Normal Distribution Of Returns Research Articles

Empirical Evidence on the Validity of the Conditional Higher Moment Capm in the Bombay Stock Exchange

Various researchers criticized the assumption of a normal distribution of returns in the traditional Capital Asset Pricing Model (CAPM), leading to the recognition of the non-normal distribution of returns and the mean and variance were insufficient to characterize the distribution of returns completely which led the researchers to look for higher order moments. The present study investigated empirical validity of the conditional higher order moment Capital Asset Pricing Model (CAPM) in the context of the Indian stock market, specifically the Bombay Stock Exchange utilizing the data from sectoral indices for the period from April 2011 to March 2021. To test the four moment CAPM empirically, the specification derived from Bollerslev et al. (1988) has been used in the study. The study's findings reveals that the higher moment i.e. conditional coskewness and cokurtosis hold significant pricing implications and do have an impact on returns in the Indian stock market. The explanatory power of the model showed an increase as compared to the unconditional CAPM as the R-square value obtained was greater than the latter model. The results of the study were found to be mixed and inconclusive. Hence, the conditional higher moment CAPM cannot be validated in the Indian context.

Empirical Evidence on the Validity of the Unconditional Higher Moment CAPM in the Bombay Stock Exchange

The traditional Capital Asset Pricing Model (CAPM) assumed a normal distribution of returns, which was criticized by various researchers who recognized the non-normal distribution of returns in the Sharpe-Lintner CAPM. The introduction of unconditional higher moments, namely co-skewness and co-kurtosis, as additional measures of systematic risk may enhance the model's explanatory power, especially when the distribution function of stock returns is asymmetric. The present study empirically investigates the applicability of unconditional higher order moment CAPM and the impact of higher moments in the Indian stock market i.e. Bombay Stock Exchange using the data of sectoral indices for the period from April 2011 to March 2021. To test the four moment CAPM empirically, the specification given by Fang and Lai has been used in the study. The findings of the present study revealed that the higher moments (coskewness and cokurtosis) are significantly priced and have impact on the returns in the Indian stock market. The market risk premium for covariance was found to be insignificant. Further the hypothesis related to intercept term was accepted and market risk premiums were rejected. The results showed an increase in the explanatory power of the model as compared to the unconditional CAPM as the R-square value of the model was obtained better than the latter model. The mixed and inconclusive findings contradicted the model in the Indian context. The alternate models like the Fama-French three factor and five factor model should be exploited in the Indian context as those models have been rarely used in the Indian context.

Application of descriptive statistics methods to Kazakhstan stock market

The article discusses the methods of descriptive statistics and its application for the most liquid shares of the Kazakhstan stock market KASE (trading codes: HSBK, KZTK, CCBN, KZTO, KCEL, KEGC, KZKAK) from 2007 to 2022. Descriptive statistics are calculated for each stock, such as average return, variance, sample range, standard deviation, coefficient of variation, kurtosis, and skewness. The coefficients of the regression equation and the coefficient of determination R2 are estimated, which show the relationship between the market return and the return on an individual stock. The comparative analysis of index growth of stocks was carried out from 2013 to 2022. The histogram of the frequency distribution of KASE market returns based on annual and monthly periods and its empirical distribution function have been constructed. The hypothesis about the normal distribution of returns according to the Pearson criterion was tested. The distributions of returns for many stocks do not fully correspond to the normal distribution and have a positive skewness, i.e. have a long tail on the right side. The market curve has been constructed that characterizes the dependence of stock returns on the level of risk. According to this curve, it is established that HSBK stocks are the most risky of all the considered stocks, undervalued and attractive investments for investors using an aggressive investment strategy. KEGC stocks are less risky and very attractive investments for investors who use a more conservative investment strategy in the Kazakhstani stock market.

A fuzzy multifactor asset pricing model.

This paper introduces a new approach of multifactor asset pricing model estimation. This approach assumes that the monthly returns of financial assets are fuzzy random variables and estimates the multifactor asset pricing model as a fuzzy linear model. The fuzzy random representations allows us to incorporate bias on prices induced by the market microstructure noise and to reflect the intra-period activity in the analysis. The application of fuzzy linear regression enables the uncertainty assessment in an alternative way to confidence interval or hypothesis testing, which is subjected the binding assumption of normal distribution of returns. However, it is well known that the distribution of many asset returns deviates significantly from the normal assumption. We illustrate this estimation in the particular case of the Fama and French’s (J Financ Econ 33:3–56, 1993) three factor model. Finally, empirical studies based on Fama and French’s portfolios and risk factors, historical dataset highlight the effectiveness of our estimation method and a comparative analysis with the ordinary least square estimation shows its ability to be applied for an optimal decision decision making in the financial market.

Practical Implementation of the Kelly Criterion: Optimal Growth Rate, Number of Trades, and Rebalancing Frequency for Equity Portfolios

We develop a general framework to apply the Kelly criterion to the stock market data, and consequently, to portfolio optimization. Under few conditions, using Monte Carlo simulations with different scenarios we prove that the Kelly criterion beats any other approach in many aspects. In particular, it maximizes the expected growth rate and the median of the terminal wealth. We also show that, under a normal distribution of returns, the Kelly criterion has the best performance in the long run. Next, we optimize a portfolio with the Kelly criterion with no leverage and no short selling conditions and show that this portfolio lays in the mean-variance efficient frontier and has higher expected return and higher variance, although it is less diversified, respect to the tangent portfolio optimized under the Markowitz approach. Finally, we implement a dynamic strategy applied on the European stock market data and compare the results between the tangent and the optimal Kelly portfolios. In a dynamic setting, the rolling Kelly portfolio outperforms competitors particularly in the case of rebalanced portfolios optimized with a 2-years window width.

Performance Appraisal of Open-ended Equity Funds in Pakistan: An alternative Approaches of Performance Measure

There are more than one hundred portfolio performances, which have been proposed in financial literature, (Cogneau and Hubner, 2009), but extensively used performance measure is a Sharpe ratio and in Pakistan Asset Management Companies (AMCs) also prefer to exhibit their performance in Sharpe ratio. However, financial literature has ample of evidence that recommend Sharpe ratio is valid under normal distribution of returns. The financial returns are not distributed normally as result of which standard deviation may not adequately measure risk (Bodie et al., 2009). Whereas, standard deviation of negatively skewed distribution underestimates and positively skewed overestimates volatility that would be misleading Sharpe index. In this study, we concluded that for skewed and non-normal distribution Omega ratio or Sharpe-Omega are alternative performance measures.

A New Approach on Effective Control regarding E-commerce Risks in Stock Exchange Market Focusing on Cloud Computing (A Caste Study: Tehran Stock Market)

Effective risk control is as significant as maximising the return of investment for investors; therefore, it is essential to examine patterns and tools of the effective risk control for investors. By using local and global optimisation algorithms, the approach minimises risk-based control patterns, draws and compares its performance boundaries. For this purpose, after testing the normal distribution of returns of selected companies, parametric nonlinear planning patterns are introduced for all three effective risk control models. Using optimisation algorithms and Matlab software, the optimisation function is based on fuzzy hierarchical analysis method, fuzzy DEA and an intelligent optimisation model in the cloud environment to do the comparison. For economic users of the stock exchange, the cloud environment was an unsafe media, and the cloud itself is considered a risk. The study findings show the fuzzy optimised model can better understand the cloud environment and the risks more effectively than other models.

A big data Bayesian approach to earnings profitability in the S&P 500

A big data Bayesian approach to earnings profitability in the S&P 500

Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization

We consider general nonlinear programming problems with cardinality constraints. By relaxing the binary variables which appear in the natural mixed-integer programming formulation, we obtain an almost equivalent nonlinear programming problem, which is thus still difficult to solve. Therefore, we apply a Scholtes-type regularization method to obtain a sequence of easier to solve problems and investigate the convergence of the obtained KKT points. We show that such a sequence converges to an S-stationary point, which corresponds to a local minimizer of the original problem under the assumption of convexity. Additionally, we consider portfolio optimization problems where we minimize a risk measure under a cardinality constraint on the portfolio. Various risk measures are considered, in particular Value-at-Risk and Conditional Value-at-Risk under normal distribution of returns and their robust counterparts under moment conditions. For these investment problems formulated as nonlinear programming problems with cardinality constraints we perform a numerical study on a large number of simulated instances taken from the literature and illuminate the computational performance of the Scholtes-type regularization method in comparison to other considered solution approaches: a mixed-integer solver, a direct continuous reformulation solver and the Kanzow–Schwartz regularization method, which has already been applied to Markowitz portfolio problems.

ROZKŁAD NORMALNY STÓP ZWROTU Z AKCJI WCHODZĄCYCH W SKŁAD NASTĘPUJĄCYCH INDEKSÓW GIEŁDOWYCH: WIG20, MWIG40 I SWIG80

The article verified the hypothesis regarding normal distribution of returns of shares - components of the following Warsaw Stock Exchange indexes Keywords: normal distribution, return rates, stock indices, ranking of companies

Comparing large-sample maximum Sharpe ratios and incremental variable testing

Most existing results on the distribution of the maximum Sharpe ratio depend on the assumption of multivariate normal return distributions. We use recent results from the literature to provide an analytical representation of the distribution of the difference between two maximum Sharpe ratios for much less restrictive distributional assumptions, both with and without short sales. Knowing the distribution of the difference enables us to test ex ante whether or not the inclusion of additional variables leads to a significant improvement in the maximum Sharpe ratio. In addition, we characterize the optimal long-only solution and provide conditions for global optimality.

Biased Risk Parity with Fractal Model of Risk

For the past two decades investors have observed long memory and highly correlated behavior of asset classes that does not fit into the framework of Modern Portfolio Theory. Custom correlation and standard deviation estimators consider normal distribution of returns and market efficiency hypothesis. It forced investors to search more universal instruments of tail risk protection. One of the possible solutions is a naive risk parity strategy, which avoids estimation of expected returns and correlations. The authors develop the idea further and propose a fractal distribution of returns as a core. This class of distributions is more general as it does not imply strict limitations on risk evolution. The proposed model allows for modifying a rule for volatility estimation, thus, enhancing its explanatory power. It turns out that the latter improves the performance metrics of an investment portfolio over the ten year period. The fractal model of volatility plays a significant protective role during the periods of market abnormal drawdowns. Consequently, it may be useful for a wide range of asset managers which incorporate innovative risk models into globally allocated portfolios.

Measuring Portfolio Performance: Sharpe, Alpha, or the Geometric Mean?

The most popular portfolio performance measures are the Sharpe ratio and alpha. While the Sharpe ratio is optimal under the CAPM assumptions of normal return distributions and unlimited borrowing at the risk-free rate, we find that it is not well aligned with investors’ preferences in more realistic settings. Alpha is a poor measure under both the theoretical and the realistic settings. For investors with typical borrowing constraints, the geometric mean provides an alternative measure that is much better than both the Sharpe ratio and alpha. It may very well be the most important single number to consider in portfolio selection.

Weak Dependence of CRRA on Standard Deviation in the Case of Truncated Normal Distribution of Returns

This paper analyzes the dependence of the Certainty Equivalent Return of a Constant Relative Risk Aversion, CER[CRRA], on the Standard Deviation of the Return with the hypothesis of a Truncated Normal distribution of returns and for some level of Relative Risk Aversion (RRA) parameter. The paper compares this dependence with those detected by an Annualized Geometrical Return (AGR) and by CER of the Quadratic Utility Function, CER[Q]. The behavior of CER[CRRA] is more similar to AGR than CER[Q] and only for a higher value of RRA is it possible to find substantial differences, even if in this case we find values of Standard Deviation that have discontinuity points for the concavity. Using a ranking criteria equal to the one introduced by Morningstar for a set of Funds, the paper shows that, in a wide range for monthly Standard Deviation and Mean of the Returns, the ranking done by CER[CRRA] is similar to the one induced by AGR, and that a CER[Q] has essentially different behavior. It will be shown that Morningstar ranking may be considered a particular case of the CER[CRRA] and thus all the considerations can be applied to the well-known Morningstar Rating methodology. An application is made to Italian Pension Funds.

Ex Ante Comparison of Maximum Sharpe Ratios and Incremental Variable Testing

A typical decision problem faced by investors is whether or not to include additional asset classes or return drivers into their portfolios. In the context of classical portfolio theory and in the presence of a riskless asset, the relevant criterion is a significant increase in the maximum Sharpe ratio (the Sharpe ratio of the tangency portfolio) achievable when enlarging the investment opportunity set. Although not all assumptions of classical portfolio theory are always satisfied by empirical return distributions, the Sharpe ratio is widely used in practice as a risk-adjusted performance measure and a benchmark. Most existing results on the distribution of the maximum Sharpe ratio depend on the assumption of multivariate normal return distributions. We generalize results from the literature to provide an analytical representation of the distribution of the difference between two maximum Sharpe ratios for much less restrictive distributional assumptions, both with and without short sales. For the long-only case, we provide conditions that allow us to check whether any local optima that may be encountered indeed correspond to the global optimum. Knowing the distribution of the difference enables us to test ex ante whether or not the inclusion of additional variables leads to a significant improvement in the maximum Sharpe ratio. High estimation uncertainty regarding means of asset returns makes it difficult to distinguish statistically between tangency portfolios at conventional significance levels.

Income Inequality and Stock Market Volatility

This study tests whether income inequality creates greater volatility in stock prices across a broad sample of countries. Contrary to the idea that inequality creates social and political uncertainty that is reflected in higher volatility, we instead find a negative association between inequality and volatility. To make stronger causal inferences, we use the passing of the French “Millionaire Tax” as a natural experiment. Results show that volatility of French stocks increases after the Tax implementation. Besides finding that inequality stabilizes stock prices, we also find that countries with greater inequality also have stocks with more normal return distributions.

'Reverse Spreads Find the Trends', Futures Magazine (Modern Trader)

There is a conventional division of trading techniques into two classes: mean reversion and trend-following. The trader can create a synthetic instrument that wittingly possesses trend behavior attributes along with memory as well as a prevailing noise component. In this case, a trader controls a synthetic market structure. Two relationship models can be assigned to each type of trading class: partnership “win-win” (mean-reversion) strategies or competition “win-lose” (trend-following) strategies. Instruments based on the “win-win” model mostly demonstrate market efficiency with a normal distribution of returns. A different class of instruments are based on the “win-lose” competitive model. Spread composition based on the “win-lose” model can be a possible way to follow the trend effectively. Imagine a Google/Apple (APPL) stock spread, formed by the shares of these two high-tech competitors. Under certain conditions, this instrument is distinguished by long-term trends and short-lived flat periods.Correlation analysis is one of the simplest and most affordable approaches used for identifying asset relationships. A negative correlation between two assets shows that a mean basic asset growth corresponds to a drop in the other asset and vice versa. As a result, a small fluctuation of a basic asset, caused by some systematic market factors, should develop into a spread trend. A flat movement pattern becomes unstable for reverse spreads because a small fluctuation leads to amplifying volatility. The effect in the related asset may change based on the inertia property of the asset itself. Here, we may define a reverse spread as a spread instrument of inverse statistical relationship between two underlying assets.

Value-at-risk estimates of the stock indices in developed and emerging markets including the spillover effects of currency market

This study derives the quantiles of the standardized generalized t (GT) in terms of a nonlinear equation which contains a regularized incomplete beta function. Then the quantiles are evaluated by utilizing Secant numerical approach to solve this nonlinear equation. Subsequently, the exponential generalized autoregressive conditional heteroskedasticity (EGARCH) model with GT distribution is utilized to estimate the corresponding volatility, and further estimate the value-at-risk (VaR) of seven stock indices in the developed and emerging markets. Empirical results show that, the stylized facts that appeared in most financial assets are seized effectively by this model and negative return and volatility spillover effects significantly subsist from the currency markets to stock markets. Moreover, the stock indices in emerging market have the higher return and the higher risk. As to VaR performance comparison, the modified historical simulation (MHS) and the EGARCH volatility specification significantly affect the VaR forecast performance for stock indices in the emerging market as compared with the developed market. Moreover, the VaR forecast performance of all models with GT is superior to that with normal return distribution only for stock indices in the developed market and only for 99% level. Turning to the whole market, the VaR forecast performance is almost the same as that for the emerging market. Finally, the MHS-EGARCH model with GT distribution is the optimal model to forecast the VaR among these eight models, irrespective of which of the three markets are used. This finding can provide the financial institutions to select an appropriate model to forecast and further control the market risk they faced.

Portfolio selection in a two-regime world

Abstract Standard mean-variance analysis is based on the assumption of normal return distributions. However, a growing body of literature suggests that the market oscillates between two different regimes – one with low volatility and the other with high volatility. In such a case, even if the return distributions are normal in both regimes, the overall distribution is not – it is a mixture of normals. Mean-variance analysis is inappropriate in this framework, and one must either assume a specific utility function or, alternatively, employ the more general and distribution-free Second degree Stochastic Dominance (SSD) criterion. This paper develops the SSD rule for the case of mixed normals: the SSDMN rule. This rule is a generalization the mean-variance rule. The cost of ignoring regimes and assuming normality when the distributions are actually mixed normal can be quite substantial – it is typically equivalent to an annual rate of return of 2–3 percent.

Bayesian Nonparametric Modelling of the Return Distribution with Stochastic Volatility

This paper presents a method for Bayesian nonparametric analysis of the return distribution in a stochastic volatility model. The distribution of the logarithm of the squared return is flexibly modelled using an infinite mixture of Normal distributions. This allows efficient Markov chain Monte Carlo methods to be developed. Links between the return distribution and the distribution of the logarithm of the squared returns are discussed. The method is applied to simulated data, one asset return series and one stock index return series. We find that estimates of volatility using the model can differ dramatically from those using a Normal return distribution if there is evidence of a heavy-tailed return distribution.