Prospect theory–based portfolio optimization: an empirical study and analysis using intelligent algorithms

The behaviourally based portfolio selection problem with investor’s loss aversion and risk aversion biases in portfolio choice under uncertainty is studied. The main results of this work are: developed heuristic approaches for the prospect theory model proposed by Kahneman and Tversky in 1979 as well as an empirical comparative analysis of this model and the index tracking model. The crucial assumption is that behavioural features of the prospect theory model provide better downside protection than traditional approaches to the portfolio selection problem. In this research the large-scale computational results for the prospect theory model have been obtained for real financial market data with up to 225 assets. Previously, as far as we are aware, only small laboratory tests (2–3 artificial assets) have been presented in the literature. In order to investigate empirically the performance of the behaviourally based model, a differential evolution algorithm and a genetic algorithm which are capable of dealing with a large universe of assets have been developed. Specific breeding and mutation, as well as normalization, have been implemented in the algorithms. A tabulated comparative analysis of the algorithms’ parameter choice is presented. The prospect theory model with the reference point being the index is compared to the index tracking model. A cardinality constraint has been implemented to the basic index tracking and the prospect theory models. The portfolio diversification benefit has been found. The aggressive behaviour in terms of returns of the prospect theory model with the reference point being the index leads to better performance of this model in a bullish market. However, it performed worse in a bearish market than the index tracking model. A tabulated comparative analysis of the performance of the two studied models is provided in this paper for in-sample and out-of-sample tests. The performance of the studied models has been tested out-of-sample in different conditions using simulation of the distribution of a growing market and simulation of the t-distribution with fat tails which characterises the dynamics of a decreasing or crisis market.     

High-dimensional index tracking based on the adaptive elastic net

When a portfolio consists of a large number of assets, it generally incorporates too many small and illiquid positions and needs a large amount of rebalancing, which can involve large transaction costs. For financial index tracking, it is desirable to avoid such atomized, unstable portfolios, which are difficult to realize and manage. A natural way of achieving this goal is to build a tracking portfolio that is sparse with only a small number of assets in practice. The cardinality constraint approach, by directly restricting the number of assets held in the tracking portfolio, is a natural idea. However, it requires the pre-specification of the maximum number of assets selected, which is rarely practicable. Moreover, the cardinality constrained optimization problem is shown to be NP-hard. Solving such a problem will be computationally expensive, especially in high-dimensional settings. Motivated by this, this paper employs a regularization approach based on the adaptive elastic-net (Aenet) model for high-dimensional index tracking. The proposed method represents a family of convex regularization methods, which nests the traditional Lasso, adaptive Lasso (Alasso), and elastic-net (Enet) as special cases. To make the formulation more practical and general, we also take the full investment condition and turnover restrictions (or transaction costs) into account. An efficient algorithm based on coordinate descent with closed-form updates is derived to tackle the resulting optimization problem. Empirical results show that the proposed method is computationally efficient and has competitive out-of-sample performance, especially in high-dimensional settings.     

A dynamic model of active portfolio management with benchmark orientation

This paper studies optimal dynamic portfolios for investors concerned with the performance of their portfolios relative to a benchmark. Assuming that asset returns follow a multi-linear factor model similar to the structure of Ross (1976) [Ross, S., 1976. The arbitrage theory of the capital asset pricing model. Journal of Economic Theory, 13, 342–360] and that portfolio managers adopt a mean tracking error analysis similar to that of Roll (1992) [Roll, R., 1992. A mean/variance analysis of tracking error. Journal of Portfolio Management, 18, 13–22], we develop a dynamic model of active portfolio management maximizing risk adjusted excess return over a selected benchmark. Unlike the case of constant proportional portfolios for standard utility maximization, our optimal portfolio policy is state dependent, being a function of time to investment horizon, the return on the benchmark portfolio, and the return on the investment portfolio. We define a dynamic performance measure which relates portfolio’s return to its risk sensitivity. Abnormal returns at each point in time are quantified as the difference between the realized and the model-fitted returns. Risk sensitivity is estimated through a dynamic matching that minimizes the total fitted error of portfolio returns. For illustration, we analyze eight representative mutual funds in the U.S. market and show how this model can be used in practice.     

Diversification and generalized tracking errors for correlated non‐normal returns

Abstract

The probability distribution for the relative return of a portfolio constructed from a subset n of the assets from a benchmark, consisting of N assets whose returns are multivariate normal, is completely characterized by its tracking error. However, if the benchmark asset returns are not multivariate normal then higher moments of the probability distribution for the portfolio's relative return are not related to its tracking error. We discuss the convergence of generalized tracking error measures as the size of the subset of benchmark assets increases. Assuming that the joint probability distribution for the returns of the assets is symmetric under their permutations we show that increasing n makes these generalized tracking errors small (even though n « N). For n » 1 the probability distribution for the portfolio's relative return is approximately symmetric and strongly peaked about the origin. The results of this paper generalize the conclusions of Dynkin et al (Dynkin L, Hyman J and Konstantinovsky V 2002 Sufficient Diversification in Credit Portfolios (Lehman Brothers Fixed Income Research)) to more general underlying asset distributions.     

Do low-carbon investments in emerging economies pay off? Evidence from the Brazilian stock market

Climate change has created both challenges and opportunities for investors worldwide. Investing in carbon-efficient assets, for instance, may reduce investors' climate risks while contributing to global efforts for climate change mitigation. Investors need updated and robust information on the financial performance of low-carbon investments, especially in emerging markets, where climate finance initiatives are still scattered. In this work, we provide a first insight into the financial performance of a portfolio of shares from Brazilian carbon-efficient companies. To that end, we use as reference the Carbon Efficient Index (ICO2) and assess its financial performance from 2010 to 2019 through the lens of several classic and modern portfolio metrics. We find that the index outperformed both the Brazilian market benchmark and the country's broad sustainability index, and provided competitive risk-adjusted returns compared with other sectorial indices. The results thus indicate that investing in carbon-efficient companies in Brazil has so far positively contributed to portfolio performance while offering investors an opportunity to reduce climate risk exposure in stock markets.     

基于支持向量机的金融市场指数追踪技术研究

本文研究了指数型基金管理和指数套利中最核心的指数追踪问题。依据结构风险最小化思想,建立了基于支持向量机的指数追踪模型,并利用OR-Library中的测试数据之恒生指数历史数据进行了实证检验。数据结果表明本文提出的新方法能够提高样本外的追踪效果,同时也说明该方法具有良好的鲁棒性,从而具有较高的理论和实用价值。     

Cardinality versus q-norm constraints for index tracking

Index tracking aims at replicating a given benchmark with a smaller number of its constituents. Different quantitative models can be set up to determine the optimal index replicating portfolio. In this paper, we propose an alternative based on imposing a constraint on the q-norm (0?<?q?<?1) of the replicating portfolios’ asset weights: the q-norm constraint regularises the problem and identifies a sparse model. Both approaches are challenging from an optimization viewpoint due to either the presence of the cardinality constraint or a non-convex constraint on the q-norm. The problem can become even more complex when non-convex distance measures or other real-world constraints are considered. We employ a hybrid heuristic as a flexible tool to tackle both optimization problems. The empirical analysis of real-world financial data allows us to compare the two index tracking approaches. Moreover, we propose a strategy to determine the optimal number of constituents and the corresponding optimal portfolio asset weights.     

The diversification effects of volatility-related assets

We examine whether investors can improve their investment opportunity sets through the addition of volatility-related assets into various groupings of benchmark portfolios. By first analyzing the weekly returns of three VIX-related assets over the period 1996-2008 and then applying mean-variance spanning tests, we find that adding VIX-related assets does lead to a statistically significant enlargement of the investment opportunity set for investors. Our empirical findings are robust and have two implications. First, there is scope for the further development of financial products relating to volatility indexes. Second, hedge fund managers can utilize VIX futures contracts or VIX-squared portfolios to enhance their equity portfolio performance, as measured by the Sharpe ratio.     

Long-term U.S. infrastructure returns and portfolio selection

Our understanding of the long-term return behavior and portfolio characteristics of public infrastructure investments is limited by a relatively short history of empirical data. We re-construct U.S. listed infrastructure index returns by mapping their monthly performance to received systematic and industry risk factors from 1927 through 2010. Our findings reveal that the infrastructure returns in recent years may understate the tail-risk that investors could experience over the long-term, however, this tail-risk is commensurate with holding a broad portfolio of U.S. stocks. For mean-variance and mean-CVaR investors, we report the benefits of holding public infrastructure assets in investment portfolios.     

Linear programing models for portfolio optimization using a benchmark

We consider the problem of constructing a perturbed portfolio by utilizing a benchmark portfolio. We propose two computationally efficient portfolio optimization models, the mean-absolute deviation risk and the Dantzig-type, which can be solved using linear programing. These portfolio models push the existing benchmark toward the efficient frontier through sparse and stable asset selection. We implement these models on two benchmarks, a market index and the equally-weighted portfolio. We carry out an extensive out-of-sample analysis with 11 empirical datasets and simulated data. The proposed portfolios outperform the benchmark portfolio in various performance measures, including the mean return and Sharpe ratio.     

Portfolio Optimization under Solvency Constraints: A Dynamical Approach

We develop portfolio optimization problems for a nonlife insurance company seeking to find the minimum capital required that simultaneously satisfies solvency and portfolio performance constraints. Motivated by standard insurance regulations, we consider solvency capital requirements based on three criteria: ruin probability, conditional Value-at-Risk, and expected policyholder deficit ratio. We propose a novel semiparametric formulation for each problem and explore the advantages of implementing this methodology over other potential approaches. When liabilities follow a Lognormal distribution, we provide sufficient conditions for convexity for each problem. Using different expected return on capital target levels, we construct efficient frontiers when portfolio assets are modeled with a special class of multivariate GARCH models. We find that the correlation between asset returns plays an important role in the behavior of the optimal capital required and the portfolio structure. The stability and out-of-sample performance of our optimal solutions are empirically tested with respect to both the solvency requirement and portfolio performance, through a double rolling window estimation exercise.     

A semiparametric graphical modelling approach for large-scale equity selection

We propose a new stock selection strategy that exploits rebalancing returns and improves portfolio performance. To effectively harvest rebalancing gains, we apply ideas from elliptical-copula graphical modelling and stability inference to select stocks that are as independent as possible. The proposed elliptical-copula graphical model has a latent Gaussian representation; its structure can be effectively inferred using the regularized rank-based estimators. The resulting algorithm is computationally efficient and scales to large data-sets. To show the efficacy of the proposed method, we apply it to conduct equity selection based on a 16-year health care stock data-set and a large 34-year stock data-set. Empirical tests show that the proposed method is superior to alternative strategies including a principal component analysis-based approach and the classical Markowitz strategy based on the traditional buy-and-hold assumption.     

Portfolio optimization for heavy-tailed assets: Extreme Risk Index vs. Markowitz

Using daily returns of the S&P 500 stocks from 2001 to 2011, we perform a backtesting study of the portfolio optimization strategy based on the Extreme Risk Index (ERI). This method uses multivariate extreme value theory to minimize the probability of large portfolio losses. With more than 400 stocks to choose from, our study seems to be the first application of extreme value techniques in portfolio management on a large scale. The primary aim of our investigation is the potential of ERI in practice. The performance of this strategy is benchmarked against the minimum variance portfolio and the equally weighted portfolio. These fundamental strategies are important benchmarks for large-scale applications. Our comparison includes annualized portfolio returns, maximal drawdowns, transaction costs, portfolio concentration, and asset diversity in the portfolio. In addition to that we study the impact of an alternative tail index estimator. Our results show that the ERI strategy significantly outperforms both the minimum-variance portfolio and the equally weighted portfolio on assets with heavy tails.     

On Indexing Commercial Real Estate Properties and Portfolios

Commercial real estate indices play an important role in performance evaluation and overall investment strategy. However, the issue of how representative they are of the returns on portfolios of commercial properties is an open issue. Our study addresses this topic by analyzing a sample of 12,427 repeat sales transactions between Q4 2000 and Q2 2011. We find that the aggregate real estate indices (Moody’s REAL CPPI) do a good job of tracking real returns when portfolios of more than 20 properties are considered. At this level, tracking is somewhat less effective than our benchmark of the S&P500 and its component stocks. Compared to the average root mean squared deviation (RMSD) from one asset, randomly selected portfolios with 20 assets reduce the RMSD by 75 % for the S&P500 compared to 66 % for the aggregate index. These results suggest that the aggregate indices can be effective in hedging and evaluating the performance of direct real estate investment. We further find that tracking at the property type level provides little benefit over using an aggregate index. However, indexing using a property type and location matched index provides lower tracking error for any level of diversification.     

The diversification cost of large,concentrated equity stakes. How big is it? Is it justified?

While the hypothesis that ownership concentration can affect the value of a company has seen a lot of empirical study, little light has been shed on a complementary problem, that these concentrated owners have a cost of their position due to an undiversified portfolio. Using a unique data set of the actual diversification of all Norwegian equity owners, we show that the largest owners of a corporation in fact have very undiversified equity portfolios, and that such owners have significant costs to their concentrated portfolios. At the level of risk of a benchmark portfolio, if they were to move from their present portfolio composition in risky assets to a well diversified portfolio, their returns would have increased by about 13 percentage points in annual terms. We ask whether this cost can be explained by estimated benefits of ownership concentration (private benefits), and show that extant estimates of private benefits are too low to offset our cost estimates.     

Benchmarking the performance of recommended allocations to equities,bonds, and cash by international investment houses

We conduct performance tests of the recommended asset allocations made by a panel of international investment houses (the “Houses”) from 1982 through 2005. We compare the returns and Sharpe Ratios from the recommended-weight portfolio against those of several benchmark portfolios and to a set of 10,000 returns and Sharpe Ratios from randomly shuffled-weight and shuffled-weight change portfolios. We find that the Houses generally fail to outperform the benchmarks. The shuffled-weight change benchmark exhibits a robust “style-preserving” property in that the average portfolio standard deviation is nearly equal to the portfolio standard deviation from the actual recommended weights.     

Applying Time Series Decomposition to Construct Index-Tracking Portfolio

This study proposes a new method for creating an index-tracking portfolio using time series decomposition. First, we construct index-tracking portfolios of stocks chosen because their price movements mimic that of the Dow-Jones Industrial Average. Our method utilizes similarities of constituent stocks to the benchmark that are assessed by distances of time series trends derived from decomposing original series. Although the portfolios chosen by our method reasonably tracked the performance of the benchmark, they did not surpass the clustering approach discussed in earlier studies. Therefore, we examined what causes tracking error and found that two causes for deficiencies in our similarity-based method, which are unintended irregular movements of holding stocks and highly correlated relationships within stocks in the portfolio. To overcome them and to improve tracking performance, we propose a similarity-balanced approach that is another index-tracking method with alternate use of similarity. Doing so improved the tracking performance by avoiding the problem of high correlation among the stocks chosen under the initial method.     

Hedge funds and optimal asset allocation: Bayesian expectations and spanning tests

This paper analyzes the contribution of hedge funds to optimal asset allocations between 1993 and 2010. The preferences of specific institutional investors are captured by implementing a Bayesian asset allocation framework that incorporates heterogeneous expectations regarding hedge fund alpha. Mean-variance spanning tests are used to infer the ability of hedge funds to significantly enhance the mean-variance efficient frontier. Further, a novel democratic variance decomposition procedure sheds light on the dynamics in the co-movement of hedge fund returns with a set of common benchmark assets. The empirical findings indicate that portfolio benefits of hedge funds are time-varying and strongly depend on investor optimism regarding hedge funds’ ability to generate alpha. In general, allocations to hedge funds improve the global minimum variance portfolio even after controlling for short-selling restrictions and minimum diversification constraints. However, due to dynamics underlying the composition of the aggregate hedge fund universe, the factor structure of hedge fund returns has become more similar to the benchmark assets over time.     

Non-fully invested derivative-free bond index replication

The problem we address here is the replication of a bond benchmark when only a fraction of the portfolio is invested for the replication. Our methodology is based on a minimization of the tracking error subject to a set of constraints, namely (1) the fraction invested for the replication, (2) a no-short-selling constraint, and (3) a null-active-duration constraint, the last of which may be relaxed. The constraints can also be adapted to accommodate the use of interest rate and bond futures. Our main contribution, however, is our derivative-free approach to replication, which should prove very useful for managing assets when the use of derivatives is prohibited, for instance, by certain investors. We can, thus, still benefit from replicating a traditional investment in a bond index with a fraction of the portfolio according to our risk appetite. The rest of the portfolio can be invested in alpha-portable strategies. An analysis without the use of derivatives over the period from January 1, 2008 to October 3, 2011 shows that with 70–90 % invested for the replication, the annualized ex-ante tracking error can range from 0.41 to 0.07 %. We use principal component analysis to extract the main drivers of the size of the tracking error, namely, the volatility of and the differential between the yields in the objective function’s covariance matrix of spot rates. These results highlight our contribution of a generic and intuitive yet robust approach to bond index replication.