2.1 Two-step valuation with a quadratic loss function

Solvency regulations require a fair valuation of assets and liabilities, that is, their value should correspond to the amount for which they could be transferred to another company or exchanged on the market.Footnote 2 For this reason, the valuation of a contingent claim strictly depends on whether it is tradable on the financial market.

Similar to Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017), we denote by $\mathcal{C}^h\subseteq \mathcal{C}$ the class of claims perfectly hedgeable on the market. For any $S\in \mathcal{C}^h$ , it is possible to find a strategy $\boldsymbol{\beta}\in\mathcal{B}$ such that $S=\boldsymbol{\beta}\cdot \mathbf{Y}$ . In this case, the fair value of the liability S is simply given by the value of the trading strategy at time 0, $\boldsymbol{\beta}\cdot\mathbf{y}$ .

Moreover, we denote by $\mathcal{C}^\perp\subseteq \mathcal{C}$ the class of claims independent of the vector of traded assets $\mathbf{Y}=\left(Y_0,Y_1,\dots, Y_n\right)$ . For such claims, the position of the insurer cannot be hedged in the financial market and therefore the fair value of $S\in\mathcal{C}^\perp$ is calculated by an actuarial premium principle. In a solvency framework, with capital requirements calculated according to the ${{VaR}}$ risk measure, the standard choice is the cost-of-capital premium principle, see for instance European Commission (2009) and Pelsser (Reference Pelsser2011). It is defined as follows:

(2.1) \begin{equation} \pi(S)=e^{-r}\ \mathbb{E}\left[S\right]+e^{-r}\ i\ ({{VaR}}_\alpha (S)-\mathbb{E}\left[S\right]),\end{equation}

where $i\in(0,1)$ is the cost-of-capital rate and

\[{{VaR}}_\alpha(X)=\inf\{x\in{\mathbb R}\ |{\mathbb P}(X\leq x)\geq \alpha\}, \quad {for any risk } X{\in\mathcal{C}}\ {and } \alpha\in(0,1). \]

Thus, $\pi(S)$ is understood as the expected value (net premium) of S, loaded by the cost of the capital required to hold the liability $S-\mathbb{E}[S]$ . Specifically, it is assumed that the insurer’s shareholders require a return i on ${{VaR}}_\alpha (S-\mathbb{E}\left[S\right])$ , that is, on the assets required to support S, net of the expected value. We note that the premium principle (2.1) is not appropriate for claims that depend on traded assets as this would neglect the hedging opportunities.

Many claims that insurance companies face are not perfectly hedgeable, but nevertheless not independent of the payoffs of the traded assets. We call these claims hybrid claims when $S\in \mathcal{C}\backslash(\mathcal{C}^h\cup \mathcal{C}^\perp)$ and these are the focus of our paper.

In that case, some (generally imperfect) hedging of S by $\mathbf Y$ is possible and typically a two-step approach is followed (see Möhr, Reference Möhr2011 and Albrecher et al., Reference Albrecher, Bauer, Embrechts, Filipović, Koch-Medina, Korn, Loisel, Pelsser, Schiller, Schmeiser and Wagner2018). First, the insurer determines a hedging portfolio that is as close as possible to the liability S. To measure “closeness” the quadratic loss function is generally used, providing a trading strategy $\boldsymbol{\theta}_S=(\theta_0,\ldots,\theta_n)$ that minimises the $L^2$ -distance between the liability and the hedging portfolio (see Pelsser and Schweizer, Reference Pelsser and Schweizer2016):

(2.2) \begin{equation}\boldsymbol{\theta}_S=\arg \min_{\boldsymbol{\beta}\in\mathcal{B}} \mathbb{E}\left[(S-\boldsymbol{\beta} \cdot \mathbf{Y})^2\right]. \end{equation}

For the rest of the paper, $\boldsymbol{\theta}_S$ will always denote the trading strategy associated with the quadratic loss function and the index S will be dropped when no confusion is possible. Standard least-squares arguments provide the solution to problem (2.2), $\boldsymbol{\theta}=(\mathbb{E}\left[\mathbf{Y}^\intercal \cdot\mathbf{Y}\right])^{-1}\mathbb{E}\left[S\cdot\mathbf{Y}^\intercal\right]$ Footnote 3 and ensure that the expected value of the hedging portfolio matches the expected value of the liability:

(2.3) \begin{equation} \mathbb{E}\left[\boldsymbol{\theta} \cdot \mathbf{Y}\right]= \mathbb{E}\left[S\right].\end{equation}

Second, the insurer values the residual risk $R(S,\boldsymbol{\theta}):=S-\boldsymbol{\theta} \cdot \mathbf{Y}$ , which could not be hedged, by the cost-of-capital principle $\pi$ . Following such a method, the fair value $\phi(S)$ is calculated as the sum of the cost of the hedging portfolio and the premium principle of the residual risk (see Dhaene et al., Reference Dhaene, Stassen, Barigou, Linders and Chen2017):

(2.4) \begin{align}\phi(S)&=\boldsymbol{\theta}\cdot \mathbf{y}+\pi (R(S,\boldsymbol{\theta})) \nonumber\\[3pt] &=\boldsymbol{\theta}\cdot \mathbf{y}+e^{-r} i {{VaR}}_\alpha(S-\boldsymbol{\theta}\cdot \mathbf{Y}),\end{align}

where we used the property (2.3).

The fair value (2.4) can be seen as a generalisation of the premium principle (2.1), where (a) the net premium $\mathbb E[S]$ is replaced by the cost of the hedging strategy $\boldsymbol{\theta}\cdot \mathbf{y}$ , which once again matches on average the liability S and (b) the cost of capital is calculated on the residual $S-\boldsymbol{\theta}\cdot \mathbf{Y}$ , rather than $S-\mathbb E[S]$ .

2.2 Valuation with general loss functions

The valuation approach we just discussed relies on the use of a quadratic loss function $\ell(x)=x^2$ to penalise deviations of the hedging portfolio payoff from the liability. Here, we generalise the two-step valuation approach, addressing two specific concerns:

• The quadratic loss function penalises losses and gains equally. As an insurer, the major concern is to avoid a shortfall, namely situations where $S>\boldsymbol{\theta} \cdot \mathbf{Y}$ . Various authors (for instance Föllmer and Leukert, Reference Föllmer and Leukert2000 and François et al., Reference François, Gauthier and Godin2014) proposed alternative penalty functions that only penalise losses or penalise losses and gains asymmetrically.

• The total level of assets that the insurer has to hold with respect to their liabilities is typically given by a risk measure, for example in the case of Solvency II, ${{VaR}}_{0.995}$ . It is then not obvious why a quadratic hedging strategy should be used, which results in a residual risk with mean zero, rather than a hedging strategy that produces a ${{VaR}}_{0.995}$ -neutral portfolio.

To elaborate on these points, consider a convex loss function $\ell:{\mathbb R}\to[0,+\infty)$ , with $\ell(x)=0$ if and only if $x=0$ . The resulting hedging strategy $\boldsymbol{\xi}^{(\ell)}_S=(\xi^{(\ell)}_0,\ldots,\xi^{(\ell)}_n)$ is defined as the minimiser of the expected loss function

(2.5) \begin{equation}{\boldsymbol{\xi}^{(\ell)}_S}=\arg \min_{\boldsymbol{\beta}\in\mathcal{B}} \mathbb{E}\left[\ell(S-\boldsymbol{\beta} \cdot \mathbf{Y})\right]. \end{equation}

Again, we drop the subscript to write ${\boldsymbol{\xi}^{(\ell)}}$ , if no confusion ensues. Different choices of the loss function lead to the risk neutrality (or unbiasedness) of the residual risk with respect to different risk measures. Specifically, under mild conditions we have that (see Thm 3.2 in Rockafellar et al., Reference Rockafellar, Uryasev and Zabarankin2008)

\[\Gamma^{(\ell)}(S-{\boldsymbol{\xi}^{(\ell)}}\cdot \mathbf{Y})=0,\]

where $\Gamma^{(\ell)}$ is the risk measure given by

(2.6) \begin{equation}\Gamma^{(\ell)}(X)=\arg \min_{c \in \mathbb R } \mathbb{E}\left[\ell(X-c)\right],\quad {for any }X\in\mathcal{C}. \end{equation}

Slightly different versions of risk functionals as defined in (2.6) are treated in the literature using a first order condition to find the minimiser in (2.5), see for instance the zero-utility premium principles discussed in Deprez and Gerber (Reference Deprez and Gerber1985), the class of shortfall risk measures introduced by Föllmer and Schied (Reference Föllmer and Schied2002), the generalised quantiles investigated by Bellini et al. (Reference Bellini, Klar, Müller and Gianin2014), the optimised certainty equivalent in Ben-Tal and Teboulle (Reference Ben-Tal and Teboulle2007), the elicitable functionals studied in Gneiting (Reference Gneiting2011) and more recently the class of convex hedgers by Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017). The implications of different choices of $\ell$ are elaborated on in detail by Rockafellar and Uryasev (Reference Rockafellar and Uryasev2013). Three important examples are:

1. 1. If $\ell(x)=x^2$ , we have that $\Gamma^{(\ell)}(X)=\mathbb E[X]$ and ${\boldsymbol{\xi}^{(\ell)}}= \boldsymbol{\theta}$ , as seen before.

2. 2. Let $\ell(x):= \ell_{\alpha}(x)$ , where

   (2.7) \begin{equation} \ell_{\alpha}(x)=\frac{\alpha}{1-\alpha}x_++x_-,\qquad \alpha\in(0,1) \end{equation}
   with
   \begin{align*}x_{+}&=\max(x,0)\\[3pt] x_{-}&=\max(-x,0),\end{align*}
   is the normalised Koenker–Bassett loss function, see Koenker (Reference Koenker2005). Then, $\Gamma^{( \ell_{\alpha})}(X)=\mathbb {{VaR}}_\alpha(X)$ . We henceforth denote the trading strategy associated with this loss function by ${\boldsymbol{\xi}^{(\ell_\alpha)}}:= \boldsymbol{\xi}$ . This strategy satisfies
   (2.8) \begin{equation}{{VaR}}_\alpha \left(S-\boldsymbol{\xi} \cdot \mathbf{Y}\right)=0.\end{equation}
   This case is important to us, given the desired feature that the VaR of the residual risk is zero, indicating that sufficient assets have been allocated to satisfy regulatory requirements.

3. 3. Alternatively, with slight abuse of notation, consider the loss function $\ell(x):=\ell_\tau(x)$ where

   \begin{equation*} \ell_{\tau}(x)=\tau (x_+)^2 + (1-\tau)(x_-)^2, \quad \tau \in (0,1). \end{equation*}
   The resulting risk measure
   \[\Gamma^{(\ell_\tau)}(X)=\arg \min_{c \in \mathbb R } \mathbb{E}\left[\tau ((X-c)_+)^2 + (1-\tau)((X-c)_-)^2\right] \]
   is the $\tau$ -expectile. Expectiles generalise the mean (which is obtained by setting $\tau=0.5$ ) and, for $\tau\geq 0.5$ , are coherent risk measures, thus addressing a common criticism of VaR, while remaining within the tractable class of shortfall risk measures (see for instance Delbaen et al., Reference Delbaen, Bellini, Bignozzi and Ziegel2016). One can see hedging with $\ell_\tau(x)$ as a modification of quadratic hedging, where, for $\tau>0.5$ additional weight is given to the downside risk. Expectile regression was introduced by Newey and Powell (Reference Newey and Powell1987) and then further generalised to M-quantiles by Breckling and Chambers (Reference Breckling and Chambers1988). We return to expectile hedging strategies in Section 2.3.

By changing the hedging objective from a quadratic to an asymmetric loss function, we move the focus on positive deviations of the hedging portfolio from the liability. As a result, the residual risk changes from having zero mean to having a zero VaR or Expectile. Specifically, by (2.8), if we set up the strategy $\boldsymbol{\xi}$ for $\alpha=0.995$ , by construction, the portfolio will cover the liability S with probability $\alpha=0.995$ . In this article, we focus on the use of the quantile hedging strategy

(2.9) \begin{equation} \boldsymbol{\xi} = \arg \min_{\boldsymbol{\beta}\in\mathcal{B}} \mathbb{E}\left[ \ell_{\alpha}(S-\boldsymbol{\beta} \cdot \mathbf{Y})\right],\end{equation}

where $\ell_{\alpha}(x)$ is given in (2.7). The regression problem (2.9) is well-known as the quantile regression pioneered by Koenker and Bassett (Reference Koenker and Bassett1978). As we use it for hedging objectives, we refer to this minimisation as quantile hedging; this should not be confused with the quantile hedging of Föllmer and Leukert (Reference Föllmer and Leukert1999) which targets a different objective.

Moreover, as the quantile hedging strategy is a risk minimiser with respect to a convex loss function, we will avoid situations where capital requirements can be reduced by shifting losses to the extreme tails, beyond the VaR level. Hence one of the key criticisms of VaR, see for example Section 4.4 of Danelsson (Reference Danelsson2002), is addressed, as illustrated in the following example.

Example 1. We consider a simple example, where there is only one risky asset correlated with the liability S. Assume that the risk measure used is ${{VaR}}_{0.9}$ and that the risk-free asset has return 1 (zero interest rates, in particular $y_0\,{=}\,Y_0\,{=}\,1$ ). The liability S follows a Lognormal distribution with parameters $\mu=0.1$ and $\sigma=0.3$ , such that ${{VaR}}_{0.9}(S)=1.623$ . The asset $Y_1$ is a derivative on S, with a price of $y_1=1$ and pay-off:

\[Y_1= 1.5 \cdot \textbf{1}_{\{S\leq {{VaR}}_{0.9}(S)\}} - 3 \cdot \textbf{1}_{\{S> {{VaR}}_{0.9}(S)\}}.\]

Hence, the derivative offers a high return when S is less than ${{VaR}}_{0.9}(S)$ , but produces an even larger loss when S exceeds ${{VaR}}_{0.9}(S)$ . Investment in such an asset would be capital efficient, as it moves the loss beyond the 90% confidence level, thus making it “invisible” to ${{VaR}}_{0.9}$ . On the other hand, prudent risk management would require the holder of S to short $Y_1$ , in order to be able to hedge their tail risk.

To make these considerations precise, we look at two investment strategies:

$\boldsymbol{A}$ Invest $\beta_0= 0$ in the risk-free asset $Y_0$ and $\beta_1={{VaR}}_{0.9}(S)/1.5=1.082$ in $Y_1$ . The resulting portfolio VaR is:

\begin{align*} &{{VaR}}_{0.9}(S-\beta_1Y_1)\\[2pt] &\quad={{VaR}}_{0.9}\left(S-{{VaR}}_{0.9}(S)\left(\textbf{1}_{\{S\leq {{VaR}}_{0.9}(S)\}} - 2 \cdot \textbf{1}_{\{S> {{VaR}}_{0.9}(S)\}}\right)\right)\\[2pt] &\quad= {{VaR}}_{0.9}\left(S-{{VaR}}_{0.9}(S)+3{{VaR}}_{0.9}(S) \textbf{1}_{\{S> {{VaR}}_{0.9}(S)\}}\right)\\[2pt] &\quad={{VaR}}_{0.9}\left(S-{{VaR}}_{0.9}(S)\right)+3{{VaR}}_{0.9}(S) {{VaR}}_{0.9}\left(\textbf{1}_{\{S> {{VaR}}_{0.9}(S)\}}\right)\\[2pt] &\quad=0, \end{align*}

where the 3rd equality is by comonotonic additivity of VaR.

$\boldsymbol{B}$ Invest $\boldsymbol{\xi} =(\xi_0,\xi_1)$ in the assets $(Y_0, Y_1)$ , where $\boldsymbol{\xi}$ is the quantile hedging strategy at level $\alpha=0.9$ . By construction we have

\begin{align*} {{VaR}}_{0.9}(S-\xi_0-\xi_1Y_1)&=0. \end{align*}

The corresponding optimal positions are

\begin{align*} \xi_0&=1.697,\\ \xi_1&=-0.174. \end{align*}

Hence, it is indeed seen that a negative exposure to $Y_1$ is produced. Furthermore, the cost of this strategy is equal to $\xi_0+\xi_1=1.523 >1.082=\beta_1$ . Hence the Strategy B is more expensive, while achieving the same zero VaR for the portfolio as Strategy A.

We compare these two strategies with respect to the cumulative distribution function (cdf) of their residual risks in Figure 1. The persistent tail risk arising from Strategy A is clearly visible, indicating how this strategy, while cost efficient, reflects poor (unethical even) risk management. This is not a problem we face with the quantile hedging Strategy B.

Figure 1: Cumulative distribution functions of the residual risks $R(S,\boldsymbol{\beta})$ (left) and $R(S,\boldsymbol{\xi} )$ (right), corresponding, respectively, to the Strategies A and B of Example 1.

2.3 Two-step valuation with quadratic and quantile hedging

We have considered different loss functions to yield a portfolio that is more suitable for fair valuation in a solvency context, compared to quadratic hedging. Valuation still needs to take into account that only a fraction of capital requirements is borne by policyholders. Here, we show that the hedging strategy based on a general loss function can be obtained also by a two-step approach, where quadratic hedging is used as a first step and hedging based on a convex loss function is subsequently applied on the residual liability (see Lemma 2.4 for more details). The two-step approach is designed based on the idea that assets can be raised from a capital investor, if sufficient return can be provided. Hence, the value of the liabilities is not equal to the full cost of the hedging strategy $\boldsymbol{\xi}$ , but needs instead to cover shareholders’ capital costs for funding this strategy. Hence, quantile hedging ensures that regulatory requirements are satisfied, while the decomposition of the hedging costs to a part that is fully borne by policyholders and a “cost-of-capital” part, reflects the source of funding of the strategy. As a result we obtain a generalisation of (2.4).

Before discussing in depth about this approach, we briefly recall the definition of a valuation and its properties as introduced in Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017).

Definition 2.1 (Valuation) A valuation is a mapping

\begin{equation*}\rho\,{:}\,\mathcal{C}\to {\mathbb R},\ S\mapsto \rho(S),\end{equation*}

that is normalised $\rho(0)=0$ and translation invariant

\[\rho(S+a)=\rho(S)+e^{-r}a,\quad \forall S\in\mathcal{C},\ a\in{\mathbb R}.\]

Here, we list some properties that a valuation may satisfy and which will be discussed in the following. For any $S,S_1,S_2\in\mathcal{C}$ we say that a valuation $\rho$ is

1. 1. Positive homogeneous if

   \[\rho(\lambda S)=\lambda \rho(S),\qquad {for any } \ \lambda\geq 0;\]

2. 2. Market-Consistent if

   \[\rho(S+S^h)=\rho(S)+\boldsymbol{\beta}\cdot \mathbf{y},\]
   for any hedgeable payoff: $S^h=\boldsymbol{\beta}\cdot \mathbf{Y}$ ;

3. 3. Actuarial if

   \begin{equation*}\rho (S^\perp)=e^{-r} \mathbb{E}[S^\perp]+{RM}(S^\perp),\end{equation*}
   for any claim $S^\perp \in\mathcal{C}^\perp $ , where ${RM}:\mathcal{C}^\perp\to {\mathbb R}$ is a mapping that does not depend on current asset prices $\mathbf{y}$ ;

4. 4. Fair if $\rho$ is market-consistent and actuarial.

The market-consistency property means that the valuation is marked-to-market for any hedgeable part of a liability, while a valuation is actuarial if it is marked-to-model for any claim which is independent of the financial market. The definition of fair valuation was recently introduced by Dhaene et al. (Reference Dhaene, Stassen, Barigou, Linders and Chen2017) to formalise the valuation of hybrid claims as required by solvency regulation (see also Barigou et al., Reference Barigou, Chen and Dhaene2019; Delbaen et al., 2019b).

Definition 2.2 (Two-step valuation) Consider the liability S and a convex loss function $\ell:\mathbb{R}\to [0,+\infty)$ , such that $\ell(x)=0$ if and only if $x=0$ . Let $\boldsymbol{\theta}$ be the quadratic hedging strategy for S and let $\boldsymbol{\eta}^{(\ell)}=(\eta^{(\ell)}_0, ,\ldots,\eta^{(\ell)}_n)$ be the hedging strategy for the residual risk $R(S,\boldsymbol{\theta})=S-\boldsymbol{\theta} \cdot \mathbf{Y}$ with loss function $\ell$ , that is, the strategy that minimises the expected loss:

(2.10) \begin{equation}\boldsymbol{\eta}^{(\ell)}:=\arg \min_{\boldsymbol{\beta}\in\mathcal{B}} \mathbb{E}\left[ \ell \big(R(S,\boldsymbol{\theta})-\boldsymbol{\beta} \cdot \mathbf{Y}\big)\right].\end{equation}

Then, the two-step valuationFootnote ⁴ for S is defined as

(2.11) \begin{equation}\rho^{(\ell)}(S):= \boldsymbol{\theta}\cdot \mathbf{y} + i \ \boldsymbol{\eta}^{(\ell)} \cdot \mathbf{y}.\end{equation}

Using the translation invariance property of the ordinary least square regression, it is immediate to verify that the two-step valuation is indeed a valuation according to Definition 2.1. The following lemma shows that performing convex hedging on the liability or performing quadratic hedging on the liability and then convex hedging on the residual risk leads to the same hedging strategy. Hence, our two-step approach is consistent with constructing an asset portfolio that “convex-hedges” the liability S. We recall from (2.5) that $\boldsymbol{\xi}^{(\ell)}_S$ is the hedging strategy that minimises the expected loss function for the liability S ( ${\boldsymbol{\xi}^{(\ell)}}=\arg \min_{\boldsymbol{\beta}\in\mathcal{B}} \mathbb{E}\left[\ell(S-\boldsymbol{\beta} \cdot \mathbf{Y})\right]$ ), while $\boldsymbol{\eta}^{(\ell)}$ is the hedging strategy for the same expected loss function applied to the residual risk as defined in (2.10).

1. (a) If $\boldsymbol{\xi}^{(\ell)}$ is the unique minimiser of the expected loss for the function $\ell$ and liability S as defined in (2.5), and if $\boldsymbol{\theta}$ and $ \boldsymbol{\eta}^{(\ell)}$ are the hedging strategies as defined in (2.2) and (2.10), then we have

   \begin{equation*} \boldsymbol{\xi}^{(\ell)}=\boldsymbol{\theta}+\boldsymbol{\eta}^{(\ell)}.\end{equation*}

2. (b) If the hedging strategy for S is not unique, denote by $\{\boldsymbol{\xi}^{(\ell),j}\}_{j\in\mathcal{A}} $ the set of minimisers of (2.5). Then, for any $j\in\mathcal{A}$ , we have that

   \begin{equation*}\boldsymbol{\xi}^{(\ell),j}=\boldsymbol{\theta}+\boldsymbol{\eta}^{(\ell),j},\end{equation*}
   where $\boldsymbol{\eta}^{(\ell),j}\in\{\boldsymbol{\eta}^{(\ell),j}\}_{j\in\mathcal{A}} $ , is the set of hedging strategies for the residual risk $S-\boldsymbol{\theta}\cdot \mathbf{Y}$ .

Proof.

1. (a) Let us first consider the case where the hedging strategy is unique. The proof is by contradiction. Assume that $ \boldsymbol{\xi}^{(\ell)}\neq \boldsymbol{\theta}+\boldsymbol{\eta}^{(\ell)}$ and define $\boldsymbol{\eta}^{{(\ell)},*}=\boldsymbol{\xi}^{(\ell)}-\boldsymbol{\theta}$ . By (2.5) we have

   \[ \mathbb{E}\left[ \ell(S-\boldsymbol{\xi}^{(\ell)}_S \cdot \mathbf{Y})\right]=\mathbb{E}\left[\ell(S-(\boldsymbol{\theta}+\boldsymbol{\eta}^{(\ell),*} ) \cdot \mathbf{Y}\right)]< \mathbb{E}\left[ \ell(S-(\boldsymbol{\theta}+\boldsymbol{\eta}^{(\ell)} ) \cdot \mathbf{Y})\right],\]
   which contradicts the definition of $\boldsymbol{\eta}^{(\ell)}$ :
   \[ \boldsymbol{\eta}^{(\ell)}=\arg \min_{\boldsymbol{\beta}\in\mathcal{B}} \mathbb{E}\left[ \ell(S-\boldsymbol{\theta} \cdot \mathbf{Y}-\boldsymbol{\beta} \cdot \mathbf{Y})\right].\]

2. (b) For the non-uniqueness case, take $\boldsymbol{\xi}^{(\ell),j}\in\{\boldsymbol{\xi}^{(\ell),j}_S\}_{j\in\mathcal{A}}$ and define $\boldsymbol{\eta}^{(\ell),j}=\boldsymbol{\xi}^{(\ell),j}-\boldsymbol{\theta} $ . By definition, we have that

   \[ \mathbb{E}\left[ \ell(S-\boldsymbol{\xi}^{(\ell),j} \cdot \mathbf{Y})\right]= \mathbb{E}\left[ \ell(S-\boldsymbol{\theta} \cdot \mathbf{Y}-\boldsymbol{\eta}^{(\ell),j }\cdot \mathbf{Y})\right].\]
   Therefore, $\boldsymbol{\eta}^{(\ell),j}$ should be a convex hedging strategy for the residual risk, otherwise this would contradict the optimality of $\boldsymbol{\xi}^{(\ell),j}_S $ . Analogously, if $\boldsymbol{\eta}^{(\ell),j}$ is a convex hedging strategy for the residual risk then $\boldsymbol{\xi}^{(\ell),j}_S=\boldsymbol{\theta}+\boldsymbol{\eta}^{(\ell),j}$ must be a convex hedging strategy for S.

The two-step valuation equals the cost of the quadratic hedging strategy $\boldsymbol{\theta}$ plus the cost of capital of the hedging strategy $\boldsymbol{\eta}^{(\ell)}$ necessary to cover the residual risk. From an economic standpoint, the insurer uses the hedging strategy $\boldsymbol{\xi}^{(\ell)}$ to cover the regulatory requirements and shares its cost between the policyholders and shareholders. From an asset–liability perspective, the interpretation is as follows: let $\boldsymbol{\xi}^{(\ell)}_S\cdot \boldsymbol{y}= (\boldsymbol{\theta}+\boldsymbol{\eta}^{(\ell)})\cdot\boldsymbol{y}$ be the cost of the convex hedging strategy at time 0 where $\boldsymbol{\theta}\cdot \boldsymbol{y}$ is the cost of the quadratic hedging at time 0. The two-step insurance valuation assumes that the quadratic hedging cost $\boldsymbol{\theta}\cdot \boldsymbol{y}$ is borne by policyholders and the residual cost $\boldsymbol{\eta}^{\ell} \cdot\boldsymbol{y}$ to achieve a VaR-neutral portfolio is borne by shareholders. Shareholders require a return for the capital they provide (the so called cost-of-capital) $i\cdot\boldsymbol{\eta}^{(\ell)}\cdot \boldsymbol{y}$ which is also paid by the policyholders. Thus, what we define as the fair value (the quadratic hedging cost plus the cost-of-capital risk margin) is $\rho(S)=\boldsymbol{\theta} \cdot \boldsymbol{y}+i\cdot\boldsymbol{\eta}^{\ell}\cdot \boldsymbol{y}$ . Note that we assume that assets are invested according to the hedging strategy $\xi^{(\ell)}_S$ , which covers the full liability. The split into the strategies $\boldsymbol{\theta} $ and $\eta^{\ell} $ is notional, with the specific purpose of apportioning the cost of hedging to different stakeholders. Our valuation is thus closely related to the two-step approach of Möhr (Reference Möhr2011) who also considers the apportionment of hedging costs to policyholders and shareholders in a very similar way. Möhr (Reference Möhr2011) employs a different acceptability condition, interpreting the cost-of-capital rate as the expected excess return required by the shareholders, while, in our case, we implicitly assume that shareholders bear the residual risk of deviation from the mean-quantile hedging strategy.

The remainder of the article focuses on the two-step valuation where the quantile hedging strategy is considered in the second step, called the mean-quantile valuation. From now on, we drop the upper-script $\ell$ if we consider quantile hedging in the second step. We also briefly discuss the two-step valuation with the expectile loss function (3) in the second step, which we call the mean-expectile valuation.

In the cost-of-capital approach for (2.4), a capital $c={{VaR}}_\alpha (R(S,\boldsymbol{\theta}))$ is set up and kept risk-free until year 1 to guarantee that

\[{{VaR}}_\alpha (S-\boldsymbol{\theta} \cdot \mathbf{Y}-c)=0.\]

In the two-step valuation proposed in this section, we set up a strategy $\boldsymbol{\eta}$ such that

\[{{VaR}}_\alpha (S-\boldsymbol{\theta} \cdot \mathbf{Y}-\boldsymbol{\eta} \cdot \mathbf{Y})={{VaR}}_\alpha\Big(R\big(R(S,\boldsymbol{\theta}),\boldsymbol{\eta}\big)\Big)=0. \]

In the following result, we show that the mean-quantile valuation is positive homogeneous and fair.

Proof. The positive homogeneity is directly obtained by the positive homogeneity of $\boldsymbol{\theta}$ and $\boldsymbol{\eta}$ (see Koenker, Reference Koenker2005). To prove that the mean-quantile valuation is fair, we show that the valuation is market-consistent and actuarial.

• First, we notice that the solution of the quadratic hedging problem is additive:

  \begin{equation*}\boldsymbol{\theta}_{S+S^h}= \boldsymbol{\theta}_{S}+\boldsymbol{\theta}_{S^h}.\end{equation*}
  Since $S^h$ can be hedged with $\boldsymbol{\nu}$ , we have that $\boldsymbol{\theta}_{S^h}=\boldsymbol{\nu}$ . Therefore, we find that
  \begin{align*}\rho (S+S^h)&=\boldsymbol{\nu}\cdot \mathbf{y}+ \boldsymbol{\theta}_{S} \cdot \mathbf{y}+ i \ \boldsymbol{\eta}_{S+S^h} \cdot \mathbf{y}\\[3pt]&=\boldsymbol{\nu}\cdot \mathbf{y}+\rho (S)\end{align*}
  where $\eta_{S+S^h}$ is the quantile hedging strategy of the residual loss of $S+S^h$ :
  \begin{equation*}R(S+S^h, \boldsymbol{\theta}_{S+S^h})=S+S^h-\boldsymbol{\theta}_{S^h} \cdot \mathbf{Y}-\boldsymbol{\theta}_{S} \cdot \mathbf{Y}=S-\boldsymbol{\theta}_{S} \cdot \mathbf{Y}=R(S,\boldsymbol{\theta}_S),\end{equation*}
  which ends the proof for the market-consistency.

• By standard least-squares arguments, the quadratic hedging strategy of $S^\perp$ is

  \begin{equation*} \boldsymbol{\theta}_{S^\perp}=(\mathbb{E}[S^\perp],0,\dots,0).\end{equation*}
  Otherwise stated, if a liability is independent of risky assets, the hedging strategy only invests risk-free. Therefore, we find that
  \begin{equation*}\rho (S^\perp)=e^{-r} \mathbb{E}[S^\perp]+ i \ \boldsymbol{\eta}_{S^\perp} \cdot \mathbf{y},\end{equation*}
  where $ \eta_{S^\perp} $ is the quantile hedging strategy for $R(S^\perp, \boldsymbol{\theta}_{S^\perp})=S^\perp-\mathbb{E}[S^\perp]$ . Since $R(S^\perp, \boldsymbol{\theta}_{S^\perp})$ is independent of the risky assets, we find that the quantile hedging strategy is (cf. Theorem 4 in Dhaene et al., Reference Dhaene, Stassen, Barigou, Linders and Chen2017)
  \begin{equation*}\boldsymbol{\eta}_{S^\perp}=({{VaR}}_\alpha(S^\perp-\mathbb{E}[S^\perp]),0,\dots,0),\end{equation*}
  which implies that the two-step valuation of $S^\perp$ is
  \begin{equation*}\rho (S^\perp)=e^{-r} \mathbb{E}[S^\perp]+e^{-r} i {{VaR}}_\alpha(S^\perp-\mathbb{E}[S^\perp]).\end{equation*}
  The mean-quantile valuation corresponds then to the standard cost-of-capital principle and the mean-quantile valuation is fair.

Proof. To verify the positive homogeneity of the valuation it is sufficient to check that the expectile strategy is positive homogeneous. Note that for any $a>0,\ S\in\mathcal{C}, \boldsymbol{\beta}\in\mathcal{B}$ we have:

\[{\mathbb E}[\ell_\tau(aS-a\boldsymbol{\beta}\cdot\boldsymbol{y})]=a^2{\mathbb E}[\ell_\tau(S-\boldsymbol{\beta}\cdot\boldsymbol{Y})],\]

that implies $\arg\min_{\boldsymbol{\beta}\in\mathcal{B}}{\mathbb E}[\ell_\tau(aS-\boldsymbol{\beta}\cdot\boldsymbol{Y})]=a\arg\min_{\boldsymbol{\beta}\in\mathcal{B}}{\mathbb E}[\ell_\tau(S-\boldsymbol{\beta}\cdot\boldsymbol{Y})]$ . The fairness of the mean-expectile valuation follows exactly the same steps of the one for the mean-quantile valuation and is therefore omitted.

Hereafter, we show that applying quantile hedging to the residual risk will reduce the tail of the residual risk compared to making an investment in the risk-free asset.

Let us assume that we want to hedge $R(S,\boldsymbol{\theta})=S-\boldsymbol{\theta} \cdot \mathbf{Y}$ and the regulator imposes that ${{VaR}}_\alpha(R(S,\boldsymbol{\theta})-\boldsymbol{\beta} \cdot \mathbf{Y}) =0$ for some trading strategy $\boldsymbol{\beta}\in\mathcal{B}$ . To achieve this, there are two possibilities:

• Consider an investment in the risk-free asset equal to ${{VaR}}_\alpha(R(S,\boldsymbol{\theta} )) $ . We denote this strategy by $\boldsymbol{\nu} $ .

• Consider the quantile hedging strategy $\boldsymbol{\eta} $ such that ${{VaR}}_\alpha(R(S,\boldsymbol{\theta} )-$ $\boldsymbol{\eta} \cdot \mathbf{Y})=0$ .

The quantile hedging strategy is the minimiser of the Tail Value-at-Risk (TVaR) deviation of the residual risk among all the strategies which satisfy the VaR regulatory constraint. We recall that TVaR is a coherent risk measure defined as

${TVaR}_{\alpha}(X)=\frac{1}{1-\alpha}\int_\alpha^1{{VaR}}_u(X){d}u, \quad{for any finite mean risk }X\ {and any }\alpha\in(0,1)$ and the TVaR deviation (dTVaR) is defined as ${{dTVaR}}_\alpha(X)={{TVaR}}_\alpha(X)-{\mathbb E}(X)$ for any $X\in\mathcal{C}$ .

Lemma 2.8 Consider $\boldsymbol{\theta} $ , $\boldsymbol{\nu}$ and $\boldsymbol{\eta} $ as defined above. The quantile hedging strategy is the minimiser of the TVaR deviation of the residual risk:

\begin{equation*}{{dTVaR}}_\alpha(R(S-\boldsymbol{\theta}\cdot \mathbf{Y},\boldsymbol{\eta})) \leq {{dTVaR}}_\alpha(R(S-\boldsymbol{\theta}\cdot \mathbf{Y},\boldsymbol{\beta})),\end{equation*}

for all hedging strategies $\boldsymbol{\beta} $ such that ${{VaR}}_\alpha(R(S,\boldsymbol{\theta} )-\boldsymbol{\beta} \cdot \mathbf{Y}) =0$ . This is in particular the case for $\boldsymbol{\beta}= \boldsymbol{\nu}$ .

Proof. This is a direct application of Theorem 3.2. in Rockafellar et al. (Reference Rockafellar, Uryasev and Zabarankin2008).

Lemma 2.8 also reveals why quantile hedging, through the minimisation of a convex loss function, does avoid perverse incentives, as shown in Example 1. In the next example, we illustrate the application of our two-step valuation approach in an insurance portfolio.

We consider a portfolio of equity-linked life insurance contracts, which guarantee a survival benefit to all policyholders who are still alive at maturity time $T=1$ . The insurance liability can be expressed as

(2.12) \begin{equation}S=N\times\max\left( Y_1,K\right),\end{equation}

where N represents the number of survivors at time 1, among an initial population of 1000 policyholders and K is a guarantee level. We make the following assumptions: $N\sim Bin(1000,0.9)$ ; $Y_1 \sim LN(0.1,0.2^2)$ ; the value of the risky asset at time 0 is $y_1=1$ ; interest rates are zero, that is, $y_0=Y_0=1$ and $K=1$ . The analysis is carried out on a sample of 200,000 simulated scenarios.

On the left of Figure 2, we plot samples of S against $Y_1$ . One can see the strong (nearly deterministic) positive relationship between the two, indicating that $Y_1$ can be used to hedge S. On the right of Figure 2, we show the residuals of the quadratic hedging strategy, $R(S,\boldsymbol{\theta})$ against $Y_1$ . The residuals are clearly not independent of the tradeable asset, indicating that quantile or expectile hedging of those residuals will be meaningful. In other words, we would expect that our two-step valuation $\rho$ would give different answers than the valuation $\phi$ defined in (2.4).

Figure 2: Liability S (left) and residual $R(S,\boldsymbol{\theta})$ (right) against value of the risky asset $Y_1$ .

In Table 1 we present the following trading strategies:

1. 1. $\boldsymbol{\theta}$ : the result of quadratic hedging of S.

2. 2. $\boldsymbol{\xi}$ : the result of quantile hedging of S, with $\alpha=0.99$ .

3. 3. $\boldsymbol{\xi}^{(\ell_\tau)}$ : the result of expectile hedging of S, with $\tau=0.998$ . The value of $\tau$ was calibrated so that the Value-at-Risk and expectiles of S match, that is ${{VaR}}_\alpha(S)= \Gamma^{(\ell_\tau)}(S)$ .

4. 4. $({{VaR}}_\alpha(R(S,\boldsymbol{\theta})),0)$ : investing the VaR of the residuals $R(S,\boldsymbol{\theta})$ in the risk-free asset.

5. 5. $\boldsymbol{\eta}$ : the result of quantile hedging of $R(S,\boldsymbol{\theta})$ .

6. 6. $\boldsymbol{\eta}^{(\ell_\tau)}$ : the result of expectile hedging of $R(S,\boldsymbol{\theta})$ .

Table 1 Investment in risk-free and risky asset, from hedging strategies associated with the two-step valuation of S.

We can observe that $\boldsymbol{\xi}$ and $\boldsymbol{\xi}^{(\ell_\tau)}$ place a substantially higher investment into the risk-free asset, compared to $\boldsymbol{\theta}$ , reflecting the more stringent criterion encoded in the respective loss functions – recall that ${\mathbb E}\big[R(\boldsymbol{\theta},S)\big]=0$ , ${{VaR}}_\alpha\big(R(\boldsymbol{\xi},S)\big)=0$ , $\Gamma^{(\ell_\tau)}\big(R(\boldsymbol{\xi}^{(\ell_\tau)},S)\big)=0$ . At the same time, the investment in the risky asset is somewhat higher for $\boldsymbol{\theta}$ , reflecting a lower sensitivity to adverse movements in asset values. The remaining three strategies pertain to hedging the residuals of the first (quadratic hedging) step. We see that this second step, for quantile and expectile hedging, involves a reduction in the exposure to the risky asset. From Table 1, we also observe that the quantile hedging strategy can lead to a reduction of the total assets required at time 0 to achieve a VaR-neutral portfolio. Indeed, the cost of quantile strategy $\boldsymbol{\xi}$ is lower than the cost of the quadratic strategy and investing the residual risk risk-free.

In Figure 3, we show the densities of the residuals corresponding to the strategies of 4.–6. above. We can see that quantile and expectile hedging lead to a somewhat different shape, compared to a quadratic regression that is followed by investing the ${{VaR}}_\alpha$ of the residual risk in the risk-free asset. We quantify the difference between the three densities, by stating the corresponding TVaR deviations:

\begin{align*} {{dTVaR}}_\alpha\big(R(S,\boldsymbol{\theta})-{{VaR}}_\alpha(R(S,\boldsymbol{\theta}))\big)&= 194.2\\[3pt] {{dTVaR}}_\alpha\big(R(R(S,\boldsymbol{\theta}),\boldsymbol{\eta})\big)&= 182.5\\[3pt] {{dTVaR}}_\alpha\big(R(R(S,\boldsymbol{\theta}),\boldsymbol{\eta}^{(\ell_\tau)})\big)&= 182.6\end{align*}

Hence, the application of quantile and expectile hedging reduces the tail of the residuals, compared to the quadratic hedging case. For quantile regression, this observation is a direct implication of Lemma 2.8. Of course different conclusions may be reached, if the criterion for measuring the variability of residuals changes. For example, considering the standard deviation of residuals privileges quadratic regression, leading to $\sigma\big(R(S,\boldsymbol{\theta}) - {{VaR}}_\alpha(R(S,\boldsymbol{\theta})) \big)=49.0$ , while $\sigma\big(R(R(S,\boldsymbol{\theta}),\boldsymbol{\eta} ) \big)=50.3$ ; note though the difference between the two is small.

Figure 3: Densities of residuals, for quadratic hedging of S, followed by investing the VaR of the residual in the risk-free asset (blue); quantile hedging of S (blue); and expectile hedging of S (red).

Finally, we state the fair value of S, as calculated via the three different hedging approaches 4.-6., for cost-of-capital rate $i=0.1$ :

\begin{align*}\phi(S)&=\boldsymbol{\theta}\cdot \mathbf{y}+i \cdot {{VaR}}_\alpha\big( R(S,\boldsymbol{\theta})\big) = 972.6 \\[3pt] \rho(S)&=\boldsymbol{\theta}\cdot \mathbf{y}+i \cdot \boldsymbol{\eta}\cdot \mathbf{y} = 972.4 \\[3pt] \rho^{(\ell_\tau)}(S)&=\boldsymbol{\theta}\cdot \mathbf{y}+i \cdot \boldsymbol{\eta}^{(\ell_\tau)}\cdot \mathbf{y} = 972.\end{align*}

Hence, in this example, the impact on the valuation of S is very limited, even though the hedging strategies and, importantly, the statistical behaviour of residuals, are different.

3.1 Fair valuation by iterated two-step valuation

Consider an insurance liability S which matures at time T. The quadratic and quantile hedging strategies at time $T-1$ are determined by

\begin{align*}&\boldsymbol{\theta}(T)=\arg \min _{\boldsymbol{\beta} \in \mathcal{B}(T)} \mathbb{E}_{T-1}\left[(S-\boldsymbol{\beta}(T) \cdot \mathbf{Y}(T))^2\right],\\[3pt]&\boldsymbol{\eta}(T)=\arg \min _{\boldsymbol{\beta} \in \mathcal{B}(T)} \mathbb{E}_{T-1}\left[ \ell_{\alpha}(S-\boldsymbol{\theta}(T) \cdot \mathbf{Y}(T)-\boldsymbol{\beta}(T) \cdot \mathbf{Y}(T)) \right],\end{align*}

where $ \ell_{\alpha}$ is the Koenker–Bassett error given in (2.7). By the properties of quantile hedging, the payoff of both hedging strategies will cover the liability with a confidence level of $\alpha$ , hence satisfying the regulatory constraint:

\begin{equation*} \text{VaR}_{\alpha,T-1}\left(S-\boldsymbol{\theta}(T) \cdot \mathbf{Y}(T)-\boldsymbol{\eta}(T) \cdot \mathbf{Y}(T)\right)=0.\end{equation*}

The fair value at time $T-1$ of the liability is then defined as the cost of the quadratic hedging strategy and a cost-of-capital risk margin for the quantile hedging strategy:

\begin{equation*}\rho_{T-1}(S)=\boldsymbol{\theta}(T) \cdot \mathbf{Y}(T-1)+i\ \boldsymbol{\eta}(T) \cdot \mathbf{Y}(T-1).\end{equation*}

As in the static setting, the cost of the quadratic hedging strategy is covered by policyholders, while the cost of the quantile hedging strategy (interpreted as a capital requirement) is provided by shareholders, who require an interest i for their investment. We can now repeat iteratively the two-step valuation until we reach the fair value at time 0, at each step hedging the fair value of one period ahead. For the fair value $\rho_{t+1}(S)$ , both hedging strategies are given by

\begin{align*}\boldsymbol{\theta}(t+1)&=\arg \min _{\boldsymbol{\beta} \in \mathcal{B}(t)} \mathbb{E}_t\left[(\rho_{t+1}(S)-\boldsymbol{\beta}(t+1) \cdot \mathbf{Y}(t+1))^2\right]\\[3pt] \boldsymbol{\eta}(t+1)&=\arg \min _{\boldsymbol{\beta} \in \mathcal{B}(t)} \mathbb{E}_t\left[\ell_{\alpha}(\rho_{t+1}(S)-\boldsymbol{\theta}(t+1) \cdot \mathbf{Y}(t+1)-\boldsymbol{\beta}(t+1) \cdot \mathbf{Y}(t+1))\right].\end{align*}

Then, the fair value at time t is given by

(3.1) \begin{equation}\rho_{t}(S):=\boldsymbol{\theta}(t+1) \cdot \mathbf{Y}(t)+i\ \boldsymbol{\eta}(t+1) \cdot \mathbf{Y}(t),\end{equation}

and the yearly solvency constraints are satisfied by construction:

(3.2) \begin{align} &\text{VaR}_{\alpha,t}\left(\rho_{t+1}(S)-\boldsymbol{\theta}(t+1) \cdot \mathbf{Y}(t+1)-\boldsymbol{\eta}(t+1) \cdot \mathbf{Y}(t+1) \right)=0,\nonumber\\[3pt] &\quad \forall t\in \{0,1,\dots,T-1\}.\end{align}

We consider now the issue of the potential need for capital injections at intermediate times. Consider for example the assets at the end of the time period t, $\boldsymbol{\theta}(t)\cdot \textbf{Y}(t) +\boldsymbol{\eta}(t)\cdot \textbf{Y}(t)=\boldsymbol{\xi}(t)\textbf{Y}(t)$ . Then, shareholders need to inject more capital if $\boldsymbol{\xi}(t)\cdot \textbf{Y}(t)<\boldsymbol{\xi}(t+1)\cdot \textbf{Y}(t)$ or, conversely, capital can be released if $\boldsymbol{\xi}(t)\cdot \textbf{Y}(t)>\boldsymbol{\xi}(t+1)\cdot \textbf{Y}(t)$ . Let $RB(t) = \boldsymbol{\xi}(t+1)\cdot \textbf{Y}(t)-\boldsymbol{\xi}(t)\cdot \textbf{Y}(t)$ stand for the rebalancing cost at time t. Focussing on a quantile hedging strategy with a high confidence level $\alpha$ , VaR-neutrality implies that

(3.3) \begin{equation} \mathbb P(\boldsymbol{\theta}(t)\cdot \textbf{Y}(t)+\boldsymbol{\eta}(t)\cdot \textbf{Y}(t) -\rho_t(S)\geq 0)= \alpha \Leftrightarrow \mathbb P(\boldsymbol{\eta}(t+1)\cdot \textbf{Y}(t)(1-i) \geq RB(t) )= \alpha \end{equation}

Hence, with high probability, the funds RB(t) required to keep satisfying the regulatory requirement at time t is lower than $\eta(t+1)\cdot \textbf{Y}(t)(1-i)$ , which represents the shareholder capital needed to recapitalise the portfolio from scratch, allowing also for the cost of that capital. In the unlikely event that $\boldsymbol{\eta}(t+1)\cdot \textbf{Y}(t)(1-i) < RB(t)$ , necessary capital cannot be raised and the procedure is stopped.

Time consistency is an important concept for characterising the relationship between different static valuations. It means that the same value is assigned to a liability regardless of whether it is calculated in one step or in two steps backwards in time.

Definition 3.1 A sequence of valuations $\left(\rho_{t}\right)_{t=0}^{T-1}=\{\rho_0,\dots,\rho_{T-1}\}$ is time-consistent if:

(3.4) \begin{equation}\rho_{t}(S)=\rho_{t}\left(\rho_{t+1}(S)\right), \quad \text { for any liability } S \text { and } t=0,1, \ldots, T-2.\end{equation}

By construction, our valuation framework is time-consistent. Indeed, from (3.1) we see that the fair valuation at time t is obtained by applying the one-period two-step valuation on $\rho_{t+1}$ , which itself comes from the two-step valuation on $\rho_{t+2}$ and so on. Therefore, the time-consistency condition (3.4) is directly satisfied.

We now provide a simple example of our multi-period two-step valuation in a multivariate normal setting. In such a framework, explicit solutions for the fair valuation can be obtained.

Example 3. We consider a multi-period model with two assets only: ${\mathbf Y}(t)=(Y_{0}(t),Y_{1}(t))$ , $t=0,1,\dots, T$ . We assume that $Y_{0}(t)=1$ for all t (i.e. the risk-free asset has zero interest rate) and that the returns of the risky asset, $R_t=\frac{Y_{1}(t)}{Y_{1}(t-1)},\ t=1, \dots, T$ are i.i.d. Moreover, we write the liability development of S as

\begin{align*} S=s_0+S_1+\dots+S_T,\end{align*}

where $s_0$ is constant, $S_t$ is the liability development at time t which is $\mathcal F_t$ -measurable, $S_t$ ’s are independent, and ${\mathbb E}[S_t]=0$ for $t=1, \dots, T, $ we remark that such a decomposition was also considered in Tsanakas et al. (Reference Tsanakas, Wüthrich and Černý2013). Furthermore, $(R_t,S_t)$ are bivariate normally distributed, with constant correlation c. For the sake of brevity we define:

\begin{align*} \kappa &= \frac{{\mathbb E}[R_t-1]}{\sigma(R_t)},\\[3pt] \gamma_t &= \sigma(S_t),\end{align*}

where $\kappa$ is constant, while $\gamma_t$ is deterministic but time varying. First, we determine the quadratic hedging strategy at time $T-1$ :

\begin{equation*} \boldsymbol{\theta}(T):={\textrm{{arg min}}}_{\boldsymbol{\beta}\in\mathcal{B}(T)}{\mathbb E}_{T-1}\left[(S-\boldsymbol{\beta}\cdot {\mathbf Y}(T))^2\right].\\\end{equation*}

By standard conditional least-squares arguments, we get:

\begin{align*} \theta_{1}(T)&=\frac{\text{Cov}_{T-1}(S,Y_{1}(T))}{\text{Var}_{T-1}(Y_1({T}))},\\[3pt] \theta_{0}(T)&={\mathbb E}_{T-1}[S]-\theta_{1}(T){\mathbb E}_{T-1}[Y_{1}(T)],\end{align*}

with the property:

\begin{equation*} {\mathbb E}_{T-1}[S]={\mathbb E}_{T-1}[\boldsymbol{\theta}(T) \cdot {\mathbf Y}(T)],\end{equation*}

so that the cost of the quadratic hedging strategy is given by:

\begin{align*} \boldsymbol{\theta}(T) \cdot {\mathbf Y}(T-1)&={\mathbb E}_{T-1}[S]-\frac{\text{Cov}_{T-1}(S,Y_{1}(T))}{\text{Var}_{T-1}(Y_{1}(T))}\left({\mathbb E}_{T-1}[Y_{1}(T)]-Y_{1}(T-1)\right)\\[3pt] &=s_0+\sum_{j=1}^{T-1}S_j-\kappa \gamma_T c.\end{align*}

In a second step, we determine the risk margin by computing the quantile hedging strategy for the residual:

\begin{align*} \boldsymbol{\eta}(T)&:={\textrm{{arg min}}}_{\boldsymbol{\beta}\in\mathcal{B}(T)} {\mathbb E}_{T-1}\left[ \ell_{\alpha}(S-\boldsymbol{\theta}(T)\cdot {\mathbf Y}(T)-\boldsymbol{\beta}\cdot {\mathbf Y}(T)) \right].\end{align*}

By the normality assumption, we have that

\[(S-\boldsymbol{\theta}(T)\cdot{\mathbf Y}(T))\perp Y_{1}(T)\implies \eta_{1}(T)=0,\]

that is there is no investment in the risky asset. The related cost is then given by

\begin{align*} \boldsymbol{\eta}(T)\cdot {\mathbf Y}(T-1)&={{VaR}}_{\alpha,T-1} (S-\boldsymbol{\theta}(T)\cdot {\mathbf Y}(T)),\\[3pt] &={\mathbb E}_{T-1}[S-\boldsymbol{\theta}(T)\cdot {\mathbf Y}(T)] +\lambda \sigma_{T-1}[S-\boldsymbol{\theta}(T)\cdot {\mathbf Y}(T)],\\[2pt] &\qquad(\lambda:=\Phi^{-1}(\alpha))\\[3pt] &=\lambda \gamma_T\sqrt{1-c^2}. \end{align*}

The resulting fair value of S is then given by

\begin{align*} \rho_{T-1}(S)&=\boldsymbol{\theta}(T)\cdot {\mathbf Y}(T-1)+i\boldsymbol{\eta}(T)\cdot {\mathbf Y}(T-1)\\[3pt] &=s_0+\sum_{j=1}^{T-1}S_j-\kappa \gamma_T c+i\lambda \gamma_T\sqrt{1-c^2}.\end{align*}

By noting that with respect to $\mathcal F_{T-2}$ , the only random element in $\rho_{T-1}(S)$ is $S_{T-1}$ , we find that

\begin{align*} \rho_{T-2}(S)&= \boldsymbol{\theta}(T-1)\cdot {\mathbf Y}(T-2)+i\boldsymbol{\eta}(T-1)\cdot {\mathbf Y}(T-2) \\ &=s_0+\sum_{j=1}^{T-2}S_j-\kappa (\gamma_T+\gamma_{T-1}) c+i\lambda (\gamma_T+\gamma_{T-1})\sqrt{1-c^2}.\end{align*}

The cost of capital of all future capital requirements is used when valuing the liability at a particular time, which is in agreement with the Solvency II risk margin. An inductive argument would lead to:

\begin{equation*} \rho_{0}(S)= s_0 -\kappa c \sum_{j=1}^T \gamma_j +i\lambda \sqrt{1-c^2}\sum_{j=1}^T \gamma_j.\\ \end{equation*}

Hence, the fair value of the liability is composed of three terms: first the expected liability $s_0$ , a second term accounting for the dependence between the excess risky asset returns and the liability increments, and, third, is the cost-of-capital risk margin which takes into account the non-hedgeable risk. We note that the risk margin vanishes as $c\to1$ since the liability can be completely hedged in this case.

4.1 General algorithm for the dynamic hedging problem

We recall that the iterated dynamic hedging problem is given by

(4.1) \begin{align}\begin{split}\boldsymbol{\theta}(t+1)&:={\textrm{{arg min}}}_{\boldsymbol{\beta}\in\mathcal{B}(t+1) }{\mathbb E}_{t}\left[(\rho_{t+1}(S)-\boldsymbol{\beta} \cdot {\mathbf Y}(t+1))^2\right]\\[3pt]\boldsymbol{\eta}(t+1)&:={\textrm{{arg min}}}_{\boldsymbol{\beta}\in \mathcal{B}(t+1)} {\mathbb E}_{t}\left[ \ell_{\alpha}(\rho_{t+1}(S)-\boldsymbol{\theta}(t+1)\cdot {\mathbf Y}(t+1)-\boldsymbol{\beta}\cdot {\mathbf Y}(t+1)) \right]\\[3pt] \rho_{t}(S)&:=\boldsymbol{\theta}(t+1) \cdot \mathbf{Y}(t)+i\ \boldsymbol{\eta}(t+1) \cdot \mathbf{Y}(t)\end{split}\end{align}

for any $t=T-1,\dots,0$ , starting with $\rho_{T}(S)=S$ and $\ell_{\alpha}$ is the Koenker–Bassett error (2.7).

From now on, we assume that there exists an m-dimensional process $\boldsymbol{Z}(t)$ which drives all the processes of interest. In an insurance context, Z(t) may represent for instance the asset processes Y(t) and the number of policyholders alive at time t. The filtration $\mathcal{F}$ is generated by all observations about the process $\boldsymbol{Z}$ : $\mathcal{F}_t=\sigma(\boldsymbol{Z}(u)\mid u \leq t)$ .

To avoid path-dependence issues in the hedging framework and reduce the complexity of the dynamic hedging algorithm, we make the standard assumption that $\boldsymbol{Z}$ is Markovian. We note that many standard financial and actuarial processes do follow the Markov property.

Assumption 1. $\boldsymbol{Z}$ has the Markov property with respect to the filtration $\mathcal{F}$ , that is

\[\mathbb{P}(\boldsymbol{Z}(t+1)\leq \boldsymbol{x}\mid \mathcal{F}_t)=\mathbb{P}(\boldsymbol{Z}(t+1)\leq \boldsymbol{x}\mid \boldsymbol{Z}(t)).\]

By Assumption 1, the candidate hedging strategies in the dynamic hedging problem (4.1) can be expressed as $g(\boldsymbol{Z}(t))$ where $g: \mathbb{R}^{m}\rightarrow \mathbb{R}^{n+1}$ is a function which takes the random process at time t as inputs and outputs the hedging positions in the $(n+1)$ assets. Indeed, from the Markov property, the optimal strategy $\boldsymbol{\theta}(t+1)$ and $\boldsymbol{\eta}(t+1)$ for the period $[t,t+1]$ can only depend on the risk drivers at time t and previous observations do not provide more information. Since we cannot consider numerically any possible function g, we assume that the optimal function g belongs to a family of non-linear functions $\mathcal{G}$ which, in this paper, will correspond to a NN discussed in the next section.

Moreover, in order to approximate the expectation operator in (4.1), we use a Monte-Carlo sample by generating M random simulations. Given the $\mathbb{P}$ -dynamics of the stochastic process $\left\{ Z(t) \right\}_{t=0,\dots,T}$ , one can simulate M random observations $\boldsymbol{Z}^{(i)}(t)$ of the random process at time t, for $t=0,1, \ldots, T$ . Therefore, the dynamic hedging problem (4.1) can be expressed by the following iterative algorithm:

(4.2) \begin{align} \boldsymbol{\theta}(t+1)&:=g_{t+1}\left(\boldsymbol{Z}(t)\right) \nonumber\\[3pt] \text{with } g_{t+1}& = {\textrm{{arg min}}}_{g\in\mathcal{G}}\frac{1}{M}\sum_{i=1}^M \left( \rho^{(i)}_{t+1}(S)- g(\boldsymbol{Z}^{(i)}(t)) \cdot {\mathbf Y}^{(i)}(t+1)\right)^2\nonumber \\ \boldsymbol{\eta}(t+1)&:=h_{t+1}\left(\boldsymbol{Z}(t)\right)\nonumber\\[3pt] \text{with } h_{t+1} & = {\textrm{{arg min}}}_{g\in\mathcal{G}}\frac{1}{M}\nonumber\\ &\quad \times \sum_{i=1}^M \ell_{\alpha} \left( \rho^{(i)}_{t+1}(S)- \boldsymbol{\theta}(t+1)\cdot {\mathbf Y}^{(i)}(t+1)- g(\boldsymbol{Z}^{(i)}(t)) \cdot {\mathbf Y}^{(i)}(t+1)\right)\nonumber\\ \rho^{(i)}_{t}(S)&:=\boldsymbol{\theta}^{(i)}(t+1) \cdot \mathbf{Y}^{(i)}(t)+i\ \boldsymbol{\eta}^{(i)}(t+1) \cdot \mathbf{Y}^{(i)}(t).\end{align}

where

\begin{align*}\boldsymbol{\theta}^{(i)}(t+1)&:= g_{t+1}(\boldsymbol{Z}^{(i)}(t))\\\boldsymbol{\eta}^{(i)}(t+1)&:= h_{t+1}(\boldsymbol{Z}^{(i)}(t))\end{align*}

for any $t=T-1,\dots,0$ , starting with $\rho^{(i)}_{T}(S)=S^{(i)}$ . Hence, we observe that for each time period $[t,t+1]$ , the algorithm minimises the aggregate hedging error over all sample paths, taking into account that the hedging strategy should be $\mathcal{F}_t$ -measurable. We note that the algorithm (4.2) provides the fair valuation of S at any time t as well as the quadratic and quantile hedging strategies. Indeed, by Lemma 2.4, the quantile hedging strategy $\boldsymbol{\xi} $ for $\rho_{t+1}(S)$ is the sum of the quadratic hedging strategy $\boldsymbol{\theta} $ for $\rho_{t+1}(S) $ and the quantile hedging strategy $\boldsymbol{\eta}$ for the residual loss. Algorithm 1 presents the procedure in its compact form.

4.2 Implementation by NNs

In order to implement the Algorithm 1, we need to resort to a non-linear optimisation. In this article, we implement the algorithm by the use of NNs as these are well suited for this problem. The universal approximation theorem of Hornik et al. (Reference Hornik, Stinchcombe and White1989) states that networks can approximate any continuous function on a compact support arbitrarily well if we allow for arbitrarily many neurons $q_{1} \in \mathbb{N}$ in the hidden layer. From the universal approximation theorem, we do know that the optimal g function can be approached by a NN with sufficient layers and neurons. The number of neurons and layers that we need is rather subjective and subject to empirical studies. Here, we follow the work of Fécamp et al. (Reference Fécamp, Mikael and Warin2019) and consider three hidden layers of 10 neurons with Relu activation functions. For completeness, we briefly explain the mathematical structure of a NN in the next paragraph, see Goodfellow et al. (Reference Goodfellow, Bengio and Courville2016) for more details.

The NN takes an input of dimension m (the dimension of the risk process $\boldsymbol{Z}$ ) and outputs a vector of dimension $n+1$ (the number of units invested in the $(n+1)$ assets). The network is characterised by a number of layers $L+1 \in \mathbb{N} \backslash\{1,2\}$ with $m_{l}, l=0, \ldots, L,$ the number of neurons (units or nodes) on each layer: the first layer is the input layer with $m_{0}=m$ , the last layer is the output layer with $m_{L}=n+1,$ and the $L-1$ layers between are called hidden layers, where we choose for simplicity the same dimension $m_{l}=p, l=1, \ldots, L-1$ . The NN is then defined as the composition

\[x \in \mathbb{R}^{m} \mapsto \mathcal{N}(x)=A_{L} \circ \varphi \circ A_{L-1} \circ \ldots \circ \varphi \circ A_{1}(x) \in \mathbb{R}^{n+1}.\]

Here, $A_{l}, l=1, \ldots, L$ , are affine transformations represented by

\[A_{l}(x)=\mathcal{W}_{l} x+\beta_{l},\]

for a matrix of weights $\mathcal{W}_{l}$ and a “bias” term $\beta_{l}$ . The non-linear function $\varphi: \mathbb{R} \rightarrow \mathbb{R}$ is called the activation function and is applied component-wise on the outputs of $A_{l},$ that is, $\varphi\left(x_{1}, \ldots, x_{p}\right)=\left(\varphi\left(x_{1}\right), \ldots, \varphi\left(x_{p}\right)\right)$ . Standard examples of activation functions are the sigmoid, the ReLU, the elu and tanh.