Abstract
We show the practical values of nonlinear optimization through the illustration of three important investment theories: modern portfolio theory (Markowitz, 1959), capital market theory (Merton, 1972; Sharpe, 1963), and utility theory (Copeland & Weston, 1988). This method can be implemented using Excel in an operations research (OR) or management science (MS) course, and it is innovative because most of today's OR/MS textbooks (e.g., Stevenson & Ozgur, 2007; Winston & Venkataramanan, 2003) contain limited finance applications. And the applications discussed therein tend to focus more on corporate finance topics, not investments. For those textbooks with investment examples, the discussion is often limited to formulating and solving the portfolio optimization problem (POP) as a quadratic program-no link is made to capital market theory or utility theory. We make this link and demonstrate how these three theories relate to each other. This provides students with a better understanding of investments. Usually, investment theories are explained by mathematical derivations using calculus and linear algebra. We use Excel and its features to help students visualize these theories and thus increase their learning.
We consider an investor who wants to optimize their utility, and this depends on their level of risk aversion. Risk aversion is comprised of two components: ability to take risks and willingness to take risks. The ability to take risk is based on personal factors (e.g., age, income, wealth level, etc.) that are quantifiable whereas willingness to take risk is based on psychological factors. Thus, each investor is unique. This article combines the study of portfolio optimization with that of utility maximization, and demonstrates that the two problems can be reconciled.