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A Five-Factor Asset Pricing Model

2015 · paper · Economics

Tests a five-factor model of stock returns adding profitability and investment factors to market, size and value. The model is statistically rejected, but it adequately describes average returns for applied purposes; its main failure is small, low-profitability, high-investment stocks.

Checked: PDF from static1.squarespace.com, fetched 5 Oct 2026. Page numbers are those of that PDF.

9 findings

Substance p.14 gives the 2x2 HML–RMW correlation as '-0.04 for the 2x2 sorts', but Table 4 Panel C prints 0.04, with no sign, in both symmetric cells.
  • p. 14 Section IV, last paragraph
    -0.04 for the 2x2 sorts and 0.08 for the 2x3 sorts
  • p. 35 Table 4 Panel C, 2x2 factors, HML/RMW
    0.04

Only the authors can say what is right here.

Judge's note: On the upright p.35 image the 2x2 factors block shows HML/RMW = 0.04 and RMW/HML = 0.04. Other negative entries in the same block carry minus signs. The text and table disagree in sign, and only the authors can say which is right.

Second check: another model agreed

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Substance The Table 4 2x2x2x2 CMA_S (0.23) and CMA_B (0.06) cannot be rebuilt from the Table A1 2x2x2x2 portfolio means, which give about 0.12 and 0.16; the p.14 claim ("CMA_B is 0.06% per month (t = 0.98) and almost all the average value of CMA is from small stocks") rests on the Table 4 values.
  • p. 34 Table 4 Panel A, 2x2x2x2 factors, CMA_S and CMA_B
    0.23
  • p. 45 Table A1, 2x2x2x2 portfolios
    SLWC
  • p. 14 Section IV
    CMAB is 0.06% per month (t = 0.98)

Only the authors can say what is right here.

Judge's note: Table A1 gives small C = (1.01+1.26+1.35+1.58)/4 = 1.30 and small A = (0.73+1.16+1.26+1.57)/4 = 1.18. It gives big C = (0.89+0.93+1.01+1.26)/4 = 1.0225 and big A = (0.63+0.91+0.84+1.06)/4 = 0.86. Under the Table 4 caption's definition this gives CMA_S ≈ 0.12 and CMA_B ≈ 0.16. Both tables are internally consistent (t-statistics match their means and SDs), and both average to an overall CMA of about 0.14–0.15. So the discrepancy is in the small/big split, or in the portfolio labels in A1, and only the authors can say which table is right. HML_S/HML_B (0.40/0.20 vs 0.38/0.23) are also slightly off.

Second check: another model agreed

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Substance p.19 summary: the offending small portfolio is said to have "strong positive exposure to RMW", but the results show strongly negative RMW slopes.
  • p. 19 Section VII, summary paragraph
    strong positive exposure to RMW and strong negative exposure to CMA, suggestive of firms that invest a lot despite low profitability
  • p. 24 Section VII.C
    Both show strong negative exposures to RMW and CMA
  • p. 19 Section VII summary
    strong positive exposure to RMW and strong negative exposure to CMA
  • p. 22 Section VII.A
    -0.68 (t = -8.62)

Judge's note: Table 7 Panel B (2x2x2x2) gives the small low-B/M portfolio r = -0.68 (t = -8.62), and Table 9 gives the small high-Inv portfolio r = -0.36. On p.24 the text says 'Both show strong negative exposures to RMW and CMA'. The same summary sentence also says 'low profitability', which implies a negative RMW slope. The fix is 'negative'.

Second check: another model agreed

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Substance p.24: "(-0.36, t = -5.48, and -0.52, t = -7.48)" gives t = -5.48 for the RMW slope; Table 9 prints t(r) = -5.78 for the Small/High Inv cell.
  • p. 24 Section VII.C, para 2
    (-0.36, t = -5.48, and -0.52, t = -7.48)
  • p. 43 Table 9, t(r), Small row, High column
    -5.78

Judge's note: In Table 9 the Small/High cell has r = -0.36 with t(r) = -5.78, and c = -0.52 with t(c) = -7.48. The intercept is a = -0.40 with t(a) = -5.48. So the text has carried the intercept's t-statistic over to the RMW slope.

Second check: another model agreed

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Reference or label Huberman and Kandel is cited as '(1987)' in the text (pp.5, 18), but the reference list gives 'Huberman, Gur, and Shmuel Kandel, 1989, Mean-variance spanning. Journal of Finance 42, 873–888.'
  • p. 5 Section I
    Kandel (1987)
  • p. 18 Section VI
    In the spirit of Huberman and Kandel (1987)
  • p. 27 References
    Huberman, Gur, and Shmuel Kandel, 1989, Mean-variance spanning. Journal of Finance 42, 873–888.
  • p. 5 Section I, interpretation of the zero-intercept hypothesis
    Leaning on Huberman and Kandel (1987)

Judge's note: The page images show 1987 in both in-text citations and 1989 in the reference entry, and there is no other Huberman–Kandel entry. The mismatch is internal to the paper. Which year is correct rests on outside knowledge: that JF volume 42 is the 1987 volume.

Second check: another model agreed

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Reference or label Table 3, 2x2x2x2 SMB: "(SHRC + SHRA + SHWC + SHWA + SLRC + SLRA + SLWC + SHWA) / 8" lists SHWA twice and omits SLWA.
  • p. 33 Table 3, 2x2x2x2 SMB row
    SLRC + SLRA + SLWC + SHWA) / 8
  • p. 33 Table 3, 2x2x2x2 SMB row
    SLWC + SHWA) / 8

Judge's note: The upright page 33 image confirms the duplicated SHWA. The eight small portfolios should be SHRC, SHRA, SHWC, SHWA, SLRC, SLRA, SLWC and SLWA, so the last term should be SLWA.

Second check: another model agreed

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Reference or label Table 3, 2x2x2x2 HML: "( SLRC + SLRA + SLWC + SLWA + SLRC + BLRA + BLWC + BLWA) / 8" repeats SLRC and omits BLRC.
  • p. 33 Table 3, 2x2x2x2 HML row, second line
    ( SLRC + SLRA + SLWC + SLWA + SLRC + BLRA + BLWC + BLWA) / 8
  • p. 33 Table 3, 2x2x2x2 HML row, low B/M terms
    SLRC + SLRA + SLWC + SLWA + SLRC + BLRA
  • p. 33 Table 3, 2x2x2x2 HML
    SLWA + SLRC + BLRA + BLWC + BLWA
  • p. 33 Table 3, 2x2x2x2 HML, low-B/M part
    SLRC + SLRA + SLWC + SLWA + SLRC + BLRA + BLWC + BLWA

Judge's note: The upright page 33 image confirms the error: the fifth low-B/M term is printed as SLRC where it should be BLRC.

Second check: another model agreed

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Reference or label Table 7 Panel B, 2x3 factors block: the CMA slopes are labelled "g" and "t(g)" rather than c and t(c).
  • p. 41 Table 7 Panel B, 2x3 factors, third block
    t(g)

Judge's note: The page 41 image shows the regression equation with cCMA(t), and the 2x2x2x2 block on the same page labels the CMA slopes c / t(c). No g coefficient is defined.

Second check: another model agreed

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Reference or label In Table A1, the 2x2 Size-Inv portfolios header reads "SA SA BC BA". The first column should be SC (small conservative).
  • p. 45 Table A1, 2x2 Size-Inv portfolios header
    SA SA BC BA

Judge's note: The crop and the full image of page 45 both show the header "SA SA BC BA" for the 2x2 Size-Inv block. The other blocks use the C/A pairing (2x3 Size-Inv: SC SN SA BC BN BA). Table 4 gives the 2x2 CMA_S mean as 0.33, which equals 1.36 - 1.03, so the first column is the conservative portfolio SC. Likewise the 2x2 CMA_B mean of 0.11 equals 0.96 - 0.85. The first column header is therefore mislabelled; the data are unaffected.

Second check: another model agreed

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21 minor (typos and the like; most checked by the judge only)
Reference or label p.2: 'The predictions drawn from (3), here and below, center on the price, mt' cites equation (3) before it is introduced, while the paragraph discusses equation (1) and the per-share price m_t that appears only in (1).
  • p. 2 Paragraph after eq. (1)
    The predictions drawn from (3), here and below, center on the price, mt
  • p. 2 Introduction
    The predictions drawn from (3), here
  • p. 2 para after Eq. (1)
    The predictions drawn from (3), here and below, center on the price

Only the authors can say what is right here.

Judge's note: On the p.2 image the sentence sits in the paragraph that discusses Eq. (1). Equations (2) and (3) are introduced later on the page, and (3) is written in M_t/B_t, not m_t. 'Here and below' may be meant as a forward reference, so the authors should say whether (1) was intended.

Second check: another model agreed

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Reference or label The reference "Miller, Merton H., and Franco Modigliani, Dividend Policy, Growth, and the Valuation of Shares, Journal of Business, 34, 411-433." has no year, although the text cites Miller and Modigliani (1961).
  • p. 27 References
    Miller, Merton H., and Franco Modigliani, Dividend Policy, Growth, and the Valuation of Shares,
  • p. 2 Introduction
    Miller and Modigliani (1961)

Judge's note: On the page 27 image this entry has no year, while every other entry gives one. On p.2 the text cites '(1961)'.

Second check: another model agreed

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Wording p.6: 'In every Size row, average return increases with B/M' overstates Table 1 Panel A. Row 4 falls from 0.56 to 0.53, the Big row falls from 0.48 to 0.44, and row 2 is flat at 0.90, 0.90.
  • p. 6 Section II, para 2
    In every Size row, average return increases with B/M.
  • p. 31 Table 1 Panel A, rows 4 and Big
    0.56

Judge's note: Checked against the Table 1 image on p.31. The extreme-column spreads quoted in the text are correct (1.11-0.19 and 0.58-0.42), but the increase is not monotonic in every row.

Second check: another model agreed

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Wording p.12: 'The correlations of the 2x2x2x2 version of HML with the 2x2 and 2x3 versions are high, 0.94 and 0.96'. Table 4 Panel B shows 0.96 with the 2x2 version and 0.94 with the 2x3 version, so the numbers are listed in the reverse order.
  • p. 12 Section IV, para 3
    with the 2x2 and 2x3 versions are high, 0.94 and 0.96
  • p. 35 Table 4 Panel B, HML
    0.94

Judge's note: Checked Table 4 Panel B on the upright p.35 image: the 2x2x2x2 row of the HML block reads 0.94 (2x3), 0.96 (2x2), 1.00. The parallel order in the text pairs each number with the wrong version.

Second check: another model agreed

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Wording p.16: 'the four-factor model that excludes HML captures average returns as well as any other four-factor model considered' is presented as common to all LHS portfolios, but with 2x3 factors the HML RMW model beats RMW CMA on both GRS and |a| in Panel B (1.65/0.063 vs 1.82/0.076) and Panel D (1.97/0.112 vs 2.02/0.137).
  • p. 16 Section V
    the four-factor model that excludes HML captures average returns as well as any other four-factor model considered
  • p. 36 Table 5 Panel B, 2x3 factors
    1.65
  • p. 37 Table 5 Panel D, 2x3 factors
    0.112

Only the authors can say what is right here.

Judge's note: Checked Table 5 on the upright p.36 and p.37 images; the values are as quoted. The 2x2 and 2x2x2x2 results are mixed. The general claim overstates the table.

Second check: another model agreed

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Wording p.18: "We noted earlier that the five-factor model never improves the description of average returns from the four-factor model that drops HML" is worded more strongly than Table 5 and than the earlier statement ("much the same GRS statistics and average absolute intercepts").
  • p. 18 Section VI, para 1
    We noted earlier that the five-factor model never improves the description of average returns
  • p. 37 Table 5 Panel E, 2x3 factors
    0.093
  • p. 17 Section V (what was 'noted earlier')
    it produces much the same GRS statistics and average absolute intercepts as the four-factor model that does not include HML
  • p. 36 Table 5 Panels A–B, 2x3 factors
    0.101

Judge's note: Page 17 (text lines 548–549) says only 'much the same'. In Table 5, the five-factor model is marginally better than RMW CMA in some cells. Panel A 2x3: GRS 2.86 vs 2.87 and |a| 0.097 vs 0.101. Panel E 2x3: |a| 0.093 vs 0.098, though GRS is worse, 1.75 vs 1.70. The differences are trivial, so 'never' is an overstatement rather than an error.

Second check: another model agreed

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Typo Author footnote reads 'Amos Tuck School of Business, Dartmouth College. (French).' A stray period separates the affiliation from its author tag.
  • p. 1 Author footnote
    Dartmouth College. (French).

Judge's note: The crop of the p.1 footnote shows 'Dartmouth College.' followed on the next line by '(French).' The Fama tag has no such period.

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Typo p.2: 'Miller and Modigliani (1961) show that that the time t total market value' has a duplicated 'that'.
  • p. 2 Text after eq. (1)
    show that that the time t total market value

Judge's note: Visible on the p.2 page image.

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Typo p.2: 'In this equation Y_{t+τ}, is total equity earnings' has a stray comma between the subject and the verb.
  • p. 2 Definition below Equation (2)
    is total equity earnings for period t+τ

Judge's note: The comma is visible on the page image, so this is not an extraction artefact.

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Typo p.2: "(1961) show that that the time t total market value" duplicates 'that'; the reference entry "Gibbons, Michael R., Stephen A Ross" also lacks the period after the initial.
  • p. 27 References
    Miller, Merton H., and Franco Modigliani, Dividend Policy, Growth, and the Valuation of Shares,
  • p. 2 Text
    (1961) show that that the time t total market value
  • p. 27 References
    Stephen A Ross
  • p. 27 References
    Miller, Merton H., and Franco Modigliani, Dividend Policy

Judge's note: The page 2 image shows 'that that'. The page 27 image shows 'Stephen A Ross' with no period. This cluster also mentions the missing Miller–Modigliani year (judged under C025) and the uncited Fama–French 1992 entry (judged under C024); this verdict covers only the remaining copyedit points.

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Typo p.6: 'for the microcaps stocks in the first row' should read 'microcap stocks'.
  • p. 6 Section II, para 2
    for the microcaps stocks in the first row

Judge's note: Visible on the p.6 image.

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Typo The same factor is subscripted 'SMB_BM' on p.9 and 'SMB_B/M' on p.10.
  • p. 9 Section III
    SMBBM
  • p. 10 Section III
    SMBB/M

Judge's note: Both page crops show the two forms, so this is not an extraction artefact. It is a cosmetic inconsistency.

Second check: none (judge only)

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Typo p.15: the list of seven models runs '(i) three three-factor models ...; (ii) three four-factor models ...; and (iv) the five-factor model', skipping (iii).
  • p. 15 Section V, para 1
    and (iv) the five-factor model
  • p. 15 Section V, first paragraph
    and pairs of HML, RMW, and CMA; and (iv) the five-factor model
  • p. 15
    pairs of HML, RMW, and CMA; and (iv) the five-factor model

Judge's note: The page crop shows (iv) where (iii) belongs. The count of seven models is correct.

Second check: none (judge only)

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Typo p.17: "higher for the 2x2x4 portfolios than for the 5x5 portfolios" should read 2x4x4.
  • p. 17 Section V, last paragraph
    higher for the 2x2x4 portfolios than for the 5x5 portfolios

Judge's note: The page image shows '2x2x4'. The same paragraph, and the sentences before and after it, call the 32-portfolio sets '2x4x4 sorts'. The paper defines no 2x2x4 sort.

Second check: none (judge only)

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Typo p.17: "the p-values in the tests on the 2x4x4 portfolios are all less than 0.01 or less" is garbled wording; Table 5 Panel E (RMW CMA, 2x3) prints p = 0.010.
  • p. 17 Section V
    the p-values in the tests on the 2x4x4 portfolios are all less than 0.01 or less
  • p. 37 Table 5 Panel E, RMW CMA, 2x3 factors
    0.010
  • p. 17 Section V, last paragraph
    are all less than 0.01 or less

Judge's note: The image confirms the wording, which merges 'less than 0.01' and '0.01 or less'. In Table 5 Panels D–F the p-values are all ≤ 0.010, and Panel E RMW CMA 2x3 is exactly 0.010, so the intended reading is '0.01 or less'. This is a wording slip only.

Second check: none (judge only)

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Typo p.20: "HML, RMW¸ and CMA" has a stray cedilla-like mark where a comma should be.
  • p. 20 Section VII.A
    HML, RMW¸ and CMA

Judge's note: The page image shows the cedilla-like glyph after RMW, so this is not just a text-layer artefact. Cosmetic.

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Typo p.24: "Although the RMW and Inv slopes (-0.36, ... -0.52, t = -7.48)" calls the CMA slope an 'Inv' slope.
  • p. 24 Section VII.C
    Although the RMW and Inv slopes
  • p. 24 Section VII.C
    Although the RMW and Inv slopes (-0.36, t = -5.48, and -0.52

Judge's note: The regression in Table 9 has RMW and CMA slopes, and -0.52/-7.48 is the CMA slope c. Elsewhere the paper names the factor CMA; Inv is the sort variable.

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Typo The reference "Fama, Eugene F., and Kenneth R. French, 1992, The cross-section of expected stock returns" is listed but never cited in the text.
  • p. 27 References
    Fama, Eugene F., and Kenneth R. French, 1992, The cross-section of expected stock returns

Judge's note: A grep of main.txt finds '1992' only in the reference list (line 850). The text cites the other Fama–French works.

Second check: none (judge only)

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Typo Table 5 caption: "25 Size-OP portfolios (Panel B) 25 Size-Inv portfolios (Panel C)" is missing a comma after '(Panel B)'.
  • p. 36 Table 5 caption
    (Panel B) 25 Size-Inv portfolios (Panel C)

Judge's note: The upright page 36 image shows no comma there, while the other list items are separated by commas.

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Typo Table 9 caption reads "constructed using 2x2x2x2sorts on Size, B/M, OP, and Inv" with no space before "sorts".
  • p. 43 Table 9 caption
    2x2x2x2sorts

Judge's note: The crop of the Table 9 caption on PDF page 43 shows "2x2x2x2sorts" printed with no space, so this is not an artefact of the text layer. Every other occurrence in the paper reads "2x2x2x2 sorts", including the parallel captions of Tables 7, 8, A2 and A3. This is a cosmetic missing space.

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Typo The page footer of PDF page 47 (Table A3) is printed as "2", but the preceding appendix page (Table A1) is numbered 45.
  • p. 47 Page footer, Table A3
    2

Judge's note: The upright image of page 47 shows the folio "2" at the foot of the page. Page 45 (Table A1) shows "45". The upright image of page 46 (Table A2) shows no visible page number, so the reviewer's statement that Table A2 carries a sequential number is not borne out. Even so, "2" breaks the sequence, as page 47 would be expected. This looks like a cosmetic pagination leftover from a separately numbered appendix file.

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