Anomalies in Net Present Value, Returns and Polynomials, and Regret Theory in Decision-Making

Polynomials are a well established branch of mathematics with wide applications in finance, physics, operations research, healthcare informatics, engineering, economics and other subject areas. However, most if not all the recent research on polynomials and a large portion of associated research in related fields are based on highly inaccurate theorems and methods.

Investment decision analysis using NPV, MIRR/IRR or the mean-variance model have become the primary methods of investment evaluation. This chapter contributes to the existing literature by: (1) explaining behavioral and psychological biases inherent in financing decisions that contradict the NPV-MIRR model; (2) explaining existing and new spatio-temporal framing effects inherent in the NPV-MIRR model; (3) critiquing Kruger et al. (2015), Ang and Liu (2004), Schwab and Lusztig (1969), Merrett (1965b), and articles by Schaelemann et al. (2012), Merrett (1965a), Barcelona (2015), and Iturbe-Ormaetxe et al. (2010); (4) surveying the relevant literature on Regret Theory, which explains how it can serve as an alternative to the NPV-MIRR model for decision-making; (5) introducing framing effects inherent in the mean-variance model; (6) explaining why the NPV-IRR model (and related approaches, such as APV and NFV) do not account for Real Options, Regret, or Rejoice in decision-making. Regret Theory can help avoid the often distorting framing effects inherent in these decision models, although one current issue is that there is no generally accepted set of Regret-based decision models (unlike NPV/MIRR/APV/NFV/EVA and related models).

Although the CML (capital market line), the intertemporal capital asset pricing model (CAPM), the CAPM/SML (security market line) and the Intertemporal Arbitrage Pricing Theory (IAPT) are widely used in portfolio management, valuation, and capital markets’ financing, these theories are inaccurate and can adversely affect risk management and portfolio management processes. This chapter introduces several empirically testable financial theories that provide insights, and can be calibrated to real data and used to solve problems. It contributes to the literature by: (1) explaining the conditions under which ICAPM/CAPM, IAPT, and CML may be accurate, and why such conditions are not feasible; and explaining why the existence of incomplete markets and dynamic unaggregated markets render CML, IAPT, and ICAPM inaccurate; (2) explaining why the consumption-savings-investment-production framework is insufficient for asset pricing and analyses of changes in risk and asset values; and introducing a unified approach to asset pricing that considers six factors and the conditions under which this approach will work; (3) explaining why leisure, taxes, and housing are equally as important as consumption and investment in asset pricing; (4) introducing the Marginal Rate of Intertemporal Joint Substitution (MRIJS) to consumption, taxes, investment, leisure, intangibles, and housing—this model incorporates Regret Theory and captures features of reality that do not fit well into standard asset pricing models; this framework can support specific or very general finance theories and very complicated models; (5) showing why the elasticity of intertemporal substitution (EIS) is inaccurate and insufficient for asset pricing and analyses of investor preferences.

This chapter contributes to the existing literature by proving that the Descartes’ Sign Rule (as interpreted by most academicians—such as Oehmke (2000) and Osborne (2010))—and the Fourier-Boudan Theorem are wrong. These issues are applicable in non-linear analysis, evolutionary computation, and pattern-analysis—given the discussions in Yannacopoulos et al. (1996); Campos-Canton et al. (2015); Zheng et al. (2010); and Boyer and Goh (2007).

This chapter contributes to the existing literature by: (1) proving that the Fundamental Theorem Of Algebra (FTA) and the Binomial Theorem are wrong; (2) explaining how traditional root calculation in algebra may lead to inaccurate conclusions and introducing an alternative method for verifying real and complex roots of a polynomial; (3) solving a six-degree polynomial equation and a nine-degree polynomial equation, by introducing new classes of invariants (MN-2 invariants) and homomorphisms. Oehmke (2000), Osborne (2010), Burrus (2004), Sitton et al. (2003), and Lei et al. (1996), had concluded that such higher-order polynomials were impossible to solve. These issues are applicable in non-linear analysis, evolutionary computation, and pattern analysis—given the discussions in Yannacopoulos et al. (1996), Campos-Canton et al. (2015), Zheng et al. (2010), and Boyer and Goh (2007).

This chapter: (1) surveys the literature on anomalies in the NPV/MIRR/IRR model and related methods and shows the varied treatment of discount rates in analyses of NPV/MIRR/IRR and the recognized and unrecognized fallacy of treating discount rates as a key variable; (2) explains the conditions under which negative discount rates may be feasible and rational; (3) explains why the discount rate is often the perceived critical variable in the NPV/MIRR/IRR model and related approaches; (4) explains why the concept of plain interest rates and forward rates (unadjusted for behavioral biases, taxes, and transaction costs) may not be entirely correct.

Investment Decision Analysis using NPV has become the primary method of investment evaluation. This chapter contributes to the existing literature by: (1) simulating and proving new biases and errors inherent in the NPV-MIRR model; (2) developing the necessary and sufficient conditions for monotonic NPV (“well behaved” NPV); (3) developing the necessary and sufficient conditions for the anomalous behavior of NPV; (4) explaining the conditions under which discount rates can be negative (less than zero); (5) developing the conditions under which negative discount rates are justified; (6) proving that the power rule and the inverse function rule in differential calculus, are both wrong. NPV, modified-IRR (MIRR) and related approaches are deeply flawed and are very sensitive to the time horizon, the signs of the periodic cash flows, and discount rates that exceed 100% or are below −100%. The NPV-MIRR model does not accommodate the differences between compounded interest rates and simple interest rates and does not account for Real Options, Regret, or Rejoice in decision-making.

Investment returns are a core element of decisions in various end-uses. Investment returns are calculated in different ways and over different horizons and, regardless of the method of computation, they often contain substantial biases, which distort results in the end-uses. These biases tend to have adverse multiplier effects because investment returns are the foundation or core elements in various statistical analyses such as: (1) developing distributions of expected returns; (2) time series analysis; (3) developing options pricing models; (4) estimating various types of risk; (5) developing pricing models for insurance; (6) for the evaluation, and capital-budgeting, of large billion-dollar projects. The two most popular methods for calculating long-term returns are simple returns and compounded returns (sometimes expressed as natural logs), which yield vastly different results that have significant computational consequences. Thus, more awareness of the effects of compounding is necessary. This chapter contributes to the existing literature by: (1) explaining the biases and effects inherent in the calculation of compounded returns (distinct from human biases that can affect returns, and vice versa), which, in turn, implies that various investment approaches (such as minimum variance investing; geometric mean maximization; etc) are wrong and inaccurate; (2) some of the biases introduced herein are new types of evolutionary homomorphisms; (3) showing how biases in returns, and the calculation methods, can affect the analysis of pattern formation, chaos and adaptive systems—the discussions in Preis et al. (2012), Kenett et al. (2012), Preis (2011), Podobnik et al. (2009), Fenn, et al. (2011), Kriener et al. (2014), Hsieh (1993), Menna et al. (2002), and Hommes (2002) omitted such effects; and (4) explaining some of the effects of market volatility on compounded returns.

In financial markets in most developed and developing countries, investors’ preferences diverge substantially. Furthermore, all existing asset pricing models are inaccurate because the underlying assumptions are not realistic. The focus on consumption and price as the definition of investors’ preferences and constraints is misleading and the investors’ decision problem is much broader. The elasticity of intertemporal substitution is also very inaccurate, especially where markets are incomplete and investors’ preferences are both dynamic and multifaceted. Given the problems and inaccuracies inherent in the CSIP dichotomy, the UIWD and MRIJS are more likely to result in better policy decisions. The MRIJS is distribution free, does not require use of any specific utility functions, implicitly accounts for risk (by multifaceted wealth allocation) and provides a more unified and accurate indication/analysis of the average investor’s wealth allocation decisions. Obviously, these have important ramifications for asset management and capital budgeting.