The main goal of this thesis is to combine all of these concepts into a unified framework, evaluate its feasibility and performance, and to perform an analysis of the most common pitfalls and practical considerations. This unified framework uses both the MVPT and MA approach for asset allocation, but at the same time allows for dynamic and fat-tailed distributions of asset returns. It is implemented in approximately 1200 lines of efficient MATLAB code, which is publicly available at https://github.com/FelixAndresen/RSMentalAccounting. The application is programmed in a way that it is readily expandable to a larger number of assets and other investment approaches. The framework is also independent on the choice of assets, which is why the choice of assets for the illustration of the thesis findings was based on the availability of data.
The thesis is structured as follows. Chapter 2 reviews the current literature and theoretical concepts. More specifically, Chapter 2.1 introduces the Mental Accounting framework and the connections and differences to Markowitz’s Mean Variance Portfolio Theory. In Chapter 2.2 the most important concepts of dynamic investment management and stochastic programming are introduced. Chapter 2.3 discusses the regime switching models used to generate scenarios for the stochastic programming approach and the important topic of model selection. Chapter 2.4 gives an overview of Gaussian Mixture Models which present a tool to create the expected distribution used to optimize the asset allocation. In Chapter 3 all the pieces from the theoretical parts are brought together to formulate the dynamic programming models and the hypotheses to be tested in the thesis. The market data used to carry out the analysis is discussed in Chapter 4, as well as some necessary methodology on how to calculate, aggregate and interpret asset returns.
Chapter 5 presents exemplary and illustrative results, and discusses the strengths and weaknesses of the MVPT and MA investment approaches. The thesis closes with a summary and conclusion in Chapter 6.
Table of Contents
1 Introduction
2 Literature Review
2.1 The Mental Accounting Framework
2.1.1 Markowitz’s MVPT vs. MA framework
2.1.2 Mean-Variance Optimization of Mental Accounts
2.2 Dynamic Investment Management
2.2.1 Components of a Stochastic DP model
2.2.2 Abstraction of the stochastic programming approach
2.2.3 Scenario Generation
2.3 Regime Switching and Markov Chain Models
2.3.1 Components of a Hidden Markov Model
2.3.2 The Three Basic Problems for HMMs
2.3.3 Model Selection
2.4 Gaussian Mixture Models
2.4.1 Definition of Gaussian Mixtures
2.4.2 Gaussian Mixture Models in Stochastic Programming
2.4.3 Moments of a Gaussian Mixture Distribution
3 Mental Accounting with Regime Switching
3.1 Stochastic Programming Problem Formulation
3.1.1 Objective Function
3.1.2 Constraints
3.1.3 Definition of Stages
3.2 Scenario Generation
3.3 Decision Policy
3.4 Backtesting
3.5 Hypotheses
4 Market Data and Methodology
4.1 Market Data
4.2 Methodology
4.3 Missing Data
5 Results
5.1 HMM Calibration
5.1.1 Model Selection
5.1.2 Analysis of the selected model
5.2 Scenario Generation
5.3 Asset Allocation
5.4 Backtesting
6 Summary and Conclusion
Research Objectives & Key Themes
The primary objective of this thesis is to create a unified investment framework that integrates the Mental Accounting (MA) approach with dynamic, fat-tailed asset return distributions generated by Regime Switching models. The core research question addresses whether this dynamic programming approach can improve asset allocation and risk management compared to traditional static methods.
- Integration of Mental Accounting and Mean-Variance Portfolio Theory (MVPT) in a multi-period setting.
- Application of Hidden Markov Models (HMM) and Gaussian Mixture Models for time-varying scenario generation.
- Implementation of a stochastic programming framework to solve sequential decision problems.
- Comparative performance evaluation and backtesting of different investment strategies across market regimes.
Extract from the Book
2.1.1 Markowitz’s MVPT vs. MA framework
Das et al. established that "portfolio optimization over two moment distributions where wealth is maximized subject to reaching a threshold level of return with a given level of probability (i.e., the MA problem) is mathematically equivalent to MVPT optimization" [2]. These two problems are outlined below:
Mean Variance Portfolio Theory
MVPT minimizes the variance of a portfolio σ^2 = min_w w^TΣw, subject to achieving a specified level of expected return E = w^Tμ and being fully invested (i.e. w^T1 = 1), where w ∈ Rn is a vector of portfolio weights for n assets, Σ ∈ Rn×n is the covariance matrix of returns of the assets and μ ∈ Rn is the vector of n expected returns. Further, 1 denotes a vector consisting of ones in each component, 1 = (1, 1, . . . , 1)T ∈ Rn. Carrying out the optimization for varying E produces a set of all mean-variance efficient portfolios {w(E)}, which traces out an efficient frontier. This is displayed in Figure 2.2 using the numerical example presented in Ref. [2].
Summary of Chapters
1 Introduction: Provides an overview of portfolio theory, the transition from static MVPT to behavioral finance, and the motivation for using stochastic programming with Regime Switching models.
2 Literature Review: Details the theoretical foundations of the Mental Accounting framework, dynamic investment management, HMMs, and Gaussian Mixture Models used for return distributions.
3 Mental Accounting with Regime Switching: Describes the methodology for formulating the dynamic optimization problem, scenario generation, and the setup for backtesting hypotheses.
4 Market Data and Methodology: Outlines the data sources, return calculation techniques, and handling of missing observations for the financial indices analyzed.
5 Results: Presents findings from HMM calibration, the performance of the integrated investment model in backtesting scenarios, and an analysis of observed pitfalls.
6 Summary and Conclusion: Evaluates the framework's performance, identifies a critical fallacy within the MA framework regarding risk aversion and leverage, and suggests avenues for future research.
Keywords
Mental Accounting, Regime Switching, Hidden Markov Models, Stochastic Programming, Asset Allocation, Mean-Variance Portfolio Theory, Dynamic Programming, Gaussian Mixture Models, Portfolio Optimization, Backtesting, Risk Management, Behavioral Finance, Efficient Frontier, Scenario Generation, Return Distribution
Frequently Asked Questions
What is the core focus of this thesis?
The thesis explores a dynamic version of the Mental Accounting (MA) framework, extending traditional static portfolio optimization through the use of stochastic programming and Regime Switching models.
What are the primary investment frameworks discussed?
The work compares the traditional Mean-Variance Portfolio Theory (MVPT) with the behavioral Mental Accounting (MA) framework.
How is the asset return uncertainty modeled?
Uncertainty is captured using Hidden Markov Models (HMM) and Gaussian Mixture Models (GMM) to generate time-varying return scenarios rather than relying on stationary assumptions.
What mathematical technique is used to solve the multi-stage optimization?
The thesis employs stochastic programming, specifically leveraging Bellman’s Principle of Optimality to handle sequential investment decisions over a finite horizon.
What is the main finding regarding the Mental Accounting approach?
A significant pitfall was identified: when expected return distributions become strongly negative, the MA algorithm lowers the risk aversion coefficient to maintain the target threshold, which leads to excessive, dangerous levels of leverage.
Which indices are used to test the models?
The model is tested using data from the DAX 30 and the S&P 500, supplemented by the overnight London Interbank Offered Rate (ON Libor) as a risk-free asset.
Does the two-state HMM model sufficiently capture the market data?
Yes, analysis confirms that a two-state HMM is sufficient for modeling the return distributions of the selected indices across the considered time windows, including the 2008 financial crisis.
Why are standard questionnaires for risk aversion often considered unsatisfactory?
Financial literature suggests that individual investors struggle to specify a single risk-aversion parameter, making the MA framework's use of goal-based thresholds more intuitive.
