Question

Assuming that both the three-factor model (TFM) and the Carhart four-factor model (FFM) are used to estimate the alpha of an active portfolio. Explain which model is likely to result in alpha of higher magnitude. Also explain why might one model be preferred to another in adjusting for the risk of an active portfolio.

Answer 1

A) What Is the Fama and French Three Factor Model?

The Fama and French Three-Factor Model (or the Fama French Model for short) is an asset pricing model developed in 1992 that expands on the capital asset pricing model (CAPM) by adding size risk and value risk factors to the market risk factor in CAPM. This model considers the fact that value and small-cap stocks outperform markets on a regular basis. By including these two additional factors, the model adjusts for this outperforming tendency, which is thought to make it a better tool for evaluating manager performance.

KEY TAKEAWAYS

* The Fama French 3-factor model is an asset pricing model that expands on the capital asset pricing model by adding size risk and value risk factors to the market risk factors.

* The model was developed by Nobel laureates Eugene Fama and his colleague Kenneth French in the 1990s.

* The model is essentially the result of an econometric regression of historical stock prices.

B) How the Fama French Model Works

Nobel Laureate Eugene Fama and researcher Kenneth French, former professors at the University of Chicago Booth School of Business, attempted to better measure market returns and, through research, found that value stocks outperform growth stocks. Similarly, small-cap stocks tend to outperform large-cap stocks. As an evaluation tool, the performance of portfolios with a large number of small-cap or value stocks would be lower than the CAPM result, as the Three-Factor Model adjusts downward for observed small-cap and value stock out-performance.

The Fama and French model has three factors: size of firms, book-to-market values and excess return on the market. In other words, the three factors used are SMB (small minus big), HML (high minus low) and the portfolio's return less the risk free rate of return. SMB accounts for publicly traded companies with small market caps that generate higher returns, while HML accounts for value stocks with high book-to-market ratios that generate higher returns in comparison to the market.

C) What the Fama French Model Means for Investors

Fama and French highlighted that investors must be able to ride out the extra short-term volatility and periodic underperformance that could occur in a short time. Investors with a long-term time horizon of 15 years or more will be rewarded for losses suffered in the short term. Using thousands of random stock portfolios, Fama and French conducted studies to test their model and found that when size and value factors are combined with the beta factor, they could then explain as much as 95% of the return in a diversified stock portfolio.

Given the ability to explain 95% of a portfolio’s return versus the market as a whole, investors can construct a portfolio in which they receive an average expected return according to the relative risks they assume in their portfolios. The main factors driving expected returns are sensitivity to the market, sensitivity to size, and sensitivity to value stocks, as measured by the book-to-market ratio. Any additional average expected return may be attributed to unpriced or unsystematic risk.

D) Carhart Four-factor model

In portfolio management the Carhart four-factor model is an extension of the Fama–French three-factor model including a momentum factor for asset pricing of stocks, proposed by Mark Carhart. It is also known in the industry as the MOM factor (monthly momentum).[1] Momentum in a stock is described as the tendency for the stock price to continue rising if it is going up and to continue declining if it is going down.

The MOM can be calculated by subtracting the equal weighted average of the lowest performing firms from the equal weighed average of the highest performing firms, lagged one month (Carhart, 1997). A stock is showing momentum if its prior 12-month average of returns is positive.

The four-factor model is not as recognised as the other two models mentioned above. Mark M. Carhart wrote a paper in 1997 where he presented the model as a tool for valuating mutual funds. The paper based it work of what Fama and French did with the three-factor model in early 90´s, Carhart also based his work on Jegadeesh and Titman´s (1993) paper. Jegadeesh and Titman uncovered a tendency for good and bad performances of stocks to persist over several months, in other words a momentum effect. (Bodie et al. 2014, pp 432-433)

Carharts looked at the previous research and decided to include the momentum factor in to the three-factor model and he performed the regression analysis on mutual funds instead of stocks which Fama and French used in their paper. (Bodie et al. 2014, pp 432-433) The four-factor model in its regression form.

E) Previous studies

Much of the research that had been conducted on the three and four-factor model has mainly had its focus on the US stock market. But some researchers have been testing both the three- factor and four-factor model on other markets. Research papers has also arisen that challenge the results and method used by the original authors to the three-factor and four-factor model. Below, three research paper are listed that has in some way influenced this thesis.

Ann-Sin and Shih-Chuan (2009) presented a research paper where they concluded that the three-factor model outperforms the CAPM in the Pacific Basin markets (Japan, Hong Kong, South Korea, Malaysia, Thailand, Indonesia, and Singapore), and in addition to their results they could not find any support for the momentum effect of the Carhart´s four-factor model. Their results showed that the three-factor model performed as well or even slightly better than the four-factor model. The paper uses stock instead of mutual funds as Carhart used in his four-factor model. They also used another portfolio construction approach than Carhart, this can be an explanation to why they get so poor results from the four-factor model.

Bello (2008) did a statistical comparison between the CAPM, the three-factor model and the four-factor model in his research paper. He used mutual funds as data to evaluate the models instead of stocks. He also did an in-depth analysis about the potential multicollinearity problems that could arise when performing regression tests on the three-factor model the four- factor model. His results showed that no harmful multicollinearity was present in the dataset. In the paper he also performs a goodness of fit test, according to his results the goodness of fit difference is not significant. But, with respect to the quality of prediction, the three-factor model performs better than the CAPM and also the four-factor model performs better than the three-factor model.

F) If alpha should be zero (based on CAPM), would a FAMA-FRENCH 3 factor model explain your observations above more effectively?

Alpha is a risk-adjusted measure of active return on an investment.

The FF 3 factor model is emerging 2 classes of stock with CAPM to reflect a portfolio's theory.

r - Rf = beta3 x ( Km - Rf ) + bs x SMB + bv x HML + alpha

Alpha Coefficient can show that in an efficient market, the expected value of the alpha coefficient is zero. Therefore the alpha coefficient indicates how an investment has performed after accounting for the risk it involved:

Alpha_i < 0 : the investment has earned too little for its risk (or, was too risky for the return)

Alpha_i = 0 : the investment has earned a return adequate for the risk taken

Alpha_i > 0 : the investment has a return in excess of the reward for the assumed risk.

G) What is Alpha?

"Alpha" (the Greek letter α) is a term used in investing to describe a strategy's ability to beat the market, or it's "edge." Alpha is thus also often referred to as “excess return” or “abnormal rate of return,” which refers to the idea that markets are efficient, and so there is no way to systematically earn returns that exceed the broad market as a whole. Alpha is often used in conjunction with beta (the Greek letter β) , which measures the broad market's overall volatility or risk, known as systematic market risk.

Alpha is used in finance as a measure of performance, indicating when a strategy, trader, or portfolio manager has managed to beat the market return over some period. Alpha, often considered the active return on an investment, gauges the performance of an investment against a market index or benchmark that is considered to represent the market’s movement as a whole. The excess return of an investment relative to the return of a benchmark index is the investment’s alpha. Alpha may be positive or negative and is the result of active investing. Beta, on the other hand, can be earned through passive index investing.

Alpha is commonly used to rank active mutual funds as well as all other types of investments. It is often represented as a single number (like +3.0 or -5.0), and this typically refers to a percentage measuring how the portfolio or fund performed compared to the referenced benchmark index (i.e., 3% better or 5% worse).

Deeper analysis of alpha may also include "Jensen’s alpha." Jensen’s alpha takes into consideration the capital asset pricing model (CAPM) market theory and includes a risk-adjusted component in its calculation. Beta (or the beta coefficient) is used in the CAPM, which calculates the expected return of an asset based on its own particular beta and the expected market returns. Alpha and beta are used together by investment managers to calculate, compare, and analyze returns.

H) Optimal active portfolio management and relative performance drivers: theory and evidence:  

This paper addresses the optimal active versus passive portfolio mix in a straightforward extension of the Treynor and Black (T-B) classic model. Such a model allows fund managers to select the mix of active and passive portfolio that maximizes the (active) Sharpe ratio performance indicator. The T-B model, here adapted and made operational as a tool for performance measurement, enables one to identify the sources of fund management performance (selectivity vs market-timing). In addition, the combination of active and passive risk exposures is estimated and fund manager choice is tested against the hypothesis of optimal (active) portfolio design.  

The extended T-B model is applied to a sample of US dollar reserve management portfolios – owned by the ECB and managed by NCBs – invested in high-grade dollar denominated bonds. The best fund managers show statistically significant outperformance against the ECB-given benchmark. By far, market timing is the main driver. Positive (and statistically significant) selectivity appears to be very modest and relatively rare across fund managers. These results are not very surprising, in that low credit risk and highly liquid securities dominate portfolio selection, thus limiting the sources of profitable bond-picking activity. As far as the risk-return profile of the active portfolio is concerned, it appears that some of the best fund managers’ outperformance is realised by shorting the active portfolio (with respect to the benchmark composition). Thus, portfolios that would be inefficient (eg negative excess return) if held long can be turned into positive-alpha yielding portfolios if shorted. The ability to select long-vs-short active portfolio can be seen as an additional source of fund manager’s outperformance, beyond the skill in anticipating the return of the benchmark portfolio (market-timing contribution).  

The estimated measure of fund managers’ risk aversion turns out to be relatively high. This seems to be consistent with the fairly conservative risk-return profile of the benchmark portfolio. A relative measure of risk exposure (conditional Relative VaR) averaged across fund managers turns out to be in line with the actual risk budget limit assigned by the ECB. However, a fair amount of heterogeneity across fund managers is also found to be present. This is likely to signal a less-than-efficient use of their risk-budget by the fund managers – eg a deviation from the optimal level of relative risk accounted for by the model. At least in part, such variability might also be attributed to estimation errors. However, proper tests for RVaR statistics are sorely lacking in the risk management literature. Thus, the question remains open. This would warrant further investigation, which is left for future research.