Signal Processing - May 2017 - 16
Utility
Concave Utility Function
A Utility Function That Is
Increasing with Wealth
But Has Diminishing
Marginal Utility
Wealth
FIGURE 1. The expected utility theory assumes that people make choices
according to the expected utility E [U (W )] .
beyond stock price prediction, such as input-output relationships in certain economic systems [10]-[12], are available, and
research questions await answers. The recent surge in high-frequency trading (HFT) practices and related theoretical studies
[13] provides further opportunities for SP researchers to examine market microstructures and high-frequency system response.
Indeed, many SP models and methods share common mathematical grounds with traditional econometric analysis [14] but
present different analytical aspects. Therefore they can provide
new tools for economic system modeling, analysis, and information extraction for massive finance and economic data.
This tutorial article intends to concisely introduce the mainstream foundational concepts and frameworks in finance, economics, and marketing research to SP researchers, with conceptual
elaboration of the relationships between traditional economic research paradigms and SP methodology, and help SP researchers
understand economics and business literature and identify relevant research directions with economic significance.
First, we introduce the risks and the foundational economics
theory. Then, we formulate the fundamental market equilibrium
asset-pricing model in finance within the parameters of modern
portfolio theory. Equipped with a basic understanding of the minimum set of economics theories and principles, we introduce an
economic viewpoint and basic hypotheses within the context of
a free and competitive market-the EMH and competing behavioral economics along with the prospect theory. We elaborate on
how different economics theories factor in the individual person-
the center of any economic system-his or her choices, decision
processes, and emotions. We present a philosophical analysis of
the test on economic models with a data-joint hypothesis test. SP
perspectives are provided across the article to help readers quickly
grasp the differences.
We then move to introduce the fundamental econometrics
models and time-series analysis in comparison to the parallel
tools known in SP. We focus more on the basic concepts that
are rarely present in SP but that are critical in analyzing timeseries data in economics and business applications, such as unit
roots and causality. We then briefly summarize the relationships
16
between SP and econometric models so that SP researchers can
apply their SP knowledge to quickly take advantage of econometrics models. These fundamental economics concepts and methodology that we will introduce have won Nobel Prizes from 1990
to 2013. They are by no means comprehensive, but they encompass a skeleton and basic set of building blocks for data-based
economic studies.
We also provide a few detailed state-of-the-art examples in
reference to applying SP to economics, finance, and marketing studies. Furthermore, we focus on a few illustrative formulations of economic and business problems. In addition, we give
an empirical data analysis example to demonstrate the insight
of economic systems that SP is poised to make significant contributions. Throughout this article, we use sidebars to present
mathematical formulations and examples to further clarify and
illustrate main concepts and ideas.
Risk, risk premium, portfolio optimization,
and capital asset pricing
In this section, we introduce fundamental concepts that serve
as language and building blocks for economics and finance
theory. We begin by introducing the expected utility theory
and risk premium (RP).
Expected utility theory and RP
The expected utility theory (or hypothesis) is a cornerstone for
economics, game theory, and decision theory and pertains to
people's preferences and choice. In economics, a utility function U (w) is defined as a concave function of overall wealth w.
An example is shown in Figure 1. Such a utility function
assumes that 1) utility is increasing with wealth, i.e., U (w) is
monotonically increasing, and 2) wealth has diminishing marginal utility, represented by the concavity of U (w) . Therefore,
dU (w) dw 2 0, and d 2 U (w) dw 2 1 0. The expected utility
theory further assumes that people make rational choices
according to the overall expected utility E 6U (W )@. Here, W
represents a random variable.
Now, assume that a risk asset (e.g., a stock or an investment)
has two possible outcomes of wealth with equal probability in
the future, w 0 and w 1 and w 0 1 w 1 . The expected wealth is
E [W ] = 1 (w 0 + w 1).
2
(1)
The expected utility of this risk asset is
E [U (W )] = 1 ^E [U (w 0)] + E [U (w 1)]h .
2
(2)
Given the concavity of the utility function, we have
E [U (W )] 1 U ^E [W ]h. That is, a rational person is risk averse
and thus would prefer a riskless asset, such as cash, with guaranteed wealth of E [W ], to the risk asset with expected wealth
of the same E [W ]. The certainty equivalent (CE) represents
guaranteed wealth whose utility is equivalent to that of the risk
asset; i.e.,
IEEE Signal Processing Magazine
CE = U -1 ^E [U (W )]h .
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May 2017
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Table of Contents for the Digital Edition of Signal Processing - May 2017
Signal Processing - May 2017 - Cover1
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