Allocation Problems in Risk and Finance: Theoretical Models and Applications

Risk allocation and portfolio optimization are fundamental problems in both actuarial science and financial mathematics. This dissertation explores three distinct yet interconnected topics related to risk and capital allocation: (1) risk measure and allocation under truncated multivariate normal distributions, (2) portfolio optimization in open equity markets, and (3) a novel return-based risk measure with applications in capital allocation.

In the first part, we investigate the suitability of truncated multivariate normal distributions for actuarial risk modeling. We derive explicit formulas for quantile condition allocation (QCA) and tail conditional allocation (TCA), two widely used risk functionals in quantitative risk management. Our approach offers a deterministic alternative to traditional simulation-based methods, improving computational efficiency while maintaining accuracy.

The second part of this dissertation focuses on portfolio optimization within open markets, where the number of tradable assets evolves over time due to new stock entries and delistings. We propose a modified selection criterion that prioritizes lower-capitalization stocks with high growth potential. By leveraging tools from stochastic portfolio theory, we derive optimal investment strategies under various market conditions, addressing challenges related to leverage constraints and dynamic asset selection.

In the third part, we introduce a return-based risk measure designed to better capture risks in settings where multiplicative interactions play a crucial role. Unlike traditional measures such as Value-at-Risk (VaR) and Expected Shortfall (ES), which focus on absolute losses, our measure emphasizes proportional risk assessment. We derive its Euler allocation form and demonstrate its applications in capital allocation, portfolio optimization, and cooperative risk-sharing frameworks.

The theoretical findings presented in this dissertation are supported by numerical experiments, demonstrating their practical applicability in both actuarial and financial contexts. Our results contribute to the broader understanding of risk allocation, investment optimization, and capital distribution in uncertain environments.