Beyond the CAPM

Investment Theory

Lorenzo Naranjo

Fall 2026

From One Factor to Many

  • Beta pricing: if a portfolio Q is mean-variance efficient, expected excess returns are proportional to betas with Q.
  • The CAPM adds that Q is the market. A multifactor model says instead that Q is a combination of a few factor portfolios.
  • Each asset’s excess return follows R_{i,t} = \alpha_{i} + \beta_{i1} F_{1,t} + \cdots + \beta_{iK} F_{K,t} + e_{i,t}, \tag{1} and the model predicts \operatorname{E}(R_{i}) = \beta_{i1} \operatorname{E}(F_{1}) + \cdots + \beta_{iK} \operatorname{E}(F_{K}). \tag{2}
  • The test is again whether the alphas are zero.

Why More Than One Factor?

  • Intertemporal CAPM (Merton 1973).
    • Investors care about changes in investment opportunities, such as future interest rates or expected returns.
    • Assets that pay off when opportunities deteriorate are valuable hedges and earn lower expected returns.
  • Arbitrage pricing theory (Ross 1976).
    • If returns are driven by a few common factors plus firm-specific noise, no arbitrage implies that expected returns are approximately linear in the factor betas.
  • Neither theory says which factors to use. In practice, factors have been chosen from the anomalies.

The Three-Factor Model

  • Fama and French (1993) built factors from the size and value effects.
    • Split stocks into small and big at the NYSE median, and into low, medium and high book-to-market.
    • SMB: average of the three small portfolios minus average of the three big ones.
    • HML: average of the two high B/M portfolios minus average of the two low ones.
  • The model is R_{i,t} = \alpha_{i} + b_{i} MKT_{t} + s_{i} SMB_{t} + h_{i} HML_{t} + e_{i,t}.
  • It captures most anomalies known in the mid-1990s, except momentum.
  • Carhart (1997) added UMD, past winners minus past losers.

Example 1 (Building SMB and HML) In one month the six size and book-to-market portfolios return S/L 2.0%, S/M 1.5%, S/H 3.0%, B/L 1.0%, B/M 0.8% and B/H 1.6%. Then \begin{aligned} SMB & = \frac{2.0 + 1.5 + 3.0}{3} - \frac{1.0 + 0.8 + 1.6}{3} = 1.03\%, \\ HML & = \frac{3.0 + 1.6}{2} - \frac{2.0 + 1.0}{2} = 0.8\%. \end{aligned} Averaging over small and big makes HML roughly neutral to size.

The Valuation Identity

  • Fama and French (2015) start from the dividend discount model: \frac{M_{t}}{B_{t}} = \frac{1}{B_{t}} \sum_{\tau = 1}^{\infty} \frac{\operatorname{E}(Y_{t+\tau} - dB_{t+\tau})}{(1 + r)^{\tau}}, with market value M, book equity B, earnings Y and investment dB.
  • Holding everything else fixed, a higher expected return r goes with:
    • a higher book-to-market ratio,
    • higher expected profitability,
    • lower expected investment.
  • The identity holds whether prices are rational or not.

Five and Six Factors

  • The five-factor model adds RMW (robust minus weak profitability) and CMA (conservative minus aggressive investment): R_{i,t} = \alpha_{i} + b_{i} MKT_{t} + s_{i} SMB_{t} + h_{i} HML_{t} + r_{i} RMW_{t} + c_{i} CMA_{t} + e_{i,t}.
  • Main failure: small stocks that invest a lot despite low profitability.
  • Striking result: HML becomes redundant, its average return explained by the other four factors.
  • The five-factor model also explains low beta and low volatility (Fama and French 2016).
  • Adding UMD gives the six-factor model (Fama and French 2018).

Other Models

  • q-factor model (Hou et al. 2015), from investment-based asset pricing.
    • Firms with high discount rates invest less, and a high expected profitability relative to investment means a high discount rate.
    • Factors: market, size (ME), investment (IA) and profitability (ROE).
  • Mispricing factors (Stambaugh and Yuan 2017).
    • Average rankings across 11 anomalies in two clusters: MGMT (issuance, investment) and PERF (distress, momentum, profitability).
  • Behavioral factors (Daniel et al. 2020).
    • FIN from issuance and repurchases, PEAD from underreaction to earnings news.

Factor Premia

Mean (%) Volatility (%) Sharpe ratio t-statistic
MKT 7.57 15.77 0.48 3.53
SMB 1.44 10.44 0.14 1.01
HML 3.48 10.70 0.33 2.39
RMW 3.42 7.99 0.43 3.15
CMA 3.11 7.07 0.44 3.23
UMD 7.08 14.99 0.47 3.47
ME 2.35 10.66 0.22 1.62
IA 3.78 7.19 0.53 3.86
ROE 6.15 9.20 0.67 4.91
  • Annualized, January 1972 to December 2025; ME, IA and ROE are the q-factors. All premia are positive: ROE has the highest Sharpe ratio, SMB the lowest t-statistic.

Factor Returns over Time

Six small line charts on a logarithmic scale showing the growth of one dollar invested in T-bills plus each of the factors MKT, SMB, HML, RMW, CMA and UMD from 1972 to 2025, with the market as a gray reference line. The momentum factor ends highest. Profitability and investment grow steadily. Value grows until about 2007 and then declines. Size grows the least.

Growth of $1 invested in T-bills plus each Fama-French factor, on a log scale. The gray line is $1 in the market.

Is HML Redundant?

  • Most factor correlations are small. The exception is HML and CMA, 0.68: value firms tend to invest conservatively.
  • Fama and French warned that the redundancy of HML might be specific to their sample, which ended in 2013.
  • Regress HML on MKT, SMB, RMW and CMA:
    • July 1963 to December 2013: HML averaged 4.6% a year, alpha -0.1%.
    • January 2014 to September 2026: HML averaged -0.8% a year, alpha 0.5% (t = 0.18).
  • The value premium disappeared after 2013, and the five-factor model accounts for that too.

Testing a Model with GRS

  • Run (1) for each test asset and test whether all alphas are jointly zero: the F-test of Gibbons et al. (1989).
  • If the alphas are zero, the factors span the tangency portfolio, and adding the test assets cannot raise the maximum Sharpe ratio.
  • The GRS statistic grows with \frac{1 + \theta_{*}^{2}}{1 + \theta_{F}^{2}}, where \theta_{F} is the maximum Sharpe ratio of the factors and \theta_{*} that of the factors and test assets together.

The 25 Size and Book-to-Market Portfolios

Three scatter plots of the 25 size and book-to-market portfolios, with the model-predicted excess return on the horizontal axis and the realized average excess return on the vertical axis. Under the CAPM the points spread vertically far from the 45-degree line, since predicted returns vary little while realized returns range widely. Under the three-factor and five-factor models the points line up much more closely along the line, with the small-growth portfolio still well below it.

Average annual excess returns of the 25 size and book-to-market portfolios against the returns predicted by three models. Darker points are smaller firms.

GRS Tests

Mean abs. alpha (%) Largest abs. t GRS p-value
CAPM 2.17 3.41 3.68 < 0.001
FF3 1.15 4.83 3.30 < 0.001
Carhart 1.05 4.34 2.99 < 0.001
FF5 1.10 3.75 2.92 < 0.001
FF6 1.02 3.30 2.70 < 0.001
q-factor 1.15 3.45 3.00 < 0.001
  • Multifactor models cut the average absolute alpha by more than half, but GRS rejects every model.
  • With more than 50 years of data even small alphas are significant; small growth (S1B1) earns far less than any model predicts.
  • A model can be rejected and still be useful.

The Anomalies Revisited

CAPM FF3 Carhart FF5 FF6 q-factor
Size −0.14 −0.74 −0.79 0.08 0.03 1.01
Value 3.81 −2.28 −1.98 −0.87 −0.67 0.71
Low beta 6.36 4.31 1.33 0.30 −1.88 −1.62
Momentum 15.72 18.71 3.51 16.15 3.15 7.59
Reversal −0.65 −1.26 2.75 0.16 3.43 1.62
Profitability 6.54 5.89 5.34 0.04 −0.01 −0.03
Investment 5.69 2.84 1.64 0.21 −0.36 0.17
Accruals 3.56 3.40 2.69 3.51 2.91 3.10
Low volatility 14.52 11.60 8.58 5.18 3.12 3.30
  • Alphas of the decile spreads, % per year. Bold: |t| > 1.96.
  • Significant alphas: CAPM 6, FF3 6, FF5 3, FF6 2.

What the Models Absorb

  • The three-factor model overcorrects for value and does nothing for profitability, investment, momentum, accruals or low volatility.
  • The five-factor model absorbs profitability, investment and low beta, but leaves momentum as large as under the CAPM.
  • Adding UMD brings the momentum alpha down to 3.2% a year.
  • The q-factor model has no momentum factor, yet halves the momentum alpha: ROE has a correlation of 0.48 with UMD.
  • Accruals keep a significant alpha in almost every model.

Comparing Models by Their Sharpe Ratios

  • Every factor is a portfolio that a competing model could use as a test asset.
  • So the better model is the one whose factors attain the higher maximum Sharpe ratio (Barillas and Shanken 2018).
  • By the Treynor-Black logic, a new factor with alpha \alpha_{k} and residual volatility \sigma(e_{k}) relative to the existing factors raises the maximum squared Sharpe ratio by its squared information ratio: \theta_{\text{new}}^{2} = \theta_{\text{old}}^{2} + \left( \frac{\alpha_{k}}{\sigma(e_{k})} \right)^{2}. \tag{3}
  • A factor with a high average return but zero alpha adds nothing.

Maximum Sharpe Ratios

A horizontal bar chart of the maximum Sharpe ratio for six factor models. The CAPM is lowest at about 0.5, the three-factor model is about 0.65, the Carhart and five-factor models are near 1.0, and the six-factor and q-factor models are highest at about 1.2.

Annualized in-sample maximum Sharpe ratio attainable by combining each model’s factors.

  • These are in-sample numbers, chosen with hindsight. They are a fair comparison only because every model gets the same advantage.

Example 2 (Does a New Factor Add Anything?) A model’s factors have a maximum Sharpe ratio of 0.95. Factor A has an alpha of 2.4\% a year relative to them, with residual volatility 8\%, so its information ratio is 0.3 and by (3) the new maximum is \sqrt{0.95^{2} + 0.3^{2}} = 0.996. Factor B averages 4\% a year but has zero alpha, so adding it leaves the maximum at 0.95. This is HML in the five-factor model.

The Factor Zoo

  • Hundreds of candidate factors: a “zoo” (Cochrane 2011).
  • Are they real?
    • Hou et al. (2020): 65% of 452 anomalies fail |t| > 1.96 with NYSE breakpoints and value weights.
    • Jensen et al. (2023): most factors replicate, cluster into 13 themes, and work out of sample in 93 countries.
  • How many does a model need?
    • Kozak et al. (2020), Lettau and Pelger (2020), Kelly et al. (2019): a handful of well-chosen combinations of characteristics capture most of the information.
    • Estimation Risk and Shrinkage takes up these methods.

How Much of the Zoo Do Six Factors Explain?

Two overlaid histograms of alpha t-statistics for about 150 factors. The CAPM histogram is wide and spread far into large positive values, with many factors beyond 4. The six-factor histogram is more concentrated around zero, though a substantial number of factors still lie beyond the dashed lines at plus and minus 1.96.

Distribution of the alpha t-statistics of the U.S. factors of Jensen et al. (2023) under the CAPM and the six-factor model. The dashed lines mark \pm 1.96.

  • Significant alphas: 72% of 153 factors under the CAPM, 47% under the six-factor model.

Using Factor Models

  • Cost of equity: several exposures times their premia, instead of a single beta.
  • Risk models: the covariance matrix of many stocks follows from their loadings on a few factors.
  • Performance evaluation: a fund that tilts toward small value stocks or past winners has exposures an investor can buy cheaply.
    • Only its alpha relative to the factors measures skill.

Example 3 (A Cost of Equity) A five-factor regression gives loadings b = 1.1, s = 0.4, h = 0.3, r = 0.2 and c = -0.1, with premia of 7\%, 2\%, 3.5\%, 3\% and 3\% and a risk-free rate of 4\%. By (2), the expected return is 4\% + 1.1 \times 7\% + 0.4 \times 2\% + 0.3 \times 3.5\% + 0.2 \times 3\% - 0.1 \times 3\% = 13.85\%. With a CAPM beta of 1.15, the CAPM gives 4\% + 1.15 \times 7\% = 12.05\%, almost two points less.

References

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Carhart, Mark M. 1997. “On Persistence in Mutual Fund Performance.” Journal of Finance 52 (1): 57–82.
Cochrane, John H. 2011. “Presidential Address: Discount Rates.” The Journal of Finance 66 (4): 1047–108.
Daniel, Kent, David Hirshleifer, and Lin Sun. 2020. “Short- and Long-Horizon Behavioral Factors.” Review of Financial Studies 33 (4): 1673–736.
Fama, Eugene F., and Kenneth R. French. 1993. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33 (1): 3–56.
Fama, Eugene F., and Kenneth R. French. 2015. “A Five-Factor Asset Pricing Model.” Journal of Financial Economics 116 (1): 1–22.
Fama, Eugene F., and Kenneth R. French. 2016. “Dissecting Anomalies with a Five-Factor Model.” Review of Financial Studies 29 (1): 69–103.
Fama, Eugene F., and Kenneth R. French. 2018. “Choosing Factors.” Journal of Financial Economics 128 (2): 234–52.
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Hou, Kewei, Chen Xue, and Lu Zhang. 2020. “Replicating Anomalies.” Review of Financial Studies 33 (5): 2019–133.
Jensen, Theis Ingerslev, Bryan Kelly, and Lasse Heje Pedersen. 2023. “Is There a Replication Crisis in Finance?” The Journal of Finance 78 (5): 2465–518.
Kelly, Bryan T., Seth Pruitt, and Yinan Su. 2019. “Characteristics Are Covariances: A Unified Model of Risk and Return.” Journal of Financial Economics 134 (3): 501–24.
Kozak, Serhiy, Stefan Nagel, and Shrihari Santosh. 2020. “Shrinking the Cross-Section.” Journal of Financial Economics 135 (2): 271–92.
Lettau, Martin, and Markus Pelger. 2020. “Factors That Fit the Time Series and Cross-Section of Stock Returns.” Review of Financial Studies 33 (5): 2274–325.
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