DFA and Factor Investing

DFA and Factor Investing#

HBS Case#

Dimensional Fund Advisors, 2002 [HBS 9-203-026]#

Pages 1-5 of the case are required. Pages 6-11 discuss trading (especially adverse selection) and taxes. These sections are useful for market background, but are optional.

1. READING - DFA’s Strategy#

1. Investment philosophy.#

  • In 100 words or less, describe DFA’s belief about how to find premium in the market.

  • To what degree does their strategy rely on individual equity analysis? Macroeconomic fundamentals? Efficient markets?

  • Are DFA’s funds active or passive?

  • What do DFA and others mean by a “value” stock? And a “growth” stock?

2. Challenges for DFA’s view.#

  • What challenge did DFA’s model see in the 1980’s?

  • And in the 1990’s?

3. The market.#

  • Exhibit 3 has data regarding a universe of 5,020 firms. How many are considered ``large cap”? What percent of the market value do they account for?

  • Exhibit 6 shows that the U.S. value factor (HML) has underperformed the broader U.S. equity market in 1926-2001, including every subsample except 1963-1981. So why should an investor be interested in this value factor?

2. The Factors#

DFA believes certain stocks have higher expected excess returns. In addition to the overall market equity premium, DFA believes that there is a premium attached to a size and value factor.

Data#

Use data/dfa_analysis_data.xlsx from the course Box folder. Monthly data covers July 1926 through August 2026. Returns are stored as decimals.

  • Factors: Use all three columns, Mkt-RF, SMB, and HML, in the factors sheet for Section 2. They are already excess returns; do not subtract RF from them.

  • Risk-free return: RF is the monthly risk-free return. Use it to convert the portfolios’ total returns to excess returns in Section 3.

  • Portfolios: The portfolios (total returns) sheet contains 25 value-weighted portfolios sorted by size and book-to-market.

Source: Ken French Data Library.

1. The Factors#

Calculate the univariate performance statistics for each of the three factors:

  • annualized mean

  • annualized volatility

  • annualized Sharpe ratio

  • monthly VaR(.05), reported as the fifth percentile of returns

Report these for the following three subsamples:

  • July 1926 - December 1980

  • January 1981 - December 2001

  • January 2002 - August 2026

2.#

Based on the factor statistics above, answer the following.

  • Does each factor have a premium (positive expected excess return) in each subsample?

  • Does the premium to the size factor get smaller after 1980?

  • Does the premium to the value factor get smaller during the 1990’s?

  • How have the factors performed since the time of the case, (January 2002 - August 2026)?

3.#

The factors are constructed in such a way as to reduce correlation between them.

  • Report the correlation matrix across the three factors.

  • Does the construction method succeed in keeping correlations small?

  • Does it achieve this in each subsample?

4.#

  • Plot the cumulative returns of the three factors.

  • Create plots for the 1981-2001 subsample as well as the January 2002 - August 2026 subsample.

5.#

  • Does it appear that all three factors were valuable in 1981-2001?

  • And post-2001?

Would you advise DFA to continue emphasizing all three factors?

3. CAPM#

DFA believes that premia in stocks and stock portfolios are related to the three factors.

Let’s test 25 equity portfolios that span a wide range of size and value measures.

Footnote#

For more on the portfolio construction, see the description at Ken French’s data library. https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/tw_5_ports.html

Portfolios#

Monthly total return data on 25 equity portfolios sorted by their size-value characteristics. Denote these as \(\vec{r}^{i}\), for \(n=1, \ldots, 25\).

  • Note that while the factors were given as excess returns, the portfolios are total returns.

  • For this entire problem, focus on the January 1981 - August 2026 subsample.

1. Summary Statistics.#

For each portfolio,

  • Use the RF column in the factors sheet to convert these total returns to excess returns.

  • Calculate the (annualized) univariate statistics from 2.1; keep VaR at the monthly frequency.

2. CAPM#

The Capital Asset Pricing Model (CAPM) asserts that an asset (or portfolio’s) expected excess return is completely a function of its beta to the equity market index (Mkt-RF in this workbook).

Specifically, it asserts that, for any excess return, \(\tilde{r}^{i}\), its mean is proportional to the mean excess return of the market, \(\tilde{r}^{\text{mkt}}\), where the proportionality is the regression beta of \(\tilde{r}^{i}\) on \(\tilde{r}^{\text{mkt}}\).

\[ \mathbb{E}\left[\tilde{r}_{t}^{i}\right] = \beta^{i,\text{mkt}}\; \mathbb{E}\left[\tilde{r}_{t}^{\text{mkt}}\right] \]

Let’s examine whether that seems plausible.

For each of the \(n=25\) test portfolios, run the CAPM time-series regression:

\[ \tilde{r}_{t}^{i} = \alpha^i + \beta^{i,\text{mkt}}\; \tilde{r}_{t}^{\text{mkt}} + \epsilon_{t}^{i} \]

So you are running 25 separate regressions, each using the \(T\)-sized sample of time-series data.

  • Report the betas and alphas for each test asset.

  • Report the mean-absolute-error of the CAPM:

    \[\text{MAE} = \frac{1}{n}\sum_{i=1}^n \left|\alpha_i\right|\]

If the CAPM were true, what would we expect of the MAE?

  • Report the estimated \(\beta^{i,\text{mkt}}\), Treynor Ratio, \(\alpha^i\), and Information Ratio for each of the \(n\) regressions.

  • If the CAPM model were true, what would be true of the Treynor Ratios, alphas, and Information Ratios?

3. Cross-sectional Estimation#

Let’s test the CAPM directly. We already have what we need:

  • The dependent variable, (y): mean excess returns from each of the \(n=25\) portfolios.

  • The regressor, (x): the market beta from each of the \(n=25\) time-series regressions.

Then we can estimate the following equation:

\[ \underbrace{\mathbb{E}\left[\tilde{r}^{i}\right]}_{n\times 1\text{ data}} = \textcolor{ForestGreen}{\underbrace{\eta}_{\text{regression intercept}}} + \underbrace{{\beta}^{i,\text{mkt}}}_{n\times 1\text{ data}}~ \textcolor{ForestGreen}{\underbrace{\lambda_{\text{mkt}}}_{\text{regression estimate}}} + \textcolor{ForestGreen}{\underbrace{\upsilon}_{n\times 1\text{ residuals}}} \]

Note that

  • we use sample means as estimates of \(\mathbb{E}\left[\tilde{r}^{i}\right]\).

  • this is a weird regression! The regressors are the betas from the time-series regressions we already ran!

  • this is a single regression, where we are combining evidence across all \(n=25\) series. Thus, it is a cross-sectional regression!

  • the notation is trying to emphasize that the intercept is different than the time-series \(\alpha\) and that the regressor coefficient is different than the time-series betas.

Report

  • the R-squared of this regression.

  • the intercept, \(\eta\).

  • the regression coefficient, \(\lambda_{\text{mkt}}\).

What would these three statistics be if the CAPM were completely accurate?

4. Conclusion#

Broadly speaking, do these results support DFA’s belief in size and value portfolios containing premia unrelated to the market premium?

5. Joint Test of Alphas#

We measured how well the CAPM performs by checking the MAE of the time-series alphas.

Under classic statistical assumptions, we can test the null hypothesis that the CAPM works by calculating,

\[ H = T\left[1+\left(\text{SR}_{\text{mkt}}\right)^2\right]^{-1} \boldsymbol{\alpha}'\boldsymbol{\Sigma}_\epsilon^{-1}\boldsymbol{\alpha} \]

This test statistic has a chi-squared distribution…

\[H\sim \chi^2_n\]

Note the following:

  • \(\boldsymbol{\alpha}\) is an \(n\times 1\) vector of the individual regression alphas, \(\alpha^i\).

  • \(\boldsymbol{\Sigma}_\epsilon\) is the \(n\times n\) covariance matrix of the time-series of regression residuals, \(\epsilon^i\), corresponding to each regression.

  • \(\text{SR}_{\text{mkt}}\) is the monthly Sharpe ratio of \(\tilde{r}^{\text{mkt}}\).

The test statistic, \(H\), has a chi-squared distribution with \(n=25\) degrees of freedom. So under the null hypothesis of the CAPM holding, \(H\) should be small, and the distribution allows us to calculate the probability of seeing such a large \(H\), conditional on the CAPM being true.

  • Which is a stricter test: checking whether any of the \(n\) values of \(\alpha^i\) have a statistically significant t-test or checking whether \(H\) calculated above is significant?

  • Conceptually, how does the test-statistic \(H\) relate to checking whether \(\tilde{r}^{\text{mkt}}\) spans the tangency portfolio?

4. Beyond CAPM#

1.#

Re-do the analysis of 3.2 and 3.3, but instead of using the market return as the factor, use a new factor: the in-sample tangency portfolio of the \(n=25\) portfolios.

Use the portfolio excess returns from Section 3. The new factor is the tangency portfolio formed from these 25 portfolios.

  • Calculate \(\tilde{r}^{\text{tan}}\) by solving the MV optimization of the \(n\) excess returns.

  • Consider this to be your single factor.

Instead of testing the CAPM, you will test the tangency-factor model:

\[ \mathbb{E}\left[\tilde{r}_{t}^{i}\right] = \beta^{i,\text{tan}}\; \mathbb{E}\left[\tilde{r}_{t}^{\text{tan}}\right] \]

What do you find?

2.#

Re-do the analysis of 3.2 and 3.3, but instead of using only the Mkt-RF factor, use Mkt-RF, SMB, and HML.

Use the portfolio excess returns from Section 3. All three factors are already excess returns; do not subtract RF from them.

Thus, instead of testing the CAPM, you will be testing the Fama-French 3-Factor Model.

\[ \mathbb{E}\left[\tilde{r}_{t}^{i}\right] = \beta^{i,\text{mkt}}\; \mathbb{E}\left[\tilde{r}_{t}^{\text{mkt}}\right] + \beta^{i,\text{size}}\; \mathbb{E}\left[\tilde{r}_{t}^{\text{size}}\right] + \beta^{i,\text{val}}\; \mathbb{E}\left[\tilde{r}_{t}^{\text{val}}\right] \]
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