Probability Distribution Of Returns Research Articles

Multiportfolio Optimization: A Fairness-Aware Target-Oriented Model

Problem definition: We consider a multiportfolio optimization problem in which nonlinear market impact costs result in a strong dependency of one account’s performance on the trading activities of the other accounts. Methodology/results: We develop a novel target-oriented model that jointly optimizes the rebalancing trades and the split of market impact costs. The key advantages of our proposed model include the consideration of clients’ targets on investment returns and the incorporation of distributional uncertainty. The former helps fund managers to circumvent the difficulty in identifying clients’ utility functions or risk parameters, whereas the latter addresses a practical challenge that the probability distribution of risky asset returns cannot be fully observed. Specifically, to evaluate the quality of multiple portfolios’ investment payoffs in achieving targets, we propose a new class of performance measures, called fairness-aware multiparticipant satisficing (FMS) criteria. These criteria can be extended to encompass distributional uncertainty and have the salient feature of addressing the fairness issue with the collective satisficing level as determined by the least satisfied participant. We find that, structurally, the FMS criteria have a dual connection with a set of risk measures. For multiportfolio optimization, we consider the FMS criterion with conditional value-at-risk being the underlying risk measure to further account for the magnitude of shortfalls against targets. The resulting problem, although nonconvex, can be solved efficiently by solving an equivalent converging sequence of tractable subproblems. Managerial implications: For the multiportfolio optimization problem, the numerical study shows that our approach outperforms utility-based models in achieving targets and in out-of-sample performance. More generally, the proposed FMS criteria provide a new decision framework for operational problems in which the decision makers are target-oriented rather than being utility maximizers and issues of fairness and ambiguity should be considered. Funding: This work was supported by the Hong Kong Research Grants Council [Grants 14210821, 16204521], Leading Talent Program of Guangdong Province [Grant 2016LJ06D703], and the National Natural Science Foundation of China [Grants 71971187, 72171156, 72231002, 72331009]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2021.0363 .

Skewness-seeking behavior and financial investments

AbstractRecent theoretical and empirical advancements highlight the pivotal role played by higher-order moments, such as skewness, in shaping financial decision-making. Nevertheless, contemporary experimental research predominantly relies on limited-outcome lotteries, an oversimplified representation distant from real-world investment dynamics. To bridge this research gap, we conducted a rigorously pre-registered experiment. Our study delves into individuals’ preferences for investment opportunities, examining the influence of skewness of continuous probability distributions of returns. We document an inclination towards positively skewed outcome distributions. Furthermore, we uncovered a substitution effect between risk appetite and the sign of skewness. Finally, we unveiled a robust positive correlation between skewness-seeking behavior and a propensity for speculative behavior. Simultaneously, a distinct negative correlation surfaced between skewness-seeking behavior and the perceived risk associated with positive skewness.

Modelization and Calibration of the Power-Law Distribution in Stock Market by Maximization of Varma Entropy

Proper description of the return distribution is crucial for investment practitioners. The underestimation of the tail risk may lead to severe consequences, even for assets with moderate fluctuations. However, many empirical studies found that the distribution tails of many financial assets drop off more slowly than the Gaussian distributions. Therefore, we intend to model and calibrate the heavy tails observed in financial fluctuations in this study. By maximizing the Varma entropy with value-at-risk and expected shortfall constraints, we obtain the probability distribution of stock return and observe that the tail of stock return distribution is a power law. Since the variance of the real stock portfolio may be a random variable, using the mean-VaR-ES constraints to maximize the Varma entropy effectively avoids the problem of assuming that the variance is a constant value under the traditional mean-variance constraint. Therefore, the deduced theoretical model would be more consistent with the real market. Using high-frequency data from China’s stock markets, we calibrate our theoretical model and give the concrete form of probability density distribution p(x) for different time intervals. The calibration results show that the tail of the stock return distribution is a power law with most of the power-law orders between −2 and −7. We prove the robustness of our results by calibrating the Varma entropy for S&P 500 of the USA stock market and different stock market indices in China’s A-share market. Our research’s findings not only offer a theoretical perspective for researchers but also give investing professionals a theoretical foundation on which to base their decisions.

An Empirical Study on the Logarithmic Return of Securities

Objective: Previous studies have shown that the distribution of security returns has the characteristics of peak and fat tail, and it is not realistic to use logarithmic normal distribution to describe security returns. This paper mainly studies the specific form of security returns distribution and conducts empirical research. Methods: We first assume that the security returns follow a logarithmic t-distribution. Then, the maximum likelihood estimation and moment estimation are used to explore the degree of freedom of the distribution. Finally, the goodness of fit test is used to conduct an empirical analysis about Shanghai Securities Composite Index. Results: The specific expression formulas for the estimator of the degree of freedom are obtained. Empirical testing shows that the security returns follow a logarithmic t-distribution. Conclusion: The probability distribution of security returns is unrealistic to assume that the security returns follow the logarithmic normal distribution. It is more practical to use the logarithmic t-distribution to characterize the distribution of security returns. Our results provide theoretical assistance for exploring the pricing of financial derivatives.

Risk-Sensitive Policy with Distributional Reinforcement Learning

Classical reinforcement learning (RL) techniques are generally concerned with the design of decision-making policies driven by the maximisation of the expected outcome. Nevertheless, this approach does not take into consideration the potential risk associated with the actions taken, which may be critical in certain applications. To address that issue, the present research work introduces a novel methodology based on distributional RL to derive sequential decision-making policies that are sensitive to the risk, the latter being modelled by the tail of the return probability distribution. The core idea is to replace the Q function generally standing at the core of learning schemes in RL by another function, taking into account both the expected return and the risk. Named the risk-based utility function U, it can be extracted from the random return distribution Z naturally learnt by any distributional RL algorithm. This enables the spanning of the complete potential trade-off between risk minimisation and expected return maximisation, in contrast to fully risk-averse methodologies. Fundamentally, this research yields a truly practical and accessible solution for learning risk-sensitive policies with minimal modification to the distributional RL algorithm, with an emphasis on the interpretability of the resulting decision-making process.

Cardinality-constrained distributionally robust portfolio optimization

This paper studies a distributionally robust portfolio optimization model with a cardinality constraint for limiting the number of invested assets. We formulate this model as a mixed-integer semidefinite optimization (MISDO) problem by means of the moment-based ambiguity set of probability distributions of asset returns. To exactly solve large-scale problems, we propose a specialized cutting-plane algorithm that is based on bilevel optimization reformulation. We prove the finite convergence of the algorithm. We also apply a matrix completion technique to lower-level SDO problems to make their problem sizes much smaller. Numerical experiments demonstrate that our cutting-plane algorithm is significantly faster than the state-of-the-art MISDO solver SCIP-SDP. We also show that our portfolio optimization model can achieve good investment performance compared with the conventional robust optimization model based on the ellipsoidal uncertainty set.

Recovering subjective probability distributions

AbstractThis study proposes a direct estimation method for recovering subjective probability distributions from option prices. We find that the subjective cumulative distribution function and subjective statistics are represented as static portfolios composed of plain vanilla options. The portfolio weights of the options are determined by the levels of the reciprocal of the pricing kernel and their differentiation. Using options data, we investigate the historical behavior of subjective probability distributions of returns on the S&P 500 index. We also analyze subjective equity risk premiums and subjective variance risk premiums, which can be regarded as ex ante risk premiums.

Rentabilidad financiero-fiscal de las rentas de jubilación temporales o vitalicias aseguradas y contratadas a prima única

El objetivo de este trabajo es determinar la rentabilidad financiero-fiscal que puede obtener un asegurado al contratar una renta de supervivencia diferida y vitalicia. Se obtienen las expresiones analíticas de la distribución de probabilidad de la rentabilidad real de la operación, de la rentabilidad esperada y de los indicadores de riesgo que proponemos utilizar junto a esta rentabilidad. Se desarrolla una aplicación numérica de un caso particular de forma que el estudio realizado muestra la aplicabilidad de los resultados obtenidos permitiendo al asegurado tomar sus decisiones en ambiente aleatorio con la máxima información.

Prozesu estokastikoak fisikan eta ekonomian: higidura browndarretik finantza-merkatuen ereduetara

Azken hamarkadetan, ikerkuntza-talde anitzek, fisikan garatutako metodoak erabili dituzte merkatuen portaera aztertzeko. Izan ere, merkatuetako ezaugarri estatistiko sinple batzuen unibertsaltasuna nabarmena da: eskala-menpekotasunik gabeko fenomenoak antzeman daitezke (itzulkinen probabilitate-banaketetan, adibidez), fisikako hainbat esparrutan agertzen diren gisan. Hala, fisikarientzat, ekonomia ikertzea, ongi definitutako sistema konplexu bati buruzko datu ugari aztertzea bilakatu da. Aztertze-metodologia garrantzitsuenetariko bat, adibidez, prozesu estokastikoak erabiltzean datza. Izan ere, fisikan, ezaugarri unibertsalak dituzten sistema konplexu batzuen kasuan, eredu deterministak ez dira baliagarriak. Bi esparruen arteko paralelismotik, hortaz, ikuspegi estokastikoa ekonomiaren esparruan sartzea, eta merkatu estokastiko bat ereduztatzea zilegi litzateke. Hau da, kontzeptu matematiko beraz baliatuz, posible da eredu fisiko eta ekonomiko bat independenteki sortzea.

The Effects and Value of Financial Information Under a Power Utility CAPM

Using a distribution-free CAPM derived under power utility, we examine the parameters of the payoff distribution that have greatest effect on the market equilibrium price, cost of capital and investor welfare. Results are necessarily all numerical, and are obtained by simulating from lognormal and normal payoff distributions. Those distributions are calibrated to approximate the empirical probability distribution of returns and payoffs on the S&P500. The overriding result is that the benefits to investors of better information arise most strongly via better estimates of the mean payoff. Estimation risk surrounding the payoff risk or variance prove surprisingly much less important. That result under our CRRA CAPM replicates similar results in finance under the conventional CARA mean-variance CAPM. We also confirm the inherent disconnection under equilibrium between the market cost of capital and investor welfare.

Expected Surplus Growth Compared with Mean–Variance Optimization

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.

RETRACTED: Big data application in exchange rate financial prediction platform based on FPGA and human-computer interaction

A prediction is a statement about the financial market. The financial market prediction may lack sufficient reasons or any good stock market analysis. The financial prediction may be correct or inaccurate on any given occasion, or average, Model-based or information. The financial prediction is made by various methods, including hundreds of economic evaluation and test systems, which are Observable in the gate array. The Digital signal processing system and IoT (Internet of thing) for exchange rate finical perdition platform in the previous method. The previous method is difficulty in lower investment to reduce inflation and false value setting. The proposed method is based on Programmable Gate and learning for finical predication. A critical challenge of financial forecasting issues, along with opportunities that arise from the unique characteristics of financial data, signal-to-noise ratios, persistent predictors, predictive instability and environmental predictability resulting from competitive pressure and investors learning. The machine approaches for predicting the mean, variance, and probability distribution of asset returns. Programmable Gate Array covers how to evaluate financial forecasts, which leads to data mining concerns, taking into account the possibility that numerous forecast models are being considered.

Is Violation of the Random Walk Assumption an Exception or a Rule in Capital Markets?

Both the efficient market hypothesis and modern portfolio theory rest on the assumptions of the Gaussian probability distribution and independence of consecutive returns. This paper provides a brief excursion into the history of capital market research. A measure of long-range dependence (Hurst exponent) was applied to daily returns of selected stock indices and individual firms. The Hurst exponent was estimated using rescaled range analysis. The estimates are based on an unusually large sample of empirical-time series from capital markets. This method distinguishes whether the data-generating process follows random walk or exhibits antipersistent or persistent behavior. Both the efficient market hypothesis and modern portfolio theory assume that the data-generating process has no memory, i.e. follows Brownian motion. The random walk process is characterized by a Hurst exponent value of 0.5. Values greater than 0.5 and less than 1 indicate a persistence of local trends. Values between 0 and 0.5 indicate a process that reverts to the mean more often than a random process (mean-reverting process). The results indicated that the series of daily returns exhibit predominantly persistent or antipersistent behavior. Therefore, Brownian motion cannot be perceived as the norm for describing stock market behavior. These findings challenge the assumption of a random walk in stock prices, valuation models and assessment of risk.

Improving the Robustness of Trading Strategy Backtesting with Boltzmann Machines and Generative Adversarial Networks

This article explores the use of machine learning models to build a market generator. The underlying idea is to simulate artificial multi-dimensional financial time series, whose statistical properties are the same as those observed in the financial markets. In particular, these synthetic data must preserve the probability distribution of asset returns, the stochastic dependence between the different assets and the autocorrelation across time. The article proposes then a new approach for estimating the probability distribution of backtest statistics. The final objective is to develop a framework for improving the risk management of quantitative investment strategies, in particular in the space of smart beta, factor investing and alternative risk premia.

Better portfolios with higher moments

A toolset beyond mean–variance portfolio optimization is appropriate for those instances where higher return moments might need to be taken into account, either for individual decisions or for pricing studies. Maximizing expected log surplus utility is superior for compounding returns in excess of financial obligations. Here, it is matched with a more flexible scenario representation of the investor’s joint probability distribution of returns and with an agnostic optimization engine. We show simple examples based on extrapolating historical stock and bond returns and then extended using hypothetical option prices. We clarify how Black–Scholes implied volatility anomalies can arise in a portfolio context.

EFFECTS OF APPLYING DIFFERENT RISK MEASURES ON THE OPTIMAL PORTFOLIO SELECTION: THE CASE OF THE BELGRADE STOCK EXCHANGE

Despite its wide use in practice, Modern Portfolio Theory and Markowitz’s approach to optimization, which is based on quadratic programming and the first two moments of the probability distribution of returns as major parameters, was faced with criticism. Therefore, standard Mean-Variance approach had been modified by applying more appropriate risk measures in optimization algorithm. The aim of this paper is to indicate efficiency of these models as well as justification of their usage in managing stocks portfolio on the Belgrade Stock Exchange.

Multi-portfolio Optimization: A Fairness-aware Target-oriented Model

We consider a multi-portfolio optimization problem where nonlinear market impact costs result in a strong dependency of one account's performance on the trading activities of other accounts. We develop a novel target-oriented model that jointly optimizes the rebalancing trades and split of market impact costs. The key advantages of our proposed model include the consideration of clients' targets on investment returns and the incorporation of distributional uncertainty. The former helps the fund manager circumvent the difficulty in identifying clients' utility functions or risk parameters, while the latter addresses a practical challenge that the probability distributions of risky asset returns cannot be fully observed. Specifically, to evaluate multiple portfolios' investment payoffs achieving their targets, we propose a new type of performance measure, called the fairness-aware multi-participant satisficing (FMS) criterion. This criterion can be extended to encompass the distributional uncertainty and has the salient feature of addressing the fairness issue with the collective satisfaction level as determined by the least satisfied participant. We find that, structurally, the FMS criterion has a dual connection with a set of risk measures. For multi-portfolio optimization, we consider the FMS criterion with conditional value-at-risk, a popular risk proxy in financial studies, being the underlying risk measure to further account for the magnitude of shortfalls against targets. The resulting problem, although non-convex, can be solved efficiently by solving an equivalent converging sequence of tractable subproblems. The numerical study shows that our approach outperforms utility-based models in achieving targets and is more robust in out-of-sample performance.

Evaluation of activity meters for estrus detection: A stochastic bioeconomic modeling approach

Although estrus detection makes a relevant contribution to the reproductive performance of dairy cattle, studies on the economic evaluation of automatic estrus detection systems are rare. The objective of the present study is to provide an economic evaluation of activity meters used for estrus detection. The effect of different estrus detection rates on gross margins was modeled with SimHerd (SimHerd A/S, Viborg, Denmark). The analysis considers all costs associated with the investment in activity meters. The economic evaluation was carried out through simulation of Simmental herds with yearly milk yields of 7,000 or 9,000 kg and Holstein Friesian herds with yearly milk yields of 9,000 or 11,000 kg, each with herd sizes of 70 or 210 cows. Furthermore, we distinguished between 2 investment scenarios. In scenario 1, only cows are equipped with activity meters for estrus detection, whereas scenario 2 assumes that cows and heifers are equipped with activity meters. Because existing empirical information for some variables shows significant variability (estrus detection rates, time for estrus detection), they were modeled with distributions using the Monte Carlo method in @RISK (Palisade Corporation, Ithaca, NY), allowing us to model a probability distribution of net returns (NR) of investment in activity meters for estrus detection. The simulation results show that the average NR of investment in activity meters for estrus detection over all scenarios ranges from +€7 to +€40 per cow per year for the Simmental breed, and from +€19 to +€46 per cow per year for the Holstein Friesian breed. Generally, the NR depends on the milk production level assumed. For the Simmental breed, depending on the scenario, the simulation results show a 54 to 200% larger average NR of investment in activity meters for estrus detection with a milk yield of 9,000 kg/yr compared with 7,000 kg/yr. For the Holstein Friesian breed, the effect of the modeled milk yield on the NR is much less pronounced. Average NR of investment in activity meters are greater for larger herd sizes because of cost degression effects. An additional equipping of heifers has, on average, a positive effect on the economics of activity meters for estrus detection because of the resulting reduction in the age at first calving. Considering all scenarios, the probability of a positive NR of investment in activity meters ranges between 74 and 98% for the Simmental breed and between 85 and 99% for the Holstein Friesian breed.

Modelling demand for lotto using a novel method of correcting for endogeneity

Modelling lottery sales as a function of the mean, standard deviation and skewness of the probability distribution of returns potentially gives insights into how the design of a game could be modified to maximise net revenue. But use of OLS is problematic because the level of sales itself affects values of the moments (and insufficient instruments are available for IV regression). We draw on the concept of a rational expectations equilibrium, developing a new regression model which corrects for endogeneity where the causal impact of the dependent variable on the right-hand side variables is deterministic. We apply the model to data on lotto sales from Spain. Using the Spanish data, we show that results provide more reliable guidance to lottery agencies because accounting for endogeneity leads to significantly different results from OLS and these results have superior performance in out-of-sample forecasting of sales. More generally, results prove consistent with the Friedman-Savage explanation of why people buy lottery tickets and with evidence from racetrack data that ‘bettors love skewness’.

Forecasting dynamic return distributions based on ordered binary choice

We present a simple approach to the forecasting of conditional probability distributions of asset returns. We work with a parsimonious specification of ordered binary choice regressions that imposes a connection on sign predictability across different quantiles. The model forecasts the future conditional probability distributions of returns quite precisely when using a past indicator and a past volatility proxy as predictors. The direct benefits of the model are revealed in an empirical application to the 29 most liquid U.S. stocks. The forecast probability distribution is translated to significant economic gains in a simple trading strategy. Our approach can also be useful in many other applications in which conditional distribution forecasts are desired.