# Fama French (2000): Characteristics, Covariances and Average Returns - intuition behind sorting of the returns

I am in the middle of doing my graduate programme in Accounting and Finance. Right now, I am taking a ph.D level course in Asset Pricing, wherein our first assignment/task is to recreate some tables in the Fama French Davis paper: "Characteristics, Covariances and Average Returns", J. of Finance, Feb. 2000, pp389-406. (link)

Specifically, this is about table III-IV. I have downloaded the average return data that one can obtain, in the HML and SMB setting. This means, I have the "Small", "Medium" and "Big" firms (size) average monthly returns since 1926-2017, as well as the "Low", "Medium" and "High" average returns of the BE/ME size of the firms.

This part in the paper, however, confuses me:

[...] We form 9 portfolios (S/L, S/M, S/H, M/L, M/M, M/H, B/L, B/M and B/H) as the intersections of the three size and the three BE/ME groups. The 9 portfolios are each subdivided into three portfolios (Lh, Mh or Hh) using pre-formating HML slopes. These slopes are estimated with five years (three years minimum) of monthly returns ending in December of year t-1.

I don't understand how I should construct these 9 portfolios, as well as the "pre-formation" HML slopes. I am aware that the slope of the HML is its own factor coefficient (such as beta, in the CAPM) but what does it mean in this context?

I have a picture of the table and wording here:

Any help would be greatly appreciated!

• To "form the intersection [portfolios]" you need not just the average monthly returns of the size and BE/ME portfolios, but the actual identities (the CRSP "permno" identifiers) of the companies in these portfolios each month so you can do the intersection (i.e. find what companies are both Small and Low, Small and Medium, etc). So you need access to CRSP and have code that forms the 3 size groups and the 3 BM groups each month from 1929 to 1997. Do you have that part already done? – noob2 Feb 22 '18 at 16:30

## Preliminary

The approach in the paper of Davis/Fama/French (2000) is to give insight in four contrary explanations on the value-anomaly. Daniel/Titman (1997) trace the value-premium to the value-characteristic, not to a risk-based explanation (as do e.g. Fama/French (1993)). For empirically separate the two models, the trick in Davis/Fama/French (2000) for distinguishing them is to (p. 398):

[...] isolate variation in the HML risk loading that is independent of BE/ME.

## Construction of the size and book-to-market portfolios

For constructing the portfolios you have to follow these steps (also described in my answer here):

1. Collect book common equity (BE) and market valuation (ME) for stocks, excluding financial, transportation and public firms. ME is measured as the number of shares outstanding with the stock price at the end of December. BE is measured at the fiscal year ending.
2. Calculate cross-sectional breakpoints (33% and 67% percentile) for size (ME) and book-to-market ratio (BE/ME). The size breakpoint for year $$t$$ is the median NYSE market equity at the end of June of year $$t$$. BE/ME for June of year $$t$$ is the book equity for the last fiscal year end in $$t−1$$ divided by ME for December of year $$t−1$$.
3. Sort each stock independently by BE/ME and ME. This results in the mentioned nine portfolios.
4. Calculate the value-weighted return for each portfolio from July of year $$t$$ to end of June in year $$t+1$$.

## Pre-formation HML portfolio sort

For each of the nine portfolios above, you run the regression $$R_i - R_f = \alpha_i + b_i (R_M - R_f) + s_i SMB + h_i HML + \epsilon_i$$

where $$R_i$$ is the monthly value-weighted portfolio return, $$R_f$$ the risk-free rate of return, $$R_M$$ the market return, $$SMB$$ the return of the size-factor mimicking portfolio and equivalently $$HML$$ for the value-factor. The regression is run using monthly return data over the previous five years (minimum of three years), ending in December of year $$t-1$$, i.e. if you form the portfolios in end of June of year $$y$$, the regression uses data from January $$y-5$$ to December $$t-1$$.

The regression results in estimates for the factor-loadings $$b_i$$, $$s_i$$ and $$h_i$$ (and the intercept $$\alpha$$). What follows is, that you split up all nine portfolios into three subsamples which results in a total of 27 portfolios. So for each of the 9 portfolios, you sort each stock within a portfolio by the variable $$h_i$$ from the above regression. Just look at the monoton increasing value for $$h_i$$ in Table 3, which are the breakpoints for sorting. For each of the 27 portfolios you finally have to calculate the value-weighted return.

1. Daniel/Titman (1997) subdivide the 9 size / book-to-market portfolios into five portfolios by $$h_i$$ which results in a total of 45 portfolios. The problem is, that some of the 45 portfolios only contain a single or very few stocks, i.e. the "portfolio" is not diversified. Thats the reason for only subdividing into 3 further sorts.
HML is measured as the difference between the return of a portfolio of high BE/ME stocks and the return of a portfolio of low BE/ME stocks constructed to be neutral with respect to size. The HML portfolio is generally based on the entire stock market, but your 9 size and book-to-market portfolios (for which you are running the above regression) are much smaller subsamples. The slope just measures the portfolio specific exposure towards the HML portfolio. As you can see on the values of $$h_i$$ (and the t-values), the portfolios are far from being "its own factor coefficient".