Furman, EdwardPatgunarajah, Rishigesh2025-07-232025-07-232025-02-192025-07-23https://hdl.handle.net/10315/42961By incorporating Gumbel's bivariate exponential (BVE) distribution as the foundation, we aim to minimize losses between two business lines. Motivated by the need for a financial model, we chose Gumbel’s BVE for its thin-tailed property, serving as a great foundation for future extensions to multidimensional risk measures. We derive a closed-form expression for $H(x,y$) using the law of total probability, linking it to the value-at-risk concept represented by set $\mathcal{A}_p$. Recognizing that companies allocate capital near the boundary of $\mathcal{A}_p$, we define set $\mathcal{O}_p$ as the level $p$ curve of optimal values. A convexity analysis via a Hessian matrix and Sylvester’s Criterion provides insight into optimal capital allocation, and we apply Lagrange multipliers to prove a loss-minimization theorem. Graphs and tables illustrate our findings. Overall, this research offers practical insights into resource allocation, boosting company growth and financial stability.Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.MathematicsFinanceElectronic Thesis or Dissertation2025-07-23HessianConvexityMinimizationGumbelBVEVaRValue at RiskExponential distributionPearson's correlation