\boldsymbol{{s}}}$ consistent with the market prices). In this example, we consider the systematic valuation which was proposed in Deelstra et al. (Reference Deelstra, Devolder, Gnameho and Hieber2020). Assume that $\rho^s$ can be expressed as follows:

(4.6) \begin{equation}\rho^s\left[ S\right]=\text{e}^{-rT}\mathbb{E}\left[\varphi\times S \right], \end{equation}

for some non-negative random variable $\varphi$ , where we assume $\varphi$ to be in $\mathcal{C}^{\boldsymbol{{Y}},\boldsymbol{{Z}}}$ with $\mathbb{E}\left[ \varphi\right]=1$ . The valuation $\rho^s$ uses the expectation of the claim, but only after transforming the claim S with an appropriate ‘distortion’ $\varphi$ . The distortion allows to use a prudent valuation approach by considering the claim in a less preferable, or stressed, scenario. The value $\rho^s[S]$ can then be interpreted as the expected payout of the claim S in the stressed scenario. If we define the probability measure $\mathbb{Q}$ , equivalent to $\mathbb{P}$ using its Radon–Nikodyn derivative $\frac{\text{d}\mathbb{Q}}{\text{d}\mathbb{P}}=\varphi$ , then we can write $ \rho^s\left[ S\right]=\text{e}^{-rT}\mathbb{E}_{\mathbb{Q}}\left[ S \right].$ The liability S is priced using an expectation under a stressed probability measure. This valuation corresponds with a risk management strategy which provides a sufficient large capital buffer to absorb extreme losses which arise in less preferable systematic scenarios. In order to determine the distortion $\varphi$ , one can follow the approach of Deelstra et al. (Reference Deelstra, Devolder, Gnameho and Hieber2020), where the authors require that the valuation $\rho^s$ should be consistent with the market prices, that is, $\rho^s[Y_i]=y_i,\ \text{for}\ i=1,2,\ldots, n^f. $ One can employ the Esscher transform, which was introduced in Esscher (Reference Esscher1932) to find the $\varphi$ in this way; see also Gerber and Shiu (Reference Gerber and Shiu1995).

Example 5 (Modified standard deviation principle). Assume we aim for a valuation $\rho$ which is consistent with risk-neutral valuation for hedgeable claims and with the standard deviation principle for pure actuarial claims. We can then use the following 3-step hedge-based valuation:

(4.7) \begin{equation} \rho[S] = \boldsymbol{\theta}_S\cdot \boldsymbol{{y}} + \rho^s\left[\mathbb{E}\left[ S - \boldsymbol{\theta}_S\cdot \boldsymbol{{Y}}|\ \boldsymbol{{Y}}, \boldsymbol{{Z}}\right] + \beta \sqrt{\text{Var}\left[ S|\ \boldsymbol{{Y}}, \boldsymbol{{Z}}\right]}\right]. \end{equation}

Indeed, it is easy to verify that $\rho[S^h]=\text{e}^{-rT}\mathbb{E}_{\mathbb{Q}}\left[S^h\right]$ for $S^h \in \mathcal{C}^h$ and $\rho[S] = \text{e}^{-rT}\left(\mathbb{E}[S] + \beta \sqrt{\text{Var}[S]}\right)$ for a claim $S \in \mathcal{C}^{\perp, \boldsymbol{{Y}},\boldsymbol{{Z}}}.$

It is straightforward to see that the valuation introduced in Example 2 is a 3-step hedge-based valuation. Below, we show that the quantile hedging valuation of Barigou et al. (Reference Barigou, Bignozzi and Tsanakas2021) is also an example of a 3-step hedge-based valuation.

Example 7 (The two-step valuation of Reference Pelsser and Stadje Pelsser and Stadje (2014) ). We assume that the financial market is complete and there exists a pricing measure $\mathbb{Q}$ . The 2-step valuation is given by $\rho^{2-step}[S] =\text{e}^{-rT} \mathbb{E}_{\mathbb{Q}}\left[\pi\left[S|\ \boldsymbol{{Y}} \right] \right], $ where $\pi\left[ \cdot|\ \boldsymbol{{Y}}\right]:\;\mathcal{C} \rightarrow \mathcal{C}^{\boldsymbol{{Y}}}$ is a conditional valuation. We can rewrite this expression as follows

(4.8) \begin{equation}\rho^{2-step}[S] =\text{e}^{-rT} \mathbb{E}_{\mathbb{Q}}\left[H_S^h \right] + \text{e}^{-rT} \mathbb{E}_{\mathbb{Q}}\left[\pi\left[S-H_S^h|\ \boldsymbol{{Y}} \right] \right],\end{equation}

where $H_S^h=\boldsymbol{\theta}_S\cdot \boldsymbol{{Y}}$ and $\boldsymbol{\theta}$ is a fair hedger. This expression shows that the 2-step valuation is also a 3-step hedge-based valuation provided the conditional valuation $\pi$ is conditional model consistent. The first term of (4.8) corresponds to the price to buy the hedge for the claim S when a hedger $\boldsymbol{\theta}$ is used. The conditional valuation is used to separate the financial risks from the non-financial risks in the residual part of the claim. Since the market is assumed to be complete, the claim $\pi\left[S-H_S^h|\ \boldsymbol{{Y}} \right] $ is hedgeable and therefore the second term in (4.8) corresponds with the price to buy the investment strategy to replicate $\pi\left[S-H_S^h|\ \boldsymbol{{Y}} \right] $ . Different choices for the conditional valuation $\pi$ are considered in Example 3.4 of Pelsser and Stadje (Reference Pelsser and Stadje2014).

4.2. Characterisation

The following theorem shows that the 3-step hedge-based valuation is a fair valuation.

Proof. Consider the 3-step hedge-based valuation $\rho$ given by (4.5). Since $\boldsymbol{\theta}$ is a fair hedger, it is also a market-consistent hedger and we directly find that $\rho$ is market consistent. The hedger is also model consistent, from which we find that

\begin{align*} \rho\left[ S^{\perp}\right] = \rho^s\left[\rho^a\left[S^{\perp}|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right] \right],\ \text{for}\ S^{\perp} \in \mathcal{C}^{\perp, \boldsymbol{{Y}}, \boldsymbol{{Z}}}. \end{align*}

Since $\rho^a\left[\cdot | \ \boldsymbol{{Y}}, \boldsymbol{{Z}}\right]$ is a conditional model-consistent valuation, $\rho^a\left[S^{\perp}|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right] = \rho^a\left[S^{\perp} \right]$ . Model consistency of the valuation $\rho$ then follows from the model consistency of the valuation $\rho^a.$

Theorem 4.2. Consider a fair valuation $\rho$ . Then, there exists a model-consistent valuation $\rho^s$ and a conditional model-consistent valuation $\rho^a\left[\cdot|\ \boldsymbol{{Y}}, \boldsymbol{{Z}}\right]$ such that

\begin{align*} \rho[S]=\boldsymbol{\theta}_S\cdot \boldsymbol{{y}} + \rho^{s}\left[ \rho^a\left[S-\boldsymbol{\theta}_S\cdot \boldsymbol{{Y}}|\ \boldsymbol{{Y}}, \boldsymbol{{Z}}\right] \right].\end{align*}

Proof. Consider a fair hedger $\boldsymbol{\theta}$ . Since $\rho$ is market consistent, we can write

\begin{align*} \rho[S] = \boldsymbol{\theta}_S\cdot \boldsymbol{{y}} + \rho\left[S- \boldsymbol{\theta}_S\cdot \boldsymbol{{Y}}\right]. \end{align*}

We use Theorem 1 of Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017) which states that the fair valuation $\rho$ can be expressed as follows:

(4.9) \begin{equation}\rho[S] = \text{e}^{-rT}\mathbb{E}_{\mathbb{Q}}\left[\boldsymbol{\mu}_S\cdot \boldsymbol{{Y}}\right],\end{equation}

where $\mathbb{Q}$ is a risk neutral measure and $\boldsymbol{\mu}$ is a model-consistent hedger. Define the conditional valuation as follows:

(4.10) \begin{equation}\rho^a\left[S|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right]= \boldsymbol{\mu}_S\cdot \boldsymbol{{Y}},\end{equation}

and the valuation $\rho^s$ as follows

(4.11) \begin{equation}\rho^s\left[S \right]=\begin{cases}\text{e}^{-rT} \mathbb{E}_{\mathbb{P}}[S] & \text{if}\ S\in \mathcal{C}^{\perp,\boldsymbol{{Y}}, \boldsymbol{{Z}}},\\[5pt] \text{e}^{-rT} \mathbb{E}_{\mathbb{Q}}[S] & \text{if}\ S \neq \mathcal{C}^{\perp, \boldsymbol{{Y}}, \boldsymbol{{Z}}}.\end{cases}\end{equation}

Note that $\rho^s$ is model consistent. Since $\boldsymbol{\mu}$ is model consistent, there exists an actuarial valuation $\rho^a \in \mathcal{A}$ such that $\rho^a\left[S |\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right]=\rho^a\left[S\right]$ for $S\in \mathcal{C}^{\perp,\boldsymbol{{Y}}, \boldsymbol{{Z}}}$ and therefore it is a conditional model-consistent valuation. Combining (4.10) and (4.11) together with (4.9) proves the result.

We thank the anonymous referee to point out that the notation of the conditional valuation might be misleading, since with our choice $\rho^a\left[S|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right]$ does not depend on $\boldsymbol{{Z}}$ .

Theorem 4.2 shows that a fair valuation has to be a 3-step hedge-based valuation. Combining Theorems 4.1 and 4.2, we find that the class of fair valuations is equivalent with the class of the 3-step hedge-based valuations:

\begin{align*} \rho \ \text{is a fair valuation}\ \Leftrightarrow \rho\ \text{is a 3-step hedge-based valuation.} \end{align*}

It was shown in Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017) that the class of hedge-based valuations is equivalent with the class of fair valuations. Moreover, the 2-step valuation introduced in Pelsser and Stadje (Reference Pelsser and Stadje2014) also characterizes the class of fair valuations. We can conclude that the 3-step hedge-based valuations, the 2-step valuations and the hedge-based valuations are describing the same class of valuations:

(4.12) \begin{equation}\rho \ \text{is a 2-step valuation}\ \Leftrightarrow \rho\ \text{is a hedge-based valuation} \Leftrightarrow \rho\ \text{is a 3-step hedge-based valuation.}\end{equation}

In this section, we introduced a new valuation for hybrid claims S. Moreover, similar to the 2-step and the hedge-based valuation, this new valuation is defined on $\mathcal{C}$ . However, in our illustrations and examples, we only consider claims of the form (2.7). This approach is similar to Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017) and Barigou et al. (Reference Barigou, Bignozzi and Tsanakas2021), who also define a general hybrid valuation principle but mainly focus on product claims of the form (2.7). In this case, conditioning on the vectors $\boldsymbol{{Y}}$ and $\boldsymbol{{Z}}$ results in a diversifiable claim, and we have an actuarial valuation together with a corresponding risk management strategy that we want to employ for such a diversifiable claim. However, if we have a claim $S\in\mathcal{C}$ that is not diversifiable after conditioning on $\boldsymbol{{Y}}$ and $\boldsymbol{{Z}}$ , the choice of $\rho^a[\cdot| \boldsymbol{{Y}}, \boldsymbol{{Z}}]$ may not be intuitive clear. In this case, instead of first applying a conditional valuation, one may prefer to value the whole residual part $S-\boldsymbol{\theta}_S\cdot \boldsymbol{{Y}}$ with a single model-consistent valuation, which means we use a hedge-based valuation instead. The 3-step hedge-based valuations and the hedge-based valuations both provide a characterization for the class of fair valuations. The choice between the two valuations depends on the situation at hand.

4.3. Illustration

Consider a financial market where there is one tradeable asset and a risk-free bank account, that is, $\boldsymbol{{Y}}=\left( Y_0,Y_1\right)$ . The market prices are denoted by $(1,y_1)$ . The time-T asset value $Y_1$ can take values 0 or 1 and we assume the risk-free rate $r=0$ . The claim S is defined as follows:

(4.13) \begin{equation}S= Y_1\times \frac{\sum_{i=1}^{n^a}X_i}{n^a}.\end{equation}

The random variables $X_1, X_2,\ldots, X_{n^a}$ are assumed to be identical, but not independent. Moreover, we assume that $X_i=1$ corresponds with the survival of policyholder i at the maturity T, whereas $X_i=0$ otherwise. We assume that Z denotes the systematic longevity risk of an insurance portfolio. The joint probabilities of the random vector $(Y_1,Z,X_1)$ are listed in Table 1. There are two longevity scenarios. In the first scenario, $Z=0$ and we experience a decrease in longevity, whereas $Z=1$ corresponds with the scenario where longevity is increasing. Indeed, one can verify that

\begin{equation*}\mathbb{P}\left[X_1=1|Z=0\right] = 0.26,\ \ \text{and}\ \ \mathbb{P}\left[X_1=1|Z=1\right] = 0.76.\end{equation*}

Table 1. The joint probabilities for the random vector $(Y_1,Z,X_1)$ .

Survival probabilities are larger if $Z=1$ compared to the situation where $Z=0$ . Note that the random variable Z is dependent on the financial market. Indeed, we have that $\mathbb{P}\left[Y_1=1|Z=0\right] = 0.41$ , whereas $\mathbb{P}\left[Y_1=1|Z=1\right] = 0.51$ .

4.3.1. The hedgeable part

We assume that the hedge of a claim S is determined by the mean-variance hedger $\boldsymbol{\theta}_S$ (see also Theorem 5 of Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017)). Using the probabilities listed in Table 1, we then find that $\theta_S^{(1)}=\frac{\mathbb{E}\left[Y_1X_1\right]}{\mathbb{E}\left[Y_1\right]}= 0.5\ \text{and}\ \theta_S^{(0)}=0.$ The hybrid claim S can be decomposed in two parts:

(4.14) \begin{equation}S= \theta_S^{(1)}Y_1 + Y_1\left(\frac{1}{n^a}\sum_{i=1}^{n^a}\left( X_i-\theta_S^{(1)}\right)\right).\end{equation}

The first term is the hedgeable part of the hybrid claim and can be valuated using the corresponding hedging strategy, which consists of buying $0.5$ units of the stock $Y_1$ . The second term is the residual part, which is the part that remains after we take into account the income from the hedging strategy. Figure 2 shows a histogram of the residual part for different values of the portfolio size $n^a$ . The residual part is not diversifiable since the random variables $X_1, X_2,\ldots, X_{n^a}$ are dependent and each term includes the asset value $Y_1$ . The histogram shows that the variance does not converge to zero as we increase the portfolio size. Note also that the expected value is approximately zero.

Figure 2. Histogram of the residual part $S-\theta_S^{(1)}Y_1$ of the claim S given by (4.13). The red dotted lines correspond with the actuarial value of the residual part of the claim given that $Z=0$ and $Z=1$ .

4.3.2. 3-step valuation with the conditional standard deviation principle

The histogram shows there are three different ‘scenarios’. There is a positive probability mass in zero, which corresponds with the event where $Y_1=0$ . There is a scenario where the residual part is positive, that is, we are underhedging the claim. This scenario corresponds with the situation where $Y_1=1$ and $Z=1$ . Indeed, in this case there is a strongly increasing longevity and therefore more policyholders survive than anticipated when determining the hedge $\theta_S^{(1)}$ . The negative scenario corresponds with the situation where $Y_1=1$ and $Z=0$ . In this scenario, we are overhedging the claim since there are less survivors than initially anticipated. Note that $\mathbb{P}\left[ Y_1=1, Z=0\right]= 0.14$ whereas $\mathbb{P}\left[ Y_1=1, Z=1\right]= 0.34$ , which explains why there is more probability mass in the positive part than in the negative part of the histograms.

Given a particular scenario of the asset and the longevity risk, one has a diversifiable claim. Indeed, consider, for example, the case where $Y_1=1$ and $Z=1$ , which corresponds with the positive part of the histograms shown in Figure 2. Given that we are in this positive scenario, applying an actuarial valuation principle is justified. For example, applying the conditional standard deviation principle gives

(4.15) \begin{equation}\rho^a\left[S-\theta_S^{(1)}Y_1|\ Y_1=1, Z=1 \right] = 0.09+ \beta \sqrt{ \frac{0.24}{n^a}}.\end{equation}

For the negative scenario, applying a standard deviation principle results in

(4.16) \begin{equation}\rho^a\left[S-\theta_S^{(1)}Y_1|\ Y_1=1, Z=0 \right] = -0.21 + \beta \sqrt{ \frac{0.20}{n^a}}.\end{equation}

The vertical dashed lines in Figure 2 correspond with the location of the values (4.15) and (4.16) on the x-axis.

The 3-step hedge-based valuation takes into account that given the realization of $Y_1$ and Z, an actuarial valuation $\rho^a$ is appropriate. Otherwise stated, given the financial and the systematic scenario, one can determine an appropriate risk management strategy based on diversification for the residual claim and therefore valuate the residual claim accordingly. A valuation $\rho^s$ is used to cope with the uncertainty about the particular scenario that will eventually unfold.

We assume, for simplicity, that the safety loading for the conditional standard deviation principle is equal to one, that is, $\beta=1$ . The 3-step hedge-based valuation for S is then given by

\begin{align*} \rho[S]= \theta_S^{(1)}y_1 + \rho^s\left[\epsilon\right], \end{align*}

where $y_1$ is the market price of the traded asset, $\rho^s$ is a model-consistent valuation and the random variable $\epsilon$ is given by

\begin{eqnarray*}\epsilon & = &\begin{cases} -0.21 + \beta \sqrt{ \frac{0.20}{n^a}}& \text{if $Y_1=1, Z=0$,} \nonumber\\[5pt] 0 & \text{if $Y_1=0$,} \nonumber\\[5pt] 0.09+ \beta \sqrt{ \frac{0.24}{n^a}} & \text{if $Y_1=1, Z=1$.} \nonumber\end{cases}\end{eqnarray*}

4.3.3. Choice 1 for the systematic valuation: risk-free capital buffer

A possible choice for the valuation $\rho^s$ is a prudent valuation, such as the Value-at-Risk, Tail Value-at-Risk or another $\mathbb{P}$ -law invariant risk measure. This corresponds with a risk management strategy where one builds a sufficiently large capital buffer. In our particular example, we have that $\mathbb{P}\left[\epsilon < 0.09+ \beta \sqrt{ \frac{0.24}{n^a}} \right] = 0.66,$ from which we find that

(4.17) \begin{equation}\rho^s\left[\epsilon \right]=\text{VaR}_{0.95}\left[\epsilon\right] = 0.09+ \beta \sqrt{ \frac{0.24}{n^a}} .\end{equation}

Otherwise stated, we prepare for the worst-case scenario. One can also use a cost-of-capital approach resulting in a value for $\rho^s\left[\epsilon \right]$ which corresponds with the cost of borrowing the amount $ 0.09+ \beta \sqrt{ \frac{0.24}{n^a}} $ .

4.3.4. Choice 2 for the systematic valuation: Esscher transform

The random variable $\epsilon$ is dependent on the financial market. Therefore, one may consider a systematic valuation that takes into account the market prices of the traded assets. For example, if we use the valuation of Example 4, we consider a probability measure $\mathbb{Q}$ and define the following probabilities under this new probability measure

\begin{eqnarray*}\mathbb{Q}\left[ Y_1=0, Z=0\right] = q_1, & & \mathbb{Q}\left[ Y_1=0, Z=1\right] = q_2\\[5pt] \mathbb{Q}\left[ Y_1=1, Z=0\right] = q_3&\text{and}& \mathbb{Q}\left[ Y_1=1, Z=1\right] = q.\end{eqnarray*}

The probability measure $\mathbb{Q}$ should satisfy the following conditions:

\begin{equation*}\mathbb{E}_{\mathbb{Q}}\left[ Y_1\right] = y_1\ \ \ \text{and}\ \ \ \mathbb{E}_{\mathbb{Q}}\left[ Z\right] = \mathbb{E}\left[ Z\right] + \gamma,\end{equation*}

for some $0<\gamma< 1-\mathbb{E}[Z]$ . The first condition states that the valuation should take into account the market prices. The second condition states that the valuation of the longevity index should take into account a pre-specified longevity risk margin $\gamma$ . Solving these equations leads to $q_3=y_1-q,\ \ q_2=\mathbb{E}[Z]+\gamma-q\ \ \text{and}\ \ q_1=1-q_2-q_3-q,$ for some $q\in \left(\mathbb{E}[Z]+\gamma+y_1-1, \min\left\{y_1, \mathbb{E}[Z]+\gamma \right\}\right)$ . We then find that

(4.18) \begin{equation}\rho^s\left[ \epsilon \right] = \left(y_1-q \right) \times \left(-0.21 + \beta \sqrt{ \frac{0.20}{n^a}}\right) + q \left(0.09+ \beta \sqrt{ \frac{0.24}{n^a}} \right),\end{equation}

for some $q\in (0, y_1)$ . Note that this valuation principle is a linear combination between the best and the worst-case situation.

5.1. A closed-form expression

In this section, we assume that financial risks are independent of the non-financial risks, that is, the random vector $\boldsymbol{{Y}}$ is independent of both $\boldsymbol{{Z}}$ and $\boldsymbol{{X}}$ . We will use the mean-variance hedger, which is a fair hedger (see Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017)), to determine the hedgeable part of a hybrid claim. The mean-variance hedger is defined as follows:

(5.1) \begin{equation}\boldsymbol{\theta}_S=\arg\min_{\boldsymbol{\mu}\in \Theta}\mathbb{E}\left[ \left( S -\boldsymbol{\mu}_S\cdot \boldsymbol{{Y}}\right)^2\right].\end{equation}

Proposition 1. Assume the random vector $\boldsymbol{{Y}}$ is independent of $\boldsymbol{{Z}}$ and $\boldsymbol{{X}}$ . The mean-variance hedger $\boldsymbol{\theta}_S$ for the product claim (2.7) can be expressed as follows:

(5.2) \begin{equation}\boldsymbol{\theta}_S = \boldsymbol{\theta}_{S^f}\times \mathbb{E}\left[g \right],\end{equation}

where $\boldsymbol{\theta}_{S^f}$ is the mean-variance hedge for the financial derivative $S^f$ . The hedgeable part of the claim is given by

(5.3) \begin{equation}H_S^h = \mathbb{E}\left[ g_1\right] \times H_{S^f}^h,\end{equation}

where $H_{S^f}^h= \boldsymbol{\theta}_{S^f}\cdot \boldsymbol{{Y}}$ .

Proof. One can show that the mean-variance hedger can be determined by solving the following system of equations

(5.4) \begin{eqnarray}\sum_{m=1}^{n^f} \text{Cov}\left[ Y_k, Y_m\right]\theta_S^{(m)} = \text{Cov}\left[S,Y_k\right],\ \text{for}\ k=1,2,\ldots, n^f, \end{eqnarray}

(5.5) \begin{eqnarray} \theta_S^{(0)} = \text{e}^{-rT}\left(\mathbb{E}[S] - \sum_{i=1}^{n^f} \theta_S^{(i)} \mathbb{E}[Y_i] \right).\end{eqnarray}

We have that $\text{Cov}[S,Y_k] = \mathbb{E}[g]\text{Cov}[S^f, Y_k]. $ Denote by $\Sigma$ the matrix containing all the covariances $\text{Cov}[Y_i,Y_j]$ , that is, $\left( \Sigma\right)_{i,j}= \text{Cov}[Y_i,Y_j].$ Moreover, use the notation $\tilde{\boldsymbol{\theta}}_S = \left( \theta_S^{(1)}, \theta_S^{(2)}, \ldots, \theta_S^{(n^f)}\right)$ and $\boldsymbol{\beta}=\left( \text{Cov}[S^f, Y_1], \ldots, \text{Cov}[S^f, Y_{n^f}]\right).$ Then we can write (5.5) as $\Sigma \tilde{\boldsymbol{\theta}}_S' = \boldsymbol{\beta}'\mathbb{E}[g], $ from which we find that

\begin{equation*} \tilde{\boldsymbol{\theta}}_S' =\left(\Sigma ^{-1} \boldsymbol{\beta}\right)\mathbb{E}[g]. \end{equation*}

Note that $\Sigma ^{-1} \boldsymbol{\beta}$ corresponds with the mean-variance hedge of the financial derivative $S^f$ . Indeed, we have that

\begin{equation*} \boldsymbol{\theta}_{S^f}= \left(\theta_{S^f}^{(0)}, \tilde{\boldsymbol{\theta}}_{S^f} \right). \end{equation*}

It remains to prove that $\theta_S^{(0)}=\theta_{S^f}^{(0)}\mathbb{E}[g]$ . We can write

\begin{eqnarray*}\theta_S^{(0)} & = & \text{e}^{-rT}\left( \mathbb{E}[S] - \mathbb{E}[g]\sum_{i=1}^{n^f} \theta_{S^f}^{(i)} \mathbb{E}[Y_1]\right)\\[5pt] & = & \text{e}^{-rT}\left( \mathbb{E}[S] - \mathbb{E}[g]\left( \mathbb{E}[S^f]-\theta_{S^f}^{(0)}\text{e}^{rT}\right)\right)\\[5pt] & = & \text{e}^{-rT}\left( \mathbb{E}[S] - \mathbb{E}[g] \mathbb{E}[S^f]+\theta_{S^f}^{(0)}\text{e}^{rT}\right).\end{eqnarray*}

Taking into account that $ \mathbb{E}[S] = \mathbb{E}[g] \mathbb{E}\left[S^f\right]$ then proves the result.

In case the financial derivative in (2.7) is hedgeable, the hedgeable part can be expressed as follows $H_S^h = \mathbb{E}\left[g\right]\times S^f. $ The hedger for the claim S invests in $\mathbb{E}\left[g\right]$ units of the hedgeable financial derivative. Note also that the mean-variance hedge does not depend on the portfolio size $n^a$ . A similar result was derived in Møller (Reference Møller2001), where the authors determine the mean-variance hedger in the situation where there is a risky stock and a bank account. To be more precise, a hybrid claim where $S^f=Y_1$ is used in their example; see Section 3 in Møller (Reference Møller2001).

We can prove the following result which provides an explicit expression for the 3-step value of the product claim.

Proposition 2. Assume the random vector $\boldsymbol{{Y}}$ is independent of $\boldsymbol{{Z}}$ and $\boldsymbol{{X}}$ . The modified standard deviation principle, introduced in Example 5, for the product claim S given by (2.7) can be expressed as follows

(5.6) \begin{equation} \rho^{3-step}[S] =\text{e}^{-rT} \mathbb{E}_\mathbb{Q}\left[H_S^h \right] + \rho^s\left[ S^f\left(\mathbb{E}\left[g|\ \boldsymbol{{Z}} \right] + \beta \sqrt{\frac{\text{Var}\left[ g | \ \boldsymbol{{Z}} \right]}{n^a}} \right) - H_{S^f}^h\mathbb{E}\left[g \right]\right]. \end{equation}

Proof. The modified standard deviation principle of the claim S can be expressed using Expression (4.5) as follows:

(5.7) \begin{equation} \rho\left[ S\right] = \text{e}^{-rT}\mathbb{E}_\mathbb{Q}\left[H_S^h \right] + \rho^s\left[ \rho^a\left[S - H_S^h |\ \boldsymbol{{Y}}, \boldsymbol{{Z}}\right] \right], \end{equation}

where $H_S^h$ is given by (5.3) and

\begin{equation*} \rho^a\left[S-H_S^h \right] = \mathbb{E}\left[ S- H_S^h|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right] + \beta \sqrt{\text{Var}\left[S-H_S^h|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right]}. \end{equation*}

We can write:

\begin{eqnarray*} \mathbb{E}\left[ S- H_S^h|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right] & = & \mathbb{E}\left[ S|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right] - H_S^h \\[5pt] & = & S^f\mathbb{E}\left[ \sum_{i=1}^{n^a} g\left( \boldsymbol{{Z}}, X_i\right)|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right] - H_S^h \\[5pt] & = & S^f\mathbb{E}\left[ g|\ \boldsymbol{{Z}} \right] - H_S^h. \end{eqnarray*}

For the conditional variance, we find

\begin{eqnarray*} \text{Var}\left[S-H_S^h|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right] & = & \text{Var}\left[S|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right]\\[5pt] & = &\left(S^f \right)^2 \text{Var}\left[ \sum_{i=1}^{n^a} g\left( \boldsymbol{{Z}}, X_i\right)|\ \boldsymbol{{Y}}, \boldsymbol{{Z}} \right]\\[5pt] & = &\left(S^f \right)^2 \frac{\text{Var}\left[ g|\ \boldsymbol{{Z}} \right]}{n^a}. \end{eqnarray*}

Plugging these expressions in (5.7) gives the desired result.

The first term in (5.6) corresponds to the price we have to pay if we buy the mean-variance hedge. The amount we receive from the hedging portfolio is $H_S^h$ . However, the hedging portfolio assumes the expected number of units of $S^f$ needed is equal to $\mathbb{E}[g]$ . If we have information about the longevity risk, this number will be updated. There are two sources that cause hedging errors: systematic and unsystematic deviations from the mean $\mathbb{E}\left[ g\right]$ . The systematic part is captured by the random variable $ \mathbb{E}\left[g|\ \boldsymbol{{Z}} \right]$ whereas the unsystematic fluctuations are captured by $\frac{\text{Var}\left[ g | \ \boldsymbol{{Z}} \right]}{n^a}$ .

5.1.1. Systematic valuation with the Esscher transform

In this subsection we consider the valuation using the 3-step hedge-based valuation when an Esscher transform is used; see Example 4. In order to determine the distortion $\varphi$ , we follow the approach of Deelstra et al. (Reference Deelstra, Devolder, Gnameho and Hieber2020) and employ the Esscher transform

(5.8) \begin{equation}\text{Esscher distortion:}\ \ \varphi = \frac{\text{e}^{-\sum_{i=1}^{n^s} w_i Z_i-\sum_{i=1}^{n^f} v_i Y_i}}{\mathbb{E}\left[ \text{e}^{-\sum_{i=1}^{n^s} w_i Z_i-\sum_{i=1}^{n^f} v_i Y_i}\right]}, \end{equation}

for some constants $w_1,w_2,\ldots, w_{n^s}$ and $v_1, v_2,\ldots, v_{n^f}$ . In order to determine the valuation (4.6) with Esscher transform (5.8), we need a joint calibration of the weights $w_1,w_2,\ldots, w_{n^s}$ and $v_1, v_2,\ldots, v_{n^f}$ . In this section, we assume financial risks to be independent from $\boldsymbol{{Y}}$ and $\boldsymbol{{Z}}$ . The valuation $\rho^s$ can then be expressed as

(5.9) \begin{equation} \rho^s\left[ S\right]=\text{e}^{-rT}\mathbb{E}\left[\left(\varphi^f\times \varphi^s \right)S \right]. \end{equation}

From (5.9), we find that for financial derivatives $S^f$ , we can write $ \rho^s\left[S^f\right]=\text{e}^{-rT} \mathbb{E}\left[\varphi^f S^f \right]$ , which means that $\varphi^f$ is the financial distortion. In this paper, we assume that the distortion $\varphi^f$ is tuned in such a way that the valuation $\rho^s$ is using risk neutral valuation for all financial claims. The risk neutral measure $\mathbb{Q}$ may be estimated from available market information. However, there are only finitely many traded assets and infinitely many possible risk neutral measures. The choice of the risk neutral measure is therefore not always straightforward. The distortion $\varphi^s$ can be calibrated solely using the non-traded systematic claims. Indeed, for claims $S^s$ which are functions of only the vector $\boldsymbol{{Z}}$ , we find that $\rho^s[S^s]=\text{e}^{-rT}\mathbb{E}\left[ \varphi^s S^s \right]$ . In Zeddouk and Devolder (Reference Zeddouk and Devolder2019), for example, the authors propose to calibrate the distortion $\varphi^s$ based on a given risk margin imposed on longevity products. We conclude that in this special situation where non-traded systematic and financial risks are independent, we can determine the valuation defined in (5.9) in two steps. The traded assets will result in an estimate for $\varphi^f$ whereas one can impose conditions on the risk margin of the non-traded systematic risks to calibrate the distortion $\varphi^s$ .

It is straightforward to verify that the value of the 3-step hedge-based valuation when using the Esscher transform can be expressed as follows

\begin{equation*} \rho^{Esscher}[S] = \text{e}^{-rT}\mathbb{E}_{\mathbb{Q}}\left[ S^f\right]\left(\rho^s\left[ \mathbb{E}\left[g|\ \boldsymbol{{Z}} \right]\right] +\beta \rho^s\left[ \sqrt{\frac{\text{Var}\left[ g | \ \boldsymbol{{Z}} \right]}{n^a}} \right] \right). \end{equation*}

The 3-step hedge-based value considers the price of the financial derivative $S^f$ , which has to be determined under a pricing measure $\mathbb{Q}$ . The expression between brackets represents the number of financial derivatives needed, taking into account the information contained in $\boldsymbol{{Z}}$ . Since the Esscher transform is linear, the two terms are valuated separately. The term between brackets only depends on the vector $\boldsymbol{{Z}}$ , which means the distortion $\varphi^s$ is used to determine $\rho^s\left[ \mathbb{E}\left[g|\ \boldsymbol{{Z}} \right]\right] $ and $\rho^s\left[ \sqrt{\frac{\text{Var}\left[ g | \ \boldsymbol{{Z}} \right]}{n^a}} \right]$ . Note that in case we use the Esscher transform, the value is independent on the choice of the hedger. However, since the market is incomplete and the claim $S^f$ may be not perfectly replicable, the 3-step value depends on the choice of the risk-neutral pricing measure. In case the claim $S^f$ is hedgeable, the financial part can be determined in a model-free way using only prices of traded assets, that is, in this case we do not need to make any subjective choice about the hedger or the risk-neutral measure.

5.1.2. Systematic valuation with the VaR

A drawback of the systematic valuation explained in the previous subsection is the choice of the pricing measure $\mathbb{Q}$ in case the claim is not hedgeable. Therefore, in this subsection, we consider a prudent approach which determines an appropriate capital buffer:

(5.10) \begin{equation}\rho^{VaR}[S] = \text{e}^{-rT}\mathbb{E}_{\mathbb{Q}}\left[ H_S^h\right] + \text{e}^{-rT} \text{VaR}_p\left[S^f\left(\mathbb{E}\left[g|\ \boldsymbol{{Z}} \right] + \beta \sqrt{\frac{\text{Var}\left[ g | \ \boldsymbol{{Z}} \right]}{n^a}} \right) - H_{S^f}^h\mathbb{E}\left[g \right] \right].\end{equation}

The amount $\rho^{VaR}[S]$ can be decomposed in two parts. The first term corresponds with the price to buy the hedging portfolio. The second term is the amount of cash invested in the risk-free bank account to cover the residual part. One can also include a cost-of-capital rate $\gamma$ in the valuation, which implies the capital buffer is borrowed and the valuation only takes into account the cost to borrow this amount. In our case, only the real-world probability measure $\mathbb{P}$ is used to valuate the residual part of the claim. However, one now has to make a choice about the confidence level p. We use the VaR but one may decide to use another risk measure (e.g., Tail Value-at-Risk).

5.1.3. The valuation proposed in Møller (Reference Møller2002)

In Møller (Reference Møller2002) Section 5.2, the author considers the valuation of hybrid claims by transforming the standard deviation principle, under the assumption there are no systematic risks and the financial risks $\boldsymbol{{Y}}$ are independent of the actuarial risks $\boldsymbol{{X}}$ . The financial market is assumed to be complete and there exists a pricing measure $\mathbb{Q}$ . The distribution of $\boldsymbol{{X}}$ is not affected by the change of measure and the random vectors $\boldsymbol{{X}}$ and $\boldsymbol{{Y}}$ are independent under $\mathbb{Q}$ . Consider the product claim (2.7) and assume that we start from the valuation $\rho$ defined in Example 5 where $\boldsymbol{\theta}$ is the mean-variance hedger and $\rho^s[S]=\text{e}^{-rT}\mathbb{E}[S]$ . One can then write

(5.11) \begin{equation}\rho[S] = \text{e}^{-rT}\left(\mathbb{E}_{\mathbb{Q}}[S] + \beta\mathbb{E}\left[ \text{Var}\left[ S|Y\right]\right]\right), \end{equation}

where we used Proposition 1 together with the fact that completeness implies that $S^f$ is hedgeable. In this particular case, our 3-step hedge-based valuation coincides with the valuation proposed by Møller (Reference Møller2002) in Section 5.2.2. Since $\boldsymbol{{X}}$ and $\boldsymbol{{Y}}$ are independent (under both the real world and the risk-neutral world) and the distribution of $\boldsymbol{{X}}$ is unchanged when moving to the $\mathbb{Q}$ -measure, we can interpret the first term in (5.11) as the cost to buy the hedging portfolio for the claim S. The second term is an extra buffer because the claim is not perfectly hedgeable.

Instead of defining $\rho^s$ as the real-world expectation, one can also define $\rho^s$ as the $\mathbb{Q}$ expectation. Since $\mathbb{E}_{\mathbb{Q}}\left[ S|\boldsymbol{{Y}}\right]=\mathbb{E}\left[ S|\boldsymbol{{Y}}\right]$ , we find that the modified standard deviation can then be expressed as follows

(5.12) \begin{equation}\rho[S] = \text{e}^{-rT}\left(\mathbb{E}_{\mathbb{Q}}[S] + \beta\mathbb{E}_{\mathbb{Q}}\left[ \text{Var}\left[ S|Y\right]\right]\right).\end{equation}

Comparing (5.11) and (5.12), we observe that both valuations start from the mean-variance hedge for the claim S and adjust its value by taking into account the conditional variance of the claim. The valuation (5.11) considers the real-world expectation, implying a capital buffer is used for covering future losses. The valuation (5.12) takes into account that the market is complete and therefore $ \text{Var}\left[ S|Y\right]$ is hedgeable. We can rewrite (5.12) as $ \rho[S] = \text{e}^{-rT}\mathbb{E}_{\mathbb{Q}}[S^f]\left(\mathbb{E}[g] + \beta \text{Var}\left[\frac{g}{n^a} \right]\right). $ The mean-variance hedge for S invests in $\mathbb{E}[g]$ units of $S^f$ . The valuation (5.12) increases the number of units invested in the claim $S^f$ by $\beta \text{Var}\left[\frac{g}{n^a} \right]$ to account for the hedging error.

In Section 5.1.1, we use the Esscher transform with distortion (5.9) for the valuation $\rho^s$ . This combines the approaches discussed in (5.11) and (5.12) in that we use the risk-neutral measure for the financial part of the claim and the real-world measure for the actuarial part.

5.2. Numerical illustration

We will provide a numerical illustration of the 3-step valuation under the assumption that $X_i$ takes the value 1 if the policyholder survives until time T and zero otherwise. The policyholder i will pay an initial premium equal to P at time $t=0$ and receives the amount $S^f$ at time T provided the policyholder is alive. The claim S is then given by

(5.13) \begin{equation} S = S^f \frac{\sum_{i=1}^{n^a} X_i}{n^a}. \end{equation}

The premium is invested in a stock (or risky fund) and the amount $S^f$ the policyholder receives at time T depends on the performance of this risky fund. If we denote the time-T value of this investment by $Y_1$ , we assume that $S^f$ can be expressed as follows:

(5.14) \begin{equation}S^f=1+\alpha\left( Y_1 -P(v+1)^T\right)_+,\end{equation}

where $\alpha$ is the bonus share and v is an internal rate of return. If the policyholder survives, he will always receive the amount 1 at maturity, however, this amount can be increased if the risky fund performs well.

We assume that the stock $Y_1$ follows a geometric Brownian motion, which implies that the time-T value has a lognormal distribution:

(5.15) \begin{equation}\log \frac{Y_1}{P}\overset{d}{=}N\left(\left(\mu^f-\frac{1}{2}\left(\sigma^f\right)^2\right)T, \left(\sigma^f\right)^2T \right),\end{equation}

for some $\mu^f\in \mathbb{R}$ and $\sigma^f\geq 0.$ The spot price of the stock is equal to P, since we invest the full premium in the risky asset. We assume here that we can only trade in one risky asset to hedge the claim. We assume that the conditional survival probabilities can be expressed as follows:

(5.16) \begin{equation} \mathbb{P}\left[ X_1=1|Z\right]= \text{e}^Z. \end{equation}

The random variable Z models the longevity risk among the policyholders. If Z is large, the probability of surviving goes up for each policyholder, whereas the survival probability goes down if Z goes down. If we assume that the dynamics of the systematic longevity risk can be modeled using an Ornstein–Uhlenbeck process, we find a normal distribution for the random variable Z. However, from (5.16), we find that the random variable Z should be negative since $\text{e}^Z$ is a probability. Therefore, define the random variable $\tilde{Z}$ as follows:

(5.17) \begin{equation}\tilde{Z} \overset{d}{=}N\left(\mu^s, \left(\sigma^s\right)^2 \right),\end{equation}

for some $\mu^s\in \mathbb{R}$ and $\sigma^s\geq 0$ . Note that $\mu^s$ and $\sigma^s$ depend on the maturity T of the contract. More details are given in the online Appendix. The random variable Z is then defined as follows

(5.18) \begin{equation}Z=\begin{cases}\tilde{Z}, &\ \text{if} \; \tilde{Z}<0\\[5pt] 0, & \ \text{if}\ \tilde{Z}\geq 0.\end{cases}\end{equation}

The survival probability for a policyholder is given by $\mathbb{E}[X_1]$ and assuming (5.18) we can derive a closed-form expression for this probability.

Proposition 3. Consider the random variable Z defined in (5.18). We have that

\begin{eqnarray*}\mathbb{E}[X_1] & = & \text{e}^{\mu^s+\frac{1}{2}\left(\sigma^s\right)^2}\Phi\left( -\left(\frac{\mu^s}{\sigma^s}+\sigma^s \right)\right) + \Phi\left(\frac{\mu^s}{\sigma^s}\right), \\[5pt] \mathbb{E}\left[\left(\mathbb{E}\left[X_1|Z \right] \right)^2 \right] & = & \text{e}^{2\mu^s+2\left(\sigma^s\right)^2}\Phi\left( -\left(\frac{\mu^s}{\sigma^s}+2\sigma^s \right)\right) + \Phi\left(\frac{\mu^s}{\sigma^s} \right),\\[5pt] \mathbb{E}\left[\text{Var}\left[X_1| Z\right]\right] & = & \mathbb{E}[X_1] - \mathbb{E}\left[\left(\mathbb{E}\left[X_1|Z \right] \right)^2 \right],\end{eqnarray*}

where $\Phi$ is the distribution function of a standard normal distribution.

Proof. We have that $\mathbb{E}[X_1]= \mathbb{E}[\text{e}^Z]$ and we can write

\begin{align*} \mathbb{E}\left[\text{e}^Z\right] = \int_{-\infty}^0 \text{e}^z f_{\tilde{Z}}(z)\text{d}z + \mathbb{P}\left[ \tilde{Z}\geq 0\right], \end{align*}

where $f_{\tilde{Z}}$ is the density of $\tilde{Z}$ . The integral and the probability can be derived since $\tilde{Z}$ is normal distributed.

In order to prove the second equality, we note that $\mathbb{E}\left[\left(\mathbb{E}\left[X_1|Z \right] \right)^2 \right] = \mathbb{E}\left[\left(\text{e}^Z \right)^2 \right] $ and therefore we can write

\begin{align*} \mathbb{E}\left[\left(\mathbb{E}\left[X_1|Z \right] \right)^2 \right] = \int_{-\infty}^0 \text{e}^{2z} f_{\tilde{Z}}(z)\text{d}z + \mathbb{P}\left[ \tilde{Z}\geq 0\right]. \end{align*}

We can then derive the desired expression using the same approach as we did in the first part of this proof.

We have that $\text{Var}\left[X_1| Z\right] = \text{e}^Z\left( 1-\text{e}^Z\right)$ which proves the last equality.

5.2.1. Systematic valuation with the Esscher transform

We start with a linear valuation for the valuation $\rho^s $ . To be more precise, the valuation $\rho^s$ is given by (5.9) where $ \varphi^s = \frac{\text{e}^{-\theta^s Z}}{\mathbb{E}\left[ \text{e}^{-\theta^s Z}\right]}, $ for some $\theta^s<0$ . The financial distortion $\varphi^f$ is determined such that financial claims are valuated using a risk-neutral distribution $\mathbb{Q}$ . Therefore, we have $ \varphi^f =\text{e}^{-\left(\frac{\mu^f-r}{\sigma^f} \right)^2T -\frac{\mu^f-r}{\sigma^f} W },$ where $W= \frac{\log \frac{Y_1}{P}- \left( \mu^f-\frac{1}{2}\left(\sigma^f\right)^2\right)T}{\sigma^f}.$

If we use the modified standard deviation principle introduced in Example 5, the 3-step hedge-based valuation method gives the following value:

(5.19) \begin{equation} \rho^{Esscher}\left[S \right] = \text{e}^{-rT}\mathbb{E}_{\mathbb{Q}}\left[S^f \right]\left( \mathbb{E}\left[X_1 \right] \text{e}^{-\theta^s\left(\sigma^s \right)^2} + \beta \mathbb{E}\left[\text{e}^{-\theta^s Z} \sqrt{\frac{\text{Var}\left[ X_1 | \ \boldsymbol{{Z}} \right]}{n^a}} \right]\right). \end{equation}

A closed-form expression for $ \text{e}^{-rT}\mathbb{E}_{\mathbb{Q}}\left[S^f \right]$ can easily be obtained since the risky fund follows a lognormal distribution. The expectation $ \mathbb{E}\left[X_1 \right]$ is given in closed-form in Proposition 3. Note that in order to determine the 3-step value, one needs numerical simulation to determine the value of the diversifiable part of the claim. This term, however, tends to zero if the portfolio size increases. Therefore, in case one deals with a sufficiently large portfolio, this term may be neglected and the approximate 3-step value is given in closed-form. The parameter $\theta^s$ is assumed to be negative, since it implies that increasing longevity (and its volatility) leads to larger prices. Since we are valuating a unit-linked portfolio, the bad scenarios are those where longevity increases, that is, where there are more survivors than expected.

5.2.2. Systematic valuation with the VaR

The valuation using the VaR (see Section 5.1.2) depends on the choice of the hedger, which is assumed to be the mean-variance hedger. In order to determine the hedgeable part of the claim S, we use Proposition 1 and determine the hedge for the financial derivative $S^f$ . Note that $S^f$ is not perfectly hedgeable. The mean-variance hedge $\left(\theta_{S^f}^{(0)}, \theta_{S^f}^{(1)}\right)$ is given by

(5.20) \begin{eqnarray} \theta_{S^f}^{(1)} = \frac{\text{Cov}\left[ S^f, Y_1\right]}{\text{Var}\left[ Y_1\right]} \end{eqnarray}

(5.21) \begin{eqnarray} \theta_{S^f}^{(0)} = \text{e}^{-rT}\left( \mathbb{E}\left[S^f\right] - \theta_{S^f}^{(1)} \mathbb{E}\left[Y_1 \right] \right). \end{eqnarray}

If the financial derivative $S^f$ is given by (5.14) and the stock $Y_1$ is described by (5.15), we have that $\mathbb{E}\left[Y_1 \right] = P \text{e}^{\mu T}$ . Using standard techniques for lognormal distributions, we find

(5.22) \begin{eqnarray} \mathbb{E}\left[S^f \right] = 1 + \alpha \mathbb{E}\left[Y_1\right] \Phi\left(d_1 \right) - P(1+v)^T \Phi\left(d_2 \right) \end{eqnarray}

(5.23) \begin{eqnarray} \text{Cov}\left[S^f, Y_1 \right] = \alpha \mathbb{E}\left[Y_1\right] \left(\mathbb{E}\left[Y_1\right] \left(\text{e}^{\left(\sigma^f\right)^2 T} \Phi\left(d_3 \right) - \Phi(d_1) \right) - K \left(\Phi(d_1) - \Phi(d_2) \right)\right), \end{eqnarray}

(5.24) \begin{eqnarray} \text{Var}\left[ Y_1\right] = P^2 \text{e}^{2\mu^fT}\left( \text{e}^{\left( \sigma^f\right)^2T}-1\right), \end{eqnarray}

and

\begin{align*} d_1 = \frac{\log \frac{P}{K} + \left( \mu^f + \frac{1}{2}\left(\sigma^f\right)^2\right)T}{\sigma^f\sqrt{T}}, \ d_2 = d_1 - \sigma^f\sqrt{T},\ \text{and}\ d_3= d_1+\sigma^f\sqrt{T}, \end{align*}

where $K = P\left( 1+v\right)^T$ .

Proposition 4. Consider the claim S given by (5.13). The dynamics of the risky fund are given by (5.15). The longevity risk is independent of the risky fund and modeled using (5.18). If the hedgeable part $H_S^h$ of the hybrid claim S is determined using the mean-variance hedger, we find that

\begin{align*} H_{S}^h = \left(\boldsymbol{\theta}_{S^f} \cdot \boldsymbol{{Y}}\right)\times \ \mathbb{E}\left[X_1 \right] \end{align*}

where $\boldsymbol{\theta}_{S^f}$ is given by (5.20), (5.21) and $ \mathbb{E}\left[X_1 \right]$ is given in Proposition 3.

Proposition 4 is important because it allows to determine the hedgeable part of the claim in closed-form while explicitly taking into account that the financial derivative $S^f$ is not perfectly replicable. In order to determine the 3-step hedge-based value using Expression (5.10), we can determine the first term using Proposition 4. Indeed, we have that $\text{e}^{-rT}\mathbb{E}_{\mathbb{Q}}\left[ H_S^h\right]= \mathbb{E}\left[X_1 \right] \left(\theta_{S^f}^{(0)}+\theta_{S^f}^{(1)}P\right)$ . The second term of (5.10), however, needs to be determined using Monte Carlo simulation.

5.2.3. The hedge-based valuation

We also determine values for our unit-linked portfolio when a hedge-based valuation is used where $\pi$ is an actuarial valuation. The hedge-based valuation is given by $\rho^{HB}[S] = \text{e}^{-rT}\mathbb{E}_{\mathbb{Q}}\left[ H_S^h\right] + \pi\left[S-H_S^h \right],$ and we assume $\pi$ is the standard-deviation principle with $\beta=0.2$ . We can express the hedge-based value as follows

(5.25) \begin{equation}\rho^{HB}[S] = \text{e}^{-rT}\mathbb{E}_{\mathbb{Q}}\left[ H_S^h\right] + \text{e}^{-rT} \beta\sqrt{A},\end{equation}

where

\begin{eqnarray*}A & = & \mathbb{E}\left[\left(S^f\right)^2\right]\left(\frac{\mathbb{E}\left[ \text{Var}\left[ X_1|\ Z\right]\right]}{n^a} + \mathbb{E}\left[ \text{e}^{2Z}\right]\right)+\text{Var}\left[ H_S^h\right] + \mathbb{E}[S]^2 - 2\mathbb{E}\left[ X_1\right]\mathbb{E}\left[S^f Y_1\right].\end{eqnarray*}

Note that all terms of A are given in closed-form. Indeed, Proposition 3 can be used to determine $\mathbb{E}\left[ \text{Var}\left[ X_1|\ Z\right]\right]$ , $\mathbb{E}\left[ X_1\right]$ and $\mathbb{E}\left[ \text{e}^{2Z}\right]$ . The closed-form expression for the covariance $\text{Cov}\left[S^f,Y_1 \right]$ shown in (5.23) leads to a closed-form expression for $\mathbb{E}\left[S^f Y_1\right]$ . Taking into account $\mathbb{E}[S]=\mathbb{E}[S^f]\mathbb{E}[X_1]$ , a closed-form expression for $\mathbb{E}[S]$ can be derived by combining Proposition 3 and (5.22). Finally, $\text{Var}\left[ H_S^h\right]$ can be determined in closed-form using (5.24) since $H_S^h$ is a linear combination between the risk-free bond and the risky fund. An expression for $\rho^{HB}[S]$ was also derived in Exercise 12 of Dhaene (Reference Dhaene2022), but under the assumption the financial derivative $S^f$ is hedgeable.

The numerical values and a discussion are provided in the online appendix.