Robust Security Design with Moment Information


Student thesis: Doctoral Thesis

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Award date8 Jan 2024


We consider the optimal security design problem between a risk-neutral investor and an entrepreneur. In contrast to the existing literature that assumes common knowledge on the distribution of cash flow, we consider the scenario where the entrepreneur privately has complete information on the cash flow distribution, but the investor only has access to moment information (e.g., mean and variance). Facing such information disadvantage, the investor aims to maximize its objective under the worst-case criterion. We can explicitly characterize the structure of the robust optimal security. When the investor only has mean information, we find that a linear security is robust optimal. When the investor possesses mean-variance information, the robust optimal security becomes quadratic. Interestingly, the convexity or concavity of the robust optimal security is determined by the sign of the marginal effect of the second moment with respect to the effort exerted by the entrepreneur. In turn, this sheds light on when to incentivize the entrepreneur to be risk taking/aversion, which is consistent with security designs of different phases of firm lifecycle in practice. Furthermore, we find that quadratic security is superior to linear security upon examining the two-sided limited liability constraints. Specifically, quadratic security can achieve a mutually beneficial scenario under less stringent conditions compared to linear security. Finally, we extend our analysis to consider risk-averse entrepreneurs, illustrating that the robust optimal security takes the form of a transformed quadratic security. This transformation is dictated by the inverse of the entrepreneur's utility function. Our numerical results show that as entrepreneurs become more risk-averse, they tend to exert less effort, which ultimately leads to a reduction in the investor's worst-case utility. Overall, our findings shed light on the understanding and design of robust securities with moment information only.