Multi-factor models

information, and hold the same portfolio. Furthermore, CAPM only apply the market index when estimating expected return, while multifactor models consider several factors when estimating the appropriate return. The factors, which are thought to influence the return of a specific asset, are included in the multifactor model. The chosen factors are variables that historically seem to predict average returns well. Hence, these factors might capture risk premiums. The different variables are not constant and the factors applied depend on the characteristics of the specific asset. There could be several different macro-economic variables included in the model depending on what the given asset are most sensitive to. Common risk factors could be interest rate fluctuations, inflation or oil price, along with others (Brealey, Allen, & Myers, 2014).

APT is based on the law of one price, which states that if two assets are equal in all economically relevant respects, then they should be priced the same on the market. If the law of one price is violated, when an actual price of an asset diverges from the APT-estimated price, arbitrage will bring it to the equilibrium level. The investors will take as large position as possible if an arbitrage opportunity exists, hence the price will soon be forced down and equilibrium level will be re-established (Bodie, Kane & Marcus, 2014).

The equation of the APT model is shown below

𝑅_J = 𝑎_J+ 𝑏_Jm𝐹_m+ 𝑒_J

m

mo/

𝑅J Return on asset i

𝑎J Expected return on asset i if all factors have a value of zero 𝑏Jm Sensitivity of asset I to a change in factor j

𝐹m Value of factor j if that impacts the return on asset i

𝑒J Random error term

5.4.2. The Fama and French Three-factor model

One of the most dominant multi-factor models is the Fama and French three-factor model.

Fama and French criticize the accurateness of CAPM (Fama & French, 1992). Previous researchers e.g. (Banz, 1981; Bhandari, 1988) had detected different patterns in stock returns, which could not be explained by the CAPM and were called anomalies.

The previous findings showed that a firm’s average stock return is related to size, book-to-market-equity, earnings, price and past sales growth (Fama & French, 1996). The study of (Fama &

French, 1996) shows that the anomalies, except the continuation of short-term returns, largely disappear in the three-factor model. Therefore, Fama and French are arguing that a multifactor model is a better model for explaining stock returns. The three-factor model stem from the Jensen’s alpha model, which was developed further to include two additional factors, firm size and book value.

𝑅_JE− 𝑅_NE = 𝑎_J+ 𝛽_Jl(𝑅_lE− 𝑅_NE) + 𝛽_Jplq𝑆𝑀𝐵_E+ 𝛽_Jtlu𝐻𝑀𝐿_E+ 𝑒_JE

𝑅JE− 𝑅NE Excess return on portfolio I against the risk-free rate at time t.

𝑎J Jensen’s alpha for portfolio i at time t.

𝑅lE Return of the relevant equity benchmark at time t.

𝑆𝑀𝐵E Difference in return between small cap portfolio and large cap portfolio at time t.

𝐻𝑀𝐿E Difference in return between value portfolio and growth portfolio at time t.

𝑒JE The residual variance error term at time t.

The expected excess return on a portfolio is explained by the sensitivity of its return to three different factors. The first factor is the excess return on a market portfolio. The second is the SMB factor, which is difference between the return on a portfolio of small stocks and a portfolio of large stocks in period t. This factor was included in the model due to the pattern of a higher average of realized returns for stocks of small capitalization firms in comparison to stocks of large capitalization firms, other things equal (Bodie, Kane & Marcus, 2014).

The final one is the HML factor, the difference between the return on a portfolio of high-book-to-market stocks (value stocks) and the return on a portfolio of low-book-to-high-book-to-market stocks (growth stocks) (Fama & French, 1996). A low ratio of book-to-market value is generally a characteristic for firms, which market value stems from predictions of future growth. A higher book-to-market ratio is a general characteristic for firms, whose market values stem from the company’s assets, which are already in place. Historically, value firms have generated higher realized average returns than growth firms (Bodie, Kane & Marcus, 2014).

5.4.3. Carhart 4-factor model

A further extension of the Fama & French (1993) three-factor model is the Carhart (1997) four-factor model. Carhart argues that persistent differences in mutual fund expenses and transaction costs in combination with common factors in stock returns explain almost all of the predictability in mutual fund returns. Carhart added a forth factor, the momentum factor, on the basis of the findings of Jegadeesh & Titman (1993) to Fama-French’s three factor model. Jegadeesh & Titman (1993) presented evidence for the momentum effect in their study. The momentum effect implies that performing contrarian investment strategies, which buy stocks that have performed well in the previous period and sell stocks that have performed poorly in the past, will generate abnormal returns. The study shows that trading strategies that buy past winners and sell past losers generate significant compounded excess returns of about 12 % averagely per year over a period from 1965 to 1989. In the study performed by Carhart in 1997 similar results are presented. The strategy of buying last year’s top performing mutual funds and selling last year’s poorly performing funds yields a return of 8 percent’s per year. Carhart concludes that adding a momentum factor to the original Fama and French three-factor model would adjust for this market anomaly when evaluating abnormal performance of a portfolio. In order to create the MOM-factor, a portfolio of the 30 % best-performing stocks over the past 11 months lagged one month was constructed.

Furthermore, one portfolio of the 30 % worst performing stocks over the matching period, also lagged one month, was constructed. The stocks in his portfolio consisted of all stocks from NYSE,

AMEX and NASDAQ and the portfolio was then rebalanced every month to obtain a rolling momentum factor (Carhart, 1997).

The formula for the Carhart four-factor model is an expansion of the Fama and French three-factor model:

𝑅_JE − 𝑅_NE = 𝑎_J+ 𝛽_Jl(𝑅_lE− 𝑅_NE) + 𝛽_Jplq𝑆𝑀𝐵_E+ 𝛽_Jtlu𝐻𝑀𝐿_E+ 𝛽_Jlxl𝑀𝑂𝑀_E+𝑒_JE

𝑅JE− 𝑅NE Excess return on portfolio I against the risk-free rate at time t.

𝑎J Jensen’s alpha for portfolio i at time t.

𝑅lE Return of the relevant equity benchmark at time t.

𝑆𝑀𝐵E Difference in return between a small cap portfolio and a large cap portfolio at time t.

𝐻𝑀𝐿E Difference in return between a value portfolio and a growth portfolio at time t.

𝑀𝑂𝑀E Difference in return between last year’s winner and loser stocks at time t.

𝑒JE The residual variance error term at time t.

The results in the study of Carhart indicates that the effect of adding a forth factor significantly reduced the pricing errors in comparison to both CAPM and the factor model. The three-factor model of Fama and French improves on the average pricing errors in comparison to CAPM.

The three-factor model errors are strongly negative for last year’s portfolios of loser stock and strongly positive for last year’s portfolios of winner stock. For comparative reason, the MAE (mean absolute errors) per month is 0.35 % from the CAPM, it is 0.31 % from three-factor model and 0.14

% from the four-factor model. The study concludes that the four-factor model is superior to both the CAPM and the Fama French three-factor model in terms of MAE. Furthermore, it well describes the cross-sectional variation in average stock returns since it eliminates nearly all of the patterns in pricing errors.

5.4.4. Cortez et al. 5-factor model

This study examines SRI funds on an international level. Several preceding studies have shown a tendency of home biases for international mutual funds. This bias is documented both for SRI (Bauer, Otten, & Rad, 2006; Gregory & Whittaker, 2007)and conventional (Chan, Covrig, & Ng, 2005) mutual funds. The home bias implies that fund managers tend to invest in local stocks due to information advantages, since it would be more costly to operate distanced to the information (Engström, 2003).

(Huberman, 2001) argues that people have a habit of investing in the familiar, hence ignoring the basic principles of diversification in portfolio theory. Huberman and Lewis (1999) state that there is a tendency of large investors to prefer to invest in businesses in their home countries. Therefore,

is a fifth local factor added to the Carhart model, denoted as the Cortez fifth-factor model after Cortez et al. (2012), to test for this home bias phenomenon, in similarity to the studies of (Bauer et al., 2006; Cortez, Silva, & Areal, 2012; Leite & Cortez, 2014).