Abstract
This is Part III of a series of papers which focus on a general framework for portfolio theory. Here, we extend a general framework for portfolio theory in a one-period financial market as introduced in Part I [Maier-Paape and Zhu, Risks 2018, 6(2), 53] to multi-period markets. This extension is reasonable for applications. More importantly, we take a new approach, the “modular portfolio theory”, which is built from the interaction among four related modules: (a) multi period market model; (b) trading strategies; (c) risk and utility functions (performance criteria); and (d) the optimization problem (efficient frontier and efficient portfolio). An important concept that allows dealing with the more general framework discussed here is a trading strategy generating function. This concept limits the discussion to a special class of manageable trading strategies, which is still wide enough to cover many frequently used trading strategies, for instance “constant weight” (fixed fraction). As application, we discuss the utility function of compounded return and the risk measure of relative log drawdowns.
Highlights
This is Part III of a series of papers which focus on a general framework for portfolio theory.We laid out a general framework for portfolio theory in a one-period financial market for trading-off between reward and risk in Part I (Maier-Paape and Zhu 2018a) and addressed drawdown risk measures in Part II (Maier-Paape and Zhu 2018b)
We recognize the problem of trading-off between higher reward and lower risk using portfolio/trading strategies within four modular blocks: (a) multi-period market model; (b) trading strategies; (c) risk and utility functions; and (d) the optimization problem
Let us focus on Example 2 with the trading strategy generating function vtwr which ensures that the portfolio weights are constant after each time step
Summary
This is Part III of a series of papers which focus on a general framework for portfolio theory. (a) multi-period market model; (b) trading strategies; (c) risk and utility functions; and (d) the optimization problem. Zhu (2018a), in addition, the utility function is of more general form with some reasonable assumptions and the one-period market model is assumed to be defined on a finite probability space In both cases the optimization problem is of the form of Equation (1) as well. Properties on the mean of the logarithm of the relative drawdown are discussed in Part II (Maier-Paape and Zhu 2018b), where a one-period market model is used with a finite probability space and independent and identically distributed returns. Since it only provides the framework for the core theory later on, it might help at first reading to concentrate oneself in Section 2 solely on the definitions
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