Proportional reinsurance and investment in multiple risky assets under borrowing constraint

ABSTRACT

In this paper, we consider the ruin probability minimization of an insurance company that buys proportional reinsurance and invests in markets where borrowing is constrained. We use a diffusion approximation model for the surplus process of this company and assume that the company invests its surplus into a riskless and multiple risk assets that are modeled as geometric Brownian motions. To find the results, we introduce an auxiliary market parametrized with fictitious processes to relax the borrowing constraint and apply the techniques of stochastic optimal control. In this way, we find the optimal proportional reinsurance and investment strategy of an insurance company investing into multiple risky assets to minimize its ruin probability under the borrowing constraint. Furthermore, from our solutions, we show how our results connect to economic survival analysis and how investment and reinsurance strategies are related.

KEYWORDS:

• Ruin probability
• proportional reinsurance
• Hamilton–Jacobi–Bellman equation
• portfolio selection
• borrowing constraint

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Haluk Yener http://orcid.org/0000-0003-2654-5810

Notes

1 In our case we consider four non-linear partial differential equations to solve because there are investment constraints. Without constraints one needs to consider two equations to solve.

2 The analysis is borne out of the seminal work of Kahneman & Tversky (1979) and its results are different from what we would encounter under the classical utilitarian analysis (see for example March 1988, Roy 1995, Radner 1998).

3 We call this strategy the growth optimal strategy, and it is the strategy that is known to maximize the growth rate of a traded portfolio.

4 Per our notation ${}^{\prime }$ on vectors or matrices denotes the transpose. However, when on functions, as we will explain in the sequel, it denotes the derivative of a function with respect to its argument.

5 We do not consider no short-selling constraint as it was previously considered in Bai & Guo (2008) and its application along with borrowing constraint is straightforward. Once one finds the assets that are short-sold, one could set the investment in these assets to zero because short-selling is prohibited (see Cvitanic & Karatzas 1992). If the weights on the remaining assets violate the borrowing constraint, the solutions can then be found by extending our framework to a reduced dimension of risky assets. Furthermore, the application of the case when borrowing rate higher than the lending rate is straightforward as well and can be solved by following the framework employed in this study.

6 Here, κ is the cap on borrowing and when $\kappa =1$ the constraint becomes the borrowing prohibition.

7 For further properties of the growth optimal strategy we refer the readers to Maclean et al. (2010).

8 Here, the admissible sets ${\mathcal{A}}_c\left(x\right)$ and ${\mathcal{A}}_{\nu }\left(x\right)$ are defined analogously to the definition of the set of admissible strategies $\mathcal{A}\left(x\right)$ provided in Section 2.

9 To denote the case when the borrowing constraint binds, the notations under the unconstrained case must be utilized.

10 Showing the inequality ${\theta }_1\le {\vartheta }_1^{+}$ under the multiple asset case is not as straightforward. Note that our case is the generalization of the case considered in Luo et al. (2008); we can recover the results of Luo et al. (2008) from our results. Therefore, if ${\theta }_1\le {\kappa }_1^{+}$ under the multiple asset case, then it will definitely hold under the single asset case. If not, the finding in Luo et al. (2008) would be violated.

11 This inequality as well follows from Luo et al. (2008) for the single asset case. We use the same logic as we did previously. Since the results of the single asset case may be recovered from our generalized approach, the inequality must hold in order to have a solution for the problem.

12 Here as well, the result follows from Lemma 4.8 in Luo et al. (2008) for the single asset case. We use the same logic as we did previously. We refer the reader to the previous footnotes.

13 We omit the solutions when ${\gamma }_1\le {\theta }_1$ and for other cases as well, because they are already provided in the previous section.

14 As we explained previously, for very low surplus levels, we have $1_N^{\mathrm{\prime }}{\mathit{w}}^{\ast }=1$ for all values of λ in our analysis.

15 Equations (49(49) $a^{\ast }\left(x\right)=\left\{\begin{array}{ll}1& ifx\in (0,{\vartheta }_1^{-}]\\ \frac{\lambda }{{\sigma }_0^2}\frac{\left[\lambda -\mu -\left(r_0+\frac{D}{K}\right)x\right]+\sqrt{{\mathrm{\Delta }}_1\left(x\right)}}{\frac{{\lambda }^2}{{\sigma }_0^2}+\parallel \zeta {\parallel }^2-\frac{D^2}{K}}& ifx\in ({\vartheta }_1^{-},{\theta }_1];\\ \frac{\lambda }{{\sigma }_0^2}\frac{2\left[\lambda -\mu -r_{0}x\right]}{\frac{{\lambda }^2}{{\sigma }_0^2}+\parallel \zeta {\parallel }^2}& ifx\in ({\theta }_1,\overline{x}];\\ 0& ifx\in (\overline{x},\mathrm{\infty })\end{array}\right.$(49) ) and (50(50) ${\mathit{w}}^{\ast }\left(x\right)=\left\{\begin{array}{ll}\left({\sigma }^{\mathrm{\prime }}{)}^{-1}{\zeta }_{{\nu }^{\ast }}\right(x)\frac{\mu +\left(r_0+\frac{D}{K}\right)x-\sqrt{{\mathrm{\Delta }}_2\left(x\right)}}{x\left[\parallel \zeta {\parallel }^2-\frac{D^2}{K}\right]}& ifx\in (0,{\vartheta }_1^{-}];\\ \left({\sigma }^{\mathrm{\prime }}{)}^{-1}{\zeta }_{{\nu }^{\ast }}\right(x)\frac{\left[\lambda -\mu -\left(r_0+\frac{D}{K}\right)x\right]+\sqrt{{\mathrm{\Delta }}_1\left(x\right)}}{x\left[\frac{{\lambda }^2}{{\sigma }_0^2}+\parallel \zeta {\parallel }^2-\frac{D^2}{K}\right]}& ifx\in ({\vartheta }_1^{-},{\theta }_1];\\ ({\sigma }^{\mathrm{\prime }}{)}^{-1}\zeta \frac{2\left[\lambda -\mu -r_{0}x\right]}{x\left[\frac{{\lambda }^2}{{\sigma }_0^2}+\parallel \zeta {\parallel }^2\right]}& ifx\in ({\theta }_1,\overline{x}];\\ 0& ifx\in (\overline{x},\mathrm{\infty }),\end{array}\right.$(50) ) over the domain $(0,{\vartheta }_1^{-}]$ hold for both Cases 2 and 3.