Cumulative prospect theory and mean–variance analysis: a rigorous comparison

Nonlinear decision weights or moment-based preferences? A model competition involving described and experienced skewness

A novel portfolio selection model in a hybrid uncertain environment

Aspiration Levels and Preference for Skewness in Choice Under Risk

Introduction In deriving the Mean-Variance (M-V) rule and the Capital Asset Pricing Model (CAPM), one assumes explicitly or implicitly normal distributions of returns and risk aversion. With these assumptions, it emerges that the variance of the portfolio's return is the measure of the investment's risk. In Chapter 8, we saw that although normality of returns is statistically rejected, the economic loss from assuming normality when the distributions are actually not normal is relatively small. Moreover, in Chapter 4, we saw that even if the distributions of returns are not normal, the M-V rule can be employed as an approximation to expected utility, as long as risk aversion is assumed. Furthermore, it was shown empirically in Chapter 4 that the choices made by the M-V rule are almost fully correlated with the choices made by direct expected utility maximization for a wide set of risk-averse utility function. Considering the relatively small economic loss involved as a result of the normality assumption, in this chapter we assume that the M-V and the CAPM are derived based on this normality assumption, within the expected utility framework, rather than as an approximation to expected utility. However, even in this framework, as we saw in Chapters 9 and 10, the M-V and the CAPM are questionable for the following reasons: By the Prospect Theory (PT), people maximize a value function, which is defined on change of wealth (gains and losses) rather than on total wealth; by expected utility, the M-V rule and the CAPM are defined on total wealth. To derive the M-V rule and the CAPM, risk aversion is needed, whereas by PT, the value function has a risk-seeking segment. The risk- seeking segment of preference is steeper than the risk-aversion segment, implying loss aversion . Adopting the modified version of the PT, called Cumulative Prospect Theory (CPT), people employ cumulative decision weights . Thus, the cumulative distribution of returns is transformed, and in calculating the expected value (utility), the transformed distribution rather than the original distribution is employed. This seems to be a “deathblow” to the M-V rule and the CAPM because even if the original distribution of returns is normal, generally the transformed distribution is not. Moreover, although the normal distribution is symmetric, implying that skewness is equal to zero, the transformed distribution may be skewed, thus relying only on the mean, and variance may be misleading because there is ample evidence that investors like positive skewness and dislike negative skewness. For the preceding reasons, it would seem that the CPT paradigm and the M-V rule plus the CAPM paradigm cannot coexist. Moreover, the fact that Markowitz and Sharpe won the Noble Prize in Economics in 1990 for developing the M-V rule and the CAPM, and Kahneman won the Noble Prize in Economics in 2002 for introducing the PT, emphasizes the seeming contradiction between these two paradigms and makes it even more uncomfortable for researchers. If one adopts PT, it seems the M-V and the CAPM should be abandoned. If one employs the M-V and the CAPM, the important evidence regarding subjects’ choices, as clearly shown in PT experiments, are ignored.

How can investors construct portfolios that help them reach their goals, such as a comfortable retirement for themselves, a bequest for their children, and college education for their grandchildren? Are such portfolios on the mean–variance efficient frontier? How can advisors discuss investment risk with clients whose risk tolerance varies by goal—low for retirement and higher for bequests? These are the questions answered in this article. The authors show that advisors can help clients construct portfolios that take clients to their goals while also leaving them on the mean–variance efficient frontier. <b>TOPICS:</b>Portfolio construction, performance measurement

The skewness of commodity futures returns

The opportunity cost of mean–variance choice under estimation risk

The mean???variance model is widely acknowledged as the foundation of portfolio allocation because it provides a framework for analyzing the trade-off between risk and return for gaining diversification benefits. Despite the well-known shortcomings of the model, it is often the starting point for making asset allocation decisions. In this article, the authors briefly review mean???variance optimization and approaches for resolving its limitations by demonstrating backtest results on asset allocation. Feedback from asset managers is also included to explain how optimization methods are applied in practice.

Cumulative prospect theory (CPT) is a popular approach for modeling human preferences. It is based on probabilistic distortions and generalizes the expected utility theory. We bring the CPT to a stochastic optimization framework and propose algorithms for both estimation and optimization of CPT-value objectives. We propose an empirical distribution function-based scheme to estimate the CPT value, and then, use this scheme in the inner loop of a CPT-value optimization procedure. We propose both gradient based as well as gradient-free CPT-value optimization algorithms that are based on two well-known simulation optimization ideas: simultaneous perturbation stochastic approximation and model-based parameter search, respectively. We provide theoretical convergence guarantees for all the proposed algorithms and also illustrate the potential of CPT-based criteria in a traffic signal control application.

In order to study the effect of different risk measures on the efficient portfolios (frontier) while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios (frontier) based on risk measures such as variance, value at risk (VaR), and expected shortfall (ES) is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance (MV) efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results gener alize relevant conclusions about investment theory, and can better guide investors to make their investment decision.

In this article, the authors revisit a neglected paper written by Rubinstein in which he advocated a generalized logarithmic utility function. The profession has rather ignored his model in favor of the mean–variance approach, for reasons that are well known. The authors note that generalized logarithmic utility allows us to identify absolute risk aversion more directly and intuitively than the approaches adopted in mean–variance portfolio construction. They demonstrate that, for top-level asset allocation in which the conditions required by mean–variance are satisfied and mean–variance gives a good approximation to expected utility, generalized logarithmic utility gives essentially the same answer. In situations not suited to mean–variance, such as the inclusion of derivatives in portfolios or, more generally, when higher moments in return and utility probability distributions are unavoidable, Rubinstein’s model appears superior and seems particularly well suited to multi-asset investment. The authors’ simplified Bayesian model shows out-of-sample advantages for broad US asset allocation for the 2000 through mid-2020 period. <b>TOPICS:</b>Statistical methods, portfolio theory, portfolio construction <b>Key Findings</b> ▪ Maximizing the expected Rubinstein utility increases expected surplus growth for conservative investors and for those who do not rebalance frequently. Markowitz’s risk aversion parameter is replaced without requiring subjective judgement of aversion to return variance. ▪ Bayesian prediction can incorporate both scenario priors and observed return data in a simple grid form. This increases model flexibility and decreases sensitivity to errors in estimates of return probability distribution parameters. ▪ We demonstrate out-of-sample advantages for Rubinstein utility with simplified Bayesian prediction in broad US stock, bond, and cash allocation for the most recent decades. This is based on a prior inferred from earlier history, including the 1929 crash.

Derivation of the Mean-Gini Efficient Portfolio Frontier

A load factor based mean–variance analysis for fuel diversification

Entrepreneurs derive lower risk-adjusted returns than non-entrepreneurs, but are compensated through non-pecuniary benefits. This chapter reports on findings from survey evidence. The main findings are as follows: A key non-pecuniary benefit to entrepreneurs is achieving greater control over their working environment. Doing so leads entrepreneurs to report achieving higher affect and well-being than non-entrepreneurs. Entrepreneurs report that they are more skilled socially than non-entrepreneurs. This might provide a partial explanation for why entrepreneurs have a higher marriage rate and larger families than non-entrepreneurs. Entrepreneurs exhibit greater dispositional optimism than non-entrepreneurs. However, in the study sample, the difference is not statistically significant. In terms of preference for lottery-like outcomes, entrepreneurs find prospects offering high returns with low probability attractive, but regard control of their environment as more important than the preference for positive skewness.

Stochastic Orderings represent a relevant approach in portfolio selection for various reasons. Firstly, Stochastic Ordering are theoretically justified by Expected Utility theory. Typically, investors are classified according to their attitude toward risk. For each class of investors then, it is possible to define stochastic orderings coherent with investors' preference. Secondly, Stochastic Orderings are flexible enough to allow different definitions of efficiency suitable for each category of investors. This Thesis proposes several applications of Stochastic Orderings to portfolio selection problems. In the first chapter, an analysis of the relationship between Second order of Stochastic Dominance efficient set and Mean Variance Efficient Frontier is proposed. Not only the two sets differ under many aspects, but the Global Minimum Variance portfolio and other Mean Variance Efficient portfolios are dominated in the sense of Second order of Stochastic Dominance. Based on this fact, the chapter concludes proposing dominating strategies able to outperform the Global Minimum Variance portfolio. In the second chapter, starting from recent findings in the literature, that address the behavior of investors as non satiable, nor risk averting nor risk seeking, an extension of classic definition of Stochastic Dominance efficiency, linked to behavioral finance is given. In particular, investors' behavior changes according to market conditions. The last part of the chapter presents a methodology, based on estimation function theory, to test for portfolio efficiency with respect a general stochastic ordering. Both the analysis of efficiency for Second order of Stochastic Dominance and behavioral finance, questioned the validity of highly diversified choices. For this reason, this thesis concludes introducing Risk Diversification measures, a new class of functional quantifying the amount of idiosyncratic risk diversified among the assets in a portfolio. The Mean Risk Diversification Efficient Frontier is introduced, along with the concept of Mean Risk Diversification efficiency. The empirical analysis describes the relationship between risk aversion, Risk Diversification and classic diversification, and show how Risk Diversification based strategies perform under periods of financial distress.