Lee, Jinwook. Decision support modeling via multivariate risk measures and stochastic optimization. Retrieved from https://doi.org/doi:10.7282/T3RN365B
DescriptionWhenever we have a decision to make, there is always some risk to take. From a mathematical perspective, risk is manifested by a random variable, and a risk measure simply characterizes the random variable in a more compact form. Risk, in general and in practice, is not be adequately described by a real valued random variable, but rather requires a random vector to capture the dimensions of the problem. To this end, multivariate risk measures are crucial ingredients for decision making processes, and stochastic optimization is a natural and superior skill to find a key to the optimal decision-making. A recent paper by Pr ekopa (2012) presented results in connection with Multivariate Value-at-Risk (MVaR) that has been known for some time under the name of p-quantile or p-Level E cient Point (pLEP) and introduced a new multivariate risk measure, called Multivariate Conditional Value-at-Risk (MCVaR). Lee and Pr ekopa (2013) studied new methods for numerical calculations and mathematical properties of these multivariate risk measures, presented in Chapter 2. Another new multivariate risk measure has been constructed and presented in Chapter 3. This is especially for corporate mergers and acquisition (M&A) transactions, as the limited applicability of a coherent risk measure in the sense of Artzner et al (1999) for M&A transactions is already discussed in Kou et al (2013). A decision making scheme using that risk measure is introduced and surveyed, together with illustrative real-life numerical examples. Insurance companies typically hold their money in bonds to pay out the random liabilities in the same periods. In Chapter 4, such bond portfolio construction problem is presented using various stochastic programming problem formulations. For a financial trading business, "price-bands" can be used as an indicator for successfully buying or short-selling shares of stock. Chapter 5 presents a mathematical model for the novel construction of price-bands using a stochastic programming formulation. Numerical examples using recent US stock market intraday data are presented.