Actuarial pricing with financial methods

Abstract

The objective of this paper is twofold. On the one hand, the optimal combination of reinsurance and financial investment will be studied under a general framework. Indeed, there is no specific type of reinsurance contract, there is no specific dynamics of the involved financial instruments and the financial market does not have to be free of frictions. On the other hand, it will be pointed out how the optimal combination above may provide us with new premium principles making the insurer global risk vanish. The risk will be managed with a coherent risk measure, and the new premium principles will seem to reflect several properties, which are desirable from both the analytical and the economic perspectives. From the analytical viewpoint, the premium principles will be continuous, homogeneous and increasing. From the economic viewpoint, the premium principles will lead to cheaper prices with respect to both the insurance market and the financial one. In other words, the premium principles will make the insurer more competitive in prices under a null risk. General necessary and sufficient optimality conditions will be given, as well as closed forms for the solutions under appropriate assumptions. Several methods preventing unbounded optimization problems will warrant special attention, and one particular case will be more thoroughly studied, namely, the combination of the Black–Scholes–Merton pricing model with the conditional value at risk.

AMS CLASSIFICATIONS:

• 91G05
• 91G10
• 91G20
• 91G70

Keywords:

• Reinsurance
• financial market
• risk measure
• optimization of the unhedged risk
• insurance pricing

JEL Classifications:

• G22
• G11
• G13
• C61

Acknowledgments

The authors sincerely thank the journal editor and two anonymous reviewers whose suggestions led to significant improvements. The usual caveat applies.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Among others, see Yi et al. (2015), Zhuang et al. (2017), or Cheung et al. (2019), for recent approaches, or Albrecher et al. (2017), for a very complete overview.

2 The assumptions about the financial market and the available financial securities will be general enough and will include both problems with and without budget constraints as particular cases. If there are budget constraints, the solution of the problem will indicate whether they must be saturated.

3 A very complete analysis of this unboundedness for coherent risk measures may be found in Balbás et al. (2010a), whereas (Biagini & Pinar 2013) present similar results for the gain-loss ratio.

4 Similar notation will be used in similar situations.

5 Consider the framework of Remark 2.1, but replace $Y_0\equiv 1$ with $Y_0\equiv {\mathrm{e}}^{rt}$ for $t\in \mathcal{T}$, and r being a riskless rate. $\mathrm{\Psi }\left(w\right)$ may be understood as the quantity to be paid at T if w is bought at t = 0 but its total current price is borrowed. Alternatively, $\mathrm{\Psi }\left(w\right)$ may be understood as the quotation of a forward contract with maturity at T, underlying asset equaling w, and free of counterparty risk. Usually, the pricing rule $\mathrm{\Psi }:W\to \mathbb{R}$ will be linear, but let us extend the analysis in order to incorporate potential transaction costs (Jouini & Kallal 2001 or Stoica & Li 2010).

6 ${\mathrm{\partial }}_{\mathrm{\Psi }}$ is said to be the sub-gradient of the pricing rule at zero, though we will merely say sub-gradient of the pricing rule.

7 In other words, this risk measure  is continuous, sub-additive ($\gamma (u_1+u_2)\le \gamma \left(u_1\right)+\gamma \left(u_2\right)$), positively homogeneous ($\gamma \left(\lambda u\right)=\lambda \gamma \left(u\right)$ if $\lambda \ge 0$), translation invariant ($\gamma (u+\lambda )=\gamma \left(u\right)-\lambda$ if $\lambda \in \mathbb{R}$), decreasing ($\gamma \left(u_2\right)\le \gamma \left(u_1\right)$ if $u_2\ge u_1$), and mean dominating ($\gamma \left(u\right)\ge -\mathsf{I}\mathsf{E}\left(u\right)$). Many important examples of risk measure may be considered. Among them, the $CV@R$, the $WCV@R$ and the spectral risk measures. Another interesting example is minus the mathematical expectation, that is, $\gamma \left(u\right)=-\mathsf{I}\mathsf{E}\left(u\right)$ (Artzner et al. 1999, Rockafellar et al. 2006).

8 Once again it may be more intuitive to assume that the reinsurance contract will be paid at T. Alternatively, one can interpret that $\mathrm{\Phi }\left(u\right)$ is the quantity to be paid at T if a default-free insurer borrows the whole price of the reinsurance contract.

9 The absolute deviation, the standard deviation, the downside standard semi-deviation

10 Similar notations will be used in similar situations.

11 Recall that $j^{\ast }$ is characterized by $\mathsf{I}\mathsf{E}\left(j\right(f\left)g\right)=〈f,j^{\ast }\left(g\right)〉$ for every $f\in L^{\mathrm{\infty }}\left(\mathsf{I}\mathsf{L}\right)$ and every $g\in L^2\left(\mathsf{I}{\mathsf{P}}^{(y}\right)$ (Zeidler 1995).

12 Recall that H will be selected by the reinsurer and will indicate the maximal marginal indemnity. We also impose the constraint $H\ge h>0$ for mathematical reasons. If h did not exist, then a Slater qualification could fail (Luenberger 1969), and Theorem 3.5 could become false.

13 Optimal reinsurance problems involving more than one objective and Pareto-efficiency are usual in the literature (Asimit et al. 2017, Zhang et al. 2018, Boonen & Jiang 2022).

14 $I.e.$, as usual, the model is not risk-neutral, or the risk-neutral probabilities do not coincide with the physical ones.

15 According to the properties of the conditional expectation (Durrett 2010), for every $u\in L^2\left(\mathsf{I}\mathsf{P}\right)$ one has that $\mathsf{I}\mathsf{E}\left(z_{\mathrm{\Psi }}u\right|{\mathcal{F}}_T)=z_{\mathrm{\Psi }}\mathsf{I}\mathsf{E}(u\left|{\mathcal{F}}_T\right)$. Hence, $\mathsf{I}\mathsf{E}\left(z_{\mathrm{\Psi }}u\right)=\mathsf{I}\mathsf{E}\left(\mathsf{I}\mathsf{E}\left(z_{\mathrm{\Psi }}u|{\mathcal{F}}_T\right)\right)=\mathsf{I}\mathsf{E}\left(z_{\mathrm{\Psi }}\mathsf{I}\mathsf{E}\left(u|{\mathcal{F}}_T\right)\right)=\mathrm{\Psi }\left(\mathsf{I}\mathsf{E}\left(u|{\mathcal{F}}_T\right)\right)=\hat{\mathrm{\Psi }}\left(u\right).$

16 If the pricing rule is not linear, $\hat{\gamma }$ will be the minimum risk measure among those satisfying the conditions of Section 2.3 and such that $\hat{\gamma }\left(u\right)\ge \mathrm{M}\mathrm{a}\mathrm{x}\left\{\gamma \right(u),-\hat{\mathrm{\Psi }}(u\left)\right\}$ for every $u\in L^2\left(\mathsf{I}\mathsf{P}\right)$, whose existence was proved in Balbás & Balbás (2009). Once the risk measure is replaced by $\hat{\gamma }$, Π becomes $\mathbb{R}-$valued.

17 If the reinsurer premium principle is given by (10(10) $\mathrm{\Phi }\left(u\right)=(1+k_1)\mathsf{I}\mathsf{E}\left(u\right)+k_2\rho (-u)$(10) ) and ρ is coherent, then this premium principle is increasing.

18 In particular, binary (or digital) future options will not be feasible.

19 $I.e.$, the decision variable is a couple composed of the reinsurer marginal indemnification and the delta Greek at T of the financial investment.

20 (i), (ii), (iii) and (iv) lead to problems involving a bounded delta, but similar approaches will allow us to impose the boundedness of other parameters. For instance, if the quantity to invest cannot be higher than a budget C, one can select a pay-off $w_0$ such that $\mathrm{\Psi }\left(w_0\right)=C$ and a cone such that $\mathrm{\Psi }\left(v\right)\le 0$ for every $v\in V$. Moreover, if desired, $w_0$ may equal the riskless security.

Additional information

Funding

Research partially supported by the University Carlos III of Madrid (Project 2009/00445/003) and the Spanish Ministry of Science and Innovation (Project PID2021-125133NB-I00).