Non-normal Return Distributions Research Articles

FEATURES OF MODELING INVESTMENT PORTFOLIO RISKS UNDER CRISIS CONDITIONS

The main goal of investment activity is to minimize the level of investment risk while finding the optimal balance between return and risk. This process involves accounting for probabilistic factors that arise from the inherent uncertainty in financial activities. Investors are constantly faced with the challenge of managing these risks, particularly in the context of financial markets that exhibit significant volatility and unpredictability. In this regard, the search for models that allow for accurate risk assessment and management is a key aspect of modern portfolio theory. Currently, one of the most widely used models for solving the portfolio optimization problem is the Harry Markowitz model, often referred to as Modern Portfolio Theory (MPT). The Markowitz portfolio optimization model seeks to achieve two possible outcomes: either by minimizing the variance (or risk) of the portfolio's return at a given level of expected return, or by maximizing the expected return at a given level of variance (risk). Despite the obvious advantages of the Markowitz model, such as its widespread applicability and relative ease of implementation in practice, it has several notable limitations. One key drawback is that the model uses variance as a measure of risk, which treats deviations from the expected return symmetrically. This means that both positive (upward) and negative (downward) deviations are considered equally risky, which contradicts the typical investor's view, as they are more concerned with downside risk rather than upside potential. Another limitation of the Markowitz model is the assumption of normally distributed asset returns. In reality, financial markets do not always conform to this assumption. Extreme returns, often referred to as "fat tails," tend to occur more frequently than what would be predicted by a normal distribution. In response to these limitations, there has been growing interest in the development of alternative models that can more accurately capture the complexities of financial markets. One such approach is the use of stochastic models that take into account the time-varying nature of asset volatility and the impact of extreme events. These models seek to minimize portfolio investment risk by incorporating more realistic assumptions about market behavior, including the presence of volatility clustering and non-normal return distributions. The article presented is focused on a stochastic model aimed at minimizing portfolio investment risk. This model addresses some of the shortcomings of the traditional Markowitz framework by better accounting for the unpredictable and often turbulent nature of financial markets. In particular, it has been shown that systematic risk factors play a dominant role in shaping the expected return of an investment portfolio, especially in economies undergoing transition or transformation.

Multi-Time and Multi-Moment Nonparametric Frontier-Based Fund Rating: Proposal and Buy-and-Hold Backtesting Strategy

This contribution introduces new frontier models to rate mutual funds that can simultaneously handle multiple moments and multiple times. These new models are empirically applied to hedge fund data, since this category of funds is known to be subject to non-normal return distributions. We define a simple buy-and-hold backtesting strategy to test for the impact of multiple moments and multiple times separately and jointly. The empirical results demonstrate that the proposed frontier models perform better than most financial performance measures and existing frontier models in selecting promising funds.

Risk measures for direct real estate investments with non-normal or unknown return distributions

The volatility of returns is probably the most widely used risk measure for real estate. This is rather surprising since a number of studies have cast doubts on the view that volatility can capture the manifold risks attached to properties and corresponds to the risk attitude of investors. A central issue in this discussion is the statistical properties of real estate returns—in contrast to neoclassical capital market theory they are mostly non-normal and often unknown, which render many statistical measures useless. Based on a literature review and an analysis of data from Germany we provide evidence that volatility alone is inappropriate for measuring the risk of direct real estate.We use a unique data sample by IPD, which includes the total returns of 939 properties across different usage types (56% office, 20% retail, 8% others and 16% residential properties) from 1996 to 2009, the German IPD Index, and the German Property Index. The analysis of the distributional characteristics shows that German real estate returns in this period were not normally distributed and that a logistic distribution would have been a better fit. This is in line with most of the current literature on this subject and leads to the question which indicators are more appropriate to measure real estate risks. We suggest that a combination of quantitative and qualitative risk measures more adequately captures real estate risks and conforms better with investor attitudes to risk. Furthermore, we present criteria for the purpose of risk classification.

How do the consideration of non-normal return distributions and of higher moments influence the optimal asset allocation in Swiss pension funds?

The low interest rates that prevail on many capital markets impose great challenges for the asset management of financial organizations. They try to achieve target returns for their clients, a solid one-period funding ratio and a low one-period underfunding probability. In this summarizing contribution of Muller and Wagner (2018), we aim to study the impact of capital allocation strategies for pension funds in Switzerland. Thereby, we compare classic Markowitz theory with an extended Taylor series approach for the utility function. It is further analyzed how the assumption of normally distributed returns drives the optimal asset allocation when compared with using the distributions corresponding to the best fit of the historical data. Taking the extended utility function including the first four central moments and the alternative return distributions, we simulate the assets of a pension fund in a one-period model with the Monte Carlo method. A comparison of these results with those obtained from the classic minimum variance theory concludes that a considerable change of the portfolio weights takes place. Our research is relevant for theory and practice alike. Financial institutions can strongly profit from comparing different approaches when assessing their investment strategy.

Inefficiency and bias of modified value-at-risk and expected shortfall

Modified value-at-risk (mVaR) and modified expected shortfall (mES) are risk estimators that can be calculated without modeling the distribution of asset returns. These modified estimators use skewness and kurtosis corrections to normal distribution parametric VaR and ES formulas to obtain more accurate risk measurement for non-normal return distributions. Use of skewness and kurtosis corrections can result in reduced bias, but these also lead to inflated mVaR and mES estimator standard errors. We compare modified estimators with their respective parametric counterparts in three ways. First, we assess the magnitude of standard error inflation by deriving formulas for the large-sample standard errors of mVaR and mES using the multivariate delta method. Monte Carlo simulation is then used to determine sample sizes and tail probabilities for which our asymptotic variance formula can be reliably used to compute finite-sample standard errors. Second, to evaluate the large-sample bias, we derive formulas for the asymptotic bias of modified estimators for t-distributions. Third, we analyze the finite-sample performance of the modified estimators for normal and t -distributions using their root-mean-squared-error efficiency relative to the parametric VaR and ES maximum likelihood estimators using Monte Carlo simulation. Our results show that the modified estimators are inefficient for both normal and t-distributions: the more so for t-distributions.

Flagging potential fraudulent investment activity

Purpose This paper aims to investigate the use of the bias ratio as a possible early indicator of financial fraud – specifically in the reporting of hedge fund returns. In the wake of the 2008-2009 financial crisis, numerous hedge funds were liquidated and several cases of financial fraud exposed. Design/methodology/approach Risk-adjusted return metrics such as the Sharpe ratio and Value at Risk were used to raise suspicion for fraud. These metrics, however, assume distributional normality and thus have had limited success with hedge fund returns (a characteristic of which is highly skewed, non-normal return distributions). Findings Results indicate that potential fraud would have been detected in the early stages of the scheme’s life. Having demonstrated the credibility of the bias ratio, it was then applied to several indices and (anonymous) South African hedge funds. The results were used to demonstrate the ratio’s scope and robustness and draw attention to other metrics which could be used in conjunction with it. Results from these multiple sources could be used to justify further investigation. Research limitations/implications The traditional metrics for performance evaluation (such as the Sharpe ratio), assume distributional normality and thus have had limited success with hedge fund returns (a characteristic of which is highly skewed, non-normal return distributions). The bias ratio, which does not rely on normally distributed returns, was applied to a known fraud case (Madoff’s Ponzi scheme). Practical implications The effectiveness of the bias ratio in demonstrating potential suspicious financial activity has been demonstrated. Originality/value The financial market has come under heightened scrutiny in the past decade (2005 – 2015) as a result of the fragile and uncertain economic milieu that still (2015) persists. Numerous risk and return measures have been used to evaluate hedge funds’ risk-adjusted performance, but many fail to account for non-normal return distributions exhibited by hedge funds. The bias ratio, however, has been demonstrated to effectively flag potentially fraudulent funds.

Inefficiency of Modified VaR and ES

Modified Value-at-Risk (mVaR) and Modified Expected Shortfall (mES) are risk estimators that can be calculated without modelling the distribution of asset returns. These modified estimators use skewness and kurtosis corrections to normal distribution parametric VaR and ES formulas to reduce bias in risk measurement for non-normal return distributions. However, the use of skewness and kurtosis estimators that are needed to implement mVaR and mES can lead to highly inflated mVaR and mES estimator standard errors. To assess the magnitude of standard error inflation we derive formulas for the large sample standard errors of mVaR and mES using multivariate delta method and compare them against standard errors of parametric VaR and ES estimators, under both normal and t-distributions. Our asymptotic results show that mVaR and mES estimators can have standard errors considerably larger than those of parametric VaR and ES estimators, and small-sample Monte Carlo confirms that the asymptotic results are approximately correct in sample sizes commonly used in practice.

A Wholistic Approach to Diversification Management: The Diversification Delta Strategy Applied to Non-Normal Return Distributions

Abstract/Zusammenfassung In this paper we study a higher moment diversification measure, the so-called diversification delta (Vermorken et al. (2012)), in a dynamic portfolio optimization context. Particularly, we set up an investment strategy that dynamically maximizes the diversification delta for a given set of assets. Thus, we label the resulting optimized portfolio structure as Maximum Diversification Delta Portfolio (MDDP). Our out-of-sample empirical study reveals that considering crisis-periods, the MDDP is superior to popular investment strategies, such as Minimum-Variance-Portfolio, Risk-Parity-Portfolio and Equally-Weighted-Portfolio, in terms of risk adjusted returns, risk moments and certainty equivalents. However, in line with other diversification measures the MDDP is no longer superior in upward trending markets. Ein ganzheitlicher Ansatz fur das Diversifikationsmanagement: Die Diversifikationsdelta-Strategie angewandt auf nicht-normale Renditeverteilungen In der vorliegenden Studie unters...

Fitting Non-Normal Distributions With Calibrated Cornish-Fisher Expansions

The Cornish-Fisher expansion is a popular method to adjust value-at-risk calculations for the skewness and kurtosis of non-normal return distribution. On the other hand, it is an open secret that “modified value-at-risk” calculations produce “strange” results from time to time, under certain parameter constellations. But the phenomenon was poorly understood, and no guidance was available from academia. In this research note, we illustrate the shortcomings of the traditional Cornish-Fisher expansion, by analyzing the distribution of S&P 500 price returns. We apply insights from recent research, which turns the Cornish-Fisher expansion into a well-behaved and accurate tool for modelling empirical non-normal return distributions.

Higher moment diversification benefits of hedge fund strategy allocation

Hedge funds are often used by institutional investors as a risk reduction tool in order to decrease portfolio volatility and create more stable return patterns. Normally, the portfolio construction process utilises a mean-variance approach and does not account for non-normal return distributions. In this article, we use higher moment betas to examine the effects on portfolio volatility, skewness and kurtosis when hedge funds are added to an equity portfolio. The results show that hedge funds, in general, can lower the volatility, skewness and kurtosis of the portfolio but large variations are seen between different hedge fund strategies. Convertible Arbitrage, Equity Market Neutral, Fixed Income Arbitrage, Merger Arbitrage and Macro are identified as the most attractive strategies to include in an equity portfolio for investors who care about higher moment risks and want to limit downside risk. Positive diversification effects still exist when serial correlation is accounted for but are then less pronounced.

Optimizing Benchmark-Based Portfolios with Hedge Funds

Hedge funds typically have non-normal return distributions marked by significant positive or negative skewness and high kurtosis. Mean-variance optimization models ignore these higher moments of the return distribution. This article introduces a new stochastic programming model which incorporates Monte Carlo simulation and optimization to examine the effects on the optimal allocation to hedge funds given benchmark related investment objectives such as expected shortfall and semi-variance. The results show that a substantial allocation—approximately 20% to hedge funds is justified. Specifically, the return distributions of portfolios constructed using the stochastic programming model skew to the right relative to those of the optimal mean-variance portfolios, resulting in higher Sortino ratios.

Optimizing Benchmark-Based Portfolios with Hedge Funds

Hedge funds typically have non-normal return distributions marked by significant positive or negative skewness and high kurtosis. Mean-variance optimization models ignore these higher moments of the return distribution. If a mean-variance optimization model suggests significant allocation to hedge funds, an investor concerned about unwanted skewness and kurtosis is rightfully skeptical. We apply a new stochastic programming model which incorporates Monte Carlo simulation and optimization to examine the effects on the optimal allocation to hedge funds given benchmark related investment objectives: expected shortfall, semi-variance as well as the third and fourth moments of the shortfall. Our results show that a substantial allocation - approximately 20% - to hedge funds is justified even after taken into consideration their unusual skewness and kurtosis. Moreover, results also show that our method produces a fund of funds with desired features. Specifically, our portfolios' return distributions skew to the right relative to those of the optimal mean-variance portfolios, resulting in higher Sortino ratios. Additionally, an out-of-sample backtest shows that our optimal benchmark-based portfolio achieves its goal - beating the selected benchmark.

Stochastic Dominance Analysis of iShares

Country indices as represented by iShares exhibit non-normal return distributions with both skewness and kurtosis. Earlier studies provide procedures for determining the statistical significance of stochastic dominance measures and the Sharpe Ratio. This present study uses these refinements to compare the performance of 18 country market indices. The iShares are indistinguishable when using the Sharpe Ratio as no significant differences are found. In contrast, stochastic dominance procedures identify dominant iShares. Although the results vary over time, stochastic dominance appears to be both more robust and discriminating than the CAPM in the ranking of the iShares.

Optimal Hedge Fund Allocation with Asymmetric Preferences and Distributions

Hedge funds typically have non-normal return distributions marked by significant positive or negative skewness and high kurtosis. Mean-variance optimization models ignore these higher moments of the return distribution, and thus fail to convince investors who care about the unwanted skewness and kurtosis that hedge funds may work well in a portfolio. We use a new method which incorporates Monte Carlo simulation and optimization to solve for a variety of investment objectives and address the special issues of hedge fund allocation. We applied the new optimization model to examine the effects of semi-variance, conditional third and fourth moments on portfolio allocation with hedge funds. We show that conditional on the investor's objective, a substantial allocation to hedge funds is justified even with consideration for the highly unusual skewness and kurtosis.

Anatomy of Interim Disclosures During Bimodal Return Distributions

For over 30 years, observers of financial markets have been puzzled by the behaviour of the market model residuals. This research offers one answer to two of the questions raised by the anomalous behaviour of the residuals. The first is the failure of cumulative abnormal residuals (CARs) to be centred at zero and normally or t distributed. The second is the failure of CARs to be homoscedastic. The authors argue that, in some cases, a bimodal distribution fits the data better. During the period 1985–1993 there were 47 Helsinki Exchanges listed firms having at least one bimodal return distribution after the publication of an interim report. This frequency represents 76% of the total number of firms observed. Bimodality occurs most prominently during the first few days after the event. Internal factors, contained in the interim reports, and external factors, available exogenously, help explain the uncertainty that gives rise to expost bimodality. Sometimes bimodality disappears during the investigation period, while at other times bimodality remains. These findings should promote the use of more sophisticated methods for non-normal return distributions and longer examination periods after the event. On the practical side, these results should help managers refine their communication practices.

Generalization of the Sharpe Ratio and the Arbitrage-Free Pricing of Higher Moments

We present an extension of the traditional Sharpe ratio to allow for the evaluation of non-normal return distributions. Combining earlier work in this area with stochastic simulation, we develop a procedure that allows for the construction of a benchmark for the evaluation of the performance of funds with a non-normal return distribution, while maintaining the operational ease of the Sharpe ratio. Similar to the latter, our procedure only requires the risk-free rate of interest rate, the distribution of the market index and an assumption about the type of return distributions to be evaluated. Unlike the Sharpe ratio, however, we are not restricted to normality but are able to handle any reasonable type of distribution. Since our benchmarking procedure is based on the no-arbitrage assumption, it also provides insight into the conditional arbitrage-free value of one distributional parameter in terms of another. We show that in case of the Johnson Su distribution the relationship between skewness and mean return is more or less flat. Skewness and median return on the other hand exhibit a strong negative relationship.

Interacting biases, non-normal return distributions and the performance of tests for long-horizon event studies

We report simulations of one-, three-, and five-year abnormal buy-and-hold stock return tests. Using benchmark portfolios purged of new-listings and rebalancing biases, we find severe misspecification of most tests, due in part to skewness. Control-firm matching also results in misspecification, particularly in large samples. We document a negative relation between skewness bias and sample size, and an overlapping-horizons bias. Both biases become more severe as the holding period lengthens. The biases interact such that tests can be well-specified in one situation but not another. A two groups test using winsorized abnormal returns yields correct specification and considerable power in many situations.

Interacting Biases, Non-Normal Return Distributions and the Performance of Tests for Long-Horizon Event Studies

We report simulations, using actual stock return data, of statistical tests of long-horizon buy-and-hold stock returns. We use benchmark portfolios purged of new-listings and rebalancing biases, and find that many proposed tests are misspecified, due in part to skewness. The use of a single control firm instead of a portfolio still produces misspecification, particularly in large samples. We document an inverse relation between skewness bias and sample size, and also document an overlapping-horizons bias. Both biases worsen as the holding period lengthens. Due to interacting biases, tests can be well-specified in one testing scenario but not another, seeming similar, one. A two groups test applied to winsorized abnormal returns exhibits correct specification and considerable power most often in among the tests simulated.

Interacting Biases, Non-Normal Return Distributions and the Performance of Parametric and Bootstrap Tests for Long-Horizon Event Studies

We report simulations of abnormal buy-and-hold stock return tests. Using benchmark portfolios purged of new-listings and rebalancing biases, we find severe misspecification of parametric tests, due in part to skewness. We document a negative relation between skewness bias and sample size, and an overlapping-h orizons bias. Both biases become more severe as the holding period lengthens. The biases interact such that tests can be well-specified in one situation but not another. A two-groups test using winsorized a bnormal returns yields correct specification and considerable power in many situations. Bootstrap tests detect more negative than positive abnormal returns, but are well-specified and reasonably powerful.