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This paper digests three-factor model by exploring the average impacts of factors on portfolio returns and how factors interact with each other. To do this, we use SHapley Additive exPlanations method (SHAP) to interpret the results obtained by XGBoost. We find that the factors have different impacts on portfolio returns and interact with each other in different ways. We also find that the average impacts of factors on portfolio returns are similar before and after the publication of the three-factor model and the 2008 financial crisis but the interactions between factors vary across times.

The authors demonstrated in earlier papers that a maximal predictability portfolio (MPP) using a dynamic strategy leads to a significantly better ex-post performance than the one based on a static strategy and the index. In this paper, we will consider a maximal predictability portfolio subject to transaction cost. To reduce transaction cost, we employ turnover constraint. It will be shown that this approach leads to a significantly better performance than the standard MPP and the index.

In this paper, we will propose a practical method for improving the performance of a maximal predictability portfolio (MPP) model proposed by Lo and MacKinlay and later extended by the authors. We will employ an alternative version of MPP using absolute deviation instead of variance as a measure of fitting and apply a dynamic strategy for choosing the set of factors which fits best to the market data. It will be shown that this approach leads to a significantly better performance than the standard MPP and the index.

Traditional approaches to Arbitrage Pricing Theory (APT) propose a factor model, but empirical applications of APT are, nowadays, based on seemingly unrelated regression. I drop the factor model and assume only that the market is ergodic. This enables me to apply the theory of Hilbert spaces in a natural way. The expected return on any asset can always be approximated by an affine-linear function of its betas and we are able to estimate the relative number of assets that violate the APT equation by taking the expected returns and betas in the market into account. I present a simple sufficient condition for the APT equation in its inexact form. Further, I show that the APT equation holds true in its exact form if and only if an equilibrium market is exhaustive, which means that it must be possible to replicate the betas and idiosyncratic risk of each asset by some strategy that diversifies away all approximation errors in the market.

This paper presents an asset-pricing model for an integrated financial economy in a multi-currency framework in which three risk dimensions drive asset prices: global risk, regional risk and country-specific risk. Under this framework, all risks are common since, by trading assets across countries, agents can load on foreign risk. However, the model’s solution imposes restrictions on the loading coefficients. As a result, only the dispersion in global and regional coefficients is needed to explain currency returns. The model is tested with a linear-factor model at the currency-pair level using a sample of 42 countries located in five different regions. It is shown that, as the model predicted, regional and global factors help explain the dispersion in currency returns.

It is shown in this paper that when the true mean return vector is replaced by the inferred mean vector obtained indirectly from factor model and arbitrage pricing theory, its impact on the resulting optimal portfolio is insignificant. To achieve this goal, several assumptions are imposed: (i) the asset returns are generated from a factor model, (ii) the number of assets goes to infinity, and (iii) there is no asymptotic arbitrage opportunities. Issues related to the efficiency of the estimated optimal portfolio for high-frequency data are discussed. The portfolio constructed using the sample mean vector and using the inferred mean vector from arbitrage pricing theory are compared.

Given an N-dimensional sample of size $T,$ form a sample correlation matrix $C$. Suppose that N and T tend to infinity with $T/N$ converging to a fixed finite constant $Q>0$. If the population is a factor model, then the eigenvalue distribution of $C$ almost surely converges weakly to Marčenko–Pastur distribution such that the index is Q and the scale parameter is the limiting ratio of the specific variance to the ith variable $(i\to \infty )$. For an N-dimensional normal population with equi-correlation coefficient $\rho$, which is a one-factor model, for the largest eigenvalue $\lambda$ of $C$, we prove that $\lambda /N$ converges to the equi-correlation coefficient $\rho$ almost surely. These results suggest an important role of an equi-correlated normal population and a factor model in Laloux et al. [(2000) Random matrix theory and financial correlations, International Journal of Theoretical and Applied Finance3 (3), 391–397]: the histogram of the eigenvalue of sample correlation matrix of the returns of stock prices fits the density of Marčenko–Pastur distribution of index $T/N$ and scale parameter $1-\lambda /N$. Moreover, we provide the limiting distribution of the largest eigenvalue of a sample covariance matrix of an equi-correlated normal population. We discuss the phase transition as to the decay rate of the equi-correlation coefficient in N.

This paper seeks to identify which factors are important for estimating portfolio's expected return and standard deviation in the Taiwan stock market. We have summarized from the existing empirical literature a total of 26 factors that may have explanatory power. The results of our evaluation show that except for the trading volume, the remaining 25 factors do not seem to help explain the average stock returns during the July 1985–June 1999 period. However, the power of the trading volume to account for the expected returns on the stock is affected by any changes in the sample or by the use of a different evaluation model. We suggest three potential explanations of why all 26 factors show no stable power to explain average returns on Taiwan stocks: high volatility, selection bias, and market differences. Moreover, we find that all of the 26 factors are important in capturing the systematic covariation in stock returns.

Recent progress in forecasting emphasizes the role of nonlinear factor models. In the simplest case, the nonlinearity appears in the link function. But even in this case, the classical forecasting methods, such as principal components analysis, do not perform well. Another challenge when dealing specially with financial data is the heavy-tailedness of data. This brings another difficulty to the classical forecasting methods. There are recent works in sufficient forecasting which use the technique of sliced inverse regression and local regression methods to overcome the nonlinearity. In this paper, we first observe that for heavy-tailed data, the existing approaches fail. Then we show that a suitable combination of two known methods of kernel principal component analysis and $k$-nearest neighbor regression improves the forecasting dramatically.

In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). Determining the number of spikes is a fundamental problem which appears in many scientific fields, including signal processing (linear mixture model) or economics (factor model). Several recent papers studied the asymptotic behavior of the eigenvalues of the sample covariance matrix (sample eigenvalues) when the dimension of the observations and the sample size both grow to infinity so that their ratio converges to a positive constant. Using these results, we propose a new estimator based on the difference between two consecutive sample eigenvalues.

This study examines whether there is a strong relationship between stock liquidity, which proxies for the implicit cost of trading shares, and future stock returns in an asset-pricing context in the UK stock market. The time period, 1994–2016, includes the most recent global financial crisis that drained liquidity from financial markets worldwide. Four different measures of stock liquidity are employed; the empirical findings indicate that liquidity is a systematic pricing factor and explains a significant portion of the variation in stock returns, even after the inclusion of the other traditional risk factors. The results are robust to both forms of liquidity, either as a residual effect or in its original form as a separate risk factor. Finally, for the first time quantile regression is applied, showing that the liquidity risk factor (LIQ) absorbs a significant portion of the information content of the size and value factors, while remaining independent of the momentum factor.

We study three widely used liquidity measures and find that they all carry significant premiums beyond the size, book-to-market, and momentum effects. Although liquidity as a risk factor bears a significant return premium, it is better characterized by a characteristic-based model. Further analysis shows that (1) although the premium persists for up to five years following formation, it diminishes over time and becomes insignificant in the post-1960 period; (2) the premium is larger for stocks with higher idiosyncratic risk. Thus, the empirical results provide some evidence that supports the mispricing argument.