In this paper we introduce concepts of pseudo-triangular entropy as a supplement measure of Uncertainty in the Uncertain portfolio optimization. We first prove that logarithm entropy and triangular entropy for Uncertain variables sometimes may fail to measure the Uncertainty of an Uncertain variable. Then, we propose a definition of pseudo-triangular entropy as a supplement measure to characterize the Uncertainty of Uncertain variables and we derive its mathematical properties. We also give a formula to calculate the pseudo-triangular entropy of Uncertain variables via inverse Uncertainty distribution. Moreover, we use the pseudo-triangular entropy to characterize portfolio risk and establish some Uncertain portfolio optimization models based on different types of entropy. A genetic algorithm (GA) is implemented in MATLAB software to solve the corresponding problem. Numerical results show that pseudo-triangular entropy as a quantifier of portfolio risk outperforms logarithm entropy and triangular entropy in the Uncertain portfolio optimization.

Catastrophe bonds are among the essential instruments in providing a financial hedge for insurance companies and their policyholders. Catastrophic events are rare, and the shortage of data turns using probability theory indefensible. On the other hand, Uncertainty theory is a reliable alternative to deal with these kinds of indeterminacies. We model the problem of pricing catastrophe bonds as an Uncertain optimization problem where the maximization of the cedent insurance company’s profit is constrained to the Uncertain measure of ruin defined for the investors. Consequently, one could provide a tradeoff between being profitable for the ceding company and having reasonable protection for the investors. A solution to the optimization problem will be considered as the spread over the LIBOR, leading to a complete determination of the bond price. The results suggest the practicality of the model, especially the application of Uncertainty theory in pricing catastrophe bonds. Finally, the Uncertain ruin index is calculated for a real-world problem, and the results are compared with those obtained by probability theory.