Lower Partial Moment Research Articles

Pitfalls of Downside Performance Measures with Arbitrary Targets

AbstractDownside performance measures relate above target returns with lower partial moments. They were developed to resolve restrictive assumptions of the classical Sharpe ratio. While the Sharpe ratio evaluates whether portfolios of a mutual fund and the risk‐free asset dominate passive portfolios of the benchmark and the risk‐free asset, this characteristic cannot be transferred to downside performance measures with arbitrary targets. We show that downside performance measures assign different values to passive benchmark strategies if the target differs from the risk‐free rate. This effect can lead to reverse rankings of financial assets. Therefore, downside performance measures are only applicable in asset management if the target is set equal to the risk‐free rate.

On Competence of Lower Partial Moment of the First Order for Drawing the Efficient Frontier of Portfolios

In this paper after a general literature review on the concept of Efficient Frontier (EF), an important inadequacy of the Variance based models for deriving EFs and the high necessity for applying another risk measure is exemplified. In this regard for this study the risk measure of Lower Partial Moment of the first order is decided to replace Variance. Because of the particular shape of the proposed risk measure, one part of the paper is devoted to development of a mechanism for deriving EF on the basis of new model. After that superiority of the new model to old one is shown and then the shape of new EFs under different situations is investigated. At last it is concluded that application of LPM of the first order in financial models in the phase of deriving EF is completely wise and justifiable.

Dynamic Mean-LPM and Mean-CVaR Portfolio Optimization in Continuous-Time

Instead of controlling "symmetric" risks measured by central moments of investment return or terminal wealth, more and more portfolio models have shifted their focus to manage "asymmetric" downside risks that the investment return is below certain threshold. Among the existing downside risk measures, the lower-partial moments (LPM) and conditional value-at-risk (CVaR) are probably most promising. In this paper we investigate the dynamic mean-LPM and mean-CVaR portfolio optimization problems in continuous-time, while the current literature has only witnessed their static versions. Our contributions are two-fold, in both building up tractable formulations and deriving corresponding analytical solutions. By imposing a limit funding level on the terminal wealth, we conquer the ill-posedness exhibited in the class of mean-downside risk portfolio models. The limit funding level not only enables us to solve both dynamic mean-LPM and mean-CVaR portfolio optimization problems, but also offers a flexibility to tame the aggressiveness of the portfolio policies generated from such mean - downside risk models. More specifically, for a general market setting, we prove the existence and uniqueness of the Lagrangian multiplies, which is a key step in applying the martingale approach, and establish a theoretical foundation for developing efficient numerical solution approaches. Moreover, for situations where the opportunity set of the market setting is deterministic, we derive analytical portfolio policies for both dynamic mean-LPM and mean-CVaR formulations.

ESG Shareholder Engagement and Downside Risk

Direct institutional shareholder engagement on environmental, social and governance (ESG) issues has become increasingly important. We show that ESG shareholder engagement reduces downside risk, measured using lower partial moments and value at risk. We document this effect by exploiting proprietary access to the complete engagement database of one of the world’s largest institutional shareholder activist. We document a risk-reduction effect only for engagements where portfolio firms respond with real actions to the investor’s demands. The risk effect of ESG engagement varies across engagement themes. It is effective when governance or strategy & risk topics are addressed, and if changes in firms’ environmental policies are coupled with governance improvements.

Determining the multi-scale hedge ratios of stock index futures using the lower partial moments method

Abstract This paper considers a multi-scale future hedge strategy that minimizes lower partial moments (LPM). To do this, wavelet analysis is adopted to decompose time series data into different components. Next, different parametric estimation methods with known distributions are applied to calculate the LPM of hedged portfolios, which is the key to determining multi-scale hedge ratios over different time scales. Then these parametric methods are compared with the prevailing nonparametric kernel metric method. Empirical results indicate that in the China Securities Index 300 (CSI 300) index futures and spot markets, hedge ratios and hedge efficiency estimated by the nonparametric kernel metric method are inferior to those estimated by parametric hedging model based on the features of sequence distributions. In addition, if minimum-LPM is selected as a hedge target, the hedging periods, degree of risk aversion, and target returns can affect the multi-scale hedge ratios and hedge efficiency, respectively.

Robust Multi-Period Portfolio Selection with Asymmetrically Distributed Uncertainty Set and Return Predictability

This paper considers a robust multi-period portfolio selection problem with asymmetrically distributed uncertainty set. We introduce the mean-LPM (lower partial moment) risk measure and establish a multi-period portfolio selection model, in which the mean-LPM is used to limit the loss of portfolio. To add practicality, we allow the uncertainty set to be asymmetric and include transaction cost in the model. A computationally tractable approximation approach based on second order cone optimization is used for solving the proposed model. Comprehensive numerical comparisons with simulated data and real market data are reported. The numerical results indicate that the proposed model can obtain better expected returns and Sharpe ratios, while reduce the standard deviation and turnover ratios, compared with some well-known models in the literature.

LPM Density Functions for the Computation of the SD Efficient Set

The equivalence between partial moments and stochastic dominance dates back to Bawa [1] and Fishburn [2]. We present a test for first, second, and third degree stochastic dominance between two variables using Lower Partial Moments. The results uphold Hadar and Russell’s [3] original conclusions about the odd moments of preferred prospects. We recall Nawrocki’s [4] research comparing Mean/Variance portfolios against the continuum of risk-averse investors using Lower Partial Moments. The excess skewness of the LPM portfolios clearly demonstrates the preference of positive skewness for risk-averse investors. Finally, we provide an algorithm for efficiently determining stochastic dominance efficient sets among large numbers of variables.

One-Sided Performance Measures Under Gram-Charlier Distributions

We derive closed-form expressions for risk measures based on partial moments by assuming the Gram-Charlier (GC) density for stock returns. As a result, the lower partial moments (LPM) can be expressed as linear functions on both skewness and excess kurtosis. Under this framework, we study the behaviour of portfolio rankings with performance measures based on partial moments, that is, both Farinelli-Tibiletti and Kappa ratios. Contrary to previous results, significant differences are found in ranking portfolios between the Sharpe ratio and the FT family. We also obtain closed-form expressions for LPMs under the semi non-parametric (SNP) distribution which allows higher flexibility (in terms of third- and fourth- order moments) than the GC distribution.

Composite time-consistent multi-period risk measure and its application in optimal portfolio selection

Through the composition of two real-valued functions, we propose a new class of multi-period risk measure which is time consistent. The new multi-period risk measure is monotonous and convex when the two real-valued functions satisfy monotonicity and convexity. Based on this generic framework, we construct a specific class of time-consistent multi-period risk measure by considering the lower partial moment between the realized wealth and the target wealth at individual periods. With the new multi-period risk measure as the objective function, we formulate a multi-period portfolio selection model by considering transaction costs at individual investment periods. Furthermore, this stochastic programming model is transformed into a deterministic programming problem using the scenario tree technology. Finally, we show through empirical tests and comparisons the rationality, practicality and efficiency of our new multi-period risk measure and the corresponding portfolio selection model.

Private ESG Shareholder Engagement and Risk: Clinical Study of the Extractive Industry

The impact of private ESG shareholder engagement on firm financial performance has recently received considerable attention in the academic literature (Becht et al. 2008) and Dimson et al. 2015). Comparing with Dimson et al. study of ESG engagement effect on return across all industries in the US, we conduct analysis of the impact of ESG shareholder engagement on financial performance and further investigate the mechanisms underlying the engagement effect focusing on the extractive industry. We find that engagement target has relatively lower downside risk than its equivalent control firm. Lower partial moment LPM (0, 2) and lower partial moment LPM (0, 3) are 1.1% and 1.4% less for the engagement targets respectively. And this improved target risk performance in extractive industry does not constrain target abnormal return. With regard to the evaluation of engagement success rate effect on risk and return comparing with the controls, we find that targets with average engagement achieved milestone 3 or 4 produces in average 3.9% less downside risk than targets only achieved milestone 1 or 2 in comparison with controls. Moreover, engagement with targets chair and CSR management assists to successfully complete an engagement. In all, this paper provides clinical evidence to show that ESG shareholder engagement enhance shareholder value in a long run by controlling firm downside risk in attractive industry.

A robust set-valued scenario approach for handling modeling risk in portfolio optimization

For portfolio optimization under downside risk measures, such as conditional value-at-risk or lower partial moments, we often invoke a scenario approach to approximate the high-dimensional integral involved when calculating risk. Consequently, two types of modeling risk may arise from this procedure: uncertainty in determining the distribution of asset returns and the error caused by approximating a given distribution with scenarios. To handle these two types of modeling risk within a unified framework, we propose a mathematically tractable set-valued scenario approach. More specifically, when short selling is not permitted, the robust portfolio selection problems modeled within a minimum-maximum decision framework using several types of set-valued scenarios can be translated into linear programs or second-order cone programs. These can be efficiently solved by the interior point method. The proposed set-valued scenario approach can be used not only as a methodology to alleviate the modeling risk but also as a useful tool for evaluating the impact of modeling risk. Our simulation analysis and empirical study show that robustness does not necessarily imply conservativeness, portfolio performance is affected by the investment style characterized by the return-risk tradeoff to a large degree and modeling risk only becomes significant when an aggressive strategy is adopted.

Value Functions for Prospect Theory Investors: An Empirical Evaluation for US Style Portfolios

We employ two reward and risk measures, the Upper Partial Moment and the Lower Partial Moment, in order to maximize different value functions under the budget and the short-selling constraints. We find that agents seem to prefer small capitalization and high value stock portfolios (which are positively skewed) and reject large capitalization and growth stock portfolios (which are negatively skewed) for all types of preferences (S-shaped, reverse S-shaped, kinked convex and kinked concave value function). This behavior is consistent with the value and the size effects. Probability distortion plays a significant role in the determination of the optimal perspective value; however, it is not significant for the determination of the optimal asset allocation.

Maximum Drawdown and Risk Tolerances

Due to the numerous studies of asymmetric portfolio returns, asymmetric risk measures have widely been used in risk management with extensive uses on the methodology of n-degree lower partial moment (LPM). Unlike the initial studies, we use the risk measure of n-degree maximum drawdown, which is a special case of n-degree LPM, to investigate the reduction impacts of n-degree maximum drawdown risk on risk tolerances generated by management styles from US equity-based mutual funds. We found that skewness does not impose any significant problems on the model of n-degree maximum drawdown. Thus, the tolerance effect of maximum drawdown risk in the n-degree M-DRM models is a decrease in fund returns. The n-degree CM-DRM optimization model decreased investors' risk more than two conventional models. Thus, the M-DRM can be accommodated with risk-averse investors' approach. The efficient set of mean-variance choices from the investment opportunity set, as described by Markowitz, shows that the n-degree CM-DRM algorithms create this set with lower risk than other algorithms. It implies that the mean-variance opportunity set generated by the n-degree CM-DRM creates lower risk for a given return than covariance and CLPM.

A new methodology for deriving the efficient frontier of stocks portfolios: An advanced risk-return model

In this paper, after reviewing the concept of Efficient Frontier (EF), an important inadequacy of the Variance based models for deriving EFs and the high necessity for applying another risk measure is exemplified. To meet the challenge, the traditional risk measure of Variance is replaced with Lower Partial Moment (LPM) of the first order. Because of the particular shape of the new risk measure, one part of the paper is devoted to a methodology for deriving EF on the basis of the new model. Then the model superiority over the old one is shown and finally shape of the new EFs under different situations is investigated. At last, it is concluded that application of LPM of the first order in financial models in the phase of deriving EF is completely wise and justifiable.

Lower Partial Moment and Information Entropy-Based Dynamic Electricity Purchasing Strategy

In the electricity market environment, load-serving entities (LSEs) will inevitably face risks in purchasing electricity because there are a plethora of uncertainties involved. To maximize profits and minimize risks, LSEs need to develop an optimal strategy to reasonably allocate the purchased electricity amount in different electricity markets such as the spot market, bilateral contract market, and options market. Because risks originate from uncertainties, an approach is presented to address the risk evaluation problem by the combined use of the lower partial moment and information entropy (LPME). The lower partial moment is used to measure the amount and probability of the loss, whereas the information entropy is used to represent the uncertainty of the loss. Electricity purchasing is a repeated procedure; therefore, the model presented represents a dynamic strategy. Under the chance-constrained programming framework, the developed optimization model minimizes the risk of the electricity purchasing portfolio in different markets because the actual profit of the LSE concerned is not less than the specified target under a required confidence level. Then, the particle swarm optimization (PSO) algorithm is employed to solve the optimization model. Finally, a sample example is used to illustrate the basic features of the developed model and method.

Average drawdown risk reduction and risk tolerances

We investigate the effects of average drawdown risk reduction on US mutual funds. Due to numerous evidence of the asymmetric distribution of portfolio returns, the asymmetric risk measures have extensively been used in risk management during the recent decades with extensive usages on the n-degree lower partial moment (LPM) methodology. Unlike the previous literature, we use the n-degree average drawdown risk measure, which is a special case of n-degree LPM, to empirically investigate the impacts of n-degree average drawdown risk reduction on the risk tolerances generated by the US mutual funds.The evidence shows that skewness does not impose any significant problem on the n-degree A-DRM model. Moreover, the effect of changing the tolerances of average drawdown risk in the n-degree A-DRM models is a reduction in the fund returns. The n-degree CA-DRM optimization model reduces investors׳ risk more than other models. Thus, the A-DRM can be accommodated with risk-averse investors׳ approach. The efficient set of mean–variance choices from the investment opportunity set, as described by Markowitz, shows that the n-degree CA-DRM algorithms create this set with lower risk than other algorithms. It implies that the mean–variance opportunity set generated by the n-degree CA-DRM creates lower risk for a given return than covariance and CLPM.

Portfolio optimization in an upside potential and downside risk framework

The lower partial moment (LPM) has been the downside risk measure that is most commonly used in portfolio analysis. Its major disadvantage is that its underlying utility functions are linear above some target return. As a result, the upper partial moment (UPM)/lower partial moment (LPM) analysis has been suggested by Holthausen (1981. American Economic Review, v71(1), 182), Kang et al. (1996. Journal of Economics and Business, v48, 47), and Sortino et al. (1999. Journal of Portfolio Management, v26(1,Fall), 50) as a method of dealing with investor utility above the target return. Unfortunately, they only provide dominance rules rather than a portfolio selection methodology. This paper proposes a formulation of the UPM/LPM portfolio selection model and presents four utility case studies to illustrate its ability to generate a concave efficient frontier in the appropriate UPM/LPM space. This framework implements the full richness of economic utility theory be it [Friedman and Savage (1948). Journal of Political Economy, 56, 279; Markowitz, H. (1952). Journal of Political Economy, 60(2), 151; Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. (3rd ed., 1953), Princeton University Press], and the prospect theory of (Kahneman and Tversky (1979). Econometrica, 47(2), 263).The methods and techniques proposed in this paper are focused on the following computational issues with UPM/LPM optimization.•Lack of positive semi-definite UPM and LPM matrices.•Rank of matrix errors.•Estimation errors.•Endogenous and exogenous UPM and LPM matrices.

ANOVA Using Continuous Cumulative Distribution Functions

Analysis of Variance (ANOVA) is a statistical method used to determine whether a sample originated from a larger population distribution. We provide an alternate method of determination using the continuous cumulative distribution functions derived from degree one lower partial moment ratios. The resulting analysis is performed with no restrictive assumptions on the underlying distribution or the associated error terms.

Stochastic Dominance and Mutual Fund Benchmarking: Evidence from Sectoral Funds

Performance analysis of mutual funds operating in the Indian markets are mostly dependent on ratio aproach involving methodologies suggested by Sharpe and Treynor. The present paper makes use of non-parametric endogenous benchmarking to evaluate the performance of 16 top performing sectoral mutual fund schemes based on observations for the second half of 2010. The study uses both the ‘reward to volatility’ and ‘reward to lower partial moment framework’ for the measurement of pure technical efficiency and scale efficiency of the in-sample mutual fund schemes. For this, the study uses the concepts of input, ouput and directional distance function.

An Investigation of Beta and Downside Beta Based CAPM-Case Study of Karachi Stock Exchange

Sharpe’s (1964) Capital Asset Pricing Model (CAPM) assumes that the relationship between risk and return is positive, linear and significant. However, it is not free from controversies and one of them advocates replacing CAPM’s beta by downside beta based on investors’ preference of downside risk. Roy (1952) debates that investor care for downside risk and Hogan and Warren (1974) replace variance with semivariance in CAPM as the first official version of downside risk based CAPM. Bawa (1975), Fishburn (1977) and Bawa and Lindenberg (1977) develop and extend proxy for downside risk/beta as Lower Partial Moment. This study empirically tests beta and downside beta based CAPM (DCAPM). Conceptual and empirical problems related in testing alternative models are discussed with adoption of Fama-MacBeth (1973) procedure by making it robust. This study inspects intercept, risk-return relationship, nonlinearities and effect of residuals for both CAPM and DCAPM. Intercept results are almost similar and they follow introduction of zero-beta models as outlined by Black et al. (1972). Both models show rejection of nonlinearities and effect of residuals. However, DCAPM comes out to be strong contender compared to CAPM for risk-return relationship. These results are consistent with Estrada (2002), Ang et al.(2004) and Post and Vliet (2004).