Portfolio Return Distribution Research Articles

Is gold still a safe haven for stock markets? New insights through the tail thickness of portfolio return distributions

This study examines the safe-haven property of gold during two deep financial crashes: the global financial crisis (GFC) and the coronavirus disease 2019 (COVID-19) pandemic. Applying extreme value theory, we analyze the tail behavior of the returns of 46 stock indices sorted by geographic location and level of market development and then investigate these portfolios with various gold allocations. We focus on changes in the tail thickness of the portfolio return distribution as an effect of an increased gold allocation. The COVID-19 crisis exhibited heavier lower tails of the stock indices than the GFC. During the GFC, gold enabled heavy Fréchet-type tails to change into thin Gumbel-type tails, whereas during the COVID-19 crash, the tails remained heavy, even after hedging. Thus, gold acted as a safe-haven asset, significantly decreasing extreme downside risk during the GFC, but the same result cannot be confirmed for the COVID-19 pandemic.

A stochastic programming model for dynamic portfolio management with financial derivatives

Stochastic optimization models have been extensively applied to financial portfolios and have proven their effectiveness in asset and asset-liability management. Occasionally, however, they have been applied to dynamic portfolio problems including not only assets traded in secondary markets but also derivative contracts such as options or futures with their dedicated payoff functions. Such extension allows the construction of asymmetric payoffs for hedging or speculative purposes but also leads to several mathematical issues. Derivatives-based nonlinear portfolios in a discrete multistage stochastic programming (MSP) framework can be potentially very beneficial to shape dynamically a portfolio return distribution and attain superior performance. In this article we present a portfolio model with equity options, which extends significantly previous efforts in this area, and analyse the potential of such extension from a modeling and methodological viewpoints. We consider an asset universe and model portfolio set-up including equity, bonds, money market, a volatility-based exchange-traded-fund (ETF) and over-the-counter (OTC) option contracts on the equity. Relying on this market structure we formulate and analyse, to the best of our knowledge, for the first time, a comprehensive set of optimal option strategies in a discrete framework, including canonical protective puts, covered calls and straddles, as well as more advanced combined strategies based on equity options and the volatility index. The problem formulation relies on a data-driven scenario generation method for asset returns and option prices consistent with arbitrage-free conditions and incomplete market assumptions. The joint inclusion of option contracts and the VIX as asset class in a dynamic portfolio problem extends previous efforts in the domain of volatility-driven optimal policies. By introducing an optimal trade-off problem based on expected wealth and Conditional Value-at-Risk (CVaR), we formulate the problem as a stochastic linear program and present an extended set of numerical results across different market phases, to discuss the interplay among asset classes and options, relevant to financial engineers and fund managers. We find that options’ portfolios and trading in options strengthen an effective tail risk control, and help shaping portfolios returns’ distributions, consistently with an investor’s risk attitude. Furthermore the introduction of a volatility index in the asset universe, jointly with equity options, leads to superior risk-adjusted returns, both in- and out-of-sample, as shown in the final case-study.

Silver in Equity Portfolio Risk Optimization: Polish Investor Perspective

Purpose: The study aims to verify whether silver can help reduce the risk of the equity portfolio for a Polish investor without diminishing its return. What are optimal shares of silver and equity in such a portfolio, and does it also improve skewness and kurtosis of the portfolio return distribution compared with a stock portfolio? Do optimal shares change depend on the situation on the silver market? Design/Methodology/Approach: The author calculates descriptive statistics for silver and equity to decide if the former can diversify assets for the latter. As equity, the WIG20 Total Return index is used. Equity-silver portfolio optimization is conducted in the light of the Markowitz theory. The analysis is done for the period between January 2005 and April 2021. Findings: The research shows that including silver in an equity portfolio lets both decrease its risk and increase the return during the silver bull market and an extended period, which take part in a growing silver market trend. Conclusions do not change when the short sale is allowed. It is indicated that during silver bear markets 100% equity portfolio is not an efficient portfolio in the light of the Harry Markowitz theory. Simultaneously, the author suggests that although the portfolio diversification conducted in the study results in diminishing risk understood as variance or standard deviation, the results for skewness and kurtosis of created optimum portfolios are various. Practical Implications: The research may be interesting for institutional and individual Polish investors seeking different diversifying instruments in various market conditions. Originality/Value: The research contributes to the existing international literature by confirming that silver can be a diversifying asset and indicates its optimal shares in the silver-equity portfolio in different market conditions for a Polish investor endangered with USD/PLN currency rate risk.

Robust Conditional Variance Estimation and Value-at-Risk

A common approach to estimating the conditional volatility of short horizon asset returns is to use an exponentially weighted moving average (EWMA) of squared past returns. The EWMA estimator is based on the maximum likelihood estimator of the variance of the normal distribution, and is thus optimal when returns are conditionally normal. However, there is ample evidence that the conditional distribution of short horizon financial asset returns is leptokurtic, and so the EWMA estimator will generally be inefficient in the sense that it will attach too much weight to extreme returns. In this paper, we propose an alternative EWMA estimator that is robust to leptokurtosis in the conditional distribution of portfolio returns. The estimator is based on the maximum likelihood estimator of the standard deviation of the Laplace distribution, and is a function of an exponentially weighted moving average of the absolute value of past returns, rather than their squares. We employ the robust EWMA estimator to forecast the VaR of aggregate equity portfolios for the US, the UK and Japan using historical simulation. We find that the robust EWMA estimator offers an improvement over the standard EWMA estimator. In particular, the VaR forecasts that it generates are as accurate as those generated by the standard EWMA estimator, but are more efficient in the sense that the average level of capital required to cover against unexpected losses is lower and the root mean square deviation between the VaR forecast and actual returns is smaller. Moreover, the volatility of the VaR forecast itself is substantially lower with the robust EWMA estimator than with the standard EWMA estimator, reflecting its lower sensitivity to extreme returns.

Bayesian mean–variance analysis: optimal portfolio selection under parameter uncertainty

The paper solves the problem of optimal portfolio choice when the parameters of the asset returns distribution, for example the mean vector and the covariance matrix, are unknown and have to be estimated by using historical data on asset returns. Our new approach employs the Bayesian posterior predictive distribution which is the distribution of future realizations of asset returns given the observable sample. The parameters of posterior predictive distributions are functions of the observed data values and, consequently, the solution of the optimization problem is expressed in terms of data only and does not depend on unknown quantities. By contrast, the optimization problem of the traditional approach is based on unknown quantities which are estimated in the second step, and lead to a suboptimal solution. We also derive a very useful stochastic representation of the posterior predictive distribution whose application not only gives the solution of the considered optimization problem, but also provides the posterior predictive distribution of the optimal portfolio return which can be used to construct a prediction interval. A Bayesian efficient frontier, the set of optimal portfolios obtained by employing the posterior predictive distribution, is constructed as well. Theoretically and using real data we show that the Bayesian efficient frontier outperforms the sample efficient frontier, a common estimator of the set of optimal portfolios which is known to be overoptimistic.

Risk minimisation using options and risky assets

We consider mean-risk portfolio optimisation models, with risk measured by symmetric measures (variance) as well as downside or tail measures (lower partial moments, conditional value at risk). A framework for including index options in the universe of assets, in addition to stocks, is provided. The exercise of index options is settled in cash, making this implementable with a variety of strike prices and maturities. We use a dataset with stocks from FTSE 100 and index options on FTSE100. Numerical results show that, for low risk-low return and to medium risk-medium return portfolios, the addition of an index put further reduces the risk to a considerable extent, particularly in the case of mean-CVaR efficient portfolios, where the left tail of the portfolio return distribution is dramatically improved. For high risk-high return portfolios, the inclusion of an index call improves the right tail of the return distribution, creating thus the opportunity for considerably higher returns.

Estimating portfolio risk for tail risk protection strategies

AbstractWe forecast portfolio risk for managing dynamic tail risk protection strategies, based on extreme value theory, expectile regression, copula‐GARCH and dynamic generalized autoregressive score models. Utilizing a loss function that overcomes the lack of elicitability for expected shortfall, we propose a novel expected shortfall (and value‐at‐risk) forecast combination approach, which dominates simple and sophisticated standalone models as well as a simple average combination approach in modeling the tail of the portfolio return distribution. While the associated dynamic risk targeting or portfolio insurance strategies provide effective downside protection, the latter strategies suffer less from inferior risk forecasts, given the defensive portfolio insurance mechanics.

Black–Litterman model for continuous distributions

The Black–Litterman methodology of portfolio optimization, developed at the turn of the 1990s, combines statistical information on asset returns with investor’s views within the Markowitz mean-variance framework. The main assumption underlying the Black–Litterman model is that asset returns and investor’s views are multivariate normally distributed. However, empirical research demonstrates that the distribution of asset returns has fat tails and is asymmetric, which contradicts normality. Recent advances in risk measurement advocate replacing the variance by risk measures that take account of tail behavior of the portfolio return distribution. This paper extends the Black–Litterman model into general continuous distributions and deviation measures of risk. Using ideas from the Black–Litterman methodology, we design numerical methods (with variance reduction techniques) for the inverse portfolio optimization that extracts statistical information from historical data in a stable way. We introduce a quantitative model for stating investor’s views and blending them consistently with the market information. The theory is complemented by efficient numerical methods with the implementation distributed in the form of publicly available R packages. We conduct practical tests, which demonstrate significant impact of the choice of distributions on optimal portfolio weights to the extent that the classical Black–Litterman procedure cannot be viewed as an adequate approximation.

Model Risk and Disappointment Aversion

Extensions of expected utility theory are sensitive to the tail behavior of the portfolio return distribution and may not be approximated reliably through higher-order moment expansions. We develop a novel approach for model risk assessment based on a projection method and apply it to portfolio construction. We provide an extensive out-of-sample analysis to explore the economic gains of incorporating non-normality about financial asset returns into utility maximization with the generalized disappointment aversion (GDA) preferences. We find that the marginal utility gains of the optimal portfolio of a GDA investor are remarkably robust to misspecifications in the marginal distributions but are very sensitive to the structural assumption of stock returns implemented through a factor model.

Volatility versus downside risk: performance protection in dynamic portfolio strategies

Volatility-based and volatility targeting approaches have become popular among equity fund managers after the introduction in 1993 of the VIX, the implied volatility index on the S&P500 at the Chicago Board of Exchange (CBOE), followed, in 2004, by futures and option contracts on the VIX: since then we have assisted to an increasing interest in risk control strategies based on market signals. In January 2000 also the FTSE implied volatility index (FTSEIVI) was introduced at the London Stock Exchange. As a result, specifically in the US, portfolio strategies based on combinations of market indices and derivatives have been proposed by Stock Exchanges and investment banks: one such example is the S&P500 protective put index (PPUT). Early in 2016, relevant to the definition of optimal bond-equity strategies, CBOE launched an Index called TYVIX/VIX featuring an investment rotation strategy based jointly on signals coming from the VIX and the 10-year Treasury Yield implied volatility (TYVIX). All these are rule-based portfolio strategies in which no optimization methods are involved. While rather effective in reducing the downside risk, those index-based portfolio approaches do not allow an optimal risk-reward trade-off and may not be sufficient to control financial risk originated by extreme market drops. To overcome these limits we propose an optimization-based approach to portfolio management jointly focusing on volatility and tail risk controls and able to accommodate effectively the return payoffs associated with option strategies, whose cost as market volatility increases may become excessive. The model is based on a mean absolute deviation formulation and tested in the US equity market over the 2000–2016 period and with a focus on three periods of high volatility, in 2000, 2001 and 2008. The results confirm that optimal volatility controls produce better risk-adjusted returns if compared with rule-based approaches. Moreover the portfolio return distribution is dynamically shaped depending on the adopted risk management approach.

Diversification e ect of commodity futures on financial markets

This paper examines the portfolio diversification effect of commodity futures on financial market products introducing a comprehensive evaluation standard of risk standardization, robustly small correlation, and risk-return tradeo. Regarding risk standardization, we propose a definition of portfolio diversification as how much the distribution of portfolio returns is close to a normal distribution. It is shown by using an ɑ-stable distribution that if commodity price return distribution has the opposite sign of skewness parameter β to financial portfolio's β, commodity diversification e ect exists. The empirical studies using S & P 500, U.S. 10-year treasury notes, and DJ-AIG commodity index are conducted to investigate the portfolio diversification e ects. The parameter estimation results of portfolio return distributions, the conditional correlations using the dynamic conditional correlation model with financial exogenous variables, and the e cient frontier from the mean-CVaR portfolio optimization all suggest that commodity futures have a diversification e ect on financial markets.

Estimating Portfolio Risk for Tail Risk Protection Strategies

We forecast portfolio risk for managing dynamic tail risk protection strategies, based on extreme value theory, expectile regression, Copula-GARCH and dynamic GAS models. Utilizing a loss function that overcomes the lack of elicitability for Expected Shortfall, we propose a novel Expected Shortfall (and Value-at-Risk) forecast combination approach, which dominates simple and sophisticated standalone models as well as a simple average combination approach in modelling the tail of the portfolio return distribution. While the associated dynamic risk targeting or portfolio insurance strategies provide effective downside protection, the latter strategies suffer less from inferior risk forecasts given the defensive portfolio insurance mechanics.

On the Link between Volatilities, Regime Switching Probabilities and Correlation Dynamics

When markets are stressed, volatilities and correlations tend to increase jointly, and volatilities often react quicker than correlations. Based on this intuition, we extend the Dynamic Conditional Correlation model (Engle, 2002) in order to check whether the individual volatilities and/or the probabilities that some assets belong to a high/low volatility regime influence their correlation dynamics. We evaluate potential asymmetrical leverage effects too. We apply our methodology to MSCI Developed Markets indexes that cover twenty-three countries. The new models provide better in-sample fits and forecasts of the portfolio return distributions. Therefore, they are valuable frameworks for portfolio allocation and financial risk management.

The Diversification Delta: A Different Perspective

In a 2012 article published in The Journal of Portfolio Management, Vermorken, Medda, and Schroder introduce a new measure of diversification, the Diversification Delta (DD), based on the entropy of the portfolio return distribution. Entropy as a measure of uncertainty has been used successfully in several frameworks and takes into account the entire statistical distribution, rather than just the first two moments. In this article, the authors highlight some drawbacks of the DD measure and go on to propose an alternative measure based on exponential entropy that overcomes the identified shortcomings. The authors present the properties of this new measure and propose it as an alternative for portfolio optimization that incorporates higher moments of asset returns, such as skewness and excess kurtosis.

Where Has the Trend Gone? An Update on Momentum Returns in the U.S. Stock Market

Prior studies have found broad-based support for the efficacy of trading strategies based on momentum in stock returns. More recent studies, however, have noted a declining benefit relative to that identified in seminal studies on momentum. These recent studies, however, primarily examined time periods ending prior to or immediately following the 2008 financial crisis. Thus, we extend this line of research by exploring the profitability of momentum trading in the U.S. equity markets over the broader 1986 to 2015 time period. Our results for the earlier part of our sample period (i.e., 1986–2006) fall in line with previous studies, as we find a monotonic relationship between decile portfolios formed based on their prior six-month performance and subsequent twelve-month holding period excess returns. In contrast, when we evaluate more recent periods (i.e., 2007–2015 and 2010–2015), we find dramatically different results. In particular, alphas for the winner portfolio, which have historically been the highest, are actually negative during both subperiods. Furthermore, the curvature of the recent portfolio return distribution is clearly no longer monotonic. Rather, it follows a more inverted U-shaped curve from the winner portfolio (P1) to the loser (P10) portfolio, with excess returns cresting around the fifth decile portfolio (P5). In fact, we find that the risk-adjusted returns of the winner portfolio (P1) are significantly less than those of the middle portfolio (P5) for both recent subperiods. These results suggest that individual and institutional investors seeking to profit from traditionally based momentum trading strategies may need to rethink their approaches.

Arbor City Community Foundation (B): Managing Good Fortune

The Arbor City Community Foundation (ACCF) was a medium-sized endowment established in Illinois in the late 1970s through the hard work of several local families. The vision of the ACCF was to be a comprehensive center for philanthropy in the greater Arbor City region. ACCF had a fund balance (known collectively as “the fund”) of just under $240 million. The ACCF board of trustees had appointed a committee to oversee investment decisions relating to the foundation assets. The investment committee, under the guidance of the board, pursued an active risk-management policy for the fund. The committee members were primarily concerned with the volatility and distribution of portfolio returns. They relied on the value-at-risk (VaR) methodology as a measurement of the risk of both short- and mid-term investment losses. The questions in Part (A) of the case direct the students to analyze the risk inherent in both one particular asset and the entire ACCF portfolio. For this analysis the students need to calculate daily VaR and monthly VaR values and interpret these figures in the context of ACCF's risk management. In Part (B) the foundation receives a major donation. As a result, the risk inherent in its portfolio changes considerably. The students are asked to evaluate the risk of the fund's new portfolio and to perform a portfolio rebalancing analysis.Understanding the concept of value at risk (VaR); Calculating daily and monthly VaR by two different methods, the historical and the parametric approach; Interpreting the results of VaR calculations; Understanding the role of diversification for managing risk; Evaluating the impact of portfolio rebalancing on the overall risk of a portfolio.

Arbor City Community Foundation (A): The Foundation

The Arbor City Community Foundation (ACCF) was a medium-sized endowment established in Illinois in the late 1970s through the hard work of several local families. The vision of the ACCF was to be a comprehensive center for philanthropy in the greater Arbor City region. ACCF had a fund balance (known collectively as “the fund”) of just under $240 million. The ACCF board of trustees had appointed a committee to oversee investment decisions relating to the foundation assets. The investment committee, under the guidance of the board, pursued an active risk-management policy for the fund. The committee members were primarily concerned with the volatility and distribution of portfolio returns. They relied on the value-at-risk (VaR) methodology as a measurement of the risk of both short- and mid-term investment losses. The questions in Part (A) of the case direct the students to analyze the risk inherent in both one particular asset and the entire ACCF portfolio. For this analysis the students need to calculate daily VaR and monthly VaR values and interpret these figures in the context of ACCF's risk management. In Part (B) the foundation receives a major donation. As a result, the risk inherent in its portfolio changes considerably. The students are asked to evaluate the risk of the fund's new portfolio and to perform a portfolio rebalancing analysis.Understanding the concept of value at risk (VaR); Calculating daily and monthly VaR by two different methods, the historical and the parametric approach; Interpreting the results of VaR calculations; Understanding the role of diversification for managing risk; Evaluating the impact of portfolio rebalancing on the overall risk of a portfolio.

Arbor City Community Foundation: Executive Education Version

The Arbor City Community Foundation (ACCF) was a medium-sized endowment established in Illinois in the late 1970s through the hard work of several local families. The vision of the ACCF was to be a comprehensive center for philanthropy in the greater Arbor City region. ACCF had a fund balance (known collectively as “the fund”) of just under $240 million. The ACCF board of trustees had appointed a committee to oversee investment decisions relating to the foundation assets. The investment committee, under the guidance of the board, pursued an active risk-management policy for the fund. The committee members were primarily concerned with the volatility and distribution of portfolio returns. They relied on the value-at-risk (VaR) methodology as a measurement of the risk of both short- and mid-term investment losses. In its report for the investment committee, the ACCF risk analytics team recommended the daily VaR at 95% confidence as a measure for short-term risk and reported the corresponding numbers. It is now the task of the investment committee to interpret these figures. The case questions guide the executive students to a critical evaluation of both the reported VaR figures as well as of the VaR methodology.Understanding the concept of value at risk (VaR); Interpreting the results of VaR calculations; Evaluating the appropriateness of VaR calculations; Critical discussion of the VaR methodology.

Multivariate approximations to portfolio return distribution

This article proposes a three-step procedure to estimate portfolio return distributions under the multivariate Gram-Charlier (MGC) distribution. The method combines quasi maximum likelihood (QML) estimation for conditional means and variances and the method of moments (MM) estimation for the rest of the density parameters, including the correlation coefficients. The procedure involves consistent estimates even under density misspecification and solves the so-called ‘curse of dimensionality’ of multivariate modelling. Furthermore, the use of a MGC distribution represents a flexible and general approximation to the true distribution of portfolio returns and accounts for all its empirical regularities. An application of such procedure is performed for a portfolio composed of three European indices as an illustration. The MM estimation of the MGC (MGC-MM) is compared with the traditional maximum likelihood of both the MGC and multivariate Student’s t (benchmark) densities. A simulation on Value-at-Risk (VaR) performance for an equally weighted portfolio at 1% and 5% confidence indicates that the MGC-MM method provides reasonable approximations to the true empirical VaR. Therefore, the procedure seems to be a useful tool for risk managers and practitioners.

Where Has the Trend Gone? An Update on Momentum Returns in the U.S. Stock Market

Prior studies find broad-based support for the efficacy of trading strategies based on momentum in stock returns. More recent studies, however, note a declining benefit relative to that identified in seminal studies on momentum. These recent studies, however, primarily examine time periods ending prior to or immediately following the 2008 financial crisis. Thus, we extend this line of research by exploring the profitability of momentum trading in the U.S. equity markets over the broader 1986 to 2015 time period. Our results for the earlier part of our sample period (i.e., 1986 to 2006) fall in line with previous studies, as we find a monotonic relationship between decile portfolios formed based on prior six month performance and their subsequent twelve month holding period excess returns. In contrast, when we evaluate more recent periods (i.e., 2007-2015 and 2010-2015), we find dramatically different results. In particular, alphas for the “winner” portfolio, which have historically been the highest, are actually negative during both sub-periods. Furthermore, the curvature of the recent portfolio return distribution is clearly no longer monotonic. Rather, it follows a more inverted U-shaped curve from the “winner” portfolio (P1) to the “loser” (P10) portfolio, with excess returns cresting around the fifth decile portfolio (P5). In fact, we find that the risk adjusted returns of the “winner” portfolio (P1) are significantly less than the middle portfolio (P5) for both recent sub-periods. These results suggest that individual and institutional investors seeking to profit from traditionally-based momentum trading strategies may need to rethink their approaches.