**What exactly Fama-French three-factor model is?**

Before introducing multi-factor asset pricing models, the Capital Asset Pricing Model (CAPM) played a significant role in explaining the changes in stock returns. According to CAPM (this model has only one independent variable, which is market premium i.e. market return minus risk-free rate), the market premium can explain the stock returns, but financial experts Eugene Fama and Kenneth French proved that the CAPM faced challenges in grasping the intricacies of the market. Single-factor models, such as CAPM, simply assumed that the only important factor was the market factor and ignored other factors that played a very important role in defining stock price changes.

Therefore, in 1993, Fama and French introduced a new model. They expanded the Capital Asset Pricing Model (CAPM) by adding two factors, size and book-to-market ratio, to investigate factors beyond market risk premium. Their goal was to understand if size and book-to-market ratio factors, along with market risk premium, could jointly explain the changes in the stock returns. The reference markets in this study were NYSE, AMEX, and Nasdaq. By adding size and book-to-market ratio into their model, they sought to enhance the explanatory power of the CAPM. The study found that these three factors- size, book-to-market ratio, and market risk premium- significantly explain the stock returns. Their findings show that after adding new variables, the R-squared of the model increases in comparison to CAPM. They understood that small companies have, on average, greater returns than big companies.

**Calculating the size factor (SMB)**

Size: The market capitalization of the company has been used as a criterion to calculate the size of the company. The market capitalization of the company is obtained as follows:

Market Capitalization = outstanding shares * closing price

In the financial world, the size factor (SMB) is a critical component of the Fama-French three-factor model. It aims to capture the historical trend of small-cap stocks that consistently outperform their larger peers. Understanding how to calculate the SMB factor is essential for investors and analysts looking to implement this multifaceted model.

Delving into the intricacies of the size factor, one unravels the notion that small-cap stocks tend to unveil a unique set of risk and return characteristics in contrast to their larger counterparts, the large-cap stocks. Eugene Fama and Kenneth French, seasoned observers of financial landscapes, astutely observed the historical trend where small-cap stocks have, over time, demonstrated elevated returns.

To calculate the Size factor, or SMB (Small Minus Big), Fama and French divided companies into two categories based on median market capitalization. Companies below the median were considered small or “Small Caps,” while those above were categorized as big or “Big Caps.” This division allowed them to create two groups of companies. SMB, or Small Minus Big, represents the returns of small companies minus the returns of big companies. In essence, it quantifies the performance difference between small and large companies, forming the Size factor in the Fama-French framework.

Creating portfolios in June each year t involves categorizing companies based on their market capitalization. They used the market capitalization from year t to construct the portfolios, and this construction takes place each June. The return calculations extend from June of one year to July of the following year, ensuring a comprehensive evaluation period.

**Calculating the Book-to-Market ratio (HML)**

At the core of the HML factor, there’s this idea that stocks labeled as value stocks (those with lower price-to-book ratios) usually do better than growth stocks (the ones with higher ratios). Eugene Fama and Kenneth French noticed this happening regularly, with value stocks scoring higher returns. That’s why they decided to throw the HML factor into models that figure out how to price stuff.

To calculate the Book-to-Market Equity (BE/ME) ratio in the Fama-French framework, you need the market equity (market capitalization) in the denominator and the common equity (Book Common Equity) in the numerator. Portfolios are created in June each year, and for BE/ME in the Fama-French framework, you use the market capitalization at the end of December of the previous year.

For the numerator, the Book Common Equity in the Fama-French framework is typically derived from the Total Assets minus Total Liabilities. However, for a more precise calculation, according to the Fama-French paper of 1993, Book Common Equity is expressed by the following formula.

Book Common Equity = Book value of Equity – Book value of preferred stocks

To calculate Book Common Equity, you need accounting figures aligned with the end of the fiscal year t-1.

Fama and French categorize companies into three groups based on the Book-to-Market Ratio. The lowest 30% are classified as Low Book-to-Market Ratio, representing companies with the lowest ratios. The top 30% are designated as High Book-to-Market Ratio, indicating companies with the highest ratios. The remaining 40% fall into the Medium Book-to-Market Ratio category. To calculate the High Minus Low (HML) factor, the monthly (or daily) returns of Low Book-to-Market Ratio companies are subtracted from the returns of High Book-to-Market Ratio companies.

Six portfolios will be created from the interaction of Size (Small and Big) and Book-to-Market Ratio (Low, Medium, High): Small-Low, Small-Medium, Small-High, Big-Low, Big-Medium, and Big-High portfolios. This sorting from the interaction of two or more factors is bivariate sorting. If we have only one factor, we call it univariate.

SMB and HML Factors are calculated according to the following formulas:

SMB= ((small low+small medium+small high) / 3) – ((big low+big medium+big high) / 3)

HML= ((small high+big high) / 2) – ((small low+big low) / 2)

**Example 1:**

Imagine a simple situation with two sets of portfolios: one with small-cap stocks and the other with big-cap stocks.

- Small Cap Portfolio Returns: 10%, 4%, 7%
- Big Cap Portfolio Returns: 2%, 7%, 3%

Average Return of Small Caps = (10% + 4% + 7%) / 3 = 7%

Average Return of Big Caps = (2% + 7% + 3%) / 3 = 4%

SMB = 7% – 4% = 3%

When the SMB factor is positive, it points to small-cap stocks generally doing better than their big-cap stocks during a specific period. Now, if the SMB factor goes negative, this signals that big-cap stocks took the lead, outshining the small-cap ones.

**Example 2:**

Imagine a basic situation with two groups of portfolios. One group has value stocks, and the other has growth stocks.

- Value Portfolio Returns: 12%, 12%, 12%
- Growth Portfolio Returns: 5%, 7%, 9%

Average Return of Value Stocks = (12% + 12% + 12%) / 3 = 12%

Average Return of Growth Stocks = (5% + 7% + 9%) / 3 = 7%

HML = 12% – 7% = 5%

If the HML factor is on the positive side, it means that, overall, value stocks did better than growth stocks during a certain period. On the other hand, if the HML factor is negative, it means that growth stocks took the lead, outperforming the value ones.

**Calculating the Market Risk Premium (R**_{M}-R_{F})

_{M}-R

_{F})

The whole deal with the Market Risk factor boils down to this notion that a stock’s ups and downs are influenced by how much it nods along with the general market vibes. It’s like a sidekick that helps us figure out the usual moves a stock makes when the whole market is doing its dance. This factor gives us the lowdown on how much a stock might be at the mercy of the big-picture risks that affect everyone.

For calculating the market risk factor, you need the monthly closing prices for the reference market, and you also require the risk-free rate. Typically, when researching the U.S. market and applying the Fama-French 3-factor model, the risk-free rate is the monthly U.S. treasury bill rate.

**What is the GRS test?**

In the big world of pricing models for stuff like money and investments, there are tests people often use to check how well these models work. The Fama-MacBeth test, the Hansen-Jagannathan test, and the Gibbons-Ross-Shanken (GRS) test are some of them. Now, the GRS test is like the go-to when you’re dealing with models that think about lots of different things. Researchers like using it to figure out if a bunch of factors make a big difference in explaining why prices of things like assets go up and down, more than just looking at the basic market stuff.

The GRS test is like an upgrade to the capital asset pricing model (CAPM). It checks if we should consider more things than just the market return when looking at how assets perform. If the p-value in the GRS test is pretty small, it means these additional could significantly and jointly explain the variations of stocks.

**Critiques and Limitations of the Fama-French three-factor model**

So the critics say that the Fama-French model might miss some key risk factors, leaving stock returns with a bunch of mysteries. And then there are those who question if the size and value effects will stick around, wondering if changes in the market scene might mess with how much these factors really matter.

So, this Fama-French three-factor model isn’t flawless, just like any other financial models. It’s all about assuming people are smart with their money moves and that markets are like genius-level smart, but not everyone’s buying into that idea, especially the behavioral finance crowd.

**Different sorting methods and cutoff points**

Here pops up a question regarding portfolio construction: Why should we sort the data and build different portfolios? The answer is easy. After determining the qualified companies, portfolios are formed to eliminate the effect of size and the ratio of book value to market value on stocks. If we don’t do so, we cannot get the exact and correct results.

- In this method, we have 16 portfolios. In this way, companies are divided into four groups based on their size (market value) (big, somewhat big, somewhat small, and small), and then each of these groups is divided into four groups (low, somewhat low, somewhat high and high) based on the ratio of book value to market price. Then, 16 portfolios are formed using the intersection of these two types of classification. In this method, the 25th, 50th, and 75 are used for sorting the factors.
- In this method, size and value factors categorize companies into two segments in terms of size and three segments in terms of value. The size separation limit, its median, and the factor separation limits are the 30th and 70th percentiles. With these separation points, 3 x 2 = 6 portfolios are obtained, and the size factor is equal to the average of three small portfolios minus the average of three large portfolios according to the book-to-market ratio. The value factor is equal to the average of two portfolios with high value minus the average of two portfolios with low value.
- Another group uses the median separation limit to check the distribution of factors (size and value), that’s why we have 2 x 2 = 4 portfolios.
- Fama and French used another method in 2015. In this method, size is divided into 5 groups (from small to large). Each of these groups is independently divided into 5 different groups (low to high). In this method, the 20
^{th}, 40^{th}, 60^{th}, and 80^{th}percentiles are used. So, the final number of portfolios is 5 x 5 = 25. - There are also further methods like 10
^{th}, 20^{th}, 30^{th}, 40^{th}, 50^{th}, 60^{th}, 70^{th}, 80^{th}, and 90^{th}, which subject to 100 portfolios.

You can calculate the returns for each of these portfolios using either a value-weighted or equal-weighted approach. The choice between these methods depends on your preference or the specific requirements of your analysis. But we know that the value-weighted portfolios are better diversified, which influences the result of the analyses.