2.1. Monotonicity properties

Throughout this paper, for any two random variables $Y_1,Y_2$ the notation $Y_1\leq_{\text{st}} Y_2$ refers to first-order stochastic dominance, that is, for the respective cumulative distribution functions, we have $F_{Y_1}(x)\ge F_{Y_2}(x)$ for all $x\in{\mathbb R}$ .

We need to establish some overall soundness of the functionals $R_0$ and $V_0$ .

Let $S_1\leq_{\text{icx}}\widetilde{S}_1$ mean that $\widetilde{S}_1$ is larger than $S_1$ in increasing convex order (which is equivalent to stop-loss order, and also equivalent to $\int_{\beta}^{1}F_{S_1}^{-1}(u)du\leq \int_{\beta}^{1}F_{\widetilde{S}_1}^{-1}(u)du$ for all $\beta\in [0,1]$ , see, e.g., Denuit et al. Reference Denuit, Dhaene, Goovaerts and Kaas2005). Take $S_1\geq_{\text{icx}} 1$ . Then, for $\widetilde{w}\geq w$

\begin{align*}&\int_{\beta}^{1}F_{\widetilde{w}S_1+1-\widetilde{w}}^{-1}(u)du-\int_{\beta}^{1}F_{wS_1+1-w}^{-1}(u)du\\&\quad = (\widetilde{w}-w)\int_{\beta}^{1}F_{S_1}^{-1}(u)du+(1-\beta)(w-\widetilde{w})\\&\quad = (\widetilde{w}-w)\bigg(\int_{\beta}^{1}F_{S_1}^{-1}(u)du-(1-\beta)\bigg)\geq 0.\end{align*}

Hence, for $S_1\geq_{\text{icx}} 1$ , we see that $Z_1^w=wS_1+1-w$ increases in increasing convex order with w. The condition $S_1\geq_{\text{icx}} 1$ is easily checked. For a random variable with symmetric density, it simply means that the mode (mean if it exists) is greater than one.

Increasing convex order enables results on how the fraction of the invested amount in the risky asset affects the value of liabilities. Let $\mu_w=w\mathbb{E}[S_1]+1-w$ and let $R_0^w$ refer to the quantity $R_0$ in (2.3) under (2.5) (in particular, $R_0^0=\rho({-}X_1)$ ).

The following result says that, under mild conditions, some risky investment, that is a small $w\gt0$ compared to $w=0$ , is always beneficial in the sense that $R_0^w\lt R_0^0$ and $V_0^w\lt V_0^0$ .

Proposition 2.5. Let $\rho=\operatorname{VaR}_{\alpha}$ and take $S_1$ and $X_1$ to be nonnegative and independent. Assume that $\mathbb{E}[S_1]\gt1$ and that $X_1$ has a bounded and continuous density that is nonvanishing in a neighborhood of $R_0^0=F_{X_1}^{-1}(1-\alpha)$ . If $w\mapsto R_0^w$ is continuously differentiable, then

\begin{align*}\lim_{w\to 0}\frac{d}{dw}R_0^w=\lim_{w\to 0}\frac{d}{dw}V_0^w=-R_0^0(\mathbb{E}[S_1]-1)\lt0, \quad \lim_{w\to 0}\frac{d}{dw}C_0^w=0.\end{align*}

Remark 2.5. If $X_1=x\gt0$ is degenerate, then the assumptions in Proposition 2.5 do not hold. In this case, $0=\operatorname{VaR}_{\alpha}\!(R_0^w(wS_1+1-w)-x)=R_0^w(w(\!\operatorname{VaR}_{\alpha}\!(S_1)+1)-1)+x$ . If further $\operatorname{VaR}_{\alpha}\!(S_1)\in ({-}1,0)$ , then

\begin{align*}\lim_{w\to 0}\frac{d}{dw}R_0^w=x(\!\operatorname{VaR}_{\alpha}\!(S_1)-1)\gt0.\end{align*}

The conclusion is that there is no w which gives $R_0^w\lt R_0^0$ for all possible insurance risks $X_1$ . In order for risky investment to reduce $R_0^w$ , some uncertainty about the outcome of $X_1$ is needed. This conclusion is consistent with Theorem 1.1 in Filipović (Reference Filipović2008), although Filipović (Reference Filipović2008) focuses on convex risk measures.

Remark 2.6. We focus on cost-of-capital valuation in this paper, but we could consider replacing $C_0=\mathbb{E}[Z_{\text{sh}}]/(1+\eta)$ in (2.1) by risk-neutral valuation $C_0=\mathbb{E}^{\mathbb{Q}}[Z_{\text{sh}}]$ and still obtain the qualitative conclusion that some risky investment is always beneficial. Under the additional assumptions that $\mathbb{E}^{\mathbb{Q}}[S_1]=1$ , that independence between $S_1$ and $X_1$ holds also with respect to $\mathbb{Q}$ , and that $X_1$ has a continuous density with respect to $\mathbb{Q}$ , which is nonvanishing in a neighborhood of $R_0^0$ , the expression for the limit of the derivative for $V_0^w$ takes the form

\begin{align*}\lim_{w\to 0}\frac{d}{dw}V_0^w=R_0^0(F^{\mathbb{Q}}_{X_1}(R_0^0)-1)(\mathbb{E}[S_1]-1)\lt0,\end{align*}

where $F^{\mathbb{Q}}_{X_1}(R_0^0)=\mathbb{Q}(X_1\leq R_0^0)$ . A proof of this fact follows along the same lines as the proof of Proposition 2.5.

3.1. Value-at-risk

For $Y\sim N(\mu,\sigma^2)$ , it is well known that

(3.1) \begin{align} \operatorname{VaR}_{\alpha }\!(Y) = -\mu + \sigma \Phi ^{-1}(1 - \alpha).\end{align}

We will always assume that $\alpha\in (0,1/2)$ so that $ \Phi ^{-1}(1-\alpha)\gt0$ .

Proposition 3.1. Suppose that $X_1\sim N(\gamma,\nu^2)$ and $Z_1\sim N(\mu,\sigma^2)$ are independent with $\mu,\gamma,\sigma,\nu\gt0$ . Then,

(3.2) \begin{align} \operatorname{VaR}_{\alpha }\!(R_0Z_1-X_1)=\gamma-R_0\mu+\Phi^{-1}(1-\alpha)\sqrt{R_0^2\sigma^2+\nu^2}. \end{align}

1. (i) If $\mu\gt\sigma\Phi^{-1}(1-\alpha)$ , then

   (3.3) \begin{align} R_0=\frac{\mu\gamma+\Phi^{-1}(1-\alpha)\sqrt{\gamma^2\sigma^2+\mu^2\nu^2-\sigma^2\nu^2\Phi^{-1}(1-\alpha)^2}}{\mu^2-\sigma^2\Phi^{-1}(1-\alpha)^2} \end{align}
   is the unique $R_0\gt0$ solving Equation (2.3).

2. (ii) If $\mu\le \sigma\Phi^{-1}(1-\alpha)$ , then Equation (2.3) has no positive solution $R_0$ .

Remark 3.1. If $\gamma\neq \nu\Phi^{-1}(1-\alpha)$ , then $R_0$ in (3.3) is equivalent to

\begin{align*}R_0=\frac{\gamma^2-\nu^2\Phi^{-1}(1-\alpha)^2}{\mu\gamma-\Phi^{-1}(1-\alpha)\sqrt{\gamma^2\sigma^2+\mu^2\nu^2-\sigma^2\nu^2\Phi^{-1}(1-\alpha)^2}}.\end{align*}

This follows by multiplying both the numerator and denominator of (3.3) by

\begin{align*}\mu\gamma-\Phi^{-1}(1-\alpha)\sqrt{\gamma^2\sigma^2+\mu^2\nu^2-\sigma^2\nu^2\Phi^{-1}(1-\alpha)^2},\end{align*}

and factoring out $\mu ^2 - \sigma ^2 \Phi ^{-1}(1 - \alpha )^{2} $ from the numerator. The usefulness comes from the fact that we will consider $\gamma$ and $\nu$ fixed while allowing $\mu$ and $\sigma$ to vary as a result of considering different positions in a risky asset.

Investing a portion $0\lt w\lt 1$ of the wealth in the risky asset according to (2.5), in the Gaussian model, this simply translates into replacing the parameters $(\mu,\sigma)$ by $(\mu_{w},\,\sigma_{w})\;:\!=(w\mu+1-w,w\sigma)$ . The following result shows that, under very mild conditions, the function $w\mapsto R_0^{w}$ is strictly convex and in addition strictly decreasing for w sufficiently small.

Proposition 3.2. Suppose that $X_1\sim N(\gamma,\nu^2)$ and $S_1\sim N(\mu,\sigma^2)$ are independent with

\begin{align*}\nu\gt0,\sigma\gt0,\gamma\gt\nu\Phi^{-1}(1-\alpha) \text{ and } \mu\gt\max\!(1,\sigma\Phi^{-1}(1-\alpha)).\end{align*}

For $w\in [0,1]$ , let $R_0^{w}$ be the positive solution to

\begin{align*}\operatorname{VaR}_{\alpha}\!(R_0^w(wS_1+1-w)-X_1)=0.\end{align*}

Then, $w\mapsto R_0^{w}$ is strictly convex. Moreover, $R_0^{w}\lt R_0^0$ for all $w\in (0,\widehat{w})$ , where $\widehat{w}=1$ if $\mu\geq 1+\sigma\Phi^{-1}$ $(1-\alpha)$ , and otherwise

\begin{align*}\widehat{w}=\frac{2(\mu-1)\nu\Phi^{-1}(1-\alpha)}{(1+\sigma\Phi^{-1}(1-\alpha)-\mu)(\mu-1+\sigma\Phi^{-1}(1-\alpha))(\gamma+\nu\Phi^{-1}(1-\alpha))}.\end{align*}

Note that the quantity $\widehat{w}$ is of particular interest, since in view of Proposition 2.3, it represents the limit weight of risky assets until which the overall capital requirement is not larger than $R_0^0$ (the one for purely risk-less assets), and the needed premium $V_0^w$ is smaller than the one for $w=0$ . If $\max\!(1,\sigma\Phi^{-1}(1-\alpha))\lt$ $\mu\lt 1+\sigma\Phi^{-1}(1-\alpha)$ , then it is easily verified that $\nu\mapsto \widehat{w}$ is strictly increasing. Hence, more insurance risk allows for more risky investment without increasing the total capital requirement above the level corresponding to purely risk-less investment.

The capital $C_0$ given by (2.1) can be computed explicitly in the Gaussian setting. Together with the explicit expression for $R_0$ in Proposition 3.1, this means that all the quantities $R_0,C_0,V_0$ can be computed explicitly.

Proposition 3.3. Suppose that $X_1\sim N(\gamma,\nu^2)$ and $Z_1\sim N(\mu,\sigma^2)$ are independent with $\mu,\gamma,\sigma,\nu\gt0$ . If there exists an $R_0\gt0$ solving $\operatorname{VaR}_{\alpha}\!(R_0Z_1-X_1)=0$ , then

\begin{align*}C_0&=\frac{R_0\mu-\gamma}{1+\eta}(1+\delta(\alpha))\quad \text{and} \\V_0&=\frac{\gamma(1+\delta(\alpha))}{1+\eta}+R_0\frac{1+\eta-\mu(1+\delta(\alpha))}{1+\eta},\end{align*}

where

\begin{align*} 0\lt\delta(\alpha)\;:\!=\frac{\varphi(\Phi^{-1}(1-\alpha))}{\Phi^{-1}(1-\alpha)}-\alpha\lt\frac{\alpha}{\Phi^{-1}(1-\alpha)^2}.\end{align*}

Remark 3.4. From Proposition 3.3, it follows that the value of the limited liability option (cf. Remark 2.3) in the Gaussian model and $R_0$ determined by the risk measure $\operatorname{VaR}_{\alpha}$ is

\begin{align*}\delta(\alpha)\frac{R_0\mu-\gamma}{1+\eta}.\end{align*}

For $\alpha$ small, for example, $\alpha=0.005$ , the value of the limited liability option is very small due to the light Gaussian tails.

The following result shows that investment in the (sufficiently attractive) risky asset always makes the payoff for a capital provider more attractive compared to the case with only risk-less investment. Hence, the contribution $C_0$ to the financing of the capital requirement should increase if risky investments are allowed.

Proposition 3.4. Suppose that $X_1\sim N(\gamma,\nu^2)$ and $S_1\sim N(\mu,\sigma^2)$ are independent with $\gamma,\sigma,\nu\gt0$ and $\mu \gt \sigma \Phi ^{-1}(1-\alpha )$ . For $w\in [0,1]$ , let $R_0^{w}$ be the positive solution to

\begin{align*}\operatorname{VaR}_{\alpha}\!(R_0^w(wS_1+1-w)-X_1)=0.\end{align*}

Then, $C_{0}^{0} \lt C_{0}^{w}$ for all $w \in (0,1]$ .

If $R_0^w\lt R_0^0$ and $C_0^w\gt C_0^0$ , then $V_0^w\;:\!=R_0^w-C_0^w\lt R_0^0-C_0^0=\!:\;V_0^0$ . Hence, by combining Propositions 3.2 and 3.4, we immediately obtain the following result.

Corollary 3.5. Suppose that $X_1\sim N(\gamma,\nu^2)$ and $S_1\sim N(\mu,\sigma^2)$ are independent with $\sigma,\nu\gt0$ , $\gamma\gt\nu\Phi^{-1}(1-\alpha)$ and $\mu\gt\max\!(1,\sigma\Phi^{-1}(1-\alpha))$ . For $w\in [0,1]$ , let $R_0^{w}$ be the positive solution to

\begin{align*}\operatorname{VaR}_{\alpha}\!(R_0^w(wS_1+1-w)-X_1)=0.\end{align*}

Then, $V_0^{w}\leq V_0^0$ for all $w\in (0,\widehat{w})$ , where $\widehat{w}$ is given in Proposition 3.2.

Note that the statement of Corollary 3.5 is stronger than what we would get by combining Propositions 2.3 and 3.2, since we do not need any condition involving the cost-of-capital rate $\eta$ . The statement of Corollary 3.5 is a statement for w sufficiently small where $R_0^w\lt R_0^0$ . For w larger with $R_0^w\gt R_0^0$ , there is no hope for a simple statement to hold regardless of the parameter $\eta$ . Note that if we let $\eta\to\infty$ , then $V_0^w\to R_0^w$ and $R_0^w$ is typically not monotone, although decreasing for w sufficiently small. However, by evaluating $V_0^w$ numerically, we do observe that $V_0^{w}\leq V_0^0$ for all w for realistic parameter values, see, for example, Figure 1 in Section 4 for an illustration. We emphasize that the statement about $V_0^w$ for realistic parameter values implicitly assumes that $\mu\geq 1$ is not too small. That is, for $\mu=1$ , it follows immediately from Proposition 3.3 that

\begin{align*}V_0^w=\frac{\gamma(1+\delta(\alpha))}{1+\eta}+R_0^w\frac{\eta-\delta(\alpha)}{1+\eta}.\end{align*}

So for $\mu=1$ , we see that $V_0^{w}\gt V_0^0$ is equivalent to $R_0^{w}\gt R_0^0$ . From Remark 3.1, it follows that $R_0^{w}\gt R_0^0$ holds for all $w\in (0,1]$ if $\mu=1$ and $\gamma\gt\nu\Phi^{-1}(1-\alpha)$ .

Remark 3.5. In this paper, we restrict our attention to the setting where $S_1$ and $X_1$ are independent. If the correlation coefficient $\operatorname{Cor}(S_1,X_1)\gt0$ , then risky investments would not only generate a more favorable expectation (assuming $\mu\gt1$ ) but also partly hedge the insurance liability. One may wonder whether risky investments could be advisable even in the case $\operatorname{Cor}(S_1,X_1)\lt0$ . In this case, the expressions for $V_0$ and $C_0$ in Proposition 3.3 remain the same, but the expression for $R_0$ changes. With $c\;:\!=\operatorname{Cor}(S_1,X_1)\leq 0$ , and under the assumption that $\mu\gt\sigma\Phi^{-1}(1-\alpha)$ and $\gamma\neq\nu\Phi^{-1}(1-\alpha)$ ,

\begin{align*}C^w_0&=\frac{R^w_0\mu_w-\gamma}{1+\eta}(1+\delta(\alpha)), \\V^w_0&=\frac{\gamma(1+\delta(\alpha))}{1+\eta}+R^w_0\frac{1+\eta-\mu_w(1+\delta(\alpha))}{1+\eta}, \\R^w_0&=\frac{\gamma^2-\nu^2\Phi^{-1}(1-\alpha)^2}{h(w)},\end{align*}

where $\mu_w\;:\!=w\mu+1-w$ , $\sigma_w\;:\!=w\sigma$ , and

\begin{align*}h(w)&=\mu_w\gamma-c\sigma_w\nu\Phi^{-1}(1-\alpha)^2-\Big((\mu_w\gamma-c\sigma_w\nu\Phi^{-1}(1-\alpha)^2)^2 \\&\quad-(\mu_w^2-\sigma_w^2\Phi^{-1}(1-\alpha)^2)(\gamma^2-\nu^2\Phi^{-1}(1-\alpha)^2)\Big)^{1/2}.\end{align*}

The only difference from the case $c=0$ is that a term involving c appears twice in the expression for h(w). Straightforward computations give

\begin{align*}\frac{dR^w_0}{dw}(0)&=-(\gamma^2-\nu^2\Phi^{-1}(1-\alpha))\frac{h'(0)}{h^2(0)}\\&=-(\gamma+\nu\Phi^{-1}(1-\alpha))(\mu-1+c\sigma\Phi^{-1}(1-\alpha)).\end{align*}

We conclude that as long as $\mu\gt1-c\sigma\Phi^{-1}(1-\alpha)$ , small positions $w\gt0$ in the risky asset will still imply $R_0^w\lt R_0^0$ .

3.2. Expected shortfall

In the case of normal distributions, the expected shortfall of $Y\sim N(\mu,\sigma^2)$ at safety level $\alpha$ can simply be expressed as

\begin{align*} \text{ES}_{\alpha }(Y) = -\mu + \sigma \frac{\varphi (\Phi ^{-1}(1-\alpha ) ) }{\alpha },\end{align*}

so the constant $\Phi ^{-1}(1-\alpha)$ for the value-at-risk in (3.1) is just replaced by another constant

\begin{align*}\psi=\frac{1}{\alpha}\int_0^{\alpha}\Phi^{-1}(1-\beta)d\beta=\frac{1}{\alpha}\int_{\Phi^{-1}(1-\alpha)}^{\infty}z\varphi(z)dz=\frac{\varphi(\Phi^{-1}(1-\alpha))}{\alpha}.\end{align*}

Correspondingly, we can directly adapt the results previously obtained for the value-at-risk. For instance, Proposition 3.1 turns into the following result.

Proposition 3.2 is easily adjusted to $\operatorname{ES}_{\alpha}$ instead of $\operatorname{VaR}_{\alpha}$ . Replacing $\Phi^{-1}(1-\alpha)$ by $\psi$ gives the following result.

Proposition 3.7. Suppose that $X_1\sim N(\gamma,\nu^2)$ and $S_1\sim N(\mu,\sigma^2)$ are independent with $\nu\gt0,\sigma\gt0$ , $\gamma\gt\nu\psi$ and $\mu\gt\max\!(1,\sigma\psi)$ . For $w\in [0,1]$ , let $R_0^{w}$ be the positive solution to

\begin{align*}\operatorname{ES}_{\alpha}\!(R_0^w(wS_1+1-w)-X_1)=0.\end{align*}

Then, $w\mapsto R_0^{w}$ is strictly convex. Moreover, $R_0^{w}\lt R_0^0$ for all $w\in (0,\widehat{w})$ , where $\widehat{w}=1$ if $\mu\geq 1+\sigma\psi$ , and otherwise

\begin{align*}\widehat{w}=\frac{2(\mu-1)\nu\psi}{(1+\sigma\psi-\mu)(\mu-1+\sigma\psi)(\gamma+\nu\psi)}.\end{align*}

Proposition 3.3 is also easily adjusted to $\operatorname{ES}_{\alpha}$ instead of $\operatorname{VaR}_{\alpha}$ . Replacing $1+\delta(\alpha)$ in Proposition 3.3 by $\Phi (\psi) + {\varphi(\psi)}/{\psi}$ gives the following result.

Proposition 3.8. Suppose that $X_1\sim N(\gamma,\nu^2)$ and $Z_1\sim N(\mu,\sigma^2)$ are independent with $\mu,\gamma,\sigma,\nu\gt0$ . If there exists an $R_0\gt0$ solving $\operatorname{ES}_{\alpha}\!(R_0Z_1-X_1)=0$ , then

\begin{align*} C_{0} &= (R_0 \mu - \gamma ) \frac{\Phi (\psi ) + \varphi (\psi ) /{\psi } }{1 + \eta },\\ V_{0} &= \gamma \frac{\Phi (\psi) + {\varphi(\psi)}/{\psi}}{1+\eta} + R_0 \frac{1+\eta -\mu(\Phi (\psi) + {\varphi(\psi)}/{\psi})}{1+\eta}.\end{align*}

Finally, Corollary 3.5 has the following version for $\operatorname{ES}_{\alpha}$ instead of $\operatorname{VaR}_{\alpha}$ .

Corollary 3.9. Suppose that $X_1\sim N(\gamma,\nu^2)$ and $S_1\sim N(\mu,\sigma^2)$ are independent with $\sigma,\nu\gt0$ , $\gamma\gt\nu\psi$ and $\mu\gt\max\!(1,\sigma\psi)$ . For $w\in [0,1]$ , let $R_0^{w}$ be the positive solution to

\begin{align*}\operatorname{ES}_{\alpha}\!(R_0^w(wS_1+1-w)-X_1)=0.\end{align*}

Then, $V_0^{w}\leq V_0^0$ for all $w\in (0,\widehat{w})$ , where $\widehat{w}$ is given in Proposition 3.7.

4.1. The Gaussian model

Assuming that both the insurance risk $X_1$ and the financial risk $Z_1$ are normally distributed, we can use the explicit formulas from Section 3 to study the effect of the choice of w on the resulting requirements for insurance premium $V_0$ , solvency capital requirement $C_0$ , and their sum $R_0$ . Let us focus on the case of value-at-risk with safety level $\alpha = 0.005$ , which is the risk measure used in Solvency II. Let us further assume $\mu = 1.05$ , $\sigma =0.2$ , which may be considered a realistic return and volatility for an institutional investor, as well as the variations $\sigma=0.1$ and $\sigma=0.3$ . For the insurance risk $X_1$ , we assume a mean of $\gamma = 1$ (cf. Remark 2.4) and a standard deviation of $\nu =0.3$ (which corresponds to a coefficient of variation of the total claim size of 0.3), together with a few variations of $\nu$ . Figure 1 plots the resulting values of $R_{0}$ , $C_{0}$ and $V_{0}$ as functions of w (recall that $w=0$ corresponds to purely risk-less investment and $w=1$ corresponds to purely risky investment, cf. (2.5)).

Note that while the needed insurance premium $V_0^{w}$ reduces with increasing risky asset allocation for all w, the needed solvency requirement $C_0^{w}$ from shareholders increases correspondingly, and the minimal overall capital requirement $R_0^{w}$ for $\sigma=0.2$ is achieved for $w^*=0.083$ , for example, only 8.3% of the assets are invested in the risky asset, which then constitutes the most capital-efficient solution under the present model assumptions. One sees how this value $w^*$ decreases for increasing volatility in the risky asset, which matches the intuition. It is also insightful to see that for larger values of w the solvency requirement $C_0^w$ starts to dominate the needed insurance premium $V_0$ , more prominently for larger values of $\sigma$ . The value $\widehat{w}$ until which the overall capital requirement $R_0^{w}$ is smaller than $R_0^{0}$ indeed corresponds to the one obtained from Proposition 3.2. Figure 2 plots the corresponding results for fixed $\sigma=0.2$ , but varying standard deviation $\nu$ of the insurance risk. One observes how the overall capital requirement $R_0^w$ increases with $\nu$ , and how $C_0^w$ becomes larger than the needed insurance premium $V_0^{w}$ already for smaller values of w. One also observes from Figure 2 that the riskier the insurance risk $X_1$ is (larger value $\nu$ ), the larger fraction $w^*$ of the initial capital should be invested in the risky asset in order to minimize the total capital requirement $R_0^w$ .

Figure 2.

$R_{0}^{w}, C_{0}^{w}$ , and $V_{0}^{w}$ for the Gaussian model with $\mu = 1.05, \sigma=0.2,\gamma = 1$ , $\rho=\operatorname{VaR}_{0.005}$ and various values of $\nu$ .

Finally, at first sight, it may seem surprising that the increase of $R_0^{w}$ for increasing standard deviation $\nu$ of the insurance risk is almost exclusively swallowed by the increase of $C_0^{w}$ and leaves the premium $V_0^{w}$ virtually unchanged. The explanation is that the increased standard deviation (for fixed $\mathbb{E}[X_1]$ ) raises the value of the limited liability option. Concretely, for any absolutely continuous random variable $X_1$ (not only Gaussian!), it follows from (2.2) that

$$ C_0^{w}=R_0^{w}\frac{\mathbb{E}[Z_1]}{1+\eta}-\frac{1}{1+\eta}\mathbb{E}[(R_0^{w}Z_1\wedge X_1)] $$

Therefore,

\begin{align*} \frac{\partial C_0^{w}}{\partial R_0^{w}}&=\frac{\mathbb{E}[Z_1]}{1+\eta}-\frac{1}{1+\eta}\mathbb{E}[Z_1\unicode{x1D7D9}_{\{R_0^{w}Z_1 \lt X_1\}}]\\[3pt] &=\frac{\mathbb{E}[Z_1]}{1+\eta}-\frac{1}{1+\eta}\mathbb{E}[Z_1 \mid R_0^{w}Z_1 \lt X_1]\mathbb{P}(R_0^{w}Z_1 \lt X_1) \end{align*}

With $\rho=\operatorname{VaR}_{\alpha}$ we then get

(4.1) \begin{align} \frac{\partial C_0^{w}}{\partial R_0^{w}} &=\frac{\mathbb{E}[Z_1]}{1+\eta}-\frac{1}{1+\eta}\mathbb{E}[Z_1 \mid R_0^{w}Z_1 \lt X_1]\alpha \nonumber\\[3pt] &\geq \frac{\mathbb{E}[Z_1]}{1+\eta}-\frac{\mathbb{E}[Z_1]\alpha}{1+\eta}=\frac{\mathbb{E}[Z_1](1-\alpha)}{1+\eta}. \end{align}

As $\mathbb{E} \left[ Z_1 \right]$ is close to $1 + \eta $ and $\alpha$ is very small, most of the increase of the required capital is indeed absorbed by the shareholders’ contribution $C_0^w$ (see also Remark 2.3).

4.2. The lognormal model

Let us alternatively consider the lognormal model, which is of interest for two reasons. First, it allows to see the effects of heavier right tails for the insurance risk when compared to the normal model. Second, the lognormal assumption on $S_1$ corresponds to a Black–Scholes market assumption for the risky asset, which is not uncommon in this framework. We will choose the parameters of the lognormal model in such a way that the first two moments of both $X_1$ and $S_1$ match the ones from the normal model. That is, for $\mu=1.05$ , $\gamma=1$ and each choice of $\sigma$ and $\nu$ , we use

\begin{align*}s_s^2=\log\bigg(1+\frac{\sigma^2}{\mu^2}\bigg), \quad m_s=\log\bigg(\mu\bigg(1+\frac{\sigma^2}{\mu^2}\bigg)^{-1/2}\bigg)\end{align*}

and

\begin{align*}s_x^2=\log(1+\nu^2), \quad m_x=-\frac{1}{2}\log(1+\nu^2)\end{align*}

to determine the respective parameters of the lognormal distributions. In the absence of explicit expressions, the figures are then obtained by Monte Carlo simulation. All figures are based on two i.i.d. samples of size 1’000’000 from the standard normal distribution, where the simulated variates are suitably transformed into lognormal variates using the parameters $m_s,m_x,s_s,s_x$ .

Figure 3 is the analogue of Figure 1 under lognormality for both insurance and financial risk. One sees that the heavier tail increases $R_0^w$ , and $C_0^w\gt V_0^w$ already for smaller weights w. In all three situations depicted in Figure 3, both the optimal weight $w^*$ as well as the critical threshold $\widehat{w}$ are increased when compared to the same situation in Figure 1 with identical first two moments of $S_1$ and $X_1$ , which can also be attributed to the heavier tails. For large values of w, the safety loading $V_0^w-\mathbb{E}[X_1]$ in the premiums can become very small and even negative. The model-independent upper bound for $V_0^w$ from Remark 2.2 can be expressed as

\begin{align*}V_0^w \leq \mathbb{E}[X_1] + \frac{\eta (R_0^w-\mathbb{E}[X_1])-w(\mathbb{E}[S_1]-1)R_0^w}{1+\eta}\end{align*}

from which it follows immediately that, for $\mathbb{E}[S_1]\gt1$ ,

\begin{align*}w\gt\frac{\eta}{\mathbb{E}[S_1]-1}\frac{R_0^w-\mathbb{E}[X_1]}{R_0^w}\end{align*}

implies a negative safety loading for any model.

Figure 3.

$R_{0}^{w}, C_{0}^{w}$ , and $V_{0}^{w}$ for the lognormal model with $\mathbb{E}[S_1] = 1.05$ , $\mathbb{E}[X_1] = 1$ , $\text{std}(X)=0.3$ , $\rho=\operatorname{VaR}_{0.005}$ and various values of $\text{std}(S_1)$ .

Figure 4 zooms into the respective values of the needed premium $V_0^w$ and also depicts the bounds that were derived in Remark 2.2 (the upper bound is very tight, as a further magnification in the plots indicates). One observes that the lower bound is too coarse for practical purposes, but the upper bound is remarkably sharp in these cases. As the upper bound is obtained by ignoring the limited liability, this indicates that the value of the limited liability option is quite small (which here is due to the small value $\alpha=0.005$ and the fact that the right tail of $X_1-R_0Z_1$ beyond its $1-\alpha=0.995$ quantile value is not sufficiently heavy to significantly impact $V_0$ ).

Figure 4.

Upper and lower bounds for $V_{0}^{w}$ in Figure 3.

Figure 5 depicts the analogue of Figure 2 under lognormality for both insurance and financial risk, in this case varying the standard deviation of the insurance risk. Again, each of these situations leads to larger overall capital requirement $R_0^w$ compared to the counterparts of Figure 2, and again the solvency capital requirement $C_0^w$ absorbs most of that increase, whereas the premium $V_0^w$ stays almost unchanged for increasing standard deviation. The latter is again due to the increased value of the limited liability option, cf. (4.1) and also Example 2.1 for the case $w=0$ .

Figure 5.

$R_{0}^{w}, C_{0}^{w}$ , and $V_{0}^{w}$ for the lognormal model with $\mathbb{E}[S_1] = 1.05$ , $\mathbb{E}[X_1] = 1$ , $\text{std}(S_1)=0.2$ , $\rho=\operatorname{VaR}_{0.005}$ and various values of $\text{std}(X_1)$ .

Figure 6 zooms into $V_0^w$ and depicts the bounds that were derived in Remark 2.2. Again the sharpness of the upper bound is noteworthy.

Figure 6.

Upper and lower bounds for $V_{0}^{w}$ in Figure 5.

Figures 7 and 8 depict the analogues of Figures 3 and 5 for the situation when the expected return for the risky asset is only 2%. While the magnitudes of the quantities are very similar (visually almost indistinguishable to the case of larger $\mu$ ), one sees that the optimal weight $w^*$ and the critical threshold $\widehat{w}$ are considerably smaller with smaller expected invested return $\mu$ , that is there is less incentive to invest a larger amount of the capital into the risky asset.

Figure 7.

$R_{0}^{w}, C_{0}^{w}$ , and $V_{0}^{w}$ for the lognormal model with $\mathbb{E}[S_1] = 1.02$ , $\mathbb{E}[X_1] = 1$ , $\text{std}(X)=0.3$ , $\rho=\operatorname{VaR}_{0.005}$ and various values of $\text{std}(S_1)$ .

Figure 8.

$R_{0}^{w}, C_{0}^{w}$ , and $V_{0}^{w}$ for the lognormal model with $\mathbb{E}[S_1] = 1.02$ , $\mathbb{E}[X_1] = 1$ , $\text{std}(S_1)=0.2$ , $\rho=\operatorname{VaR}_{0.005}$ and various values of $\text{std}(X_1)$ .

4.3. Lognormal investment returns and Pareto claims

Figures 9 and 10 illustrate the possible effect of even heavier tails for the insurance risk, assuming that $X_1$ is Pareto-distributed with cumulative distribution function $F(x)=1-(x/x_m)^{-\beta}$ , $x\gt x_m\gt0$ , with the same first two moments as in the case of lognormal $X_1$ in Figures 3 and 5:

\begin{align*}\mathbb{E}[X_1]=x_m\frac{\beta}{\beta-1}, \quad \text{std}(X_1)=x_m\frac{\sqrt{\beta}}{(\beta-1)\sqrt{\beta-2}}.\end{align*}

For example, the resulting Pareto parameters in Figure 10, where $\mathbb{E}[X_1]=1$ , are given by

\begin{align*}\beta=1+\sqrt{1+1/\text{std}(X_1)^2}, \quad x_m=(\beta-1)/\beta.\end{align*}

A comparison of Figures 5 (lognormal model) and 10 (lognormal $S_1$ and Pareto-distributed $X_1$ ) shows that the main difference is that the heavier Pareto tail increases the overall capital requirement $R_0$ , and that the increase in $R_0$ is solely financed by increased capital $C_0$ (the premiums $V_0$ stay virtually unchanged). In order to see the effects of heavier tails even more significantly, in Figure 11 the expected aggregate claim size is kept at $ \mathbb{E}[X_1] = 1$ , but the parameter $\beta$ is taken more extreme ( $\beta=2$ and $\beta=1.1$ ), in which case the standard deviation does not exist anymore (but the mean still does). One observes how the capital requirement $R_0$ goes up tremendously, but the theoretical premiums $V_0$ are still virtually unchanged. It should be noted that whereas $R_0$ increases as $\beta$ decreases from $\beta=2$ to $\beta=1.1$ , this is not the case for $V_0$ due to the increasing value of the shareholders limited liability option (cf. Example 2.1).

Figure 9.

$R_{0}^{w}, C_{0}^{w}$ , and $V_{0}^{w}$ for lognormal $S_1$ and Pareto-distributed $X_1$ with $\mathbb{E}[X_1] = 1$ , $\text{std}(X_1)=0.3$ , $\mathbb{E}[S_1] = 1.05$ , $\rho=\operatorname{VaR}_{0.005}$ and various values of $\text{std}(S_1)$ .

Figure 10.

$R_{0}^{w}, C_{0}^{w}$ , and $V_{0}^{w}$ for lognormal $S_1$ and Pareto-distributed $X_1$ with $\mathbb{E}[X_1] = 1$ , $\mathbb{E}[S_1] = 1.05$ , $\text{std}(S_1)=0.2$ , $\rho=\operatorname{VaR}_{0.005}$ and various values of $\text{std}(X_1)$ .

Figure 11.

$R_{0}^{w}, C_{0}^{w}$ , and $V_{0}^{w}$ for lognormal $S_1$ and Pareto type I-distributed $X_1$ with $\mathbb{E}[X_1] = 1$ , $\mathbb{E}[S_1] = 1.05$ , $\text{std}(S_1) =0.2$ , $\rho=\operatorname{VaR}_{0.005}$ and low levels of Pareto parameter $\beta$ .

Generally, the optimal investment weight $w^*$ and the critical threshold $\widehat{w}$ are larger in the Pareto case, indicating that for heavier-tailed insurance risks a riskier investment is advantageous (in the very heavy-tailed case of $\beta=1.1$ even to the extent that full investment in the risky asset, $w=1$ , does not lead to a higher overall capital requirement $R_0$ than risk-less investment, $w=0$ ). Since the insurance risk dominates here, we gain from an attractive return on the risky investment without substantially affecting the overall capital requirement. One also observes that in comparison with lighter-tailed risks, the curves are generally flatter, that is, they are less sensitive to changes in w.

Finally, it may be of interest to see the effect of the risk measure on the result. While for the normal model, the expected shortfall can be replaced by a value at risk with a different security level, this is not the case for the lognormal model. Figures 12 and 13 depict the analogues of Figures 3 and 5 when $\operatorname{VaR}_{0.005}$ is replaced by $\operatorname{ES}_{0.01}$ (which is the risk measure in the Swiss Solvency Test). As expected, the overall capital requirement $R_0$ increases, but this increase is provided by a larger investment $C_0$ of the shareholders and the insurance premiums $V_0$ stay virtually unchanged. In this case, the optimal weights $w^*$ and the critical threshold $\widehat{w}$ are very similar to the ones obtained under the $\operatorname{VaR}_{0.005}$ criterion.

Figure 12.

$R_{0}^{w}, C_{0}^{w}$ , and $V_{0}^{w}$ for the lognormal model with $\mathbb{E}[S_1] = 1.05$ , $\mathbb{E}[X_1] = 1$ , $\text{std}(X_1)=0.3$ , $\rho=\operatorname{ES}_{0.01}$ and various values of $\text{std}(S_1)$ .

Figure 13.

$R_{0}^{w}, C_{0}^{w}$ , and $V_{0}^{w}$ for the lognormal model with $\mathbb{E}[S_1] = 1.05$ , $\mathbb{E}[X_1] = 1$ , $\text{std}(S_1)=0.2$ , $\rho=\operatorname{ES}_{0.01}$ and various values of $\text{std}(X_1)$ .

All these numerical examples illustrate that the appropriate blending of (independent) financial risk into the management of insurance risk can drive down the necessary insurance premiums according to actuarial considerations, which may be considered as a promising insight in situations where insurance premiums are considered too high. At the same time, the absolute reduction of $V_0$ through increasing w is still relatively moderate.