Risk-return Space Research Articles

Conventional and downside CAPM with higher-order moments: Evidence from emerging markets

Research background: Conventional CAPM is a well-known and tested theory on various capital markets. It was also repeatedly rejected as a model of capital pricing. This article proposes a different approach to both CAPM testing and the use of other risk measures. In addition, research is global, including emerging countries. Purpose of the article: This paper investigates the standard CAPM, and this model is based on higher moments of the return distribution for the global emerging market. In addition, this paper aims to compare the conventional and downside CAPM versions using the beta coefficient and co-moments. Methods: Contrary to the classical unconditional tests for the risk premium, conditional relationships are also estimated considering the market portfolio condition. Moreover, the studies considered conventional and downside approaches to risk measures. The cross-sectional regressions are based on the Fama-MacBeth (F-M) procedure and panel models. Findings & value added: The findings contribute to the debate on whether beta coefficient and higher order co-moments in conventional and downside approaches can explain the cross-sectional emerging indices returns. The unconditional models using all measures do not significantly describe the cross-sectional volatility of returns. The cross-sectional regressions in up and down-market based on both the classic F-M procedure and panel models show that the beta and co-kurtosis risk premium is significant and depends on market conditions. The risk premium for co-skewness is not valid, and the direction of the relationships is opposite than expected. Research also demonstrates that the test results of CAPM relationships are not robust to the presence of outliers and shocks resulting from the Covid-19 pandemic in the context of risk-return space. Research provides strong support for the importance of downside risk in the context of standard CAPM and, above all, higher co-moments.

Comparing the performance and composition of tracking error constrained and unconstrained portfolios

Active portfolios subject to tracking error constraints are bound by an ellipse in absolute risk-return space, known as the constant tracking error frontier, where the absolute risk is the standard deviation of portfolio returns. Portfolios within this space are presented: Jorion’s (2003) portfolio, which lies on the frontier and has the same risk as the benchmark, and the maximum Sharpe ratio portfolio (also on the frontier and referred to as the constant tracking error constrained market portfolio in this paper). Using historical data from an emerging market (South Africa), the performance and composition of these portfolios are explored and compared with the traditional market portfolio (the maximum Sharpe portfolio on the efficient frontier). The results illustrate how these portfolios perform during bull and bear markets and provide an indication of the positions in assets and the amount of rebalancing required to maintain these portfolios: (i) Contrary to theory, the returns and Sharpe ratio of the traditional market portfolio are substantially worse than the constrained portfolios for extended periods of time; (ii) The constrained portfolios perform similarly and require less rebalancing; (iii) There is evidence for statistically significant correlations between the compositions of the traditional market portfolio and the constrained portfolios, and further research is recommended. This work may be used to install optimal risk-adjusted active investment strategies net of transaction costs.

Portfolio performance under tracking error and asset weight constraints

Orientation: Active portfolio managers must simultaneously maximise excess returns (over benchmarks), limit risk and observe constraints on, for example, tracking errors (TRs), betas and asset weights. Research purpose: Determining the range of possible risk and returns attainable by such constrained portfolios is of interest to active portfolio managers. Weight restrictions reduce the range of achievable returns. This work demonstrates the magnitude of these reductions. Motivation for the study: This research installs and augments an approach that ascertains the effect on a TR (active) constrained portfolio in absolute risk–return space. The effects are displayed in risk–return space, demonstrating the impact on such constraints. Research approach/design and method: A theoretical approach to plot the constant TR frontier was used. Theoretical and quantitative analytical approaches to establish changes in the constant TR frontier on a simulated (but highly stylistic) market portfolios were also employed. Main findings: Considerable reduction is observed in possible investable portfolios, even for limited asset weight restrictions. This effect is amplified if multiple restrictions are imposed simultaneously, driven by both a reduced area in risk–return space enclosed by the constant TR frontier and changes in the frontier long-axis slope. Practical/managerial implications: The change in the long-axis slope sign is also a feature of changing economic conditions, thereby acting as an early warning signal with associated ramifications for asset managers. Contribution/value add: The combined effects on active portfolio performance of TR and asset weight constraints have not been investigated and demonstrated before.

Climate Change Investment Risk: Optimal Portfolio Construction Ahead of the Transition to a Lower-Carbon Economy

There is an increasing likelihood that governments of major economies will act within the next decade to reduce greenhouse gas emissions, probably by intervening in the fossil fuel markets through taxation or cap \& trade mechanisms (collectively ``carbon pricing''). We develop a model to capture the potential impact of carbon pricing on fossil fuel stocks, and use it to inform Bayesian portfolio construction methodologies, which are then used to create what we call Smart Carbon Portfolios. We find that investors could reduce ex-post risk by lowering the weightings of some fossil fuel stocks with corresponding higher weightings in lower-risk fossil fuel stocks and/or in the stocks of companies active in energy efficiency markets. The financial costs of such de-risking strategy are found to be statistically negligible in risk-return space. Robustness of the results is explored with alternative approaches.

Climate change investment risk: optimal portfolio construction ahead of the transition to a lower-carbon economy

There is an increasing likelihood that governments of major economies will act within the next decade to reduce greenhouse gas emissions, probably by intervening in the fossil fuel markets through taxation or cap & trade mechanisms (collectively “carbon pricing”). We develop a model to capture the potential impact of carbon pricing on fossil fuel stocks, and use it to inform Bayesian portfolio construction methodologies, which are then used to create what we call Smart Carbon Portfolios. We find that investors could reduce ex-post risk by lowering the weightings of some fossil fuel stocks with corresponding higher weightings in lower-risk fossil fuel stocks and/or in the stocks of companies active in energy efficiency markets. The financial costs of such de-risking strategy are found to be statistically negligible in risk-return space. Robustness of the results is explored with alternative approaches.

Is Alpha Minus Beta A Simple Rule for A Smart Beta Strategy?

Thematic investment portfolios have been of interest to portfolio managers. They can take on a variety of names including Smart Beta. One such model was suggested by Minami and Wakatsaki. They suggested an investment rule of Alpha Minus Beta as a good method to build a factor tilt portfolio which is efficient in an absolute risk-return space. This paper addresses their model by empirically examining market data. The S&P 1500 was segmented into quintiles based on Alpha Minus Beta for a fifteen year period. The suggested model had value as the Coefficient of Variation was favorable compared to the index.

A New Method of Measuring Financial Risk Aversion Using Hypothetical Investment Preferences: What Does It Say in the Case of Gender Differences?

ABSTRACTAversion to risk is one of the main factors driving investment decisions. Studies have been based on either simple decisions in a laboratory setting or real-life decisions viewed in retrospect. The study's main contribution to the literature consists of a new and elaborate method of measuring risk combined with a real-world investment task brought into a laboratory setting and show that in this controlled environment on average women are more risk averse than men. Unlike previous studies, the authors measure risk tolerance in units that naturally map into the risk-return space used by investors, giving them the missing tool to identify the optimal portfolio among the set of investment options that comprise the efficient frontier.

How to manage future groundwater resource of China under climate change and urbanization: An optimal stage investment design from modern portfolio theory

Groundwater management in China has been facing challenges from both climate change and urbanization and is considered as a national priority nowadays. However, unprecedented uncertainty exists in future scenarios making it difficult to formulate management planning paradigms. In this paper, we apply modern portfolio theory (MPT) to formulate an optimal stage investment of groundwater contamination remediation in China. This approach generates optimal weights of investment to each stage of the groundwater management and helps maximize expected return while minimizing overall risk in the future. We find that the efficient frontier of investment displays an upward-sloping shape in risk-return space. The expected value of groundwater vulnerability index increases from 0.6118 to 0.6230 following with the risk of uncertainty increased from 0.0118 to 0.0297. If management investment is constrained not to exceed certain total cost until 2050 year, the efficient frontier could help decision makers make the most appropriate choice on the trade-off between risk and return.

Consistency Regions and Frontiers: Using Density Forecasting to Find Consistent Portfolios

Our objective is to develop a methodology to detect regions in risk-return space where the out-of-sample performance of portfolios is consistent with their in-sample performance. We use the Berkowitz statistic to evaluate the accuracy of the density forecast, derived from in-sample portfolio returns, of out-of-sample portfolio returns. Defined by its co-ordinates in risk-return space, a portfolio is 'consistent' if this statistic indicates that the in-sample return density is a good predictor of out-of-sample portfolio returns. Given a track record of portfolio selections by an algorithm, we create a framework that defines a portfolio's position in risk-return space and describes a 'consistency region' bounded by a 'consistency frontier'. We argue that, for a given level of expected return, a consistent portfolio is preferable to an inconsistent one. This methodology for the creation of a consistency region is independent of the portfolio selection algorithm. We use mean-variance portfolio selection, to firstly validate the approach by using well-behaved simulated multivariate normal returns; variation of the time interval for estimation demonstrates that the consistency region behaves intuitively reasonably. Secondly, we show that with real data, the behaviour of the consistency region is driven by asset return dynamics and consequently less stable. Two investment strategies using consistency are created and compared with a minimum variance portfolio benchmark strategy. In nearly all cases, consistent portfolios lead to out-of-sample out-performance of the benchmark. We also investigate the consistency of the naive diversified (1/N) portfolio.

Alpha Minus Beta: Simple Rule for Smart Beta Strategy

Methods of getting factor premium by constructing a factor portfolio are called factor tilts. In this study, we propose a simple method to build a factor portfolio from alpha factor (score) and market beta. Simple investing rule "alpha minus beta" is a good method to build a factor portfolio which is efficient in absolute risk return space.

Portfolio frontiers with restrictions to tracking error volatility and value at risk

Asset managers are often given the task of restricting their activity by keeping both the value at risk (VaR) and the tracking error volatility (TEV) under control. However, these constraints may be impossible to satisfy simultaneously because VaR is independent of the benchmark portfolio. The management of these restrictions is likely to affect portfolio performance and produces a wide variety of scenarios in the risk-return space. The aim of this paper is to analyse various interactions between portfolio frontiers when risk managers impose joint restrictions upon TEV and VaR. Specifically, we provide analytical solutions for all the intersections and we propose simple numerical methods when such solutions are not available. Finally, we introduce a new portfolio frontier.

Evaluation of linear asset pricing models by implied portfolio performance

We present a theoretical perspective that motivates the use of the Generalized Least Squares R-Square, prominently advocated by Lewellen et al. [Lewellen, J., Nagel, S., Shanken, J., forthcoming. A skeptical appraisal of asset-pricing tests. Journal of Financial Economics], as an evaluation measure for multivariate linear asset pricing models. Adapting results from Shanken [Shanken, J., 1985. Multivariate tests of the zero-beta CAPM. Journal of Financial Economics 14, 327–348] and Kandel and Stambaugh [Kandel, S., Stambaugh, R.F., 1995. Portfolio inefficiency and the cross-section of expected returns. Journal of Finance 50, 157–184], we provide various interpretations and a graphical account in mean-variance space of this measure, facilitating a better understanding of its properties. We furthermore relate it to another leading evaluation metric, the HJ-distance of Hansen and Jagannathan [Hansen, L.P., Jagannathan, R., 1997. Assessing specification errors in stochastic discount factor models. Journal of Finance 52, 557–590]. Additionally, we present a comparison between these evaluation measures using mean-variance mathematics in risk-return space, and we provide a simple formula for calculating both model evaluation measures that involves only the parameters of the mean-variance asset and factor frontiers.

A Contingent Relationship Between Risk and Return: Toward A Behavioral Model of Decision Making

Different researchers have conceptualized the risk-return relationship as being positive, negative, or curvilinear. This paper proposes a framework where different theoretical approaches can be mapped onto a two-dimensional risk-return space in order to reconcile them. Rather than a single and fixed reference point, we propose that there are two additional reference points. Furthermore, we present a polynomial risk-return relationship that suggests a negative risk-return relationship below and above the failure and success reference points, and a positive relationship in between the two points. Practitioner as well as research implications are discussed.

The stability of bank efficiency rankings when risk preferences and objectives are different

We analyze the stability of efficiency rankings of German universal banks between 1993 and 2004. First, we estimate traditional efficiency scores with stochastic cost and alternative profit frontier analysis. Then, we explicitly allow for different risk preferences and measure efficiency with a structural model based on utility maximization. Using the almost ideal demand system, we estimate input- and profit-demand functions to obtain proxies for expected return and risk. Efficiency is then measured in this risk-return space. Mean risk-return efficiency is somewhat higher than cost and considerably higher than profit efficiency (PE). More importantly, rank–order correlation between these measures are low or even negative. This suggests that best-practice institutes should not be identified on the basis of traditional efficiency measures alone. Apparently, low cost and/or PE may merely result from alternative yet efficiently chosen risk-return trade-offs.

Time-varying risk and return characteristics of US and European bond markets: Implications for efficient portfolio allocation

This study examines return volatility among the Swiss, German, UK, and US bond markets by comparing both in-country and US-based returns and risk measures. Our findings include: (1) volatility of in-country bond market returns has been much higher in the US and UK than in Switzerland or Germany; (2) although adding to return volatility, currency returns have generally enhanced overall returns to US-based investors in European bonds; (3) low correlations between US bond market returns and European bond market returns offer potential diversification benefits; (4) overall, the risk per unit of return has been much greater in the US bond market and much lower in the German market than in the Swiss or UK markets; and (5) the risk per unit of return has been much more stable for the Swiss and German bond markets than for the UK or US markets. Our results show that global bond market investors can achieve the greatest efficiency in terms of risk per unit of return by hedging currency risk. The relatively low correlations of returns among these markets afford a global investor the opportunity to improve risk-return efficiency by diversifying among the three markets. This study offers important insights for US-based investors seeking the optimal bond portfolio allocation in risk-return space.

Chinese equity market and the efficient frontier

This study documents how China's stock market is able to contribute differently and substantially to the diversification benefits of international investment portfolios. Moreover, allowing exposure to the stock market in China allows international investors to span combinations in the risk-return space that are otherwise unavailable by investing only in developed markets such as US, UK, German, and French stock markets. In addition, it is found that the efficient frontier is not sensitive to changes in parameters.

The Stability of Efficiency Rankings When Risk-Preferences and Objectives are Different

We analyze the stability of efficiency rankings of German universal banks between 1993 and 2004. First, we estimate traditional efficiency scores with stochastic cost and alternative profit frontier analysis. Then, we explicitly allow for different risk preferences and measure efficiency with a structural model based on utility maximization. Using the almost ideal demand system, we estimate input and profit demand functions to obtain proxies for expected return and risk. Efficiency is then measured in this risk-return space. Mean risk-return efficiency is somewhat higher than cost and considerably higher than profit efficiency. More importantly, rankorder correlation between these measures are low or even negative. This suggests that best-practice institutes should not be identified on the basis of traditional efficiency measures alone. Apparently, low cost and/or profit efficiency may merely result from alternative yet efficiently chosen risk-return trade-offs.

New Frontiers in Compliance

In an era of increased regulatory oversight (not to mention litigious clients), compliance procedures need to be forward-looking, complete, scalable, and part of the order generation process. Compliance officers and fiduciaries must have simple, understandable documented evaluations of 1) where their portfolios were, and are, in risk-return space, and 2) all trades, detailing their effect upon portfolios in terms of risk and return (and taxes, if applicable) along with a statement or code regarding their relationship to any investment restrictions.

Mean-Variance Convergence Around the World

In this paper, we show (i) that the risk-return characteristics of our sample of 17 developed stock markets of the world have converged significantly toward each other during our study period 1974 - 2004, (ii) that the speed of convergence, however, varies greatly across individual markets, largely reflecting the 'initial position' of each market relative to the rest of markets in the risk-return space, and that (iii) the documented international convergence in risk-return characteristics is driven mainly by the declining 'country effect', rather than the rising 'industry effect', suggesting that the convergence is associated with international market integration. Specifically, we first compute the riskreturn distance among international stock markets based on the Euclidean distance and find that the distance thus computed has been deceasing significantly over time, implying a mean-variance convergence. In particular, the average risk-return distance has decreased by about 43% over our sample period. Lastly, we document that the risk-return characteristics of our sample of 14 emerging markets have been converging rapidly toward those of developed markets in recent years. This development notwithstanding, emerging markets still remain as a distinct asset class.

The Stability of Efficiency Rankings when Risk-Preferences are Different

In this paper we analyse bank efficiency in Germany for four cross-sections of data during the period 1995-2001. Under the assumption of cost minimisation we obtain firm-specific efficiency estimates using stochastic frontier analysis. To explicitly allow for different risk preferences when measuring efficiency we then develop a model based on utility maximisation. Using the almost ideal demand system, input- and profit demand functions are estimated and risk-preferences recovered. Efficiency is then measured in the risk-return space. Efficiency scores improve substantially and the dispersion of performance across sectors and size classes vanishes. Rank-order correlation between the two measures is low or insignificant. This suggests that best-practice institutes should not be identified only on the basis of cost efficiency. However, in terms of magnitude risk-return efficiency seems to be of less importance than cost efficiency.