Sub-optimal investment for insurers

Abstract

We consider the investment problem for a non-life insurance company seeking to minimize the ruin probability. Its reserve is described by a perturbed risk process possibly correlated with the financial market. Assuming exponential claim size, the Hamilton-Jacobi-Bellman equation reduces to a first order nonlinear ordinary differential equation, which seems hard to solve explicitly. We study the qualitative behavior of its solution and determine the Cramér-Lundberg approximation. Moreover, our approach enables to find very naturally that the optimal investment strategy is not constant. Then, we analyze how much the company looses by adopting sub-optimal constant (amount) investment strategies.

Key Words:

• Insurance risk process
• ruin probability
• investment
• sub-optimal strategies

Acknowledgments

The authors wish to thank an anonymous referee for his helpful comments on an earlier version of this paper.

Notes

1 Notice that the classical pure compound Poisson process for the reserve (i.e. (1) with η = 0), requires $c>\lambda \mathbb{E}[Y]$ (the so called positive safety loading) otherwise ruin occurs with probability one whatever is the initial reserve, due to the fact that $\underset{t\to \infty }{\lim }\mathbb{E}[S(t)/t]=\lambda \mathbb{E}[Y].$ As it is typical in model with investment, this condition is not needed because π can be chosen such that $c+\pi a>\lambda \mathbb{E}[Y].$ We further notice that the pure compound Poisson cannot be obtained as limit case, as $\eta \to 0,$ of the case treated in this paper.

2 If the bank account pays a positive interest rate then the qualitative behavior of the optimal investment policy may be significantly different (cf. Yang and Zhang, 2005).

3 The absence of bounds on the investment into the risky asset is widely assumed in literature, see, among others, Hipp and Plum (2000), Yang and Zhang (2005), Schmidli (2002).

4 The cases of perfect correlation, i.e. $\rho =\pm 1,$ need to be treated separately and cannot be obtained as limits, as $\rho \to \pm 1,$ of cases treated in this paper. Indeed, when $\rho =\pm 1,$ the insurer (investor) can perfectly hedge part of the random reserve R(t) and ${\pi }^{\ast }(0)$ and $\delta (0)$ are fixed by using arguments similar to Hipp and Plum (2000).

5 Notice that $$0\le \underset{u\to 0}{\lim }\frac{{\int }_0^u{\delta }^{\prime }\left(y\right)e^{-k(u-y)} \mathit{dy}}{{\delta }^{\prime }\left(u\right)}\le \underset{u\to 0}{\lim }\frac{{\int }_0^u{\delta }^{\prime }\left(y\right)dy}{{\delta }^{\prime }\left(u\right)}=\underset{u\to 0}{\lim }\frac{\delta \left(u\right)}{{\delta }^{\prime }\left(u\right)}=0$$ since $\underset{u\to 0}{\lim }{\delta }^{\prime }(u)$ exists and is positive due to the fact that $\delta (u)$ is positive and concave on $\{u>0\}.$ In fact, $\underset{u\to 0}{\lim }{\delta }^{\prime }\left(u\right)<+\infty ,$ otherwise an application of de l’Hospital theorem to the limit $\underset{u\to 0}{\lim }\delta \left(u\right)/{\delta }^{\prime }\left(u\right)$ would yield $$0=\underset{u\to 0}{\lim }\frac{\delta \left(u\right)}{{\delta }^{\prime }\left(u\right)}=\underset{u\to 0}{\lim }\frac{{\delta }^{\prime }\left(u\right)}{{\delta }^{″}\left(u\right)}=-\frac{1}{R_0}<0,$$ a contradiction. Hence, ${\delta }^{\prime }\left(0\right)$ exists, it is finite and $0<{\delta }^{\prime }\left(u\right)\le {\delta }^{\prime }\left(0\right),$ for all $u\ge 0.$ Moreover, the fact that both ${\delta }^{″}\left(u\right)/{\delta }^{\prime }\left(u\right)$ and ${\delta }^{\prime }\left(u\right)$ admit the above limits, as $u\to 0,$ implies that $\underset{u\to 0}{\lim }{\delta }^{″}\left(u\right)$ exists and is finite too. Finally, by using the HJB structure, it can be proved that ${\delta }^{″}\left(u\right)$ and ${\left({\delta }^{\prime }\left(u\right)\right)}^2/{\delta }^{″}\left(u\right)$ are bounded too on $[0,+\infty ).$

6 Here our analysis disagrees with Yang and Zhang (2005), Section 5, where the authors fix ${\pi }^{\ast }(0)=0.$

7 The substitution is similar to Hipp and Plum (2000), Section 5, but our analysis of the resulting equation is different.

8 The presence of a risk-less investment opportunity paying positive interest rate may significantly alter the qualitative behavior of the optimal investment policy as it is numerically showed in Yang and Zhang (2005), Example 1 and Figure 1.

9 The same qualitative approach applied to (5.2)(5) $${\psi }_{\pi }\left(u\right):=\mathbb{P}\left(X^{u,\pi }\left(t\right)<0 \text{for} \text{some} t\ge 0\right)=\mathbb{P}\left({\tau }_{\pi }\left(u\right)<\infty \right)$$(5) in Hipp and Plum (2000) enables to prove that also in their model the optimal investment policy is strictly increasing, bounded, positive (except at 0), and converges to $1/{\gamma }_1$ (in their notation) as reserve diverges to infinity.

10 Compare with Grandits (2004), Theorems 4.1 and 4.2, where the same results are proved for more general claim’s distribution by using an analytical approach different from the qualitative analysis of the relevant differential equation applied here for the particular case of exponential distribution.

11 We remark that, due to the price variation, a constant strategy in our setting will nonetheless require continuous trading to keep the amount of wealth invested in the risky financial index constant. Strategies that avoid trading are the buy-an-holds strategies, which are not considered here also because they badly suit the investment behavior of big companies such as insurance companies. Clearly, we are abstracting from transaction costs.

12 Frolova et al. (2002) assume that insurance companies choose to hold a constant fraction of wealth in the risky asset and proves that the ruin probability decreases as a power function.

13 Given two correlated Brownian motion W₁ and W₂ with correlation coefficient ρ, it holds $\alpha W_1+\beta W_2=\sqrt{{\alpha }^2+2\rho \alpha \beta +{\beta }^2}{W}_3,$ for all $\alpha ,\beta \in \mathbb{R},$ where W₃ is another Brownian motion.