2.1 Valuation

To construct a natural hedge, the insurer must first quantify the present value of the annuity and insurance liabilities. Under stochastic mortality modeling, these liabilities are random variables as their values depend on uncertain future survival outcomes. Thus, the first step of our natural hedging framework is to define and value these present value random variables as the foundation for establishing the hedge.

Let $\mathcal{S}_{x}(T)$ be the probability that an individual aged $x$ at time $0$ will survive another $T$ years. This probability can be expressed as:

(1) \begin{equation} \mathcal{S}_{x}(T) = \prod _{s=1}^{T} (1 - q_{x+s-1, s}), \end{equation}

where $q_{x, s}$ is the probability that an individual aged $x$ at the beginning of year $s$ dies during year $s$ . In a stochastic setting, $\mathcal{S}_{x}(T)$ is a random variable because $q_{x,s}$ , for $s=1,\ldots ,T$ , is uncertain due to random mortality fluctuations. We denote its expectation by $S_{x}(T) := \mathbb{E}[\mathcal{S}_{x}(T)]$ . The procedure for generating realizations of $\mathcal{S}_{x}(T)$ is provided in Appendix B of the Supplementary Material.

2.1.1 Annuity liabilities

Recall that we denote $\mathcal{A}$ as the present value random variable of the cash flows generated by the annuity portfolio under stochastic mortality. Assume that the insurer’s annuity portfolio consists of $n_A$ distinct product types (e.g., annuities issued at different ages). If a constant force of interest $\delta$ is used, then $\mathcal{A}$ is given by

\begin{equation*} \mathcal{A} = \sum _{j=1}^{n_{A}} \omega _{j}^{A} \sum _{k = \tau _{j}}^{\tau _{j} + t_{j} - 1} c_{j} \cdot e^{-\delta k} \cdot \mathcal{S}_{x_{j}}(k), \end{equation*}

where $\omega _{j}^{A}$ is the weight of the $j$ -th annuity product, $\tau _{j}$ is its deferral period, $t_{j}$ is the number of payments, $c_{j}$ is the payment size, and $x_{j}$ is the issuing age. The weights satisfy $\sum _{j=1}^{n_{A}} \omega _{j}^{A} = 1$ and $0 \lt \omega _{j}^{A} \leq 1$ for all $j$ . We remark that $\omega _j^A$ and $c_j$ serve distinct roles and allow us to distinguish between portfolio allocation decisions (captured by $\omega _j^A$ ) and contractual product specifications (captured by $c_j$ ). A similar argument applies to insurance liabilities in the next subsection.

It follows that the expectation of $\mathcal{A}$ represents the expected present value of the entire annuity portfolio liabilities, which can be written as:

(2) \begin{equation} A := \mathbb{E} \left [ \mathcal{A} \right ] = \sum _{j=1}^{n_{A}} \omega _{j}^{A} \sum _{k = \tau _{j}}^{\tau _{j} + t_{j} - 1} c_{j} \cdot e^{-\delta k} \cdot S_{x_{j}}(k). \end{equation}

The distribution of $\mathcal{A}$ and the value of $A$ can be obtained using Monte Carlo simulations under stochastic mortality modeling.

2.1.2 Insurance liabilities

Recall again that we denote $\mathcal{I}$ as the present value random variable of the cash flows generated by the insurance portfolio under stochastic mortality. If the insurer has $n_{I}$ distinct insurance product types in the portfolio and applies a constant force of interest $\delta$ , then $\mathcal{I}$ is given by

\begin{equation*} \mathcal{I} = \sum _{j = 1}^{n_I} \omega _{j}^{I} \sum _{k=0}^{t_{j}-1} b_{j} \cdot e^{-\delta (k+1)} \cdot ( \mathcal{S}_{x_{j}}(k) - \mathcal{S}_{x_{j}}(k+1) ), \end{equation*}

where $\omega _{j}^{I}$ is the weight of the $j$ -th insurance product, $t_{j}$ is its term length, $b_{j}$ is the benefit amount, and $x_{j}$ is the issuing age. The weights again satisfy $\sum _{j=1}^{n_{I}} \omega _{j}^{I} = 1$ and $0 \lt \omega _{j}^{I} \leq 1$ for all $j$ .

Similar to the annuity liabilities, the expectation of $\mathcal{I}$ is the expected present value of the entire insurance portfolio liabilities, which can be written as:

(3) \begin{equation} I := \mathbb{E}[\mathcal{I}] = \sum _{j = 1}^{n_I} \omega _{j}^{I} \sum _{k=0}^{t_{j}-1} b_{j} \cdot e^{-\delta (k+1)} \cdot ( S_{x_{j}}(k) - S_{x_{j}}(k+1) ). \end{equation}

The distribution of $\mathcal{I}$ and the value of $I$ can also be obtained using Monte Carlo simulations under stochastic mortality modeling.

2.1.3 Summary of notations

Having derived the annuity and insurance portfolios, we now summarize the notations before presenting the next two steps of the framework. The hedge ratio is $h$ , the calibrated insurance portfolio is $\mathcal{L}(h) = h \cdot \mathcal{I}$ with its expected present value being $L(h) = h \cdot I$ , and the combined portfolio (i.e., the hedged position) is $\mathcal{P}(h) = \mathcal{A} + \mathcal{L}(h)$ with its expected present value being $P(h) = A + L(h)$ .

For ease of reference, Table 1 summarizes the key notations used throughout the paper and also introduces the mean-adjusted present values that will play a central role in the rest of the paper. Note that, in Table 1, the calibrated portfolio refers to the insurance portfolio after applying the hedge ratio $h$ , the combined portfolio refers to the annuity and calibrated insurance portfolios taken together, and the mean-adjusted present values in the last column represent deviations from the corresponding expected present values.

Table 1.

Summary of the notations used in the natural hedging framework

Table 1. Long description

A table with four rows and four columns summarizing key notations used in a natural hedging framework. The columns are labeled Description, Present value (PV), Expected PV, and Mean-adjusted PV. Row 1: Annuity portfolio, A, A, A tilde equals A minus A. Row 2: Insurance portfolio, I, I, I tilde equals I minus I. Row 3: Calibrated portfolio, C(h), L(h), C hat (h) equals C(h) minus L(h). Row 4: Combined portfolio, P(h), P(h), P hat (h) equals P(h) minus P(h).

2.2 Calibration

We now turn to the calibration step of the framework. The hedge ratio $h$ determines how the insurance liabilities offset annuity liabilities, and calibrating $h$ is therefore a critical step that directly affects the effectiveness of the natural hedge. In this section, we categorize calibration methods into three groups: optimization, immunization, and Greek neutral. Before outlining a specific technique from each category to be implemented later in this paper, we briefly review the main approaches and related literature.

The first category is optimization, which calibrates $h$ by minimizing a chosen risk measure. Cupido et al. (Reference Cupido, Jevtić, Regis and Zhou2024) implemented a risk-minimization approach to construct a naturally hedged portfolio that accounts for spatial dependencies between different populations. Yang et al. (Reference Yang, Huang and Yeh2019) proposed a unified hedging framework that minimizes changes in the insurer’s profit function for a portfolio combining internal natural hedges and external index-based hedges. Zhu and Bauer (Reference Zhu and Bauer2014) applied a financial hedging approach to minimize economic capital, as originally introduced by Zhu and Bauer (Reference Zhu and Bauer2011).

The next category is immunization, which seeks to minimize portfolio liability sensitivity to changes in mortality rates. Li and Luo (Reference Li and Luo2012) proposed a key q-Duration framework that matches sensitivities in mortality rates to hedge longevity risk with q-forward contracts. Tsai et al. (Reference Tsai, Wang and Tzeng2010) introduced a conditional VaR minimization approach that optimizes the insurer’s product mix and compared it with the modified duration matching method of Wang et al. (Reference Wang, Huang, Yang and Tsai2010). Chen et al. (Reference Chen, Gao and Zhu2024) generalized the approach of Tsai et al. (Reference Tsai, Wang and Tzeng2010) to incorporate both insurance supply and demand, while Gatzert and Wesker (Reference Gatzert and Wesker2014) extended the method of Wang et al. (Reference Wang, Huang, Yang and Tsai2010) to the entire insurance perspective, building on the framework of Gatzert and Wesker (Reference Gatzert and Wesker2012).

Lastly, Greek neutral techniques aim to reduce portfolio liabilities with respect to changes in model-specific mortality quantities. Cairns (Reference Cairns2013) introduced delta-nuga hedging as a robust approach to address recalibration risk in portfolios with index-based hedging instruments. Liu and Li (Reference Liu and Li2017) expanded the delta-nuga method of Cairns (Reference Cairns2013) through a generalized state-space hedging framework. To hedge changes in the reserves of a naturally hedged portfolio, Luciano et al. (Reference Luciano, Regis and Vigna2017) developed delta-gamma hedging based on first- and second-order approximations. Jevtić and Regis (Reference Jevtić and Regis2015) applied this approach to assess the solvency of a naturally hedged portfolio from an asset-liability perspective.

We remark that the above review is not intended to be exhaustive but rather to provide a context for the calibration step of the natural hedging framework. A simple benchmark is the uncalibrated hedge, where the hedge ratio is set to $h=1$ , and the insurer makes no adjustment to the insurance liabilities. Additionally, setting $h=1$ may be appropriate when an insurer has already constructed a naturally hedged portfolio with pre-determined annuity and insurance liabilities. Lastly, under a discrete-time stochastic mortality model, the hedge ratio will generally need to be calculated using Monte Carlo simulations. In the remainder of this section, we focus on one representative technique from each category, which will later be used in our numerical implementation.

2.2.1 Optimization

The aim of optimization is to calibrate the hedge ratio $h$ by minimizing a chosen risk measure of the hedged position. Formally, this optimization problem can be written as:

\begin{equation*} \min _{h \in \mathbb{R}} \rho (\mathcal{P}(h)), \end{equation*}

where $\rho$ denotes a selected risk measure, and $\mathcal{P}(h) = \mathcal{A} + h \cdot \mathcal{I}$ is the hedged portfolio.

When $\rho$ is taken to be variance, the problem reduces to variance minimization, a well-established approach in longevity risk management. In this case, Cairns et al. (Reference Cairns, Dowd, Blake and Coughlan2014) showed that the optimal hedge ratio $h^{(VM)}$ has a closed-form solution given by

(4) \begin{equation} h^{(VM)} = -\frac {\text{Cov}(\mathcal{A},\mathcal{I})}{\text{Var}(\mathcal{I})}. \end{equation}

The corresponding hedged portfolio is denoted by $\mathcal{P}(h^{(VM)}) = \mathcal{A} + h^{(VM)}\mathcal{I}$ . This case will be used in the numerical illustrations later in the paper.

2.2.2 Immunization

Immunization was originally introduced by Redington (Reference Redington1952) for hedging interest rate risk. Wang et al. (Reference Wang, Huang, Yang and Tsai2010) adapted it for natural hedging, where the goal is to neutralize the sensitivity of annuity and insurance liabilities to changes in mortality. The mortality duration of the combined portfolio $P(h)$ with respect to the force of mortality $\mu$ is defined as:

\begin{equation*} D^{P}_{\mu } = \frac {dP(h)}{d\mu } = \frac {dA}{d\mu } + \frac {dL(h)}{d\mu } = \frac {dA}{d\mu } + h \frac {dI}{d\mu }. \end{equation*}

The optimal hedge ratio under this approach is obtained by setting $D^{P}_{\mu } = 0$ .

To approximate the mortality durations in $D^{P}_{\mu }$ , we define the modified durations of the annuity and insurance portfolios as $D^{A}_{\mu } = \frac {A^{+} - A^{-}}{2\epsilon }$ and $D^{I}_{\mu } = \frac {I^{+} - I^{-}}{2\epsilon }$ , where $\epsilon$ is a small constant, $A^{\pm }$ and $I^{\pm }$ are the expected present values of the annuity and insurance liabilities, respectively, evaluated under adjusted mortality $\mu \pm \epsilon$ . These quantities capture how much the expected present values change in response to a small shift in mortality. The resulting hedge ratio is

(5) \begin{align} h^{(DM)} = - \frac {D^{A}_{\mu }}{D^{I}_{\mu }}. \end{align}

The corresponding hedged portfolio is $\mathcal{P}(h^{(DM)}) = \mathcal{A} + h^{(DM)}\mathcal{I}$ . This portfolio matches the modified durations of the annuity and insurance portfolios, thereby aligning their sensitivities to mortality fluctuations.

2.2.3 Greek neutral

Greek neutral methods extend the notion of Greeks to mortality and longevity hedging. Luciano et al. (Reference Luciano, Regis and Vigna2012) first introduced this idea, and Zhou and Li (Reference Zhou and Li2021) further developed it for index-based longevity hedging, where the longevity delta was defined as a sensitivity measure with respect to the period effect of a stochastic mortality model. We now adapt this approach to natural hedging.

Consider the Lee–Carter (LC) model (Lee and Carter, Reference Lee and Carter1992):

\begin{equation*} \ln (m_{x,t}) = \alpha _{x} + \beta _{x}\kappa _t, \end{equation*}

where $m_{x,t}$ is the central death rate for age $x$ in year $t$ , and $\alpha _{x}$ , $\beta _{x}$ , and $\kappa _t$ are the LC parameters, with $\kappa _t$ following a random walk with drift. The longevity delta measures the first-order sensitivity of the survival probability $S_x(T)$ to changes in the current period effect $\kappa _0$ and is defined as $\Delta _x(T) := \frac {\partial S_x(T)}{\partial \kappa _0}$ . Assuming a constant force of mortality between integer ages, Zhou and Li (Reference Zhou and Li2021) showed that $\Delta _x(T)$ can be written as:

\begin{equation*} \Delta _{x}(T) = - \sum _{s = 1}^{T} \beta _{x+s-1} \, \mathbb{E}\!\left [ \exp \!\left (Y_{x}(s) - Z_{x}(T)\right ) \right ]\!, \end{equation*}

where $Y_{x}(s) = \alpha _{x+s-1} + \beta _{x+s-1}\kappa _s$ and $Z_{x}(T) = \sum _{s=1}^{T} \exp (Y_{x}(s))$ , with $\Delta _x(0)=0$ because $S_x(0)=1$ .

For the natural hedging framework, the longevity deltas of the annuity and insurance portfolios from Section 2.1 can then be derived as $\Delta ^A = \sum _{j=1}^{n_A} \omega _j^A \sum _{k=\tau _j}^{\tau _j+t_j-1} c_j e^{-\delta k} \Delta _{x_j}(k)$ and $\Delta ^I = \sum _{j=1}^{n_I} \omega _j^I \sum _{k=0}^{t_j-1} b_j e^{-\delta (k+1)} (\Delta _{x_j}(k) - \Delta _{x_j}(k+1) )$ . A delta-neutral natural hedge would require $\Delta ^A + h \Delta ^I = 0$ , and thus the hedge ratio is

(6) \begin{align} h^{(DN)} = -\frac {\Delta ^A}{\Delta ^I}. \end{align}

The corresponding delta-neutral hedged portfolio is $\mathcal{P}(h^{(DN)}) = \mathcal{A} + h^{(DN)}\mathcal{I}$ . Lastly, we note that longevity delta is model-dependent. While we have illustrated it under the LC model, the delta-neutral approach can be applied to other stochastic mortality models by deriving the corresponding longevity deltas.

2.3 Evaluation

The last step of the proposed natural hedging framework is evaluation. In this step, the insurer evaluates the effectiveness of a hedged position $\mathcal{P}(h)$ using both numerical and graphical assessment tools. We discuss both types of assessments in this subsection and consider them in the toy example shown in the next subsection.

A numerical risk measure for natural hedging can be defined in general as a mapping,

\begin{equation*} \rho \,:\, \mathcal{P}(h) \mapsto \mathbb{R}, \end{equation*}

which assigns a real-valued assessment to the hedged position $\mathcal{P}(h)$ . A fundamental example is the portfolio variance, given by $\text{Var}(\mathcal{P}(h)) = \text{Var}(\mathcal{A}) + h^{2}\text{Var}(\mathcal{I}) + 2h\,\text{Cov}(\mathcal{A},\mathcal{I})$ . If the insurer focuses on downside risk, the VaR can be considered, defined as $\text{VaR}_{\alpha }(\mathcal{P}(h)) =$ $\inf \{x \in \mathbb{R} \,:\, F_{\mathcal{P}(h)}(x) \geq \alpha \}$ at confidence level $\alpha \in (0,1)$ , where $F_{\mathcal{P}(h)}$ is the distribution function of $\mathcal{P}(h)$ . A related measure is expected shortfall (ES), given by $\text{ES}_{\alpha }(\mathcal{P}(h)) = \frac {1}{1-\alpha } \int _{\alpha }^{1} \text{VaR}_{u}(\mathcal{P}(h)) du$ , which captures the average loss in the tail beyond VaR. For a natural hedge, these numerical risk measures can provide complementary perspectives on the risk profile of the hedged position.

We make two remarks on numerical risk measures. First, beyond variance, VaR, and ES, other measures may also be considered, and they can be applied to transformations of the hedged position. For example, one may evaluate hedge effectiveness using $\text{VaR}_{\alpha }(\tilde {\mathcal{P}}(h))$ , where $\tilde {\mathcal{P}}(h)$ is the mean-adjusted hedged portfolio given in Table 1. Second, we do not aim to identify a single numerical risk measure that is universally applicable to hedged portfolios. In optimization, the chosen objective is a natural candidate, but in immunization or Greek neutral approaches no obvious measure exists. In such cases, it is important to consider multiple numerical risk measures and complementary graphical assessments.

In conjunction with numerical measures, graphical risk metrics can further provide visual insights into the distribution of the hedged position. A common approach is to plot histograms of $\mathcal{P}(h)$ or $\tilde {\mathcal{P}}(h)$ , and overlay numerical risk measures such as VaR and ES. Other useful visualizations include empirical cumulative distribution functions, Q-Q plots, or kernel density estimates. While these graphical tools are widely used, they remain limited in the depth of analysis they provide. In Section 3, we extend beyond such standard methods and propose a new graphical risk metric tailored for natural hedging.

In summary, the last step of our framework incorporates both numerical and graphical risk assessments to evaluate hedged positions, aiming to provide a comprehensive view of their risk profile. However, relying solely on numerical measures can overlook important aspects of hedge performance, and simple graphical diagnostics may also be insufficient. To illustrate these issues, the next subsection presents a toy example that demonstrates the implementation of the proposed natural hedging framework and motivates the new graphical risk metric introduced in Section 3.

2.4 Framework summary and toy example

We now summarize the proposed natural hedging framework and illustrate its implementation through a toy example. Table 2 provides a compact reference for the three steps of the framework. Under stochastic mortality modeling, each step requires future mortality projections. In practice, Monte Carlo simulations are used to compute present values in the valuation step, hedge ratios in the calibration step, and risk measures in the evaluation step. The mortality simulation methods need not be identical across steps, which in turn allows model uncertainty to be present in the natural hedge. The mortality simulation methods considered in this paper are presented in Appendix B of the Supplementary Material.

Table 2.

The proposed natural hedging framework

Table 2. Long description

A table detailing the steps in a natural hedging framework. The table has four rows and two columns. The columns are labeled Step and Details. Row 1: Step 1: Valuation, Derive and calculate the present values of annuity and insurance liabilities to be included in the hedge. Row 2: Step 2: Calibration, Determine the hedge ratio, either set to 1 for an uncalibrated hedge or calculated under a chosen calibration method. Row 3: Step 3: Evaluation, Assess the resulting hedged positions using numerical risk measures and graphical risk assessment tools. Row 4: Projected mortality, Mortality scenarios are required in each of the valuation, calibration, and evaluation steps.

To demonstrate the implementation of the proposed framework, we now present a toy example. Consider a life insurer with an annuity portfolio consisting of a single 20-year deferred 20-term life annuity issued to a 45-year-old individual, paying $20 annually at the beginning of each year after the deferral period. To hedge this liability, the insurer issues a 30-year term life insurance policy with a death benefit of $250, payable at the end of the year of death, to an individual aged either 40 or 50. For both issuing ages, the hedge ratio is set to $h=1$ , yielding two uncalibrated hedges. The interest rate is assumed to be $i=4\%$ (or $\delta =\ln (1.04)$ ). We note that all portfolios assumed are composed of a large number of identical contracts so that individual mortality randomness is fully diversified, and only systematic mortality risk arising from uncertain future mortality trends is being hedged.

Following the framework, Step 1 derives the present values of the annuity $\mathcal{A}$ and the two insurance portfolios $\mathcal{I}_1$ and $\mathcal{I}_2$ for ages 40 and 50, respectively. Step 2 constructs the calibrated insurance portfolios $\mathcal{L}_1(1)=\mathcal{I}_1$ and $\mathcal{L}_2(1)=\mathcal{I}_2$ by setting $h_1=h_2=1$ . Step 3 evaluates the hedged portfolios $\mathcal{P}_1=\mathcal{A}+\mathcal{I}_1$ and $\mathcal{P}_2=\mathcal{A}+\mathcal{I}_2$ using both numerical and graphical risk assessments. Mortality scenarios are generated using the bootstrapping method described in Appendix B.2 of the Supplementary Material, based on US male mortality data (ages 40–99, years 1970–2018) obtained from the Human Mortality Database.Footnote ¹

Table 3 reports the mean, variance, $\text{VaR}_{0.95}$ , and the excess of $\text{VaR}_{0.95}$ over the mean for the annuity portfolio $\mathcal{A}$ , the two insurance portfolios $\mathcal{L}_1$ and $\mathcal{L}_2$ , and the combined hedged portfolios $\mathcal{P}_1$ and $\mathcal{P}_2$ . The variance is greatly reduced when either $\mathcal{L}_1$ or $\mathcal{L}_2$ is added to $\mathcal{A}$ , reflecting the hedging effect of combining portfolios. Between the two hedged portfolios, $\mathcal{P}_2$ has the smaller variance and excess VaR, while $\mathcal{P}_1$ achieves the lower mean and lower VaR. The mixed results underscore the complexity of assessing hedge effectiveness in natural hedging and motivate the need for further graphical risk assessments.

Table 3.

Numerical risk measures for the annuity portfolio $\mathcal{A}$ , the two insurance portfolios $\mathcal{L}_1$ and $\mathcal{L}_2$ , and the hedged portfolios $\mathcal{P}_1$ and $\mathcal{P}_2$

Table 3. Long description

A table with five rows and five columns comparing numerical risk measures for different portfolios. The columns are labeled Portfolio, Mean, Variance, VaR0.95, and VaR0.95 - Mean. The rows are labeled A, C1, C2, P1, and P2. Row 1: Portfolio A, Mean 98.81, Variance 1.61, VaR0.95 100.84, VaR0.95 - Mean 2.03. Row 2: Portfolio C1, Mean 22.00, Variance 1.01, VaR0.95 23.69, VaR0.95 - Mean 1.68. Row 3: Portfolio C2, Mean 43.52, Variance 2.67, VaR0.95 46.27, VaR0.95 - Mean 2.75. Row 4: Portfolio P1, Mean 120.81, Variance 0.39, VaR0.95 121.83, VaR0.95 - Mean 1.01. Row 5: Portfolio P2, Mean 142.33, Variance 0.27, VaR0.95 143.19, VaR0.95 - Mean 0.86.

Figure 1

Graphical risk assessment for the hedged portfolios $\mathcal{P}_{1}$ (pink) and $\mathcal{P}_{2}$ (blue).

Figure 1 shows a graphical assessment of the hedged portfolios using histograms. Panel (a) shows the empirical distributions of $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ , with dashed lines indicating their $\text{VaR}_{0.95}$ . Consistent with the numerical results, $\mathcal{P}_{1}$ has a smaller $\text{VaR}_{0.95}$ , reflecting lower liabilities compared with $\mathcal{P}_{2}$ . Panel (b) displays the empirical distributions of the mean-adjusted portfolios $\tilde {\mathcal{P}}_{1}$ and $\tilde {\mathcal{P}}_{2}$ (defined in Table 1), again with dashed lines marking their $\text{VaR}_{0.95}$ . In this panel, $\mathcal{P}_{2}$ shows smaller deviations from its mean than $\mathcal{P}_{1}$ , confirming the lower variance observed in the numerical measures.

This toy example illustrates how different numerical and graphical risk assessments can lead to inconclusive results about hedge effectiveness. More importantly, such assessments cannot capture the underlying relationship between the annuity portfolio $\mathcal{A}$ and the insurance portfolio $\mathcal{L}$ within the hedged position $\mathcal{P}$ . For example, when the realized liability of $\mathcal{P}$ exceeds its $\text{VaR}_{0.95}$ , it is unclear whether this outcome reflects weak offsetting effect between $\mathcal{A}$ and $\mathcal{L}$ , or whether both portfolios simultaneously contribute to large losses. Answering such questions can help insurers to make more informed decisions in portfolio selection and in the design of natural hedges, which we address in Section 3.

3.1 Constructing the metric

The construction of our graphical risk metric is inspired by the work of Chan et al. (Reference Chan, Li, Zhou and Zhou2016), who introduced a visual tool for evaluating population basis risk in index-based longevity hedges. We adapt this approach to the context of natural hedging and construct a two-dimensional assessment of the joint distribution of the underlying annuity and insurance portfolios, rather than focusing solely on the hedged position. This representation enables direct visualization of the dependence structure between the annuity and insurance liabilities.

Figure 2 illustrates the result of applying the metric to the hedged position $\mathcal{P}_{1}$ from the toy example. Details on how this figure is constructed are provided in Appendix A of the Supplementary Material. The x- and y-axes represent the present values of the annuity portfolio $\mathcal{A}$ and the insurance portfolio $\mathcal{L}_1$ , respectively. Each black dot corresponds to a realization of $(\mathcal{A}, \mathcal{L}_1)$ , while the shaded pink regions denote joint prediction regions at different confidence levels. The figure reveals a clear inverse relationship between $\mathcal{A}$ and $\mathcal{L}_1$ , forming a downward-sloping pattern in both the point cloud and the shaded regions.

The graphical risk metric illustrated in Figure 2 enables a visual evaluation of the hedge effectiveness of a single portfolio. The same construction can be extended to compare multiple hedged portfolios, which is particularly useful when several hedging opportunities are available. Figure 3 illustrates such an extension by jointly displaying the annuity portfolio $\mathcal{A}$ against two insurance portfolios, $\mathcal{L}_1$ and $\mathcal{L}_2$ , corresponding to the hedged positions $\mathcal{P}_1$ and $\mathcal{P}_2$ from the toy example. This extension allows for a direct comparison of alternative natural hedges under a single visual platform.

Figure 3

Graphical risk metric illustrating the effectiveness of the hedged positions $\mathcal{P}_{1}$ (pink) and $\mathcal{P}_{2}$ (blue) from the toy example.

Panel (a) of Figure 3 plots the annuity portfolio $\mathcal{A}$ against the corresponding insurance portfolios $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ . As the annuity is identical across both portfolios, the larger insurance liabilities of $\mathcal{L}_{2}$ (relative to $\mathcal{L}_{1}$ ) result in higher total liabilities, consistent with the numerical measures reported in Table 3. This outcome is expected, as the life insurance policy from $\mathcal{P}_{2}$ is issued to a 50-year-old individual, whose higher mortality risk will lead to greater liabilities than the 40-year-old policy from $\mathcal{P}_{1}$ .

To enable a fairer comparison, Panel (b) of Figure 3 presents the mean-adjusted annuity and insurance liabilities, represented by $(\tilde {\mathcal{A}}, \tilde {\mathcal{L}}_{1})$ and $(\tilde {\mathcal{A}}, \tilde {\mathcal{L}}_{2})$ , whose joint prediction regions are centered at the origin. The solid black line with slope $-1$ represents the benchmark line, corresponding to scenarios in which deviations in the annuity and insurance liabilities perfectly offset (i.e., $\tilde {\mathcal{A}} + \tilde {\mathcal{L}} = 0$ ), resulting in a “perfect” hedge. Building on this benchmark, we next introduce an interpretative procedure to analyze potential hedging outcomes based on the mean-adjusted portfolios.

3.2 Interpreting the metric

To gain a comprehensive visual understanding of the risk profile underlying a hedged position, we now develop an interpretation aid for the proposed graphical risk metric. This aid functions as a diagnostic tool for analyzing the joint behavior of the annuity and insurance portfolios. By representing all present value realizations in the $(\tilde {\mathcal{A}}, \tilde {\mathcal{L}})$ plane, it enables visual assessment of how effectively deviations in the insurance liability offset those in the annuity liability.

Figure 4

Interpretation procedure for the proposed graphical risk metric and its application to the hedged positions $\mathcal{P}_{1}$ (pink) and $\mathcal{P}_{2}$ (blue) from the toy example.

Figure 4 Long description

Panel A: Interpretation. A scatter plot shows the relationship between mean-adjusted insurance portfolios on the vertical axis and mean-adjusted annuity portfolios on the horizontal axis. The plot is divided into four quadrants labeled ‘Not Enough Insurance Liabilities,’ ‘Too Much Insurance Liabilities,’ and ‘Adverse Hedging Effect.’ Four points labeled A, B, C, and D are plotted within these quadrants. Panel B: Application. Another scatter plot with the same axes and quadrant divisions shows shaded regions in pink and blue, representing different hedged positions. The pink and blue regions indicate areas of risk and hedging effects.

Figure 4 illustrates our interpretation procedure. Panel (a) defines the shaded regions and benchmark lines used to classify hedging outcomes, where the shading corresponds to different types of hedging outcome and the dashed lines indicate the magnitude of deviation from a perfect hedge. To clarify the interpretation of these shaded regions, we provide brief explanations of the scenarios marked by Points A, B, C, and D in Figure 4(a):

• • Point A: The annuity liability is $2 below its expected value, while the insurance liability is $2 above. The surplus in the insurance liability fully offsets the annuity shortfall, representing a perfectly hedged scenario. All points along the benchmark line correspond to this ideal outcome.

• • Point B: The annuity liability is $3 above its expectation, but the insurance liability is only $2 below. The shortfall in the insurance liability leaves a net liability surplus of $1 for the hedged position, corresponding to the case of Not Enough Insurance Liabilities (red region).

• • Point C: The annuity liability is $1 below its expected value, while the insurance liability is $2 above. The excess in the insurance liability leads to a net liability surplus of $1 for the hedged position, representing the case of Too Much Insurance Liabilities (blue region).

• • Point D: Both the annuity and insurance liabilities fall below their expectations, by $1 and $1.5 respectively. This scenario creates a combined liability deficit of $2.5, indicating the case of Adverse Hedging Effect (gray region), where losses occur simultaneously in both portfolios.

The two dashed lines in Figure 4(a) represent equal magnitudes of deviation from the expected total liability, corresponding to either a surplus or a deficit of $1. Because Point D lies below the lower dashed line, it immediately means that it has a larger magnitude of deviation than Points B and C. Although Points B and C both have a surplus of $1, they arise from different causes; Point B has insufficient insurance liabilities, whereas Point C has excessive insurance liabilities. This distinction would not be apparent in a one-dimensional graphical assessment, such as a histogram. Figure 5 extends this idea by showing simulated realizations colored by hedging outcome types.

Figure 5

Simulated outcomes of $\tilde {\mathcal{P}}_{1}$ and $\tilde {\mathcal{P}}_{2}$ , colored by hedging outcome with the corresponding proportion shown in each region; the dashed line denotes the mean-adjusted $\text{VaR}_{0.95}$ , and the boxed black percentages indicate the share of each outcome exceeding the mean-adjusted $\text{VaR}_{0.95}$ .

Figure 5 presents 20,000 simulated realizations of $(\tilde {\mathcal{A}}, \tilde {\mathcal{L}}_{1})$ and $(\tilde {\mathcal{A}}, \tilde {\mathcal{L}}_{2})$ in Panels (a) and (b), respectively. Points are colored by the type of hedging outcome. Pink and blue indicate Too Much Insurance Liabilities, leading to a surplus and deficit in total liability, respectively. Green and teal denote Not Enough Insurance Liabilities, leading to a deficit and surplus in total liability, respectively. Red and gold represent Adverse Hedging Effect, where both the insurance and annuity portfolios deviate in the same direction. The dashed lines mark the mean-adjusted $\text{VaR}_{0.95}$ of the corresponding hedged positions. Any realizations above these lines exceed the respective $\text{VaR}_{0.95}$ values reported in Table 3. Lastly, we caution that the proportion of realizations in each colored region reflects the probability of the corresponding hedging outcome, but does not directly indicate the magnitude of the associated financial gain or loss.

Comparing the simulated hedging outcomes in Figure 5, $\tilde {\mathcal{P}}_{1}$ shows more realizations associated with Not Enough Insurance Liabilities (green and teal) and Adverse Hedging Effect (red and gold) than $\tilde {\mathcal{P}}_{2}$ . Conversely, $\tilde {\mathcal{P}}_{2}$ exhibits a higher frequency of Too Much Insurance Liabilities (pink and blue), indicating that its insurance exposure tends to overcompensate for annuity deviations. Among the outcomes that exceed the mean-adjusted $\text{VaR}_{0.95}$ as marked by the dashed lines, those from $\tilde {\mathcal{P}}_{1}$ primarily arise from Not Enough Insurance Liabilities (teal) and Adverse Hedging Effect (gold), while those from $\tilde {\mathcal{P}}_{2}$ are mostly driven by Too Much Insurance Liabilities (pink).

We now return to Figure 4(b), which applies the interpretation aid to compare $\tilde {\mathcal{P}}_{1}$ and $\tilde {\mathcal{P}}_{2}$ by overlaying their joint prediction regions. The dashed lines above and below the solid benchmark line represent equal magnitudes of surplus and deficit in the hedged position. It is clear that the joint prediction region of $\tilde {\mathcal{P}}_{1}$ extends further into the area associated with Not Enough Insurance Liabilities, while $\tilde {\mathcal{P}}_{2}$ covers more of the Too Much Insurance Liabilities area. More importantly, the joint prediction region of $\tilde {\mathcal{P}}_{1}$ extends above the upper dashed line, indicating more severe (liability surplus) scenarios than that of $\tilde {\mathcal{P}}_{2}$ .

In conclusion, the proposed graphical risk metric complements numerical evaluation by exposing the underlying hedging behavior. Specifically, $\mathcal{P}_{2}$ achieves stronger offsets between annuity and insurance liabilities but is more prone to excess insurance exposure, whereas $\mathcal{P}_{1}$ faces greater downside risk from insufficient offsetting. These findings highlight how the graphical risk metric reveals important behavioral differences between hedged positions. In the next section, we apply this graphical framework to a broader set of numerical illustrations to further explore its practical use and interpretive value.

4.1 Illustration 1: insurance portfolio selection

Our first illustration investigates how to identify an effective insurance portfolio for natural hedging. Consider an insurer managing an annuity portfolio issued to individuals aged 40–60, with ${\$}$ 10,000 annual payments starting at age 65. The insurer considers three candidate insurance portfolios, each providing a death benefit of ${\$}$ 750,000 payable at the end of the year of death: (1) whole life policies issued to individuals aged 40–60 ( $\mathcal{I}_{1}$ ), (2) whole life policies issued to individuals aged 40–49 ( $\mathcal{I}_{2}$ ), and (3) 20-year term policies issued to individuals aged 40–49 ( $\mathcal{I}_{3}$ ). The annual interest rate is assumed to be 4%, and the limiting age is set at 100. Table 4 summarizes how the natural hedging framework is applied to this illustration. The three steps are applied consistently, with the only difference arising from the features of the insurance portfolios.

Table 5 reports the calibrated hedge ratios and numerical risk measures for the three hedged portfolios. Based on the variance and the mean-adjusted $\text{VaR}_{0.95}$ , $\mathcal{P}_{1}$ outperforms $\mathcal{P}_{2}$ and $\mathcal{P}_{3}$ , whereas the $\text{VaR}_{0.95}$ indicates that $\mathcal{P}_{3}$ performs best. To complement these numerical results and reveal the underlying hedging behavior, Figure 6 presents the graphical risk metric for the three hedged portfolios. In Panel (a), the unadjusted version shows that the three portfolios differ in the magnitude of their calibrated insurance liabilities, with $\mathcal{P}_{1}$ having the largest and $\mathcal{P}_{3}$ the smallest. The mean-adjusted version in Panel (b) shows that $\tilde {\mathcal{P}}_{1}$ aligns most closely with the benchmark line, indicating stronger offsetting between annuity and insurance liabilities, $\tilde {\mathcal{P}}_{2}$ exhibits greater variability with more outcomes falling into the Not Enough Insurance Liabilities and Too Much Insurance Liabilities regions, and $\tilde {\mathcal{P}}_{3}$ displays the widest dispersion and more frequent Adverse Hedging Effect.

Figure 6

Graphical risk metric applied to the hedged positions $\mathcal{P}_{1}$ (yellow), $\mathcal{P}_{2}$ (blue), and $\mathcal{P}_{3}$ (pink) from Illustration 1.

Figure 6 Long description

Panel A: A scatter plot shows the relationship between the Annuity Portfolio and the Insurance Portfolio. The x-axis represents the Annuity Portfolio with values ranging from 68000 to 74000. The y-axis represents the Insurance Portfolio with values ranging from 20000 to 60000. The data points are color-coded in yellow, blue, and pink, indicating different clusters or groups. Panel B: Another scatter plot displays the mean-adjusted values of the Annuity Portfolio and the Insurance Portfolio. The x-axis represents the mean-adjusted Annuity Portfolio with values ranging from -2000 to 2000. The y-axis represents the mean-adjusted Insurance Portfolio with values ranging from -2000 to 2000. The data points are color-coded similarly in yellow, blue, and pink, and the plot includes dashed lines indicating trends or boundaries.

In summary, this illustration demonstrates how the proposed framework and graphical risk metric together facilitate the selection of an appropriate insurance portfolio for natural hedging. Consistent with expectations, $\mathcal{P}_{1}$ with whole life policies issued to the same age range as the annuity portfolio provides the most effective hedge. Restricting the insurance portfolio to younger issuing ages, as in $\mathcal{P}_{2}$ , increases variability and weakens the offset between annuity and insurance liabilities. Further shortening the policy term to 20 years, as in $\mathcal{P}_{3}$ , amplifies this effect and leads to more outcomes with Adverse Hedging Effect. These results highlight that when issuing ages or policy terms deviate from those of the underlying annuity portfolio, the effectiveness of natural hedging reduces and the residual risk becomes more pronounced.

4.2 Illustration 2: hedge calibration

The second illustration examines how different hedge calibration methods affect the effectiveness of a natural hedge. Consider an insurer managing an annuity portfolio issued to individuals aged 40–60, with ${\$}$ 10,000 annual payments starting at age 65 for 20 years. The insurer constructs an insurance portfolio of whole life policies issued to the same age range and applies three different calibration methods for determining the hedge ratio: variance-minimization (VM), duration-matching (DM), and delta-neutral (DN). Table 6 summarizes how the natural hedging framework is applied to this illustration. The valuation and evaluation steps remain identical across the three methods, with the only difference lying in the calibration step.

Table 6.

Natural hedging framework applied to Illustration 2

Table 6. Long description

A table with four rows and two columns. The columns are labeled Step and Details. Row 1: Step 1: Valuation, Derive the present values of the annuity portfolio (issuing ages 40-60, deferred to 65, 20 annual payments of 10,000) and the insurance portfolio of whole life policies with a 750,000 death benefit issued to individuals aged 40-60. Row 2: Step 2: Calibration, Calibrate the hedge ratio using three different methods: h subscript VM from the variance-minimization method, h subscript DM from the duration-matching method, and h subscript DN from the delta-neutral method. Row 3: Step 3: Evaluation, Construct the hedged portfolios P subscript VM, P subscript DM, and P subscript DN and evaluate them comparatively using both numerical risk measures and the proposed graphical risk metric. Row 4: Projected mortality, Mortality scenarios are generated using the Lee-Carter model described in Appendix B.1 of the Supplementary Material, applied consistently across the valuation, calibration, and evaluation steps.

Table 7 reports the calibrated hedge ratios and numerical risk measures for the three portfolios. The hedge ratios range between 0.189 and 0.211, with the smallest from duration matching and the largest from delta neutral. As expected, $\mathcal{P}^{(VM)}$ achieves the lowest variance while $\mathcal{P}^{(DN)}$ exhibits the highest risk across all measures. Figure 7 compares the three hedged portfolios using the graphical risk metric. In Panel (a), the unadjusted portfolios show that $\mathcal{P}^{(DM)}$ requires the least amount of insurance liabilities, while $\mathcal{P}^{(DN)}$ requires the most, consistent with their respective hedge ratios. Panel (b) displays the mean-adjusted portfolios, where $\tilde {\mathcal{P}}^{(VM)}$ lies closest to the benchmark line, indicating balanced offsetting between annuity and insurance liabilities. In contrast, $\tilde {\mathcal{P}}^{(DM)}$ tilts toward the Not Enough Insurance Liabilities region, and $\tilde {\mathcal{P}}^{(DN)}$ toward the Too Much Insurance Liabilities region, revealing systematic under- and over-hedging tendencies.

Table 7.

Calibrated hedge ratios and numerical risk measures for Illustration 2

Table 7. Long description

The table presents calibrated hedge ratios and numerical risk measures for three portfolios. It has four rows and five columns. The columns are labeled ‘Portfolio’, ‘Hedge ratio’, ‘Variance’, ‘VaR0.95’, and ‘Mean-adjusted VaR0.95’. The rows are labeled with different portfolio types and their corresponding values. Row 1: Portfolio P(IM), Hedge ratio 0.198, Variance 4,816, VaR0.95 108,670, Mean-adjusted VaR0.95 113. Row 2: Portfolio P(DM), Hedge ratio 0.189, Variance 5,744, VaR0.95 106,614, Mean-adjusted VaR0.95 123. Row 3: Portfolio P(DN), Hedge ratio 0.211, Variance 6,642, VaR0.95 111,690, Mean-adjusted VaR0.95 132.

Figure 7

Graphical risk metric applied to the hedged positions $\mathcal{P}^{(VM)}$ (yellow), $\mathcal{P}^{(DM)}$ (blue), and $\mathcal{P}^{(DN)}$ (pink) from Illustration 2.

In summary, this illustration shows how the choice of hedge ratio calibration method influences both numerical performance and hedging behavior. The variance-minimization method achieves the most balanced hedge, with the lowest variance and a symmetric distribution of hedging outcomes. The duration-matching approach tends to under-hedge, whereas the delta-neutral approach tends to over-hedge, leaving asymmetric residual risk exposure in the hedged position. The numerical and graphical results jointly provide a coherent view of the trade-offs underlying different calibration methods.

4.3 Illustration 3: model risk

Our final illustration examines the impact of model risk on natural hedging effectiveness. Suppose an insurer uses the LC model to calibrate a natural hedge but is concerned that actual mortality may not be accurately described by it. To investigate the risk that the mortality model assumed for calibration differs from the one generating actual future mortality, the insurer considers two additional models for evaluation: the Cairns–Blake–Dowd (CBD) model and a non-parametric bootstrapping (BS) model. The goal is to assess whether the hedged position constructed by the LC model is robust to alternative mortality generating methods.

Table 8.

Natural hedging framework applied to Illustration 3

Table 8. Long description

The table has four rows and two columns. The columns are labeled Step and Details. Row 1: Step 1: Valuation, Derive the present values of the annuity portfolio (issuing ages 40-60, deferred to age 65, 35 annual payments of 10,000) and the whole-life insurance portfolio with a $750,000 death benefit issued to the same ages. Row 2: Step 2: Calibration, Calibrate the hedge ratio using the variance-minimization method with mortality scenarios generated from the LC model. Row 3: Step 3: Evaluation, Construct and compare the hedged portfolios P subscript LC, P subscript CBD, and P subscript BS, which are evaluated using mortality scenarios generated by the LC, CBD, and BS models, respectively. Row 4: Projected mortality, Mortality scenarios are generated from the LC model in the calibration step, and from the LC, CBD, and BS models in the evaluation step. The mortality generating processes are provided in Appendix B of the Supplementary Material.

Table 8 summarizes how the natural hedging framework is applied to this setting. The annual interest rate is again assumed to be 4%, and the limiting age is set at 100. Before presenting the numerical results, we briefly remark on model risk in general. A mortality model is suitable for a given application if it can adequately capture the aspects of mortality dynamics that are relevant to the question at hand. In particular, a model used for long-term stochastic projections may require different features from one used for short-term projections. This distinction is especially relevant to natural hedging, where the annuity and insurance liabilities involve different projection horizons. For illustration purposes, we only consider two well-known parametric models and one non-parametric model, which might lead to a misrepresentation of the true level of model risk. We therefore caution the reader that model risk can be substantial but can also be mitigated through careful model selection.

Table 9 reports the calibrated hedge ratio and numerical risk measures for the three portfolios. The hedge ratio is identical across all portfolios ( $h = 0.310$ ). When evaluated by the same model used in the calibration step, $\mathcal{P}^{(LC)}$ exhibits the lowest variance and mean-adjusted $\text{VaR}_{0.95}$ . However, when mortality is generated from an alternative model, $\mathcal{P}^{(CBD)}$ shows substantially higher variability, while $\mathcal{P}^{(BS)}$ lies between $\mathcal{P}^{(LC)}$ and $\mathcal{P}^{(CBD)}$ in both variance and mean-adjusted $\text{VaR}_{0.95}$ . These results suggest that when a natural hedge is both calibrated and evaluated under the LC model, its hedge effectiveness may be significantly overestimated.

Table 9.

Calibrated hedge ratio and numerical risk measures for Illustration 3

Table 9. Long description

A table with four rows and five columns. The columns are labeled Portfolio, Hedge ratio, Variance, VaR0.95, and Mean-adjusted VaR0.95. The row labels are P(LC), P(CBD), and P(I5). Row 1: P(LC), 0.310, 1,628, 143,093, 67. Row 2: P(CBD), 0.310, 245,611, 142,538, 697. Row 3: P(I5), 0.310, 66,711, 143,290, 350. The table compares the calibrated hedge ratio and numerical risk measures for three portfolios, showing identical hedge ratios but varying risk measures.

Figure 8

Graphical risk metric applied to the hedged positions $\mathcal{P}^{(LC)}$ (yellow), $\mathcal{P}^{(CBD)}$ (blue), and $\mathcal{P}^{(BS)}$ (pink) in Illustration 3.

Figure 8 compares the three portfolios using the graphical risk metric. In Panel (a), the unadjusted version shows that $\mathcal{P}^{(LC)}$ exhibits the smallest joint prediction region, while $\mathcal{P}^{(CBD)}$ and $\mathcal{P}^{(BS)}$ display much wider dispersion in both the annuity and insurance portfolios. The mean-adjusted version in Panel (b) reveals that $\tilde {\mathcal{P}}^{(LC)}$ lies closest to the benchmark line, whereas both $\tilde {\mathcal{P}}^{(CBD)}$ and $\tilde {\mathcal{P}}^{(BS)}$ tilt toward the Too Much Insurance Liabilities region. This suggests that the LC-calibrated natural hedge tends to require an excessive amount of insurance liabilities, resulting in suboptimal hedging outcomes when actual mortality deviates from the LC model. This is an insight that cannot be easily inferred from the numerical results alone.

From Figure 8, we see that $\tilde {\mathcal{P}}^{(CBD)}$ extends beyond the dashed lines, while $\tilde {\mathcal{P}}^{(BS)}$ remains within them, indicating that the CBD model implies more volatile and extreme hedging outcomes than the non-parametric BS approach. To further illustrate their differences, Figure 9 plots simulated realizations of $\tilde {\mathcal{P}}^{(CBD)}$ and $\tilde {\mathcal{P}}^{(BS)}$ , colored by hedging outcome. Both portfolios show a concentration of realizations in the Too Much Insurance Liabilities region. However, when compared against their respective mean-adjusted $\text{VaR}_{0.95}$ thresholds (marked by the dashed lines), the exceedances for $\tilde {\mathcal{P}}^{(CBD)}$ occur mainly due to Too Much Insurance Liabilities, whereas for $\tilde {\mathcal{P}}^{(BS)}$ , the exceedances arise across all three hedging outcome types.

Figure 9

Simulated outcomes of $\tilde {\mathcal{P}}^{(CBD)}$ and $\tilde {\mathcal{P}}^{(BS)}$ , colored by hedging outcome with the corresponding proportion shown in each region; the dashed line denotes the mean-adjusted $\text{VaR}_{0.95}$ , and the boxed black percentages indicate the share of each outcome exceeding the mean-adjusted $\text{VaR}_{0.95}$ .

Figure 9 Long description

Panel A: Scatter plot showing the relationship between mean-adjusted insurance and annuity portfolios with hedging outcomes. The horizontal axis represents the mean-adjusted annuity portfolio, and the vertical axis represents the mean-adjusted insurance portfolio. Data points are colored by hedging outcome, with corresponding proportions shown in each region. The dashed line denotes the mean-adjusted hedged position, and the boxed percentages indicate the share of each outcome exceeding the mean-adjusted hedged position. Panel B: Scatter plot showing the relationship between mean-adjusted insurance and annuity portfolios with hedging outcomes. The horizontal axis represents the mean-adjusted annuity portfolio, and the vertical axis represents the mean-adjusted insurance portfolio. Data points are colored by hedging outcome, with corresponding proportions shown in each region. The dashed line denotes the mean-adjusted hedged position, and the boxed percentages indicate the share of each outcome exceeding the mean-adjusted hedged position.

In summary, this illustration demonstrates that model risk can significantly distort the perceived effectiveness of a natural hedge. When the mortality model used for evaluation differs from the one used in calibration, the hedged position exhibits suboptimal performance and excess volatility under our natural hedging framework. The proposed graphical risk metric complements traditional numerical measures by revealing how these distortions arise from differing mortality dynamics and by exposing variations in hedging outcome types that numerical summaries alone would overlook.