Appendix — Solutions — PS 5.3 (Momentum Portfolio Replication and the Transaction-Cost Critique)

Problem set: book/weeks/week-05/ps5.3.md (PS 5.3, Week 5). Chapter: Ch 5.3 — Reader’s Guide: Jegadeesh & Titman (1993), with an inference callback to Ch 2.4 (HAC / Newey–West). Paper: Jegadeesh, N., & Titman, S. (1993). Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency. Journal of Finance, 48(1), 65–91.

These are full worked solutions, not just answers. Notation follows CONVENTIONS.md and locks to Ch 5.3: the J/K formation/holding sort; the equal-weighted zero-cost winner-minus-loser (WML) portfolio with return r^{\text{WML}} = r^{W} - r^{L}; the overlapping monthly-reformation (calendar-time) construction; testing a mean by regressing the series on a constant; the classical / i.i.d. versus HAC (Newey–West) standard error with lag L; portfolio turnover and a per-round-trip cost in basis points (1 bp = 0.01\% = 0.0001); the break-even cost; and the qualitative robustness checks. Every number in this sheet is illustrative — chosen for clean arithmetic, not transcribed from the paper. The single claim about the real paper is the Ch 5.3 order-of-magnitude one: the 6/6 WML spread is positive and roughly \sim 1\%/month over a sample near the original. All numerical results below were confirmed in Python; verifying notes appear where useful.

Problem 1 — Build a winner-minus-loser portfolio by hand (14 points)

(a) (4 pts) Sort on the past-six-month return column (the formation signal) — not on the next-month column, which is the outcome we are not allowed to peek at when ranking. Descending order:

\underbrace{D\,(+61)}_{\text{rank 1}},\; \underbrace{J\,(+47)}_{2},\; A\,(+38),\; G\,(+24),\; I\,(+15),\; C\,(+9),\; E\,(+2),\; H\,(-4),\; \underbrace{B\,(-12)}_{9},\; \underbrace{F\,(-27)}_{\text{rank 10}}.

The winner portfolio is the top two, W = \{D, J\}. The loser portfolio is the bottom two, L = \{F, B\}. The realized next-month returns play no role here: ranking uses only past prices, which is exactly the one input weak-form efficiency swears is useless — the whole point of the test.

(b) (4 pts) Equal-weight within each leg, using the next-month table:

r^{W} = \tfrac{1}{2}\big(r_D + r_J\big) = \tfrac{1}{2}(2.0 + 2.5) = \mathbf{2.25\%}, r^{L} = \tfrac{1}{2}\big(r_F + r_B\big) = \tfrac{1}{2}(4.0 + 1.0) = \mathbf{2.50\%}.

The zero-cost spread is

r^{\text{WML}} = r^{W} - r^{L} = 2.25\% - 2.50\% = \mathbf{-0.25\%}.

(Verified in Python: ranked, sliced top-2 / bottom-2, averaged — r^W=2.25, r^L=2.50, WML =-0.25.) Note the spread came out negative this month — momentum “lost” — because the loser F bounced hard. That is deliberate; see part (d).

(c) (3 pts) It is zero-cost because you fund the $1 long position in winners with the $1 proceeds from short-selling losers: long and short legs are equal in dollars, so your net cash outlay is (notionally) zero and there is no capital base to divide by — the strategy’s “return” is a pure spread. The spread r^{W}-r^{L}, not r^{W} alone, is the right object to test because the subtraction cancels whatever common movement hits winners and losers alike — most importantly the market return. If the whole market rose 2% this month, both legs inherit that 2% and it nets out; what is left is the differential between past winners and past losers, which is the only thing an efficiency violation could be hiding in.

(d) (3 pts) Momentum is a claim about the average of the spread across many months and many stocks, not about any single name in any single month. F bouncing +4\% is one realization of one stock in one month — pure idiosyncratic noise that the two-stock loser leg cannot diversify away. With ten stocks and a one-month hold, the spread is dominated by such noise, which is exactly why our toy number came out negative. Jegadeesh & Titman re-form deciles (hundreds of names per leg) and re-do the sort every month for decades, so the idiosyncratic bounces average out and what survives is the systematic continuation — the \sim 1\%/month average spread. One bad month no more refutes momentum than one coin landing tails refutes a fair coin’s 50% rate.

Problem 2 — Read the J\times K returns grid like a referee (16 points)

The illustrative grid (percent/month):

J \backslash K	K=3	K=6	K=9	K=12
J=3	0.72	0.68	0.55	0.41
J=6	1.08	0.99	0.80	0.52
J=9	1.12	0.95	0.71	0.33
J=12	0.94	0.74	0.40	-0.06

(a) (4 pts) The 6/6 cell (row J=6, column K=6) is \mathbf{0.99\%/\text{month}}. That sits right on the Ch 5.3 order-of-magnitude claim — positive and \approx 1\%/month. Annualizing roughly, 0.99\% \times 12 \approx \mathbf{11.9\%/\text{year}} (this ignores compounding; it is an order-of-magnitude statement, not a precise annual return — a real annualization would compound and would also have to be quoted net of costs).

(b) (5 pts) The surface is a medium-horizon ridge that decays toward the long-horizon corner. (i) Holding J=6 fixed and lengthening the hold K from 3 to 12, the spread falls monotonically: 1.08 \to 0.99 \to 0.80 \to 0.52 — the profit is strongest at short holds and bleeds away as you hold longer, because the continuation is a medium-horizon phenomenon, not a permanent one. (ii) Reading into the bottom-right corner (J=12,\,K=12) the spread decays all the way to -0.06 — it goes negative. That sign flip foreshadows long-run reversal (De Bondt & Thaler, 1985): at long formation-and-holding horizons, past winners start to under-perform past losers as prices that overshot eventually correct. Momentum lives in the medium-horizon gap between short-run reversal and this long-run reversal.

(c) (4 pts) Two reasons the “pick the biggest cell” reading is wrong:

Statistical (data-snooping). The 1.12 at (9,3) is the maximum over sixteen correlated tests. Crowning the largest cell and reporting its t-statistic is the multiple-comparisons trap from Week 1 — the maximum of sixteen tries is upward-biased, so its apparent significance is overstated. You sharpen this in Problem 5(a).
A region beats a cell. The evidence for momentum is not one starred cell; it is that a broad contiguous region of the grid is positive and similar in magnitude (the entire upper-left block is \approx 0.7–1.1\%). A single hand-picked extreme could be luck; a whole coherent region moving together is a pattern, far harder to dismiss and far less sensitive to which exact (J,K) you trade. Reporting the surface, not the peak, is the honest move.

(d) (3 pts) The grid shows only point estimates; what must sit next to every cell is its standard error (and the t-statistic), and specifically the HAC / Newey–West version, not the naive i.i.d. one. Omitting it produces the failure mode of overstated significance: because the overlapping construction induces positive serial correlation (Problem 3), the naive standard error is too small and the t-statistic too large, so cells that look “significant” may not be once the overlap is respected. A grid of bare point estimates is not yet evidence.

Problem 3 — Overlapping holding periods, induced serial correlation, and why HAC (18 points)

(a) (5 pts) With K=6, the return recorded in calendar month t is the equal-weighted average of the six cohorts currently live — those formed in months t, t-1, \dots, t-5. The return recorded in month t+1 is the average of the cohorts formed in months t+1, t, t-1, \dots, t-4. Line them up:

Month t’s cohorts: \{t,\, t-1,\, t-2,\, t-3,\, t-4,\, t-5\}.
Month t+1’s cohorts: \{t+1,\, t,\, t-1,\, t-2,\, t-3,\, t-4\}.

They share five of the six cohorts (t through t-4). So even if the momentum payoff to each freshly formed cohort were i.i.d., the two reported monthly returns are averages built from 5/6 of the same underlying ingredients — they are mechanically correlated. The same overlap argument applies at larger gaps: month t and month t+\ell share 6-\ell cohorts for \ell = 1,2,\dots,5, and share none once \ell \ge 6. So the induced autocorrelation persists over lags \ell = 1 through \ell = K-1 = \mathbf{5} and then dies out. This is precisely the overlapping-observations mechanism from Week 2: overlapping windows manufacture serial correlation out of thin air.

(b) (5 pts) Sam regresses the spread series on a constant; the single coefficient \hat\beta_0 is the mean spread, and its standard error is the standard error of the mean. The classical / i.i.d. formula computes that standard error as (roughly) s/\sqrt{T}, which assumes all T monthly observations are independent draws. They are not: with positive serial correlation, each new month echoes its neighbors, so it carries less fresh information than an independent draw would. The Moulton-style intuition from Ch 2.4 §5.3: the effective sample size — the number of genuinely independent observations — is well below T. The naive formula divides by an effective T that is too large, so it reports a standard error that is too small and a t-statistic that is too large. The direction of the error is therefore toward over-confidence: Sam will think momentum is more decisively significant than the data actually warrant.

(c) (4 pts) The fix is the HAC (heteroskedasticity-and-autocorrelation-consistent) standard error, a.k.a. Newey–West. For the 6/6 strategy set the lag length L = K - 1 = \mathbf{5}: from part (a) that is exactly how many lags of induced autocorrelation the overlap creates before it vanishes, so L=5 absorbs all of it (rounding up further does no harm). Arithmetically, HAC adds to the “middle of the sandwich” the cross-products of residuals from nearby months, \hat\varepsilon_t\hat\varepsilon_{t-\ell} for \ell = 1,\dots,L, down-weighted by the Bartlett kernel — terms the classical formula assumes are zero. Crucially the point estimate is unchanged: the mean spread is still the mean spread regardless of which standard error you compute; only the standard error (hence the t-statistic) moves.

(d) (4 pts) Mean spread = 1.00\%, classical SE = 0.083\%, HAC (L=5) SE = 0.122\%.

The two t-statistics: t_{\text{classical}} = \frac{1.00}{0.083} \approx \mathbf{12.05}, \qquad t_{\text{HAC}} = \frac{1.00}{0.122} \approx \mathbf{8.20}. (Verified in Python. The HAC SE is 0.122/0.083 \approx 1.47\times the classical — “roughly half-again larger,” consistent with the reader’s guide.)
Yes, it survives. Both far exceed 1.96; momentum is overwhelmingly significant here, and the switch from classical to HAC merely shrinks an enormous t to a still-enormous t. A t of 8 does not become insignificant when inflated by a factor of 1.47.
The switch would flip the conclusion when the original t is marginal — say a naive t \approx 2.5 that, divided by the same \approx 1.47 inflation, lands near 1.7 and falls below 1.96. That is exactly the case Ch 5.3 tells a referee to check first: a comfortable t of 6 survives the overlap correction, but a marginal starred result can be entirely an artifact of the too-small naive standard error — so the first question on the referee’s notepad is “are these t-statistics overlap-corrected?”

Problem 4 — The transaction-cost critique and the break-even cost (22 points)

Cost model and illustrative inputs: \text{net}(c) = \bar g - c\,\bar\tau_L - (c+\text{sp})\,\bar\tau_S, \qquad \bar g = 100\text{ bps},\;\; \bar\tau_L = 0.30,\;\; \bar\tau_S = 0.30,\;\; \text{sp} = 40\text{ bps}.

(a) (4 pts) You multiply the cost by turnover — the fraction of each leg that actually changes hands on re-ranking — because you only pay trading costs on the shares you trade, not on the shares you simply continue to hold. A name that stays in the winner decile from one month to the next costs nothing to “keep”; only the slice that enters or exits the decile incurs the round-trip bid-ask + commission. Charging the cost against the whole book would massively overstate the bill. The short leg gets the extra premium because shorting beaten-down small-cap losers is physically more expensive: those names are illiquid and low-priced with wide bid-ask spreads, and you must borrow the shares to short them, paying a borrow / stock-loan fee that the long leg of large-cap winners never incurs. The leg the reader’s guide warns is the dangerous one is precisely this short leg.

(b) (6 pts) At c = 50 bps: \text{long-leg cost} = c\,\bar\tau_L = 50 \times 0.30 = 15\text{ bps}, \qquad \text{short-leg cost} = (c+\text{sp})\,\bar\tau_S = (50+40)\times 0.30 = 27\text{ bps}, \text{net}(50) = 100 - 15 - 27 = \mathbf{58\text{ bps/month}}. At c = 100 bps: \text{long-leg cost} = 100 \times 0.30 = 30\text{ bps}, \qquad \text{short-leg cost} = (100+40)\times 0.30 = 42\text{ bps}, \text{net}(100) = 100 - 30 - 42 = \mathbf{28\text{ bps/month}}. The strategy is net-profitable at both cost levels (58 bps and 28 bps respectively) — unsurprising, since both lie below the break-even computed next. (Verified in Python.)

(c) (6 pts) Set \text{net}(c)=0 and solve for c: \bar g - c\,\bar\tau_L - (c+\text{sp})\,\bar\tau_S = 0 \;\Longrightarrow\; \bar g - \text{sp}\,\bar\tau_S = c\,(\bar\tau_L + \bar\tau_S) \;\Longrightarrow\; c^\star = \frac{\bar g - \text{sp}\,\bar\tau_S}{\bar\tau_L + \bar\tau_S}. The +\text{sp}\,\bar\tau_S term moves to the gross side because the short premium is a fixed charge on the short leg that eats into the gross profit before the per-round-trip cost c even applies. Plugging in: c^\star = \frac{100 - 40\times 0.30}{0.30 + 0.30} = \frac{100 - 12}{0.60} = \frac{88}{0.60} \approx \mathbf{146.7\text{ bps}}. (Verified in Python: \text{net}(146.7) \approx 0.) As a check on which leg does the damage, set \text{sp}=0: c^\star_{\text{no sp}} = \frac{100}{0.60} \approx 166.7\text{ bps}. So the short-leg borrow premium lowers the break-even by 166.7 - 146.7 = 20 bps — the short leg makes the strategy break even at a 20-bp-lower round-trip cost than it otherwise would, confirming it is the more fragile leg.

(d) (3 pts) Verdict. At a break-even of \approx 147 bps round-trip, this illustrative 6/6 strategy does survive realistic costs on the liquid long leg alone — large-cap winners trading at 20–30 bps round-trip leave the gross \sim 1\%/month largely intact. The danger is the short leg: the beaten-down small-cap losers can cost several times the winners’ rate, and once their effective round-trip cost (bid-ask + price impact + borrow) approaches and exceeds the break-even neighborhood, the net profit collapses. So the honest conclusion mirrors the reader’s guide’s contested verdict: momentum’s gross profit is real, but a meaningful share of it is trading friction, and the short leg in small, illiquid losers is what kills it — which is exactly why a gross-return result is not yet an efficiency violation. Whether momentum is a true anomaly or a paper profit depends on whether you can actually trade the losers at the costs your break-even can tolerate.

(e) (3 pts) Subtracting the monthly cost lowers the mean of the net series (gross mean 100 bps \to net mean 58 bps at c=50, etc.) but barely touches its standard deviation — the cost is a roughly steady monthly drag, not a new source of volatility, since turnover is fairly stable month to month. The t-statistic is mean/SE, so shrinking the numerator while the denominator holds steady drags the t down even though the net mean stays positive. A strategy can thus be “profitable on average but no longer statistically distinguishable from zero”: the net mean is positive but small relative to the unchanged month-to-month noise, so you can no longer reject the hypothesis that the true net spread is zero. This is the bridge between Problems 3 and 4 — the cost critique can sink significance even when it does not sink the average.

Problem 5 — What’s vulnerable: data-snooping, January, and risk vs. behavioral (16 points)

(a) (6 pts) Data-snooping. (i) The grid is sixteen correlated significance tests; crowning the single most-significant cell and reporting its t-statistic is the multiple-comparisons / multiple-testing problem from Week 1 (Ch 1.5, revisited under the false-discovery rate later). The maximum over sixteen tries is upward-biased even under the null — run enough correlated tests and one will star by chance — so a “t=2.5 that is the max over sixteen cells” is worth far less than a t=2.5 chosen in advance; a reader should mentally Bonferroni-discount it. (ii) Two mitigating facts: first, the result is a broad region of positive cells, not one isolated lucky cell (Problem 2b) — a coherent surface is much harder to produce by snooping than a single extreme. Second, momentum replicated out of sample: in other countries, other asset classes, and the decades after 1989 that the original authors could not have peeked at. Out-of-sample survival is the strongest answer to a snooping charge because you cannot snoop data that did not exist when the hypothesis was formed — post-sample confirmation is evidence the effect is real rather than a fit to one historical window.

(b) (5 pts) January / seasonality. (i) Momentum tends to lose money in January and earn its keep in the other eleven months — an anti-January pattern. Mechanism: the loser leg is full of beaten-down, small, low-priced stocks; these bounce hard in January (tax-loss-selling pressure releases at year-end and the names rebound), so the short leg gets hurt precisely in January. (ii) An anti-January result is harder to dismiss than a January-concentrated one because the era’s known anomalies (size, value) were concentrated in January, so a profit living in January would look like just another seasonal artifact; a profit that survives despite a January loss cannot be that artifact. Yet the reader’s guide still files it under “vulnerable” because the seasonality is a clue to mechanism, not a clean robustness pass: a strategy whose annual profit is “eleven good months minus a January loss” is exposed to anything that shifts that seasonal structure (changes to tax-loss-selling rules, the rise of index investing), so the seasonal dependence is itself a fragility, not a checkbox.

(c) (5 pts) Risk vs. behavioral. The behavioral story is a two-stage under/over-reaction: investors underreact to news so prices drift toward fundamentals only slowly (this is the medium-horizon continuation momentum profits from), and then later overreact, pushing prices past fundamentals so they eventually correct (this is the long-run reversal seen in the bottom-right of the grid and in De Bondt–Thaler). The risk-based story says momentum loads on some priced, time-varying macro risk and the profit is compensation for bearing it — and the evidence cited is the 2009 momentum crash, when the beaten-down losers rocketed in the recovery and the short leg blew up, exhibiting the kind of rare, severe crash risk an investor would demand to be paid for. Jegadeesh & Titman (1993) cannot adjudicate between these because it predates the evidence on both sides — the long-run-reversal and crash-risk data accumulated only later. The honest write-up therefore reports the anomaly as real and labels the interpretation as open, taking no premature side.

Problem 6 — Replication design (pointing at nb5.3) (14 points)

(a) (5 pts) Construction recipe (6/6 cell). At the end of each month t: 1. Ranking variable: compute each stock’s cumulative return over the past J=6 months (months t-5 through t). 2. Split: sort all eligible stocks on that past return into ten deciles. 3. Form legs (equal-weighted): long the top decile (winners W), short the bottom decile (losers L), equal dollars per leg so the position is zero-cost. 4. Overlapping reformation: repeat this every month; in any calendar month you hold K=6 overlapping cohorts at once (formed this month back through five months ago) and average across them. 5. Record: one number per month — the cohort-averaged WML spread r^{W}-r^{L}.

Data: CRSP monthly returns, restricted in the original to NYSE/AMEX common stocks over 1965–1989 (per Jegadeesh & Titman 1993; for the modern replication expect the magnitudes to shift, so report order-of-magnitude only rather than matching a table decimal). Licensed-data rule (CONVENTIONS §5): CRSP stays on GMU infrastructure (read-only on Hopper / WRDS Cloud), and you must pin the CRSP snapshot date in the notebook.

(b) (4 pts) Report two standard errors side by side for every cell: the naive i.i.d. (classical) and the Newey–West (HAC) — because the overlapping construction induces serial correlation that makes the naive SE too small (Problem 3), and showing both makes the t-shrinkage visible. For the 6/6 cell set the lag length L = K-1 = 5. Expected result the reader’s guide licenses: 6/6 should be positive and on the order of \sim 1\%/month over a window near the original. The one thing you must not do: claim to match a specific decimal from the paper’s table — magnitudes shift with sample and data vintage; assert order of magnitude only.

(c) (3 pts) Transaction-cost deliverable. Plot net momentum profit as a function of the round-trip cost c (in bps), with turnover measured from the actual portfolio weights and an extra premium on the short leg. Locate the break-even cost c^\star (the c where net profit crosses zero — Problem 4c). The figure must support a one-paragraph verdict on whether momentum survives realistic costs and which leg kills it; expect the short leg (illiquid small-cap losers) to dominate the cost.

(d) (2 pts) Robustness checks. (1) Data-snooping: an out-of-sample re-estimation — split into in-sample (near 1965–1989) versus out-of-sample (1990–present, including the 2009 crash window) and show 6/6 persists; the demonstrating figure is the spread (and worst drawdown) in each era. (2) Seasonality: a January-versus-rest-of-year decomposition of the 6/6 spread; the demonstrating split is mean spread in January vs. the other eleven months, showing the anti-January pattern.

End of solutions for PS 5.3. The by-hand portfolio (Problem 1), the two t-statistics (Problem 3d), and the net-spread and break-even arithmetic (Problem 4b–c) were each confirmed in a few lines of Python: break-even c^\star = (100 - 40\times0.30)/0.60 = 146.7 bps, dropping to 166.7 bps with no short premium; \text{net}(50)=58, \text{net}(100)=28 bps; t_{\text{classical}}=12.05, t_{\text{HAC}}=8.20. The grid reading (Problem 2), the HAC machinery (Problem 3), and the cost sweep (Problem 4) are all built in notebooks/week-05/nb5.3-momentum-backtest.ipynb, where the Newey–West t-statistic prints next to the naive one and the break-even line appears on the net-profit plot.

# Solutions — PS 5.3 (Momentum Portfolio Replication and the Transaction-Cost Critique) **Problem set:** `book/weeks/week-05/ps5.3.md` (PS 5.3, Week 5). **Chapter:** Ch 5.3 — Reader's Guide: Jegadeesh & Titman (1993), with an inference callback to Ch 2.4 (HAC / Newey–West). **Paper:** Jegadeesh, N., & Titman, S. (1993). *Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency.* Journal of Finance, 48(1), 65–91. These are full worked solutions, not just answers. Notation follows `CONVENTIONS.md` and locks to Ch 5.3: the $J/K$ formation/holding sort; the equal-weighted zero-cost winner-minus-loser (WML) portfolio with return $r^{\text{WML}} = r^{W} - r^{L}$; the overlapping monthly-reformation (calendar-time) construction; testing a mean by regressing the series on a constant; the classical / i.i.d. versus HAC (Newey–West) standard error with lag $L$; portfolio turnover and a per-round-trip cost in basis points (1 bp $= 0.01\% = 0.0001$); the break-even cost; and the qualitative robustness checks. **Every number in this sheet is illustrative** — chosen for clean arithmetic, not transcribed from the paper. The single claim about the real paper is the Ch 5.3 order-of-magnitude one: the 6/6 WML spread is positive and roughly $\sim 1\%$/month over a sample near the original. All numerical results below were confirmed in Python; verifying notes appear where useful. --- ## Problem 1 — Build a winner-minus-loser portfolio by hand (14 points) **(a) (4 pts)** Sort on the **past-six-month return** column (the formation signal) — *not* on the next-month column, which is the outcome we are not allowed to peek at when ranking. Descending order: $$ \underbrace{D\,(+61)}_{\text{rank 1}},\; \underbrace{J\,(+47)}_{2},\; A\,(+38),\; G\,(+24),\; I\,(+15),\; C\,(+9),\; E\,(+2),\; H\,(-4),\; \underbrace{B\,(-12)}_{9},\; \underbrace{F\,(-27)}_{\text{rank 10}}. $$ The **winner portfolio** is the top two, $W = \{D, J\}$. The **loser portfolio** is the bottom two, $L = \{F, B\}$. The realized next-month returns play no role here: ranking uses *only past prices*, which is exactly the one input weak-form efficiency swears is useless — the whole point of the test. **(b) (4 pts)** Equal-weight within each leg, using the **next-month** table: $$ r^{W} = \tfrac{1}{2}\big(r_D + r_J\big) = \tfrac{1}{2}(2.0 + 2.5) = \mathbf{2.25\%}, $$ $$ r^{L} = \tfrac{1}{2}\big(r_F + r_B\big) = \tfrac{1}{2}(4.0 + 1.0) = \mathbf{2.50\%}. $$ The zero-cost spread is $$ r^{\text{WML}} = r^{W} - r^{L} = 2.25\% - 2.50\% = \mathbf{-0.25\%}. $$ (*Verified in Python: ranked, sliced top-2 / bottom-2, averaged — $r^W=2.25$, $r^L=2.50$, WML $=-0.25$.*) Note the spread came out **negative this month** — momentum "lost" — because the loser F bounced hard. That is deliberate; see part (d). **(c) (3 pts)** It is **zero-cost** because you fund the \$1 long position in winners with the \$1 proceeds from short-selling losers: long and short legs are equal in dollars, so your *net* cash outlay is (notionally) zero and there is no capital base to divide by — the strategy's "return" is a pure spread. The spread $r^{W}-r^{L}$, not $r^{W}$ alone, is the right object to test because the subtraction **cancels whatever common movement hits winners and losers alike** — most importantly the market return. If the whole market rose 2% this month, both legs inherit that 2% and it nets out; what is left is the *differential* between past winners and past losers, which is the only thing an efficiency violation could be hiding in. **(d) (3 pts)** Momentum is a claim about the **average of the spread across many months and many stocks**, not about any single name in any single month. F bouncing $+4\%$ is *one realization of one stock in one month* — pure idiosyncratic noise that the two-stock loser leg cannot diversify away. With ten stocks and a one-month hold, the spread is dominated by such noise, which is exactly why our toy number came out negative. Jegadeesh & Titman re-form deciles (hundreds of names per leg) and re-do the sort **every month for decades**, so the idiosyncratic bounces average out and what survives is the systematic continuation — the $\sim 1\%$/month *average* spread. One bad month no more refutes momentum than one coin landing tails refutes a fair coin's 50% rate. --- ## Problem 2 — Read the $J\times K$ returns grid like a referee (16 points) The illustrative grid (percent/month): | $J \backslash K$ | $K=3$ | $K=6$ | $K=9$ | $K=12$ | |:---:|:---:|:---:|:---:|:---:| | **$J=3$** | $0.72$ | $0.68$ | $0.55$ | $0.41$ | | **$J=6$** | $1.08$ | $0.99$ | $0.80$ | $0.52$ | | **$J=9$** | $1.12$ | $0.95$ | $0.71$ | $0.33$ | | **$J=12$** | $0.94$ | $0.74$ | $0.40$ | $-0.06$ | **(a) (4 pts)** The **6/6 cell** (row $J=6$, column $K=6$) is $\mathbf{0.99\%/\text{month}}$. That sits right on the Ch 5.3 order-of-magnitude claim — positive and $\approx 1\%$/month. Annualizing *roughly*, $0.99\% \times 12 \approx \mathbf{11.9\%/\text{year}}$ (this ignores compounding; it is an order-of-magnitude statement, not a precise annual return — a real annualization would compound and would also have to be quoted net of costs). **(b) (5 pts)** The surface is a **medium-horizon ridge that decays toward the long-horizon corner**. (i) Holding $J=6$ fixed and lengthening the hold $K$ from 3 to 12, the spread falls monotonically: $1.08 \to 0.99 \to 0.80 \to 0.52$ — the profit is strongest at short holds and *bleeds away* as you hold longer, because the continuation is a medium-horizon phenomenon, not a permanent one. (ii) Reading into the bottom-right corner ($J=12,\,K=12$) the spread decays all the way to $-0.06$ — it goes *negative*. That sign flip foreshadows **long-run reversal** (De Bondt & Thaler, 1985): at long formation-and-holding horizons, past winners start to *under*-perform past losers as prices that overshot eventually correct. Momentum lives in the medium-horizon gap *between* short-run reversal and this long-run reversal. **(c) (4 pts)** Two reasons the "pick the biggest cell" reading is wrong: 1. *Statistical (data-snooping).* The $1.12$ at $(9,3)$ is the **maximum over sixteen correlated tests**. Crowning the largest cell and reporting its $t$-statistic is the multiple-comparisons trap from Week 1 — the maximum of sixteen tries is upward-biased, so its apparent significance is overstated. You sharpen this in Problem 5(a). 2. *A region beats a cell.* The evidence for momentum is not one starred cell; it is that a **broad contiguous region** of the grid is positive and similar in magnitude (the entire upper-left block is $\approx 0.7$–$1.1\%$). A single hand-picked extreme could be luck; a whole coherent region moving together is a *pattern*, far harder to dismiss and far less sensitive to which exact $(J,K)$ you trade. Reporting the surface, not the peak, is the honest move. **(d) (3 pts)** The grid shows only point estimates; what must sit *next to* every cell is its **standard error (and the $t$-statistic), and specifically the HAC / Newey–West version, not the naive i.i.d. one**. Omitting it produces the failure mode of **overstated significance**: because the overlapping construction induces positive serial correlation (Problem 3), the naive standard error is too small and the $t$-statistic too large, so cells that look "significant" may not be once the overlap is respected. A grid of bare point estimates is not yet evidence. --- ## Problem 3 — Overlapping holding periods, induced serial correlation, and why HAC (18 points) **(a) (5 pts)** With $K=6$, the return recorded in calendar month $t$ is the equal-weighted average of the **six cohorts currently live** — those formed in months $t, t-1, \dots, t-5$. The return recorded in month $t+1$ is the average of the cohorts formed in months $t+1, t, t-1, \dots, t-4$. Line them up: - Month $t$'s cohorts: $\{t,\, t-1,\, t-2,\, t-3,\, t-4,\, t-5\}$. - Month $t+1$'s cohorts: $\{t+1,\, t,\, t-1,\, t-2,\, t-3,\, t-4\}$. They **share five of the six cohorts** ($t$ through $t-4$). So even if the momentum payoff to each freshly formed cohort were i.i.d., the two reported monthly returns are averages built from $5/6$ of the *same* underlying ingredients — they are mechanically correlated. The same overlap argument applies at larger gaps: month $t$ and month $t+\ell$ share $6-\ell$ cohorts for $\ell = 1,2,\dots,5$, and share **none** once $\ell \ge 6$. So the induced autocorrelation persists over lags $\ell = 1$ through $\ell = K-1 = \mathbf{5}$ and then dies out. This is precisely the overlapping-observations mechanism from Week 2: overlapping windows manufacture serial correlation out of thin air. **(b) (5 pts)** Sam regresses the spread series on a constant; the single coefficient $\hat\beta_0$ *is* the mean spread, and its standard error is the standard error of the mean. The classical / i.i.d. formula computes that standard error as (roughly) $s/\sqrt{T}$, which assumes all $T$ monthly observations are **independent draws**. They are not: with positive serial correlation, each new month *echoes* its neighbors, so it carries less fresh information than an independent draw would. The Moulton-style intuition from Ch 2.4 §5.3: the **effective** sample size — the number of genuinely independent observations — is *well below* $T$. The naive formula divides by an effective $T$ that is **too large**, so it reports a standard error that is **too small** and a $t$-statistic that is **too large**. The direction of the error is therefore toward **over-confidence**: Sam will think momentum is more decisively significant than the data actually warrant. **(c) (4 pts)** The fix is the **HAC (heteroskedasticity-and-autocorrelation-consistent) standard error, a.k.a. Newey–West**. For the 6/6 strategy set the lag length $L = K - 1 = \mathbf{5}$: from part (a) that is exactly how many lags of induced autocorrelation the overlap creates before it vanishes, so $L=5$ absorbs all of it (rounding up further does no harm). Arithmetically, HAC adds to the "middle of the sandwich" the **cross-products of residuals from nearby months**, $\hat\varepsilon_t\hat\varepsilon_{t-\ell}$ for $\ell = 1,\dots,L$, down-weighted by the Bartlett kernel — terms the classical formula assumes are zero. Crucially the **point estimate is unchanged**: the mean spread is still the mean spread regardless of which standard error you compute; only the standard error (hence the $t$-statistic) moves. **(d) (4 pts)** Mean spread $= 1.00\%$, classical SE $= 0.083\%$, HAC ($L=5$) SE $= 0.122\%$. (i) The two $t$-statistics: $$ t_{\text{classical}} = \frac{1.00}{0.083} \approx \mathbf{12.05}, \qquad t_{\text{HAC}} = \frac{1.00}{0.122} \approx \mathbf{8.20}. $$ (*Verified in Python. The HAC SE is $0.122/0.083 \approx 1.47\times$ the classical — "roughly half-again larger," consistent with the reader's guide.*) (ii) **Yes, it survives.** Both far exceed $1.96$; momentum is overwhelmingly significant here, and the switch from classical to HAC merely shrinks an enormous $t$ to a still-enormous $t$. A $t$ of 8 does not become insignificant when inflated by a factor of $1.47$. (iii) The switch would **flip** the conclusion when the original $t$ is **marginal** — say a naive $t \approx 2.5$ that, divided by the same $\approx 1.47$ inflation, lands near $1.7$ and falls below $1.96$. That is exactly the case Ch 5.3 tells a referee to check first: a comfortable $t$ of 6 survives the overlap correction, but a marginal starred result can be entirely an artifact of the too-small naive standard error — so the first question on the referee's notepad is *"are these $t$-statistics overlap-corrected?"* --- ## Problem 4 — The transaction-cost critique and the break-even cost (22 points) Cost model and illustrative inputs: $$ \text{net}(c) = \bar g - c\,\bar\tau_L - (c+\text{sp})\,\bar\tau_S, \qquad \bar g = 100\text{ bps},\;\; \bar\tau_L = 0.30,\;\; \bar\tau_S = 0.30,\;\; \text{sp} = 40\text{ bps}. $$ **(a) (4 pts)** You multiply the cost by **turnover** — the fraction of each leg that actually *changes hands* on re-ranking — because **you only pay trading costs on the shares you trade**, not on the shares you simply continue to hold. A name that stays in the winner decile from one month to the next costs nothing to "keep"; only the slice that enters or exits the decile incurs the round-trip bid-ask + commission. Charging the cost against the whole book would massively overstate the bill. The short leg gets the extra premium because **shorting beaten-down small-cap losers is physically more expensive**: those names are illiquid and low-priced with wide bid-ask spreads, and you must *borrow the shares to short them*, paying a borrow / stock-loan fee that the long leg of large-cap winners never incurs. The leg the reader's guide warns is the dangerous one is precisely this short leg. **(b) (6 pts)** At $c = 50$ bps: $$ \text{long-leg cost} = c\,\bar\tau_L = 50 \times 0.30 = 15\text{ bps}, \qquad \text{short-leg cost} = (c+\text{sp})\,\bar\tau_S = (50+40)\times 0.30 = 27\text{ bps}, $$ $$ \text{net}(50) = 100 - 15 - 27 = \mathbf{58\text{ bps/month}}. $$ At $c = 100$ bps: $$ \text{long-leg cost} = 100 \times 0.30 = 30\text{ bps}, \qquad \text{short-leg cost} = (100+40)\times 0.30 = 42\text{ bps}, $$ $$ \text{net}(100) = 100 - 30 - 42 = \mathbf{28\text{ bps/month}}. $$ The strategy is **net-profitable at both** cost levels (58 bps and 28 bps respectively) — unsurprising, since both lie below the break-even computed next. (*Verified in Python.*) **(c) (6 pts)** Set $\text{net}(c)=0$ and solve for $c$: $$ \bar g - c\,\bar\tau_L - (c+\text{sp})\,\bar\tau_S = 0 \;\Longrightarrow\; \bar g - \text{sp}\,\bar\tau_S = c\,(\bar\tau_L + \bar\tau_S) \;\Longrightarrow\; c^\star = \frac{\bar g - \text{sp}\,\bar\tau_S}{\bar\tau_L + \bar\tau_S}. $$ The $+\text{sp}\,\bar\tau_S$ term moves to the gross side because the short premium is a *fixed* charge on the short leg that eats into the gross profit before the per-round-trip cost $c$ even applies. Plugging in: $$ c^\star = \frac{100 - 40\times 0.30}{0.30 + 0.30} = \frac{100 - 12}{0.60} = \frac{88}{0.60} \approx \mathbf{146.7\text{ bps}}. $$ (*Verified in Python: $\text{net}(146.7) \approx 0$.*) As a check on which leg does the damage, set $\text{sp}=0$: $$ c^\star_{\text{no sp}} = \frac{100}{0.60} \approx 166.7\text{ bps}. $$ So the short-leg borrow premium **lowers the break-even by $166.7 - 146.7 = 20$ bps** — the short leg makes the strategy break even at a $20$-bp-*lower* round-trip cost than it otherwise would, confirming it is the more fragile leg. **(d) (3 pts) Verdict.** At a break-even of $\approx 147$ bps round-trip, this illustrative 6/6 strategy *does* survive realistic costs **on the liquid long leg alone** — large-cap winners trading at $20$–$30$ bps round-trip leave the gross $\sim 1\%$/month largely intact. The danger is the **short leg**: the beaten-down small-cap losers can cost several times the winners' rate, and once their effective round-trip cost (bid-ask + price impact + borrow) approaches and exceeds the break-even neighborhood, the net profit collapses. So the honest conclusion mirrors the reader's guide's contested verdict: momentum's *gross* profit is real, but a meaningful share of it is trading friction, and **the short leg in small, illiquid losers is what kills it** — which is exactly why a gross-return result is not yet an efficiency violation. Whether momentum is a true anomaly or a paper profit depends on whether you can actually trade the losers at the costs your break-even can tolerate. **(e) (3 pts)** Subtracting the monthly cost lowers the **mean** of the net series (gross mean $100$ bps $\to$ net mean $58$ bps at $c=50$, etc.) but barely touches its **standard deviation** — the cost is a roughly steady monthly drag, not a new source of volatility, since turnover is fairly stable month to month. The $t$-statistic is mean/SE, so shrinking the numerator while the denominator holds steady **drags the $t$ down even though the net mean stays positive**. A strategy can thus be "profitable on average but no longer statistically distinguishable from zero": the net mean is positive but small relative to the unchanged month-to-month noise, so you can no longer reject the hypothesis that the *true* net spread is zero. This is the bridge between Problems 3 and 4 — the cost critique can sink significance even when it does not sink the average. --- ## Problem 5 — What's vulnerable: data-snooping, January, and risk vs. behavioral (16 points) **(a) (6 pts) Data-snooping.** (i) The grid is **sixteen correlated significance tests**; crowning the single most-significant cell and reporting its $t$-statistic is the **multiple-comparisons / multiple-testing problem** from Week 1 (Ch 1.5, revisited under the false-discovery rate later). The *maximum* over sixteen tries is upward-biased even under the null — run enough correlated tests and one will star by chance — so a "$t=2.5$ that is the max over sixteen cells" is worth far less than a $t=2.5$ chosen in advance; a reader should mentally Bonferroni-discount it. (ii) Two mitigating facts: first, the result is **a broad region of positive cells, not one isolated lucky cell** (Problem 2b) — a coherent surface is much harder to produce by snooping than a single extreme. Second, momentum **replicated out of sample**: in other countries, other asset classes, and the decades *after* 1989 that the original authors could not have peeked at. Out-of-sample survival is the strongest answer to a snooping charge because you cannot snoop data that did not exist when the hypothesis was formed — post-sample confirmation is evidence the effect is real rather than a fit to one historical window. **(b) (5 pts) January / seasonality.** (i) Momentum tends to **lose money in January and earn its keep in the other eleven months** — an *anti*-January pattern. Mechanism: the loser leg is full of beaten-down, small, low-priced stocks; these bounce hard in January (tax-loss-selling pressure releases at year-end and the names rebound), so the short leg gets hurt precisely in January. (ii) An anti-January result is **harder** to dismiss than a January-concentrated one because the era's known anomalies (size, value) were *concentrated in* January, so a profit living in January would look like just another seasonal artifact; a profit that *survives despite* a January loss cannot be that artifact. Yet the reader's guide still files it under "vulnerable" because the seasonality is a **clue to mechanism, not a clean robustness pass**: a strategy whose annual profit is "eleven good months minus a January loss" is exposed to anything that shifts that seasonal structure (changes to tax-loss-selling rules, the rise of index investing), so the seasonal dependence is itself a fragility, not a checkbox. **(c) (5 pts) Risk vs. behavioral.** The **behavioral** story is a two-stage under/over-reaction: investors *underreact* to news so prices drift toward fundamentals only slowly (this is the medium-horizon continuation momentum profits from), and then later *overreact*, pushing prices past fundamentals so they eventually correct (this is the long-run reversal seen in the bottom-right of the grid and in De Bondt–Thaler). The **risk-based** story says momentum loads on some priced, time-varying macro risk and the profit is compensation for bearing it — and the evidence cited is the **2009 momentum crash**, when the beaten-down losers rocketed in the recovery and the short leg blew up, exhibiting the kind of rare, severe crash risk an investor would demand to be paid for. Jegadeesh & Titman (1993) **cannot adjudicate** between these because it *predates the evidence on both sides* — the long-run-reversal and crash-risk data accumulated only later. The honest write-up therefore **reports the anomaly as real and labels the interpretation as open**, taking no premature side. --- ## Problem 6 — Replication design (pointing at nb5.3) (14 points) **(a) (5 pts) Construction recipe (6/6 cell).** At the end of each month $t$: 1. **Ranking variable:** compute each stock's cumulative return over the past $J=6$ months (months $t-5$ through $t$). 2. **Split:** sort all eligible stocks on that past return into ten deciles. 3. **Form legs (equal-weighted):** long the top decile (winners $W$), short the bottom decile (losers $L$), equal dollars per leg so the position is zero-cost. 4. **Overlapping reformation:** repeat this *every* month; in any calendar month you hold $K=6$ overlapping cohorts at once (formed this month back through five months ago) and average across them. 5. **Record:** one number per month — the cohort-averaged WML spread $r^{W}-r^{L}$. **Data:** **CRSP monthly returns**, restricted in the original to **NYSE/AMEX common stocks** over **1965–1989** (per Jegadeesh & Titman 1993; for the modern replication expect the magnitudes to shift, so report order-of-magnitude only rather than matching a table decimal). **Licensed-data rule (CONVENTIONS §5):** CRSP stays on GMU infrastructure (read-only on Hopper / WRDS Cloud), and you must **pin the CRSP snapshot date** in the notebook. **(b) (4 pts)** Report **two standard errors side by side for every cell: the naive i.i.d. (classical) and the Newey–West (HAC)** — because the overlapping construction induces serial correlation that makes the naive SE too small (Problem 3), and showing both makes the $t$-shrinkage visible. For the 6/6 cell set the **lag length $L = K-1 = 5$**. Expected result the reader's guide licenses: **6/6 should be positive and on the order of $\sim 1\%$/month** over a window near the original. The one thing you must *not* do: **claim to match a specific decimal from the paper's table** — magnitudes shift with sample and data vintage; assert order of magnitude only. **(c) (3 pts) Transaction-cost deliverable.** Plot **net momentum profit as a function of the round-trip cost $c$ (in bps)**, with turnover measured from the *actual* portfolio weights and an extra premium on the short leg. Locate the **break-even cost $c^\star$** (the $c$ where net profit crosses zero — Problem 4c). The figure must support a one-paragraph verdict on whether momentum survives realistic costs and *which leg* kills it; expect the **short leg** (illiquid small-cap losers) to dominate the cost. **(d) (2 pts) Robustness checks.** (1) *Data-snooping:* an **out-of-sample re-estimation** — split into in-sample (near 1965–1989) versus out-of-sample (1990–present, including the 2009 crash window) and show 6/6 persists; the demonstrating figure is the spread (and worst drawdown) in each era. (2) *Seasonality:* a **January-versus-rest-of-year decomposition** of the 6/6 spread; the demonstrating split is mean spread in January vs. the other eleven months, showing the anti-January pattern. --- *End of solutions for PS 5.3. The by-hand portfolio (Problem 1), the two $t$-statistics (Problem 3d), and the net-spread and break-even arithmetic (Problem 4b–c) were each confirmed in a few lines of Python: break-even $c^\star = (100 - 40\times0.30)/0.60 = 146.7$ bps, dropping to $166.7$ bps with no short premium; $\text{net}(50)=58$, $\text{net}(100)=28$ bps; $t_{\text{classical}}=12.05$, $t_{\text{HAC}}=8.20$. The grid reading (Problem 2), the HAC machinery (Problem 3), and the cost sweep (Problem 4) are all built in `notebooks/week-05/nb5.3-momentum-backtest.ipynb`, where the Newey–West $t$-statistic prints next to the naive one and the break-even line appears on the net-profit plot.*