# Modular portfolio theory : a general framework with risk and utility measures as well as trading strategies on multi-period markets

• Modulare Portfoliotheorie: Ein allgemeines Rahmenwerk mit Risiko- und Nutzen-Maßen sowie Handelsstrategien auf Mehrperioden-Märkten

Platen, Andreas; Maier-Paape, Stanislaus (Thesis advisor); Cramer, Erhard (Thesis advisor); Zhu, Qiji Jim (Thesis advisor)

Aachen (2018, 2019)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2018

Abstract

The following four modular building blocks are crucial in the context of portfolio theory: (a) the market model, (b) the "trading strategy" of the investor, (c) the risk and utility function and (d) the optimization problem. The setting of the so called "modern portfolio theory" by [Markowitz, John Wiley & Sons, Hoboken, NJ, 1959] and the capital asset pricing model (CAPM) by [Sharpe, The Journal of Finance, 19(3):425-442, 1964] consist of a one-period market model for (a) with the standard deviation as risk and the expected return as utility for (c). The trading strategies/portfolios are then just elements in the vector space ℝ^n whose entries represent the corresponding asset in the portfolio, i.e. there is nothing to do in block (b). The optimization problem, in general, is parameter depend, because of a trade-off between risk and utility. It is known in the modern portfolio theory, that the risk-utility values of the solutions, i.e. of the so-called efficient portfolios, for different parameters are on the boundary of a convex set within the two-dimensional risk-utility space. In the CAPM they even lie on a straight line. A generalization of such results is studied in [Rockafellar et al., Journal of Banking & Finance, 30(2):743-778, 2006] and [Maier-Paape and Zhu, Risks, 6(2):53, 2018]. In both literatures efficient portfolios are studied similar as in the modern portfolio theory and CAPM. In [Rockafellar et al., 2006] they use the same linear utility function but the risk is allowed to be a so-called generalized deviation measure, which e.g. requires convexity and also positive homogeneity. In [Maier-Paape and Zhu, 2018] a similar situation is discussed but the risk is defined with less assumptions, e.g. positive homogeneity is not required anymore. In addition, more general (concave) utility functions are allowed, which, however, still must be of a special form. Therein, block (a) again as one-period market and blocks (c) and (d) are studied. Again, the portfolios/trading strategies are just vectors in ℝ^n because a one-period model allows no trading strategies. This theory is now extended by the subdivision into the four modular building blocks (a) to (d) for general multi-period market models. The portfolios then are no longer represented by vectors in ℝ^n but can be complex time dependent trading strategies for investing the wealth. Examples are the buy and hold strategy or a strategy which reallocates the invested money after each day to ensure constant weights. Reasonable properties of (a) and (b), which are required for (c) and (d), are studied as well. Furthermore, the (concave) utility now is of general form. With just a few reasonable assumptions on the market model, the trading strategy and the risk and utility functions, the existence and uniqueness of efficient portfolios and many similar results as in the references above are shown here for the modular setup as well. The multi-period model is required e.g. to define risk functions based on the so-called drawdown. The drawdown of an equity curve is the (absolute or relative) difference between the maximum of this curve and its last value. Since this is an important value in praxis, the application of the optimization problem from above is studied also for a drawdown risk measure. However, the drawdown is difficult to evaluate. Therefore, some reasonable properties are derived in more detail for the absolute drawdown in case of an equity curve modeled by a Markov chain. For a general case of a Markov chain it is shown that the limit distribution for the absolute drawdown after N time steps as N goes to infinity exists if and only if the expectation of a win or loss after each time step is positive. For some special cases, the corresponding distribution can even be expressed at least with an implicit formula. In the random walk case, an explicit expression can be given for the distribution after just N time steps as well as in the limit case.