Problem Set 5.2 — FF93 Factor Construction & Time-Series Regressions

Covers Chapter 5.2 (Reader’s Guide: Fama & French (1993)). This set is a replication-flavored walk through the machinery of Fama and French (1993), “Common Risk Factors in the Returns on Stocks and Bonds.” Methods through Ch 5.2 only: the 2×3 size × book-to-market sort and the six value-weighted corner portfolios; the construction of the size factor SMB (“small minus big”) and the value factor HML (“high minus low”) from those six portfolios; the time-series factor regression of a test portfolio’s excess return on the three factors, R_{it} - R_{ft} = \alpha_i + b_i\,(R_{Mt} - R_{ft}) + s_i\,\text{SMB}_t + h_i\,\text{HML}_t + \varepsilon_{it}, read two ways — the R^2 for common variation and the intercept \alpha_i for pricing error; the CAPM-vs-FF3 comparison; and the GRS joint test (Gibbons, Ross & Shanken 1989) of the null that all test-portfolio intercepts are simultaneously zero. Notation follows the Conventions: R_{it} is portfolio i’s gross return in month t, R_{ft} the risk-free rate, R_{Mt} - R_{ft} the market excess return; b_i, s_i, h_i are factor loadings and \hat\alpha_i the pricing error.

Six problems, escalating, 100 points total. Every numerical input is supplied so you can work the arithmetic by hand — no computer needed. The companion notebook nb5.2 (notebooks/week-05/ nb5.2-ff93-factor-regressions.ipynb) lets you check the construction and the regressions on data, and the official Fama–French factors used there are free and public — Ken French has hosted them for decades, so this is one replication you can do with no licensed CRSP/Compustat access at all.

The grading rule of this set: a number reported without saying what it means — an R^2 without the words “common variation,” an \alpha without the words “pricing error,” a loading without naming which characteristic it tracks — earns half credit at most. As everywhere in this book, the reading of the regression is the skill, not the arithmetic. Whenever you name a vulnerability, name the design choice or test that addresses it (spec discipline, Conventions §4).

A standing warning on numbers. Every figure in this problem set is illustrative — invented to make the arithmetic clean and the reasoning visible. They are not the numbers in Fama and French’s tables. Do not quote them as if they were. The qualitative facts (positive SMB/HML premia, high R^2s on the test portfolios, small intercepts, small-growth as the weak spot) are well established and safe to state; any precise coefficient belongs in your own replication in nb5.2, not in memory.

Several problems share one small world, defined once here so you can carry the numbers between them.

The illustrative six-portfolio month (the working world). At the end of June, all stocks are split into two size groups (Small / Big, at the NYSE-median market-cap breakpoint) and independently into three book-to-market groups (Low / Medium / High BE/ME, at the NYSE 30th and 70th percentiles). Intersecting gives six value-weighted portfolios. Here are their excess returns in one illustrative month (percent per month — invented, clean):

	Low BE/ME (growth)	Medium BE/ME	High BE/ME (value)
Small	S-Low = 0.30	S-Med = 0.95	S-High = 1.50
Big	B-Low = 0.45	B-Med = 0.70	B-High = 0.85

Read the grid the FF way before you compute anything: down each column (Small to Big) the size effect says small should out-earn big; across each row (Low to High BE/ME) the value effect says value should out-earn growth. Both patterns are visible in this single illustrative month. The factor formulas (Ch 5.2 §2) are \text{SMB} = \tfrac{1}{3}\big(\text{S-Low}+\text{S-Med}+\text{S-High}\big) - \tfrac{1}{3}\big(\text{B-Low}+\text{B-Med}+\text{B-High}\big), \text{HML} = \tfrac{1}{2}\big(\text{S-High}+\text{B-High}\big) - \tfrac{1}{2}\big(\text{S-Low}+\text{B-Low}\big).

Problem 1 — Construct SMB and HML by hand from the 2×3 sort (18 points)

This is the factory floor. Build the two factors from the six corner portfolios and see why the construction is shaped the way it is.

(a) [6 pts] Using the working-world grid, compute SMB for this month. Show the average of the three small portfolios, the average of the three big portfolios, and the difference. Then state in one sentence what SMB is, in plain words — what bet does an investor holding SMB hold?

(b) [6 pts] Compute HML for this month. Show the average of the two high-BE/ME portfolios, the average of the two low-BE/ME portfolios, and the difference. State in one sentence what HML is as a bet.

(c) [4 pts] The construction averages across the other characteristic before differencing — SMB averages over the three value buckets before differencing small minus big; HML averages over the two size buckets before differencing value minus growth. Explain in two or three sentences why this averaging is deliberate: what would go wrong if, instead, you built a “size factor” as just S-High minus B-High (small-value minus big-value)? Use the words “neutral” and “contaminate,” and connect the idea to Frisch–Waugh–Lovell (Ch 2.3) — partialling one effect out to isolate the other.

(d) [2 pts] Compute the naive size bet S-High - B-High and the naive size bet S-Low - B-Low from the grid. Note that they disagree (one is much bigger than the other). State in one sentence what this disagreement illustrates about why FF average over the value dimension rather than picking a single column.

Problem 2 — Set up and read a single time-series factor regression (20 points)

Now move from constructing factors to using them. One test portfolio, one regression, read three ways.

A researcher forms a small-value test portfolio (it is not one of the six factor-ingredient portfolios — keep that distinction sharp; the six build the factors, the test portfolios are the patients the model must cure) and runs, over T months, R_{it} - R_{ft} = \alpha_i + b_i\,(R_{Mt} - R_{ft}) + s_i\,\text{SMB}_t + h_i\,\text{HML}_t + \varepsilon_{it}. The fitted results are (illustrative): \hat\alpha_i = 0.04,\quad \hat b_i = 1.00,\quad \hat s_i = 0.80,\quad \hat h_i = 0.40,\quad R^2 = 0.94, with the intercept’s t-statistic t(\hat\alpha_i) = 0.6.

(a) [4 pts] Write down the full empirical specification in the Conventions §4 form: name the outcome, the regressors, the sample, and state in one sentence the question the regression answers. (No fixed effects or clustering here — this is plain time-series OLS, Ch 2.1, on one portfolio’s monthly stream.)

(b) [5 pts] The R^2 reading — common variation. Interpret R^2 = 0.94 in the specific language of this paper. What fraction of what does it describe, and why is the phrase “common variation” — as opposed to just “fit” — the right one? In one sentence, say what a low R^2 here would have told you about whether SMB and HML are genuine common factors.

(c) [6 pts] The loadings reading. Interpret \hat b_i, \hat s_i, \hat h_i. For each, say which characteristic it tracks and whether its sign and magnitude make sense for a small-value portfolio. (Hint: think about how \hat s_i should look for a small portfolio versus a big one, and how \hat h_i should look for a value portfolio versus a growth one — the loadings should march with the sort.)

(d) [5 pts] The intercept reading — pricing error. Interpret \hat\alpha_i = 0.04 with t(\hat\alpha_i) = 0.6. What does the intercept mean in an asset-pricing time-series regression — why is it called the pricing error, and what would \hat\alpha_i = 0 assert about the model? Is this portfolio’s pricing error economically and statistically distinguishable from zero? State the two-column rule from Ch 5.2 (§4) in one sentence: what does high R^2 plus zero \alpha together certify?

Problem 3 — CAPM vs FF3 on the same portfolio (16 points)

The whole point of FF93 is that adding SMB and HML prices portfolios that CAPM cannot. Make that concrete with the same small-value portfolio.

Take the long-run average premia (illustrative, percent per month): \mathbb{E}[R_M - R_f] = 0.50,\qquad \mathbb{E}[\text{SMB}] = 0.25,\qquad \mathbb{E}[\text{HML}] = 0.40, and suppose the portfolio’s average excess return over the sample is \overline{R_i - R_f} = 0.90. The FF3 loadings are the Problem-2 values \hat b_i = 1.00, \hat s_i = 0.80, \hat h_i = 0.40. A CAPM-only regression of the same portfolio gives a market loading \hat b_i^{\text{CAPM}} = 1.10.

(a) [5 pts] CAPM pricing error. Under CAPM, the model-predicted average excess return is \hat b_i^{\text{CAPM}}\cdot\mathbb{E}[R_M - R_f]. Compute it, then compute the CAPM alpha \alpha_i^{\text{CAPM}} = \overline{R_i - R_f} - \hat b_i^{\text{CAPM}}\cdot\mathbb{E}[R_M - R_f]. Report the number.

(b) [5 pts] FF3 pricing error. Under FF3, the predicted average excess return is \hat b_i\,\mathbb{E}[R_M - R_f] + \hat s_i\,\mathbb{E}[\text{SMB}] + \hat h_i\,\mathbb{E}[\text{HML}]. Compute it term by term, then compute the FF3 alpha \alpha_i^{\text{FF3}} = \overline{R_i - R_f} - (\text{predicted}). Report the number, and confirm it matches the \hat\alpha_i = 0.04 given in Problem 2.

(c) [4 pts] Lay \alpha_i^{\text{CAPM}} and \alpha_i^{\text{FF3}} side by side. CAPM leaves a large unexplained chunk of the average return; FF3 leaves almost nothing. State, in two sentences, which two pieces SMB and HML added to the predicted return, and why a small-value portfolio in particular is exactly the kind of asset CAPM mis-prices and FF3 rescues.

(d) [2 pts] In a replication you would also see the R^2 rise from the CAPM regression to the FF3 regression on this portfolio. Say in one sentence which of the two readings — common variation or pricing — that R^2 increase speaks to, and why a rise in R^2 is logically separate from the shrinkage in \alpha (you can have one without the other).

Problem 4 — The GRS joint test: what it tests and why joint (16 points)

A single portfolio’s intercept is one number with one t-statistic. FF93’s real test asks about all the test portfolios at once. This problem pins down the GRS test (Gibbons, Ross & Shanken 1989) — we use it, we do not derive it.

A researcher runs the three-factor regression on each of the 25 size × BE/ME test portfolios (a 5×5 sort), obtaining 25 intercepts \hat\alpha_1, \dots, \hat\alpha_{25} and their t-statistics.

(a) [5 pts] State the null hypothesis the GRS test evaluates, symbolically and in words. What single English sentence about the model is true if and only if this null holds? (Tie it back to the two-column rule: GRS is the formal version of “the \alpha column is zero.”)

(b) [5 pts] Why joint, not 25 separate t-tests? Explain in three or four sentences why you cannot simply scan the 25 individual t-statistics and declare victory if most are insignificant. Name the Week 1 problem this is (Ch 1.5), and state what would go wrong with the family-wise error rate if you ran 25 tests at the 5% level and asked whether any rejected. Then add the second reason the test must be joint: the 25 portfolios’ residuals are correlated (they hold overlapping stocks), so the intercepts are not independent — and say in one sentence why a procedure that ignored that correlation would mis-state the joint significance.

(c) [4 pts] The GRS statistic is a single F-statistic. State the decision rule in asset-pricing terms: a model “passes” when GRS fails to reject the joint null. Explain why that is the opposite of the usual “we hope to reject the null” mindset from a treatment-effect regression — i.e., here the researcher is rooting for the null. In one sentence, say what a rejection of the GRS null would tell you about the model.

(d) [2 pts] FF93’s three-factor model dramatically out-performs CAPM on the GRS test, but the joint test is not always a clean pass — one corner of the 5×5 grid is the stubborn culprit. Name that corner portfolio and state in one sentence what its persistently nonzero intercept signals (a tolerable blemish vs. a missing factor — you need not resolve the debate, just name the cell and the tension).

Problem 5 — What’s vulnerable: risk vs. mispricing, the factor zoo, look-ahead (18 points)

A model this influential invites hard questions. This problem is pure reasoning — it tests whether you can push on FF93 the way a referee would, using the vocabulary of Weeks 1 and 2.

(a) [6 pts] Risk or mispricing? FF93 frames SMB and HML as compensation for risk; a rival camp reads the premia as the correction of mispricing (investors over-pay for glamour growth firms, under-pay beaten-down value firms). Explain in three or four sentences why the time-series regression machinery — high R^2, zero \alpha — cannot distinguish these two stories. Then name one piece of additional evidence that would push you toward the risk reading (Hint: think about when HML pays off — is it in good states or bad states of the world? — or about an out-of-sample / out-of-country replication).

(b) [6 pts] The factor zoo and multiple testing. The FF93 recipe — sort on a characteristic, build a long-short portfolio, check the alpha — has since been run on hundreds of candidate characteristics, producing the “factor zoo.” Explain, using the Week 1 multiple-testing problem (Ch 1.5), why reporting only the characteristics with significant alphas out of a large search is a data-snooping trap: what happens to the number of “significant” factors you expect to find by pure chance as the number of characteristics tested grows? Then state the retrospective worry this raises about FF93 itself, and give the one honest counter-argument in its favor (Hint: out-of-sample and out-of-country behavior).

(c) [4 pts] Look-ahead in factor construction. FF93 matches accounting data from fiscal year-end in calendar year y{-}1 to returns from July of year y onward — a deliberate lag. Explain in two or three sentences what bias this lag is guarding against (what would you be assuming the investor knew, and when, if you skipped it?), and name the specific Compustat feature — historical backfilling of data for firms only after they became established — that can sneak the bias back in even when you think you have lagged correctly. State in one clause what a sloppy lag would manufacture: a premium no real-time investor could have earned.

(d) [2 pts] For each of the three vulnerabilities above, name in a phrase the discipline or check that a careful replicator uses to address it (one per vulnerability). This is spec discipline (Conventions §4) pointed at a factor model: name the threat, then name the guard.

Problem 6 — Replication design: from this problem set to nb5.2 (12 points)

You will run this for real in nb5.2. This problem makes you design the replication before you touch the keyboard — stating the spec, the data, and the checks you would commit to in advance.

(a) [4 pts] Data and access. Name the two data sources FF93 uses for the stock side (one for returns and market caps, one for book equity), and state which is licensed and which is free. Then state the key fact that makes this replication unusually accessible for a high-school camp: Ken French hosts the official monthly factor returns and the 25 portfolios free and public. Explain in one sentence how you would use those free series as a check on factors you built yourself (what would you correlate with what, and roughly how high should the correlation be?).

(b) [4 pts] The two-column table. Sketch the table you would produce from the 25 time-series regressions: name the columns (loadings \hat b, \hat s, \hat h; intercept \hat\alpha with its t; and R^2) and state the three qualitative facts you would verify reading it — one about the R^2 column, one about how the loadings should move across the sort, and one about the intercepts (including which cell to watch). This is the §4 reading order turned into a deliverable.

(c) [4 pts] A sensitivity / robustness check. FF93’s factors depend on construction choices — the 2×3 cut, NYSE breakpoints, value-weighting, the 30/70 cutoffs, the June rebalance. Design one concrete robustness check: name one construction choice you would change (e.g., NYSE breakpoints → all-stock breakpoints, or value-weighting → equal-weighting), state what you would re-compute after the change (SMB, HML, the loadings, the GRS statistic), and state in one sentence what a robust finding would look like versus a fragile one. Tie this to the §6 critique: “the” size and value premia are really “the premia under these specific rules.”

End of Problem Set 5.2. Solutions: Appendix E, E-w5-ps5.2-solutions.md. The companion notebook nb5.2 (notebooks/week-05/nb5.2-ff93-factor-regressions.ipynb) lets you build the six 2×3 portfolios and the factors, run the 25 time-series regressions and assemble the two-column R^2/\alpha table, contrast CAPM with FF3, and compute the GRS statistic for both — checking your hand arithmetic from Problems 1–4 against real, free Ken French data, with a “Your Turn” extension that bolts a momentum factor onto the model. The single discipline to carry forward: a working factor model is high R^2 (common variation) plus zero \alpha (no pricing error), tested jointly — and even the best one, FF93, leaves the small-growth corner unhealed.

# Problem Set 5.2 — FF93 Factor Construction & Time-Series Regressions **Covers Chapter 5.2 (Reader's Guide: Fama & French (1993)).** This set is a *replication-flavored* walk through the machinery of Fama and French (1993), "Common Risk Factors in the Returns on Stocks and Bonds." Methods through Ch 5.2 only: the **2×3 size × book-to-market sort** and the six value-weighted corner portfolios; the construction of the size factor **SMB ("small minus big")** and the value factor **HML ("high minus low")** from those six portfolios; the **time-series factor regression** of a test portfolio's excess return on the three factors, $$ R_{it} - R_{ft} = \alpha_i + b_i\,(R_{Mt} - R_{ft}) + s_i\,\text{SMB}_t + h_i\,\text{HML}_t + \varepsilon_{it}, $$ read **two ways** — the $R^2$ for *common variation* and the intercept $\alpha_i$ for *pricing error*; the **CAPM-vs-FF3 comparison**; and the **GRS joint test** (Gibbons, Ross & Shanken 1989) of the null that all test-portfolio intercepts are simultaneously zero. Notation follows the Conventions: $R_{it}$ is portfolio $i$'s gross return in month $t$, $R_{ft}$ the risk-free rate, $R_{Mt} - R_{ft}$ the market excess return; $b_i, s_i, h_i$ are factor **loadings** and $\hat\alpha_i$ the **pricing error**. Six problems, escalating, **100 points total**. Every numerical input is supplied so you can work the arithmetic by hand — no computer needed. The companion notebook **nb5.2** (`notebooks/week-05/ nb5.2-ff93-factor-regressions.ipynb`) lets you check the construction and the regressions on data, and the official Fama–French factors used there are **free and public** — Ken French has hosted them for decades, so this is one replication you can do with no licensed CRSP/Compustat access at all. **The grading rule of this set:** a number reported *without saying what it means* — an $R^2$ without the words "common variation," an $\alpha$ without the words "pricing error," a loading without naming which characteristic it tracks — earns half credit at most. As everywhere in this book, the *reading* of the regression is the skill, not the arithmetic. Whenever you name a vulnerability, name the design choice or test that addresses it (spec discipline, Conventions §4). > **A standing warning on numbers.** Every figure in this problem set is **illustrative** — invented to > make the arithmetic clean and the reasoning visible. **They are not the numbers in Fama and French's > tables.** Do not quote them as if they were. The *qualitative* facts (positive SMB/HML premia, high > $R^2$s on the test portfolios, small intercepts, small-growth as the weak spot) are well established > and safe to state; any precise coefficient belongs in your own replication in nb5.2, not in > memory. Several problems share one small world, defined once here so you can carry the numbers between them. > **The illustrative six-portfolio month (the working world).** At the end of June, all stocks are > split into **two size groups** (Small / Big, at the NYSE-median market-cap breakpoint) and > independently into **three book-to-market groups** (Low / Medium / High BE/ME, at the NYSE 30th and > 70th percentiles). Intersecting gives **six value-weighted portfolios**. Here are their **excess > returns in one illustrative month** (percent per month — invented, clean): | | Low BE/ME (growth) | Medium BE/ME | High BE/ME (value) | |:---|:---:|:---:|:---:| | **Small** | S-Low $= 0.30$ | S-Med $= 0.95$ | S-High $= 1.50$ | | **Big** | B-Low $= 0.45$ | B-Med $= 0.70$ | B-High $= 0.85$ | > Read the grid the FF way before you compute anything: down each column (Small to Big) the size effect > says small should out-earn big; across each row (Low to High BE/ME) the value effect says value > should out-earn growth. Both patterns are visible in this single illustrative month. The factor > formulas (Ch 5.2 §2) are > $$ > \text{SMB} = \tfrac{1}{3}\big(\text{S-Low}+\text{S-Med}+\text{S-High}\big) - \tfrac{1}{3}\big(\text{B-Low}+\text{B-Med}+\text{B-High}\big), > $$ > $$ > \text{HML} = \tfrac{1}{2}\big(\text{S-High}+\text{B-High}\big) - \tfrac{1}{2}\big(\text{S-Low}+\text{B-Low}\big). > $$ --- ## Problem 1 — Construct SMB and HML by hand from the 2×3 sort (18 points) This is the factory floor. Build the two factors from the six corner portfolios and see *why* the construction is shaped the way it is. **(a)** [6 pts] Using the working-world grid, compute **SMB** for this month. Show the average of the three small portfolios, the average of the three big portfolios, and the difference. Then state in one sentence what SMB *is*, in plain words — what bet does an investor holding SMB hold? **(b)** [6 pts] Compute **HML** for this month. Show the average of the two high-BE/ME portfolios, the average of the two low-BE/ME portfolios, and the difference. State in one sentence what HML *is* as a bet. **(c)** [4 pts] The construction averages *across* the other characteristic before differencing — SMB averages over the three value buckets before differencing small minus big; HML averages over the two size buckets before differencing value minus growth. Explain in two or three sentences why this averaging is deliberate: what would go wrong if, instead, you built a "size factor" as just S-High minus B-High (small-value minus big-value)? Use the words "neutral" and "contaminate," and connect the idea to Frisch–Waugh–Lovell (Ch 2.3) — partialling one effect out to isolate the other. **(d)** [2 pts] Compute the *naive* size bet S-High $-$ B-High and the *naive* size bet S-Low $-$ B-Low from the grid. Note that they disagree (one is much bigger than the other). State in one sentence what this disagreement illustrates about why FF average over the value dimension rather than picking a single column. --- ## Problem 2 — Set up and read a single time-series factor regression (20 points) Now move from constructing factors to *using* them. One test portfolio, one regression, read three ways. A researcher forms a **small-value** test portfolio (it is *not* one of the six factor-ingredient portfolios — keep that distinction sharp; the six build the factors, the test portfolios are the patients the model must cure) and runs, over $T$ months, $$ R_{it} - R_{ft} = \alpha_i + b_i\,(R_{Mt} - R_{ft}) + s_i\,\text{SMB}_t + h_i\,\text{HML}_t + \varepsilon_{it}. $$ The fitted results are (illustrative): $$ \hat\alpha_i = 0.04,\quad \hat b_i = 1.00,\quad \hat s_i = 0.80,\quad \hat h_i = 0.40,\quad R^2 = 0.94, $$ with the intercept's $t$-statistic $t(\hat\alpha_i) = 0.6$. **(a)** [4 pts] Write down the full empirical specification in the Conventions §4 form: name the **outcome**, the **regressors**, the **sample**, and state in one sentence the question the regression answers. (No fixed effects or clustering here — this is plain time-series OLS, Ch 2.1, on one portfolio's monthly stream.) **(b)** [5 pts] **The $R^2$ reading — common variation.** Interpret $R^2 = 0.94$ in the specific language of this paper. What fraction of *what* does it describe, and why is the phrase "common variation" — as opposed to just "fit" — the right one? In one sentence, say what a *low* $R^2$ here would have told you about whether SMB and HML are genuine common factors. **(c)** [6 pts] **The loadings reading.** Interpret $\hat b_i, \hat s_i, \hat h_i$. For each, say which characteristic it tracks and whether its sign and magnitude *make sense* for a small-value portfolio. (Hint: think about how $\hat s_i$ should look for a *small* portfolio versus a big one, and how $\hat h_i$ should look for a *value* portfolio versus a growth one — the loadings should march with the sort.) **(d)** [5 pts] **The intercept reading — pricing error.** Interpret $\hat\alpha_i = 0.04$ with $t(\hat\alpha_i) = 0.6$. What does the intercept *mean* in an asset-pricing time-series regression — why is it called the pricing error, and what would $\hat\alpha_i = 0$ assert about the model? Is this portfolio's pricing error economically and statistically distinguishable from zero? State the **two-column rule** from Ch 5.2 (§4) in one sentence: what does high $R^2$ *plus* zero $\alpha$ together certify? --- ## Problem 3 — CAPM vs FF3 on the same portfolio (16 points) The whole point of FF93 is that adding SMB and HML *prices* portfolios that CAPM cannot. Make that concrete with the same small-value portfolio. Take the long-run average premia (illustrative, percent per month): $$ \mathbb{E}[R_M - R_f] = 0.50,\qquad \mathbb{E}[\text{SMB}] = 0.25,\qquad \mathbb{E}[\text{HML}] = 0.40, $$ and suppose the portfolio's **average excess return** over the sample is $\overline{R_i - R_f} = 0.90$. The FF3 loadings are the Problem-2 values $\hat b_i = 1.00, \hat s_i = 0.80, \hat h_i = 0.40$. A **CAPM-only** regression of the same portfolio gives a market loading $\hat b_i^{\text{CAPM}} = 1.10$. **(a)** [5 pts] **CAPM pricing error.** Under CAPM, the model-predicted average excess return is $\hat b_i^{\text{CAPM}}\cdot\mathbb{E}[R_M - R_f]$. Compute it, then compute the CAPM alpha $\alpha_i^{\text{CAPM}} = \overline{R_i - R_f} - \hat b_i^{\text{CAPM}}\cdot\mathbb{E}[R_M - R_f]$. Report the number. **(b)** [5 pts] **FF3 pricing error.** Under FF3, the predicted average excess return is $\hat b_i\,\mathbb{E}[R_M - R_f] + \hat s_i\,\mathbb{E}[\text{SMB}] + \hat h_i\,\mathbb{E}[\text{HML}]$. Compute it term by term, then compute the FF3 alpha $\alpha_i^{\text{FF3}} = \overline{R_i - R_f} - (\text{predicted})$. Report the number, and confirm it matches the $\hat\alpha_i = 0.04$ given in Problem 2. **(c)** [4 pts] Lay $\alpha_i^{\text{CAPM}}$ and $\alpha_i^{\text{FF3}}$ side by side. CAPM leaves a large unexplained chunk of the average return; FF3 leaves almost nothing. State, in two sentences, *which two pieces* SMB and HML added to the predicted return, and why a small-value portfolio in particular is exactly the kind of asset CAPM mis-prices and FF3 rescues. **(d)** [2 pts] In a replication you would also see the $R^2$ rise from the CAPM regression to the FF3 regression on this portfolio. Say in one sentence which of the two readings — common variation or pricing — that $R^2$ increase speaks to, and why a rise in $R^2$ is logically *separate* from the shrinkage in $\alpha$ (you can have one without the other). --- ## Problem 4 — The GRS joint test: what it tests and why joint (16 points) A single portfolio's intercept is one number with one $t$-statistic. FF93's real test asks about *all* the test portfolios at once. This problem pins down the GRS test (Gibbons, Ross & Shanken 1989) — we use it, we do not derive it. A researcher runs the three-factor regression on each of the **25** size × BE/ME test portfolios (a 5×5 sort), obtaining 25 intercepts $\hat\alpha_1, \dots, \hat\alpha_{25}$ and their $t$-statistics. **(a)** [5 pts] State the **null hypothesis** the GRS test evaluates, symbolically and in words. What single English sentence about the model is true if and only if this null holds? (Tie it back to the two-column rule: GRS is the formal version of "the $\alpha$ column is zero.") **(b)** [5 pts] **Why joint, not 25 separate $t$-tests?** Explain in three or four sentences why you **cannot** simply scan the 25 individual $t$-statistics and declare victory if most are insignificant. Name the Week 1 problem this is (Ch 1.5), and state what would go wrong with the family-wise error rate if you ran 25 tests at the 5% level and asked whether *any* rejected. Then add the second reason the test must be joint: the 25 portfolios' residuals are **correlated** (they hold overlapping stocks), so the intercepts are *not* independent — and say in one sentence why a procedure that ignored that correlation would mis-state the joint significance. **(c)** [4 pts] The GRS statistic is a single $F$-statistic. State the decision rule in asset-pricing terms: **a model "passes" when GRS *fails to reject*** the joint null. Explain why that is the opposite of the usual "we hope to reject the null" mindset from a treatment-effect regression — i.e., here the researcher is rooting *for* the null. In one sentence, say what a *rejection* of the GRS null would tell you about the model. **(d)** [2 pts] FF93's three-factor model dramatically out-performs CAPM on the GRS test, but the joint test is not always a clean pass — one corner of the 5×5 grid is the stubborn culprit. Name that corner portfolio and state in one sentence what its persistently nonzero intercept signals (a tolerable blemish vs. a missing factor — you need not resolve the debate, just name the cell and the tension). --- ## Problem 5 — What's vulnerable: risk vs. mispricing, the factor zoo, look-ahead (18 points) A model this influential invites hard questions. This problem is pure reasoning — it tests whether you can push on FF93 the way a referee would, using the vocabulary of Weeks 1 and 2. **(a)** [6 pts] **Risk or mispricing?** FF93 frames SMB and HML as compensation for *risk*; a rival camp reads the premia as the correction of *mispricing* (investors over-pay for glamour growth firms, under-pay beaten-down value firms). Explain in three or four sentences why the time-series regression machinery — high $R^2$, zero $\alpha$ — **cannot distinguish** these two stories. Then name *one* piece of additional evidence that would push you toward the risk reading (Hint: think about when HML pays off — is it in good states or bad states of the world? — or about an out-of-sample / out-of-country replication). **(b)** [6 pts] **The factor zoo and multiple testing.** The FF93 recipe — sort on a characteristic, build a long-short portfolio, check the alpha — has since been run on *hundreds* of candidate characteristics, producing the "factor zoo." Explain, using the Week 1 multiple-testing problem (Ch 1.5), why reporting only the characteristics with significant alphas out of a large search is a **data-snooping** trap: what happens to the number of "significant" factors you expect to find by pure chance as the number of characteristics tested grows? Then state the retrospective worry this raises about FF93 *itself*, and give the one honest counter-argument in its favor (Hint: out-of-sample and out-of-country behavior). **(c)** [4 pts] **Look-ahead in factor construction.** FF93 matches accounting data from fiscal year-end in calendar year $y{-}1$ to returns from July of year $y$ onward — a deliberate **lag**. Explain in two or three sentences what bias this lag is guarding against (what would you be assuming the investor knew, and when, if you skipped it?), and name the specific Compustat feature — historical **backfilling** of data for firms only after they became established — that can sneak the bias back in even when you think you have lagged correctly. State in one clause what a sloppy lag would manufacture: a premium *no real-time investor could have earned*. **(d)** [2 pts] For *each* of the three vulnerabilities above, name in a phrase the discipline or check that a careful replicator uses to address it (one per vulnerability). This is spec discipline (Conventions §4) pointed at a factor model: name the threat, then name the guard. --- ## Problem 6 — Replication design: from this problem set to nb5.2 (12 points) You will run this for real in nb5.2. This problem makes you design the replication *before* you touch the keyboard — stating the spec, the data, and the checks you would commit to in advance. **(a)** [4 pts] **Data and access.** Name the two data sources FF93 uses for the stock side (one for returns and market caps, one for book equity), and state which is licensed and which is free. Then state the key fact that makes this replication unusually accessible for a high-school camp: Ken French hosts the official monthly factor returns and the 25 portfolios **free and public**. Explain in one sentence how you would use those free series as a *check* on factors you built yourself (what would you correlate with what, and roughly how high should the correlation be?). **(b)** [4 pts] **The two-column table.** Sketch the table you would produce from the 25 time-series regressions: name the columns (loadings $\hat b, \hat s, \hat h$; intercept $\hat\alpha$ with its $t$; and $R^2$) and state the **three qualitative facts** you would verify reading it — one about the $R^2$ column, one about how the loadings should move across the sort, and one about the intercepts (including which cell to watch). This is the §4 reading order turned into a deliverable. **(c)** [4 pts] **A sensitivity / robustness check.** FF93's factors depend on *construction choices* — the 2×3 cut, NYSE breakpoints, value-weighting, the 30/70 cutoffs, the June rebalance. Design one concrete robustness check: name *one* construction choice you would change (e.g., NYSE breakpoints → all-stock breakpoints, or value-weighting → equal-weighting), state what you would re-compute after the change (SMB, HML, the loadings, the GRS statistic), and state in one sentence what a *robust* finding would look like versus a *fragile* one. Tie this to the §6 critique: "the" size and value premia are really "the premia *under these specific rules*." --- *End of Problem Set 5.2. Solutions: Appendix E, `E-w5-ps5.2-solutions.md`. The companion notebook **nb5.2** (`notebooks/week-05/nb5.2-ff93-factor-regressions.ipynb`) lets you build the six 2×3 portfolios and the factors, run the 25 time-series regressions and assemble the two-column $R^2$/$\alpha$ table, contrast CAPM with FF3, and compute the GRS statistic for both — checking your hand arithmetic from Problems 1–4 against real, free Ken French data, with a "Your Turn" extension that bolts a momentum factor onto the model. The single discipline to carry forward: a working factor model is **high $R^2$ (common variation) plus zero $\alpha$ (no pricing error), tested jointly** — and even the best one, FF93, leaves the small-growth corner unhealed.*