Critically discussing the application of multi-factor asset pricing models

Overview

The development of asset pricing models in the financial industry demonstrates the continuous necessity of better understanding and predicting asset returns. The limitations presented by the CAPM[1], - which set the groundwork by first quantifying the market risk - developed the need to explain the market anomalies through more advanced multifactor models. By incorporating additional factors that influence returns, these models captured the effects not previously reflected in CAPM.

Firstly, the Fama-French three-factor model accounted for extra size and value factors, whereas Carhart’s four-factor model and Fama-French five- and six-factor models integrated momentum and other factors such as investment strategies and profitability factors, reflecting the historical behavior of the stocks as well as the long/short position risk exposure for different investors that need to be compensated for. Recently, machine learning techniques have advanced to reflect the non-linear relationship between patterns within financial data.

This literature review explores the developments of these models, analyzing their components, empirical validations, and their applications in modern finance and portfolio management.

Overview of multifactor asset pricing models

Initially developed by Sharpe (1964) based on Markowitz’s (1952) diversification theory, the CAPM considered beta risk and the market factor. This left space for introducing short sales and higher borrowings into the theory, as Black (1972) suggested in his zero-beta CAPM model. These developments built the foundations of traditional single-period CAPM theory.

The introduction of ICAPM[2] by Merton (1973) and its extensions[3], who expanded CAPM into multiperiod frameworks, marked a significant advancement in developing a multifactor approach. Many scholars believe that this model has paved the way for the development of multifactor models (Kolari et al., 2023).

Addressing the practical limitations of the CAP-M, such as reliance on i) single period frameworks, ii) addressing only systematic risk factors, and iii) the avoidance of risk appetition in investors' behaviors (Domian et al., 2007 & Ward and Muller, 2012), researchers delved into new capital asset pricing models that involved more than one risk factor in the equation. Aiming to provide a more accurate and inclusive result, these models are spread into two groups: i) discretional multifactor models and ii) machine learning models. The first group is based on the researcher’s judgmental view of factors, whereas the second corresponds to using AI[4] to avoid human error.

Based on long/short risk factors, Ross (1976) developed the APT[5], which emphasizes the unidentified risk factors of zero-investment portfolios[6], providing a solid theoretical foundation for the following models in the first group. By identifying more long/short relations that explained average returns, researchers developed three-, four-, five-, and six-factor models, discussed in the following section. These models benefited from the redundancy of long/short relations and suitable data gathered, whereas they suffered from the “factor zoo” problem[7] (Cochrane, 2011).

The second group of multifactor models aims to reduce human error by optimizing the process through trial-and-error and removing the researcher’s discretion (Gu et al., 2020).?

Popular asset pricing models in the financial industry

Driven by the APT model and the inconsistencies of CAPM, Fama and French (1992) proposed the three-factor asset pricing model. This model explained asset pricing better by adding the CAPM with two extra factors: market capitalization and book-to-market value factors. The first factor accounts for small-cap companies outperforming large-cap companies in the long run due to higher growth potential and risk, necessitating higher returns to compensate for the investors needs. ?The second factor shows the value premium, indicating the spread of returns between value stocks and growth stocks.

According to Kolari et al. (2023), another popular multifactor model in the financial markets is Carhart's Four-factor model, which augments the FF-3 model with a momentum factor. Driven by the findings of Jegadeesh et al. (1993), this model was validated through U.S. mutual funds data from 1963-1993, depicting its returns accurately (Carhart, 1997).?

Fama and French (2015) further developed their asset pricing model into a five-factor model by adding the profitability and investment factors, hence incorporating insights from Novi-Marx (2013) and Titman et al. (2004). The profitability factor accounts for the difference in returns between high-profit and low-profit firms, reflecting the stability and competitive advantages of firms with strong earnings and cash flows, as these firms are better positioned to endure economic downturns and seize growth opportunities. The investment factor captures the return disparity between firms with conservative and aggressive capital investments. Firms with conservative investments demonstrate more discipline, avoiding overexpansion and excessive risk, whereas aggressive investment may indicate overconfidence, potentially leading to overvaluation and underperformance.

Another widely used model is the Six-factor model. Fama and French (2018) added a momentum factor to their five-factor model, accounting for the tendency of those stocks with strong past performance to continue performing well in the short term, while those with poor past performance tend to keep underperforming. In this manner, they accounted for the fact that well-performing stocks attract more investors, driving up prices, while poorly performing stocks exhibit reduced interest and prices.

Lately, machine learning models have been expanded in the market. Fama and French (2020) used AI to solve their cross-sectional regression equation. Additionally, Lettau and Pelger (2020) also introduced their PCA[8] model to generate asset pricing factors based on past data trial-and-error runs. Yi et al. (2022) illustrate the machine learning process build-up and compositions.

Empirical evidence and validation of multifactor models

The multifactor models mentioned in the previous section have been developed based on empirical tests to show their applicability to market valuations. ?Evidence from the Chinese market during 1995-2008 shows that the FF3 model accurately described the cross-section of stock returns, in line with market speculation activity, devaluing the market beta. Nevertheless, the model failed to capture the small-size firm effect in this set of data (Li & Dempsey, 2018). Based on the findings from Jiao and Lilti (2017), the FF3 model could generate accurate results for 90% of the time-series variations on the Chinese A-share stock market in the 2010-2015 period. In contrast, the same was not reflected in the US market data. Due to market segmentation, Brooks et al. (2009) also fail to fully integrate the risk factors for the US stock market, whereas he could integrate them fully for the Chinese markets. Comparing this model with CAPM performance, when applied in the Indian markets in 2003-2019, Sehrawat et al. (2020) show that FF3 provided a better result.

According to Kolari et al. (2023), even though there is empirical evidence supporting the effects that momentum brings to the returns, the implementation of the momentum factor in Carhart’s model lacks the theoretical foundation and results in high volatility during economic recessions. When applied in the French equity markets, evidence shows that this model explains the cross-section of returns for medium and long-term investment horizons (Trimech and Kortas, 2009). The same stands for the Polish markets, as Zaremba et al. (2019) lists Carhart’s model as the best-performing when compared to FF3 and FF5.

Fama and French (2015) claim that the FF5 model is superior to the previous ones, statistically showing how this model better explains the cross-sectional return for the US stocks with new and longer data from 1967 to 2013. When applied in the UK markets (FTSE350[9]), the FF5 model prevailed among others (Nichol and Dowling, 2014). In addition, Zaremba et al. (2017) pointed out that the FF5 model outperforms other models in emerging markets, better explaining the returns of all capitalization-weighted and many equal-weighted cross-sectional patterns. However, when tested in the Japanese market with data set from 1978 to 2014, Kubota et. al (2018) show that this model is not the best benchmark pricing model, finding that profitability and investments’ factor betas are weakly correlated with the cross-sectional variations of stock returns.

When testing the FF-6 model in the TSM[10], evidence from Do?an et al. (2022) implied that FF-6 outperforms the previous models. In addition, Mesut et al. (2022) tested the validity of this model through a data set of Turkish stocks from 2013 to 2022, categorized as per Fama and French criteria applied in the US stock dataset. They concluded that this model outperforms the previous ones. When applied to the German market, O’Connell (2023) shows that adding the momentum factors offers a dominant explanatory model for the stock returns, analyzed by a data set from 1991 to 2021.

Ye et al. (2024) offer an empirical review of the asset pricing ML[11] models, exploring a new field of finance AI. This study shows that the focus on stock return generated from the previous models composes just a fraction of ML’s potential. Through different sets of models[12] generated from ML techniques, they conclude that this approach will be supreme in the future. When applied to the U.S equities, Gu, Kelly, and Xiu (2020) derived that the neural network models offered a higher effectiveness in generating the right return value, showing that it can improve the Sharpe ratio to 0.71, compared to 0.51 that was previously, for a 30-year sample period for S&P500. Likewise, Abe and Nakagawa (2020) concluded the same when applying the neural DL[13] models to the global markets.

Application of multifactor models in portfolio management

Another means of validating these models is checking their continuous applicability in portfolio management companies. These models have gained substantial traction among institutional investors, hedge funds, and mutual funds, reflecting their tendency to examine risk factors better and adopt more sophisticated investment strategies. The adoption rate has continuously increased since their initial application (CFA Institute, 2022).

Hedge funds are known for their advanced and proprietary models. Many hedge funds use multifactor models to identify arbitrage opportunities and construct long/short portfolios that exploit factor premiums. Fang and Almeida (2019) provided an investigative view of the applicability of first-order models[14] to hedge funds returns. They conclude that MKT[15] is the most important risk factor considered in the hedge fund equity risk analysis. However, Blocher and Molyboga (2017) reverse this claim, finding that CAPM is investors' preference in hedge funds. They mention that CAPM’s alpha correlates with managerial skills and predicts performance better.

Pension funds and endowments optimize asset allocation and risk management processes using multifactor models. There is increased evidence for using FF3- and FF6- factor models widely used to evaluate portfolio performance (CFA Institute, 2022).

Analyzes from Andrew Ang (2014) show that mutual funds have also integrated multifactor models into their investment strategies. This approach is enhanced in those focused on quantitative and systematic approaches, such as the creation of factor-based ETF’s[16], allowing investors to target specific factors like value or size within their portfolios.

Moreover, Lekander (2024) shows that multifactor models optimize equity and fixed-income allocations for the Norwegian Sovereign Wealth Fund. The fund's strategy involves decomposing portfolio risk into various factors, enabling better risk management and more efficient capital allocation.

The ML multifactor models have also demonstrated an increased trend of application in the industry. However, they lack a theoretical foundation and are more mechanically constructed (Kolari et al., 2023).

Advantages and disadvantages of multifactor models in portfolio management

This section will critically discuss the advantages and disadvantages of the most popular multifactor asset pricing models. Firstly, an enhanced explanatory power is generated from these multifactor models. Incorporating other-than-traditional factors such as size, value, momentum, profitability, or investment strategies provides a broader spectrum of analysis and accounts for the anomalies in the market. Liew and Vassalou (1999) show that incorporating size and value offers a better ability to predict future economic growth in developed countries, as they serve as variables that predict future changes in investment opportunities. In addition, Fama and French (2015) support this claim by finding that, whereas the value of the β coefficient can explain 70% of the expected return, the other part is attributable to other factors introduced by FF3-and FF5 models. Moreover, this claim was also supported by Paliienko (2020), which focus on the explanatory power of the FF5 model when addressing the controversies and anomalies in small-size and large-size US companies using ANOVA and STARR[17] tests. They concluded that the higher explanatory power is higher in small-size companies rather than large-sized ones.

However, the benefits of applying these models are highly context-based. Subrahmanyam (2010) and Blanco (2012) first prove their context-based nature when explaining the expected return for different data sets and periods.

Multifactor asset pricing models also have the advantage of covering diverse factors. Pioneering other models in the introduction of the momentum factor, Carhart’s four-factor model served as a good basis for capturing the effects of momentum in mutual fund returns. This effect resulted in a lower Jensen’s alpha calculation and predicted correct returns for U.S. mutual funds from 1963 to 1993, hence empowering the predictive feature of the model (Kolari et al., 2023).

Furthermore, ML models serve as powerful practices, showing their advantages and serving as comparison benchmarks. By combining a suitable set of factors, they capture the non-linearity of cause-and-effect relationships between the factors. Such models can navigate through data noise and process large data sets designed properly among criteria (Gu, Kelly, and Xiu, 2020).

Being data-driven, ML models can adapt to new data and improve accuracy over time by incorporating many factors beyond market risk and can be tailored to specific objectives (Dutta and Sinha, 2022).??

On the contrary, these models also have disadvantages, offset as they evolve, optimizing and testing the previous versions. Researchers commonly claim that these models are developed to operate under the assumption of homogeneous expectations, meaning that all investors have the same perception regarding future cash flows and risks of assets when analyzing the financial performance of different financial instruments. This leads to increased standard error margins (CFA Institute, 2022). This is also referred to as the static nature of these models.

Another concern, which is being addressed in ML models, relates to the linear relation that the multifactor models possess. Not all relationships between factors are treated as linear, and a non-linear treatment is requested to explain the expected return better. However, according to Lettau and Pelger (2020), implementing ML models is highly costly.

Findings

The evolution from CAPM to multifactor and ML models in asset pricing validates the dynamic nature of financial research and practice. From Fama-French FF3 to ML-driven approaches, each model has reflected the deficiencies of the previous ones, providing accurate and comprehensive tools for understanding asset returns. These models have demonstrated solid empirical validations for additional factors such as size, value premium, profitability, and investment factors. Furthermore, the implementation of ML techniques marks a forward-looking self-fixing approach, enhancing computational power to avoid human error and enhance precision. The application improves theoretical understanding and finds practical implications in investment strategies and risk procedures. As advancements proceed and in order to adapt to new data and emerging market trends, the ongoing refinement of these models will remain essential to address it. This evolving trend to better capture the complexities of asset returns through new models within the financial industry will contribute to more informed decision-making processes and, consequently, to higher financial stability.?

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References

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[1] Capital Asset Pricing Method

[2] Intertemporal CAPM

[3] ICAPM produced several extensions, such as the international asset pricing model (Solnik, 1974), consumption CAPM (Lucas, 1978 & Breeden, 1979), production CAPM (Cochrane, 1991), and conditional CAPM (Hansen and Hodrick, 1980).

[4] Artificial Intelligence

[5] Arbitrage pricing theory

[6] A zero-investment portfolio consists of investments in no-weighted assets (i.e., selling borrowed assets, which consists of selling short other assets from long position)

[7] Cochrane (2011) described the introduction of more long/short relations under the “factor zoo problem” which fogged up the asset pricing picture

[8] Principal Component Analysis

[9] FTSE 350, the largest 350 UK equities by market cap

[10] Turkish Stock Market

[11] Machine Learning

[12] AI- Augmented, Traditional factor, Temporal and Spatio-temporal models.

[13] Deep learning

[14] FF-3, FF-5, Carhart’s Four factor model

[15] Excess market return factor

[16] For example, iShares MSCI USA Size Factor ETF

[17] Stable tail adjusted return ratio