Should investors include Bitcoin in their portfolios? A portfolio theory approach

Many papers in recent years have examined the benefits of adding alternative assets to traditional portfolios containing stocks and bonds. Bitcoin has emerged as a new alternative investment for investors which has attracted much attention from the media and investors alike. However relatively little is known about the investment benefits of Bitcoin and therefore this paper examines the benefit of including Bitcoin in a traditional benchmark portfolio of stocks and bonds. Specially, we employ data up to June 2018 and analyse the potential out-of-sample portfolio benefits resulting from including Bitcoin in a stock-bond portfolio for a range of eight popular asset allocation strategies. The out-of-sample analysis shows that, across all different asset allocation strategies and risk aversions, the benefits of Bitcoin are quite considerable with substantially higher risk-adjusted returns. Our results are robust to rolling estimation windows, the incorporation of transaction costs, the inclusion of a commodity portfolio, alternative indices, short-selling as well as two additional optimization techniques including higher moments with (and without) variance-based constraints (VBCs). Therefore, our results suggest that investors should include Bitcoin in their portfolio as it generates substantial higher risk-adjusted returns.     

Optimal hedge fund portfolios under liquidation risk

We use an expected utility framework to integrate the liquidation risk of hedge funds into portfolio allocation problems. The introduction of realistic investment constraints complicates the determination of the optimal solution, which is solved using a genetic algorithm that mimics the mechanism of natural evolution. We analyse the impact of the liquidation risk, of the investment constraints and of the agent's degree of risk aversion on the optimal allocation and on the optimal certainty equivalent of hedge fund portfolios. We observe, in particular, that the portfolio weights and their performance are significantly affected by liquidation risk. Finally, tight portfolio constraints can only provide limited protection against liquidation risk. This approach is of special interest to fund of hedge fund managers who wish to include the hedge fund liquidation risk in their portfolio optimization scheme.     

Optimizing a portfolio of mean-reverting assets with transaction costs via a feedforward neural network

Optimizing a portfolio of mean-reverting assets under transaction costs and a finite horizon is severely constrained by the curse of high dimensionality. To overcome the exponential barrier, we develop an efficient, scalable algorithm by employing a feedforward neural network. A novel concept is to apply HJB equations as an advanced start for the neural network. Empirical tests with several practical examples, including a portfolio of 48 correlated pair trades over 50 time steps, show the advantages of the approach in a high-dimensional setting. We conjecture that other financial optimization problems are amenable to similar approaches.     

The cross-section of industry equity returns and global tactical asset allocation across regions and industries

This study investigates which index characteristics predict returns in the cross-section of local industry indexes in six regions. The results show that geographical origin and market capitalization of indexes critically determine the predictive ability of characteristics. We find that industry indexes of any market capitalization with high earnings-to-price ratio yield higher expected returns in the US, Europe, and Asia-Pacific. Recent winner (loser) portfolios in Europe have a tendency to outperform (underperform) recent loser (winner) portfolios in the near future for all groups of market capitalization. Small portfolios with high idiosyncratic volatility in Asia-Pacific earn an idiosyncratic volatility premium. Dividend yield is positively related to future returns of small European portfolios. These results are robust to the inclusion of transaction costs and control variables and have implications for portfolio managers following a global tactical asset allocation policy.     

Learning risk preferences from investment portfolios using inverse optimization

The level of risk an investor can endure, known as risk-preference, is a subjective choice that is tightly related to psychology and behavioral science in decision making. This paper presents a novel approach of measuring risk preference from existing portfolios using inverse optimization on mean–variance portfolio allocation framework. Our approach allows the learner to continuously estimate real-time risk preferences using concurrent observed portfolios and market price data. We demonstrate our methods on robotic investment portfolios and real market data that consists of 20 years of asset pricing and 10 years of mutual fund portfolio holdings. Moreover, the quantified risk preference parameters are validated with two well-known risk measurements currently applied in the field. The proposed methods could lead to practical and fruitful innovations in automated/personalized portfolio management, such as Robo-advising, to augment financial advisors’ decision intelligence in a long-term investment horizon.     

Multi-asset portfolio optimization and out-of-sample performance: an evaluation of Black–Litterman,mean-variance,and naïve diversification approaches

The Black–Litterman model aims to enhance asset allocation decisions by overcoming the problems of mean-variance portfolio optimization. We propose a sample-based version of the Black–Litterman model and implement it on a multi-asset portfolio consisting of global stocks, bonds, and commodity indices, covering the period from January 1993 to December 2011. We test its out-of-sample performance relative to other asset allocation models and find that Black–Litterman optimized portfolios significantly outperform naïve-diversified portfolios (1/N rule and strategic weights), and consistently perform better than mean-variance, Bayes–Stein, and minimum-variance strategies in terms of out-of-sample Sharpe ratios, even after controlling for different levels of risk aversion, investment constraints, and transaction costs. The BL model generates portfolios with lower risk, less extreme asset allocations, and higher diversification across asset classes. Sensitivity analyses indicate that these advantages are due to more stable mixed return estimates that incorporate the reliability of return predictions, smaller estimation errors, and lower turnover.     

Estimating a covariance matrix for market risk management and the case of credit default swaps

We analyze covariance matrix estimation from the perspective of market risk management, where the goal is to obtain accurate estimates of portfolio risk across essentially all portfolios—even those with small standard deviations. We propose a simple but effective visualisation tool to assess bias across a wide range of portfolios. We employ a portfolio perspective to determine covariance matrix loss functions particularly suitable for market risk management. Proper regularisation of the covariance matrix estimate significantly improves performance. These methods are applied to credit default swaps, for which covariance matrices are used to set portfolio margin requirements for central clearing. Among the methods we test, the graphical lasso estimator performs particularly well. The graphical lasso and a hierarchical clustering estimator also yield economically meaningful representations of market structure through a graphical model and a hierarchy, respectively.     

Household portfolio optimization with XTFs? An empirical study using the SHS-base

XTFs are plain-vanilla Exchange Traded Funds (ETFs) which replicate a broad, internationally diversified market index. We question, if XTFs can optimize the performance of households’ portfolios when taking multiple relevant asset classes into account, not only stocks. As opposed to most existing studies, we apply representative household portfolio data to estimate households’ portfolios. Households’ portfolios in our sample show similar compositions and can be grouped into one of three stylized portfolio compositions which exhibit asset class concentrations on cash/savings, mutual funds and individual stocks. For each stylized portfolio, we first investigate if an easily investable 60/40 stock/bond XTF portfolio which is risk-adjusted (including (de-)leverage costs) to the risk of the stylized portfolios, achieves higher returns than the stylized portfolios. This is the case for all stylized portfolios, even those with concentrations on cash/savings or mutual funds. Second, we examine risk/return-changes when replacing the entire risky assets of the stylized portfolios with the 60/40 stock/bond XTF portfolio including transaction costs. This leads to return enhancements in all stylized portfolios and particularly in the portfolio with high stock concentrations to risk reductions. Overall, we find that XTFs are generally suitable to optimize the performance of households’ portfolios under consideration of multiple relevant asset classes.     

Parameter uncertainty in portfolio selection: Shrinking the inverse covariance matrix

The estimation of the inverse covariance matrix plays a crucial role in optimal portfolio choice. We propose a new estimation framework that focuses on enhancing portfolio performance. The framework applies the statistical methodology of shrinkage directly to the inverse covariance matrix using two non-parametric methods. The first minimises the out-of-sample portfolio variance while the second aims to increase out-of-sample risk-adjusted returns. We apply the resulting estimators to compute the minimum variance portfolio weights and obtain a set of new portfolio strategies. These strategies have an intuitive form which allows us to extend our framework to account for short-sale constraints, transaction costs and singular covariance matrices. A comparative empirical analysis against several strategies from the literature shows that the new strategies often offer higher risk-adjusted returns and lower levels of risk.     

Two-sided coherent risk measures and their application in realistic portfolio optimization

By using a different derivation scheme, a new class of two-sided coherent risk measures is constructed in this paper. Different from existing coherent risk measures, both positive and negative deviations from the expected return are considered in the new measure simultaneously but differently. This innovation makes it easy to reasonably describe and control the asymmetry and fat-tail characteristics of the loss distribution and to properly reflect the investor’s risk attitude. With its easy computation of the new risk measure, a realistic portfolio selection model is established by taking into account typical market frictions such as taxes, transaction costs, and value constraints. Empirical results demonstrate that our new portfolio selection model can not only suitably reflect the impact of different trading constraints, but find more robust optimal portfolios, which are better than the optimal portfolio obtained under the conditional value-at-risk measure in terms of diversification and typical performance ratios.     

A Note on the Three–Portfolios Matching Problem

A typical problem arising in financial planning for private investors consists in the fact that the initial investor's portfolio, the one determined by the consulting process of the financial institution and the universe of instruments made available to the investor have to be matched/optimised when determining the relevant portfolio choice. We call this problem the three–portfolios matching problem. Clearly, the resulting portfolio selection should be as close as possible to the optimal asset allocation determined by the consulting process of the financial institution. However, the transition from the investor's initial portfolio to the final one is complicated by the presence of transaction costs and some further more specific constraints. Indeed, usually the portfolios under consideration are structured at different aggregation levels, making portfolios comparison and matching more difficult. Further, several investment restrictions have to be satisfied by the final portfolio choice. Finally, the arising portfolio selection process should be sufficiently transparent in order to incorporate the subjective investor's trade–off between the objectives 'optimal portfolio matching' and 'minimal portfolio transition costs'. In this paper, we solve the three–portfolios matching problem analytically for a simplified setting that illustrates the main features of the arising solutions and numerically for the more general situation.     

Transaction costs and crowding

We use industry data to determine whether crowding of the investment space is caused by portfolio construction processes typical to the investment community. In particular, this paper examines the extent that transaction cost models cause crowding of the investment space, even when the investment models are completely unrelated to one another. We find that as transaction costs become more significant in the portfolio creation process as portfolios increase in size from $500 million to $5 billion, crowding actually declines for long-only portfolios and mainly declines, but sometimes increases for market neutral portfolios. This research sheds more light on how crowding develops through actions by players within the financial system.     

Linear predictability vs. bull and bear market models in strategic asset allocation decisions: evidence from UK data

Most papers in the portfolio choice literature have examined linear predictability frameworks based on the idea that simple but flexible Vector Autoregressive (VAR) models can be expanded to produce portfolio allocations that hedge against the bull and bear dynamics typical of financial markets through careful selection of predictor variables that capture business cycles and market sentiment. Yet, a distinct literature exists that shows that non-linear econometric frameworks, such as Markov switching, are also natural tools to compute optimal portfolios arising from the existence of good and bad market states. This paper examines whether and how simple VARs can produce portfolio rules similar to those obtained under a simple Markov switching, by studying the effects of expanding both the order of the VAR and the number/selection of predictor variables included. In a typical stock-bond strategic asset allocation problem for UK data, we compute the out-of-sample certainty equivalent returns for a wide range of VARs and compare these measures of performance with those of non-linear models. We conclude that most VARs cannot produce portfolio rules, hedging demands or (net of transaction costs) out-of-sample performances that approximate those obtained from simple non-linear frameworks.     

Asset allocation and investment horizon

Quarterly recommendations by national brokerage firms since the third quarter of 1989 provide an opportunity to compare different approaches to asset allocation. To follow a brokerage firm's recommendation every quarter is to practice tactical asset allocation. Both the length of the investor's decision horizon and brokerage commissions that are incurred when portfolio changes are made impact investment performance, and both contribute to the risk experienced by investors. Buy-and-hold and strategic asset allocation would have served investors better than tactical asset allocation during the first half of the 1990s.     

Does sophistication of the weighting scheme enhance the performance of long-short commodity portfolios?

The commodity pricing literature advocates the design of long-short portfolios based on equal weights. Relaxing the assumption of naive diversification, this article studies the benefits of applying sophisticated weighting schemes to the construction of long-short momentum and term structure portfolios. Weighting schemes based on risk minimization and risk timing are found to dominate the naive allocation and the weighting schemes based on utility maximization. This conclusion is not challenged by concerns pertaining to transaction costs, illiquidity, data mining, sub-periods, and model parameters and robustly persists when we consider as sorting signals hedging pressure, speculative pressure and, to a lower extent, basis-momentum.     

In search of a theory of debt management

The complete market approach to government debt management argues that a portfolio of non-contingent bonds at different maturities should be chosen so that fluctuations in market value offset changes in expected future deficits. However, this approach recommends huge fluctuations in positions, enormous changes in portfolios for minor changes in maturities and no presumption it is always optimal to issue long and invest short term in a wide array of model specifications. These extreme, volatile and unstable features are undesirable for two reasons. Firstly fragility of portfolios to small changes in assumptions means that it is often better to follow a balanced budget rather than issue the optimal debt portfolio under some possibly misspecified model. Secondly for even miniscule transaction costs, governments prefer a balanced budget rather than the large positions complete markets recommends. The complete market recommendations conflict with a number of features we believe are integral to bond market incompleteness, e.g. transaction costs, liquidity effects, robustness, etc. and which need to be explicitly incorporated into the portfolio problem.     

Islamic and conventional portfolios optimization under investor sentiment states: Bayesian vs Markowitz portfolio analysis

This paper investigates the portfolio optimization under investor’s sentiment states of Hidden Markov model and over a different time horizon during the period 2004–2016. To compare the efficient portfolios of the Islamic and the conventional stock indexes, we have employed two approaches: the Bayesian and Markowitz mean-variance. Our findings reveal that the Bayesian efficient frontier of Islamic and conventional stock portfolios is affected by the investor’s sentiment state and the time horizon. Our findings also indicate that the investor’s sentiment regimes change the Islamic and the conventional optimal diversified portfolios.Moreover, the results show that the potential diversification benefits seem to be more important when using the Bayesian approach than when applying the Markowitz approach. This finding is valid for the bearish, depressed, bullish and calm states in Islamic stock markets. However, the diversification of potential portfolios is significant only for the bullish and the bubble states in the conventional financial markets.The findings of the study provided additional evidence for investors to exploit googling investor sentiment states to evaluate the portfolio performance and make an optimal portfolio allocation.     

Mean-variance optimization using forward-looking return estimates

Despite its theoretical appeal, Markowitz mean-variance portfolio optimization is plagued by practical issues. It is especially difficult to obtain reliable estimates of a stock’s expected return. Recent research has therefore focused on minimum volatility portfolio optimization, which implicitly assumes that expected returns for all assets are equal. We argue that investors are better off using the implied cost of capital based on analysts’ earnings forecasts as a forward-looking return estimate. Correcting for predictable analyst forecast errors, we demonstrate that mean-variance optimized portfolios based on these estimates outperform on both an absolute and a risk-adjusted basis the minimum volatility portfolio as well as naive benchmarks, such as the value-weighted and equally-weighted market portfolio. The results continue to hold when extending the sample to international markets, using different methods for estimating the forward-looking return, including transaction costs, and using different optimization constraints.

     

Optimal mean-reversion strategy in the presence of bid-ask spread and delays in capital allocations

A portfolio optimization problem for an investor who trades T-bills and a mean-reverting stock in the presence of proportional and convex transaction costs is considered. The proportional transaction cost represents a bid-ask spread, while the convex transaction cost is used to model delays in capital allocations. I utilize the historical bid-ask spread in US stock market and assume that the stock reverts on yearly basis, while an investor follows monthly changes in the stock price. It is found that proportional transaction cost has a relatively weak effect on the expected return and the Sharpe ratio of the investor's portfolio. Meantime, the presence of delays in capital allocations has a dramatic impact on the expected return and the Sharpe ratio of the investor's portfolio. I also find the robust optimal strategy in the presence of model uncertainty and show that the latter increases the effective risk aversion of the investor and makes her view the stock as more risky.     

On the investment credentials of Bitcoin: A cross-currency perspective

We examine diversification capabilities of Bitcoin for a global portfolio spread across six asset classes from the standpoint of investors dealing in five major fiat currencies namely US Dollar, Great Britain Pound, Euro, Japanese Yen and Chinese Yuan. Considering the period of prolonged decline in Bitcoin’s value throughout 2018, we employ modified conditional value-at-risk and standard deviation as measures of risk to perform portfolio optimisations across three asset allocation strategies. Results show that portfolios denominated in Japanese Yen, Chinese Yuan and US Dollar account for greater proportion of optimal investment in Bitcoin and exhibit higher improvement in risk-adjusted returns due to investment in Bitcoin. We also perform a comprehensive risk-adjusted evaluation of portfolios with and without Bitcoin to reinforce striking variation in degree of diversification benefits of Bitcoin in a cross-currency context. Taken together, our findings provide insights into sharp disparity in Bitcoin trading volumes across national currencies from a portfolio theory perspective.