# Factor Models Concept Card: [Your course, fund or case] > A concept card for the investments elective: multi-factor models of returns. It covers the single-index model, the > arbitrage pricing theory, the Fama-French three-factor and five-factor models and the Carhart momentum factor, how > factors are built, how to run and read a time-series factor regression, how to use factor models for expected > returns, risk decomposition and manager evaluation, and the statistical traps of the "factor zoo". It builds on > mba/mba-capm-wacc-card (CAPM and beta) and mba/mba-regression-for-managers-card (OLS, standard errors, R-squared); > read those first, because this card does not re-teach them. It is for second-year MBA students in investments and > asset management electives, buy-side interview candidates, and investment committee members who hire managers. > Worked example A tests whether a fictional fund's alpha survives the Fama-French model, Worked example B (half > solved) decomposes expected return and risk, and Exercise C is left for you, with a separate answer key. Source > basis: Sharpe (1963), Ross (1976), Fama and French (1993, 2015), Carhart (1997) and the related literature, written up > in original words. All data are invented *Example* values, not real factor returns. Replace every [bracketed] value > and every row marked *Example*. This is a learning aid, not investment advice. ## Document control | Field | Value | |---|---| | Course, fund or case | [Investments elective, session 5 / manager review] | | Learner or analyst | [Your name] | | Analysis date | YYYY-MM-DD | | Fund return series and fee basis | [Gross or net of fees, share class, currency] | | Factor data source, region and download date | [Source, region, file name, date] | | Risk-free rate series | [One-month bill or equivalent, source] | | Estimation window and frequency | [60 months, monthly] | | Reviewed by | [Study partner, tutor or manager-research reviewer] | | Version | [0.1] | | Last updated | YYYY-MM-DD | ## Contents - [When to use it](#when-to-use-it) - [The idea in plain language](#the-idea-in-plain-language) - [Formulas and notation](#formulas-and-notation) - [The main factors](#the-main-factors) - [Assumptions and limits](#assumptions-and-limits) - [Worked example A (fully solved): does Fund I have alpha?](#worked-example-a-fully-solved-does-fund-i-have-alpha) - [Worked example B (half solved): expected return and risk from factors](#worked-example-b-half-solved-expected-return-and-risk-from-factors) - [Exercise C (unsolved): a momentum fund](#exercise-c-unsolved-a-momentum-fund) - [Process diagram](#process-diagram) - [How investment managers use factor models](#how-investment-managers-use-factor-models) - [Connections to other cards](#connections-to-other-cards) - [Common mistakes and misconceptions](#common-mistakes-and-misconceptions) - [Self-test questions](#self-test-questions) - [My notes and next steps](#my-notes-and-next-steps) - [Sources and further reading](#sources-and-further-reading) - [Answer key](#answer-key) - [Revision history](#revision-history) ## When to use it **One-sentence rule:** explain a portfolio's excess return with its exposures to a few common factors, call what is left over alpha, and only believe an alpha that survives the factors the portfolio could have bought cheaply. | Use when | Use something else, or add to it, when | |---|---| | Judging whether an active manager's return is skill or a factor tilt | Comparing allocation and selection decisions against a benchmark (use Brinson attribution, finance/performance-attribution-report) | | Building a covariance matrix for many securities | Few assets with long, stable histories (a sample covariance matrix may do, mba/mba-portfolio-theory-card) | | Estimating a cost of equity that reflects size or value exposure | A single CAPM estimate is the agreed convention, as in many corporate settings (mba/mba-capm-wacc-card) | | Designing or reviewing a smart-beta or factor product | Bond portfolios, where duration and spread factors replace equity factors (mba/mba-fixed-income-card) | ## The idea in plain language **From one factor to several.** The CAPM says one factor, the market, prices all assets. Sharpe (1963) had already proposed the single-index model as a practical shortcut: if stocks are related only through the market, the covariance of any two stocks is the product of their betas times the market variance. That cuts the inputs for 500 stocks from 125,250 variances and covariances to 1,502 parameters. The cost is the assumption that nothing else is shared. **Arbitrage pricing theory.** Ross (1976) showed that if returns are driven by a few common factors plus diversifiable noise, and there are no arbitrage opportunities, then expected returns must be approximately linear in the factor loadings. The APT does not say which factors; it says that any priced factor must carry a premium. **Factors that worked in the data.** Banz (1981) found that small firms earned more than the CAPM predicted, and Fama and French (1992) found that the book-to-market ratio and size, not beta, explained the cross-section of US average returns. Fama and French (1993) turned these into tradable factor portfolios: **SMB** (small minus big) and **HML** (high minus low book-to-market), added to the market excess return. Jegadeesh and Titman (1993) documented momentum, and Carhart (1997) added a momentum factor when he studied mutual fund persistence. Fama and French (2015) added **profitability** (RMW, robust minus weak) and **investment** (CMA, conservative minus aggressive), and found that HML becomes largely redundant once these are included in their US sample. **Loadings and alpha.** Regress a portfolio's excess return on the factor returns. The slopes are the **loadings**: how much the portfolio behaves like each factor. The intercept is **alpha**: the average return not explained by the factors. If a manager's CAPM alpha disappears once size and value are added, the manager was mostly holding small value stocks, which an investor could do with an index fund at a fraction of the fee. **Priced factor versus risk factor.** A factor can explain a lot of variance without earning a premium (industry factors), and a characteristic can predict returns without being a covariance risk. Whether the premiums on size, value and momentum are compensation for risk or the result of investor behaviour is still debated. For a manager, the practical point does not depend on the answer: a return available from a cheap factor portfolio is not skill. ## Formulas and notation Time-series factor regression (Fama-French three-factor; add $m_i \, MOM_t$ for Carhart, and $r_i \, RMW_t + c_i \, CMA_t$ for the five-factor model): $$ R_{i,t} - R_{f,t} = \alpha_i + b_i (R_{m,t} - R_{f,t}) + s_i \, SMB_t + h_i \, HML_t + \varepsilon_{i,t} $$ Expected excess return implied by a factor model (alpha zero), and the single-index covariance: $$ E[R_i] - R_f = \sum_{k=1}^{K} \beta_{ik} \, \lambda_k \qquad \sigma_{ij} = \beta_i \beta_j \sigma_m^2 \quad (i \neq j) $$ Variance decomposition with factor covariance matrix $\Omega$ (diagonal if the factors are uncorrelated): $$ \sigma_i^2 = \boldsymbol{\beta}_i^\top \Omega \, \boldsymbol{\beta}_i + \sigma_{\varepsilon,i}^2 \qquad R^2 = \frac{\boldsymbol{\beta}_i^\top \Omega \, \boldsymbol{\beta}_i}{\sigma_i^2} $$ Testing and judging alpha: $$ t_\alpha = \frac{\hat{\alpha}}{\operatorname{SE}(\hat{\alpha})} \qquad \text{Appraisal ratio} = \frac{\hat{\alpha}}{\hat{\sigma}_\varepsilon} \times \sqrt{12} \quad \text{(monthly data)} $$ | Symbol | Meaning | Unit | |---|---|---| | $R_{i,t} - R_{f,t}$ | Excess return of portfolio $i$ in period $t$ | % per period | | $R_{m,t} - R_{f,t}$ (Mkt-RF) | Market excess return | % per period | | $SMB_t$, $HML_t$ | Size and value factor returns (long-short portfolios) | % per period | | $RMW_t$, $CMA_t$, $MOM_t$ | Profitability, investment and momentum factor returns | % per period | | $b_i, s_i, h_i$, $\beta_{ik}$ | Factor loadings (slopes) | Number | | $\alpha_i$ | Intercept: average return not explained by the factors | % per period | | $\lambda_k$ | Premium (expected return) of factor $k$ | % per year | | $\Omega$ | Covariance matrix of the factors | %² | | $\varepsilon_{i,t}$, $\sigma_{\varepsilon,i}$ | Residual return, residual (idiosyncratic) volatility | % per period | | $t_\alpha$ | t-statistic of alpha | Number | ## The main factors | Factor | Long side | Short side | How it is built (Fama-French style) | Source | |---|---|---|---|---| | Mkt-RF | Value-weighted market | Risk-free bill | Market return minus bill return | Sharpe (1964) | | SMB | Small stocks | Big stocks | 2 x 3 sorts on size and book-to-market, average of small portfolios minus average of big | Fama and French (1993) | | HML | High book-to-market (value) | Low book-to-market (growth) | Average of the two value portfolios minus the two growth portfolios | Fama and French (1993) | | MOM (also UMD, WML) | Past 12-month winners, skipping the last month | Past losers | Sorts on prior returns | Jegadeesh and Titman (1993); Carhart (1997) | | RMW | High operating profitability | Low profitability | 2 x 3 sorts on size and profitability | Fama and French (2015) | | CMA | Low asset growth (conservative) | High asset growth (aggressive) | 2 x 3 sorts on size and investment | Fama and French (2015) | Other widely studied factors include low beta, betting against beta (Frazzini and Pedersen 2014), quality, and the q-factor model's investment and return-on-equity factors (Hou, Xue and Zhang 2015). Asness, Moskowitz and Pedersen (2013) found value and momentum premiums across countries and asset classes, which makes data mining a less likely explanation for those two. **Data (dated fact).** As of October 2026 the Kenneth R. French Data Library publishes free monthly and daily returns for the three-factor model, the five-factor (2 x 3) model and a momentum factor, for the US and international regions (check the current files and their construction notes at mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html before you rely on them). ## Assumptions and limits 1. **Linear and stable loadings.** Managers change style; a five-year regression averages over changes. Use rolling windows to see drift. 2. **The right factors are included.** An omitted factor that the portfolio loads on shows up as alpha. The reverse also holds: an irrelevant factor costs degrees of freedom. 3. **The factor portfolios are investable.** Academic factors are long-short, rebalanced, before costs and include small, illiquid stocks; a real investor pays to replicate them. Compare alphas with what a low-cost factor fund would have delivered. 4. **Short samples are noisy.** The standard error of alpha shrinks only with the square root of the number of periods; a genuine alpha of 2% a year with residual volatility of 6% a year needs about $(2 \times 6 / 2)^2 = 36$ years for a t-statistic of 2. 5. **Multiple testing.** Hundreds of factors have been published. Harvey, Liu and Zhu (2016) argue that a new factor should clear a t-statistic of about 3.0, not 2.0. The same logic applies when a committee screens many managers: some will look skilled by luck (Fama and French 2010). 6. **Testing the model.** Fama and MacBeth (1973) estimate factor premiums with cross-sectional regressions period by period; Gibbons, Ross and Shanken (1989) test whether a set of alphas is jointly zero. Roll (1977) warned that the CAPM cannot be tested without the true market portfolio, a caution that carries over to any proxy. ## Worked example A (fully solved): does Fund I have alpha? **Situation (fictional).** The investment committee of the Tallow Creek Medical Center foundation (see mba/mba-investments-guide) is reviewing Fund I, an actively managed equity fund that has beaten the market by a wide margin over 24 months. Monthly excess returns (in %, net of fees) for the fund and three factors are below. Twenty-four months is far too short for a real decision; it keeps the example readable. | *Example* month | Mkt-RF | SMB | HML | Fund I excess | |---|---|---|---|---| | 1 | 4.65 | 4.94 | -0.13 | 7.53 | | 2 | -1.13 | 5.82 | -0.36 | 1.08 | | 3 | -1.84 | -0.69 | -1.88 | -3.96 | | 4 | -4.44 | 2.84 | -2.23 | -1.94 | | 5 | -4.35 | -0.70 | 8.82 | -0.55 | | 6 | 11.32 | 0.17 | 3.06 | 12.45 | | 7 | 1.16 | -1.71 | 0.50 | 0.54 | | 8 | 1.07 | 1.45 | 4.59 | 3.63 | | 9 | 4.24 | 2.71 | -1.32 | 5.63 | | 10 | 1.82 | 2.09 | -0.18 | 0.90 | | 11 | -3.29 | 1.28 | 1.56 | -2.23 | | 12 | -3.54 | -1.60 | -2.94 | -3.89 | | 13 | 2.38 | -0.12 | 5.31 | 5.67 | | 14 | -2.23 | 1.28 | 4.85 | 0.30 | | 15 | 4.50 | 1.16 | 2.12 | 6.26 | | 16 | -1.54 | -1.69 | -0.04 | -2.23 | | 17 | -3.26 | 2.48 | 2.54 | -0.78 | | 18 | -0.47 | 4.14 | -3.13 | 0.63 | | 19 | -2.87 | 0.46 | -2.78 | -4.45 | | 20 | 0.38 | 3.10 | -2.39 | 1.21 | | 21 | -8.16 | 0.82 | -0.02 | -7.47 | | 22 | -1.55 | -2.57 | 1.66 | -3.77 | | 23 | 6.23 | -0.08 | -0.26 | 6.38 | | 24 | -1.20 | 4.27 | 1.70 | 0.17 | | **Mean** | **-0.0883** | **1.2438** | **0.7937** | **0.8796** | **Step 1: CAPM regression.** `=INTERCEPT(E2:E25,B2:B25)` = 0.9702 and `=SLOPE(E2:E25,B2:B25)` = 1.0262; `=RSQ(E2:E25,B2:B25)` = 0.8642. `=LINEST(E2:E25,B2:B25,TRUE,TRUE)` gives the standard errors: 0.3549 for alpha and 0.0867 for beta, so $t_\alpha = 2.73$ (p = 0.0121, `=T.DIST.2T(2.73,22)`). The CAPM alpha is 0.97% a month, about 11.64% a year (x 12), with a 95% interval from 0.23% to 1.71% a month. On this evidence Fund I looks highly skilled. **Step 2: three-factor regression.** `=LINEST(E2:E25,B2:D25,TRUE,TRUE)` returns the coefficients in reverse column order (HML, SMB, Mkt-RF, intercept) in its first row and their standard errors in the second. | *Example* coefficient | CAPM | t | Three-factor | t | |---|---|---|---|---| | Alpha (% a month) | 0.9702 | 2.73 | 0.0408 | 0.17 | | Mkt-RF | 1.0262 | 11.83 | 0.9863 | 19.89 | | SMB | | | 0.4910 | 5.21 | | HML | | | 0.3971 | 5.63 | | R-squared | 0.8642 | | 0.9602 | | | Residual volatility (% a month) | 1.7385 | | 0.9871 | | | Degrees of freedom | 22 | | 20 | | The three-factor alpha is 0.04% a month (0.49% a year) with p = 0.8701 and a 95% interval from -0.47% to 0.55%. **Step 3: where did the return come from?** In an OLS regression with an intercept, the mean of the dependent variable equals alpha plus each loading times the factor mean. Fund I's mean excess return of 0.8796% a month splits into: | *Example* source | CAPM view (% a month) | Three-factor view (% a month) | |---|---|---| | Alpha | 0.9702 | 0.0408 | | Market | -0.0906 | -0.0871 | | Size (SMB) | | 0.6107 | | Value (HML) | | 0.3152 | | **Total** | **0.8796** | **0.8796** | ```vega-lite { "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "description": "Decomposition of Fund I mean monthly excess return under CAPM and the three-factor model (Example data)", "width": 380, "height": 220, "data": {"values": [ {"model": "CAPM", "source": "Alpha", "pct": 0.9702}, {"model": "CAPM", "source": "Market", "pct": -0.0906}, {"model": "Three-factor", "source": "Alpha", "pct": 0.0408}, {"model": "Three-factor", "source": "Market", "pct": -0.0871}, {"model": "Three-factor", "source": "Size (SMB)", "pct": 0.6107}, {"model": "Three-factor", "source": "Value (HML)", "pct": 0.3152} ]}, "mark": "bar", "encoding": { "x": {"field": "source", "type": "nominal", "title": "Source of return", "sort": ["Alpha", "Market", "Size (SMB)", "Value (HML)"]}, "xOffset": {"field": "model"}, "y": {"field": "pct", "type": "quantitative", "title": "Contribution (% a month)"}, "color": {"field": "model", "type": "nominal", "title": "Model"} } } ``` **Conclusion.** Fund I's apparent skill is almost entirely a tilt to small and value stocks during a period in which both factors did well and the market was flat. The appraisal ratio falls from 1.93 under the CAPM to 0.14 under the three-factor model (annualised, `=alpha/residual_sd*SQRT(12)`). The committee should ask whether it wants a small-value tilt and, if it does, whether a low-cost factor fund delivers it more cheaply. It should also extend the sample to at least 60 months, test with momentum, profitability and investment factors, and look at holdings to confirm the tilt. ## Worked example B (half solved): expected return and risk from factors **Situation (fictional).** A Northfield Outdoor Co. analyst wants an expected return and a risk breakdown for a small-cap equity portfolio held in the company pension plan. Annual *Example* inputs: risk-free rate 3.5%; factor premiums Mkt-RF 5.0%, SMB 2.0%, HML 3.0%; factor volatilities 16%, 10% and 11%; factors assumed uncorrelated; residual volatility 8%. Loadings: market 0.95, SMB 0.40, HML 0.30. | *Example* factor | Loading | Premium | Contribution to expected excess return | Factor variance (%²) | Contribution to variance (%²) | |---|---|---|---|---|---| | Mkt-RF | 0.95 | 5.0% | 4.75% | 256 | 231.04 | | SMB | 0.40 | 2.0% | [ ] | 100 | [ ] | | HML | 0.30 | 3.0% | 0.90% | 121 | 10.89 | | Residual | | | | | 64.00 | | **Total** | | | [ ] | | [ ] | 1. Complete the SMB row and the totals. Excel: `=SUMPRODUCT(B2:B4,C2:C4)` for the expected excess return. 2. Expected total return = 3.5% + [ ] = [ ]. 3. Total volatility = $\sqrt{\text{total variance}}$ = [ ]; R-squared = systematic variance / total variance = [ ]. 4. A CAPM regression of the same portfolio gives a beta of 1.10. The CAPM expected return is 9.00%. Explain the gap between the two estimates. 5. Which assumption in this example is least realistic, and how would you relax it in Excel? ## Exercise C (unsolved): a momentum fund **Situation (fictional).** A consultant presents Fund J to the committee with a Carhart four-factor regression on 60 monthly returns, **gross of fees**. The fund charges 1.20% a year. | *Example* coefficient | Estimate | Standard error | |---|---|---| | Alpha (% a month) | 0.18 | 0.15 | | Mkt-RF | 1.02 | 0.04 | | SMB | -0.05 | 0.07 | | HML | -0.10 | 0.08 | | MOM | 0.40 | 0.08 | | Residual volatility (% a month) | 1.90 | | | R-squared | 0.91 | | 1. Compute the t-statistics for alpha and for the momentum loading. Which are significant at 5% (55 degrees of freedom, critical value 2.0040)? 2. Annualise the alpha (x 12) and subtract the fee. What is left? 3. Compute the annualised appraisal ratio. 4. Holding alpha and residual volatility fixed, roughly how many months of data would make the alpha's t-statistic reach 2? 5. What is the fund's main style, and what would you ask the consultant next? ## Process diagram The flow below is a manager-review routine with factor models, from data to a hiring recommendation. ```mermaid flowchart TD A["Fund returns: gross or net, currency, at least 60 months"] --> B["Matching factor data: region, frequency, risk-free series"] B --> C["CAPM regression"] C --> D["Three-factor, then Carhart and five-factor regressions"] D --> E{"Alpha significant and stable across models and windows?"} E -->|No| F["Return is mainly factor exposure"] F --> G["Compare with a low-cost factor or index alternative"] E -->|Yes| H["Check holdings, rolling loadings, t above about 3, fees"] H --> I{"Survives the checks?"} I -->|No| G I -->|Yes| J["Candidate for hiring, with a monitoring plan"] ``` ## How investment managers use factor models - **Manager selection and monitoring.** Committees and consultants run factor regressions to separate skill from style (finance/manager-due-diligence-report). Carhart (1997) found that most persistence in US mutual fund returns was explained by momentum and expenses. - **Performance attribution.** Factor attribution complements Brinson-style allocation and selection analysis (finance/performance-attribution-report, mba/mba-performance-evaluation-card). - **Risk models.** Commercial equity risk models estimate exposures to dozens of style and industry factors and build the covariance matrix from them; portfolio managers use them to control tracking error and unintended bets. - **Factor investing.** Smart-beta and factor funds sell systematic exposure to value, momentum, quality, low volatility or size at fees between index funds and active funds (finance/fund-factsheet). - **Cost of equity.** Some practitioners and regulators use factor models for the cost of equity, though the CAPM remains more common in corporate practice (mba/mba-capm-wacc-card). - **Active management arithmetic.** The fundamental law of active management (Grinold 1989) links the information ratio to skill and breadth; factor models define the benchmark against which that skill is measured. ## Connections to other cards - mba/mba-capm-wacc-card: the one-factor model and beta. - mba/mba-regression-for-managers-card and academic/stats-linear-regression: OLS, standard errors, R-squared and multicollinearity, all used in every factor regression. - mba/mba-portfolio-theory-card: factor models supply the covariance matrix the optimiser needs. - mba/mba-performance-evaluation-card: Jensen's alpha, the information ratio and the appraisal ratio in depth. - mba/mba-fixed-income-card: the bond equivalents (level, slope and credit spread factors). - mba/mba-investments-guide: places this card in sessions 5 and 6 and uses it in the Tallow Creek mini-case. - academic/stats-hypothesis-testing: p-values, power and the multiple-testing problem behind the factor zoo. ## Common mistakes and misconceptions 1. **Using raw returns instead of excess returns.** Subtract the risk-free rate from the fund and the market; SMB, HML and other long-short factors are already excess returns. 2. **Calling a CAPM alpha "skill" without testing other factors.** Small, value or momentum tilts often explain it. 3. **Mismatching regions or frequencies.** A European fund regressed on US factors, or monthly fund returns on daily factors, gives meaningless loadings. 4. **Reading LINEST output in the wrong order.** Excel returns coefficients in reverse column order. 5. **Ignoring fees.** A gross-of-fee alpha must be compared with the fee; net-of-fee alpha is what the investor gets. 6. **Trusting a t of 2 from a large search.** Screening many funds or factors guarantees some false positives; demand more evidence. 7. **Treating loadings as constant.** Managers drift; check rolling windows and holdings. 8. **Confusing statistical and economic significance.** A tiny alpha can be significant in a long sample and still not cover costs; a large alpha can be insignificant in a short one. 9. **Assuming factor premiums are guaranteed.** Value had long periods of negative returns; premiums are expected, not certain. 10. **Annualising inconsistently.** Multiply monthly alpha by 12, but volatility by $\sqrt{12}$. 11. **Forgetting implementation costs.** Academic long-short factors are not free to replicate. ## Self-test questions 1. What does the single-index model assume about the covariance of two stocks, and what does it save? 2. What does the APT say, and what does it not say? 3. Describe how SMB and HML are built. 4. A stock has loadings 1.1 (market), 0.3 (SMB) and -0.2 (HML). With premiums 5%, 2% and 3%, what is its expected excess return? 5. A fund has a significant CAPM alpha and a zero three-factor alpha with a large positive HML loading. What do you conclude? 6. A fund has an alpha of 0.25% a month and residual volatility of 2.0% a month. What is its annualised appraisal ratio, and roughly what t-statistic would 36 months give? 7. Why do Harvey, Liu and Zhu (2016) suggest a higher t-statistic threshold for new factors? 8. Name the two factors Fama and French (2015) added and what each measures. 9. Why might a factor explain much of a portfolio's variance but contribute nothing to its expected return? 10. Interleaving: which tool answers each question? (a) Did the manager add value through sector weights or stock picks within sectors? (b) Did the manager add value beyond cheap factor exposures? (c) What discount rate should the firm use for a new project? ## My notes and next steps | Item | My entry | |---|---| | Confidence with factor regressions (1 = lost, 5 = could teach it) | [ ] | | Factor data source and region I will use, with download date | [ ] | | A fund or product I would like to test | [ ] | | Mistakes logged in mba/mba-error-log | [Date and short description] | | Next review date (spacing: +2 days, +1 week, +1 month) | YYYY-MM-DD | | Next card to study | [mba/mba-performance-evaluation-card / mba/mba-portfolio-theory-card] | ## Sources and further reading - Sharpe, W. F. (1963). A simplified model for portfolio analysis. *Management Science*, 9(2), 277-293. - Sharpe, W. F. (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. *The Journal of Finance*, 19(3), 425-442. - Jensen, M. C. (1968). The performance of mutual funds in the period 1945-1964. *The Journal of Finance*, 23(2), 389-416. - Fama, E. F., and MacBeth, J. D. (1973). Risk, return, and equilibrium: empirical tests. *Journal of Political Economy*, 81(3), 607-636. - Ross, S. A. (1976). The arbitrage theory of capital asset pricing. *Journal of Economic Theory*, 13(3), 341-360. - Roll, R. (1977). A critique of the asset pricing theory's tests. Part I: on past and potential testability of the theory. *Journal of Financial Economics*, 4(2), 129-176. - Banz, R. W. (1981). The relationship between return and market value of common stocks. *Journal of Financial Economics*, 9(1), 3-18. - Gibbons, M. R., Ross, S. A., and Shanken, J. (1989). A test of the efficiency of a given portfolio. *Econometrica*, 57(5), 1121-1152. - Grinold, R. C. (1989). The fundamental law of active management. *The Journal of Portfolio Management*, 15(3), 30-37. - Fama, E. F., and French, K. R. (1992). The cross-section of expected stock returns. *The Journal of Finance*, 47(2), 427-465. - Fama, E. F., and French, K. R. (1993). Common risk factors in the returns on stocks and bonds. *Journal of Financial Economics*, 33(1), 3-56. - Jegadeesh, N., and Titman, S. (1993). Returns to buying winners and selling losers: implications for stock market efficiency. *The Journal of Finance*, 48(1), 65-91. - Carhart, M. M. (1997). On persistence in mutual fund performance. *The Journal of Finance*, 52(1), 57-82. - Fama, E. F., and French, K. R. (2010). Luck versus skill in the cross-section of mutual fund returns. *The Journal of Finance*, 65(5), 1915-1947. - Asness, C. S., Moskowitz, T. J., and Pedersen, L. H. (2013). Value and momentum everywhere. *The Journal of Finance*, 68(3), 929-985. - Frazzini, A., and Pedersen, L. H. (2014). Betting against beta. *Journal of Financial Economics*, 111(1), 1-25. - Fama, E. F., and French, K. R. (2015). A five-factor asset pricing model. *Journal of Financial Economics*, 116(1), 1-22. - Hou, K., Xue, C., and Zhang, L. (2015). Digesting anomalies: an investment approach. *The Review of Financial Studies*, 28(3), 650-705. - Harvey, C. R., Liu, Y., and Zhu, H. (2016). ... and the cross-section of expected returns. *The Review of Financial Studies*, 29(1), 5-68. - Kenneth R. French Data Library, Tuck School of Business at Dartmouth (factor return files; named as a data source, no values quoted; checked October 2026). - Textbooks (by name only): Bodie, Z., Kane, A., and Marcus, A. J., *Investments*; Ang, A., *Asset Management: A Systematic Approach to Factor Investing*; Grinold, R. C., and Kahn, R. N., *Active Portfolio Management*. - Open resource: OpenStax, *Principles of Finance* (CC BY-NC-SA 4.0), chapters on risk, return and the CAPM. ## Answer key **Worked example B.** (1) SMB contribution $0.40 \times 2.0\% = 0.80\%$; variance contribution $0.40^2 \times 100 = 16.00$. Expected excess return $4.75\% + 0.80\% + 0.90\% = 6.45\%$; total variance $231.04 + 16.00 + 10.89 + 64.00 = 321.93$. (2) $3.5\% + 6.45\% = 9.95\%$. (3) $\sqrt{321.93} = 17.94\%$; $R^2 = 257.93 / 321.93 = 0.8012$. (4) The CAPM estimate (9.00%) gives the size and value exposures no premium of their own; part of their effect is absorbed into a higher market beta (1.10 rather than 0.95). If size and value carry premiums, the CAPM understates the expected return of a small-value portfolio; if they do not, the factor estimate overstates it. (5) Uncorrelated factors: in real data the factors are correlated. Estimate the factor covariance matrix and use `=MMULT(TRANSPOSE(beta),MMULT(Omega,beta))` for systematic variance. **Exercise C.** (1) $t_\alpha = 0.18 / 0.15 = 1.20$ (p = 0.2353, `=T.DIST.2T(1.2,55)`): not significant. $t_{MOM} = 0.40 / 0.08 = 5.00$: highly significant. The market loading ($t = 25.5$) is significant; SMB ($-0.71$) and HML ($-1.25$) are not. (2) $0.18 \times 12 = 2.16\%$ a year gross; minus the 1.20% fee leaves 0.96%, itself not distinguishable from zero. (3) $0.18 / 1.90 \times \sqrt{12} = 0.3282$. (4) $n \approx (2 \times 1.90 / 0.18)^2 = 445.7$ months, about 37.1 years: the data will never settle the question. (5) A market-beta fund with a strong momentum tilt and a slight growth lean. Ask what a low-cost momentum product would have returned net of fees, whether the momentum exposure is intended and controlled, how turnover and trading costs are handled, and for net-of-fee results over a longer period. **Self-test.** 1. Stocks co-move only through the market, so $\sigma_{ij} = \beta_i \beta_j \sigma_m^2$. It cuts the inputs from $n(n+1)/2$ variances and covariances to $3n + 2$ parameters (1,502 instead of 125,250 for 500 stocks). 2. With a few common factors and no arbitrage, expected returns are approximately linear in factor loadings. It does not say which factors, or how many. 3. Stocks are split by size (median) and by book-to-market (30th and 70th percentiles), giving six value-weighted portfolios. SMB is the average of the three small portfolios minus the average of the three big ones; HML is the average of the two high book-to-market portfolios minus the two low ones. 4. $1.1 \times 5\% + 0.3 \times 2\% - 0.2 \times 3\% = 5.5\%$. 5. The CAPM alpha came from a value tilt, not from skill beyond what a value index provides. 6. $0.25 / 2.0 \times \sqrt{12} = 0.433$. With 36 months, $t \approx 0.25 / (2.0 / \sqrt{36}) = 0.75$: not significant. 7. Hundreds of factors have been tested on the same data; with many tests, a 5% threshold produces many false discoveries, so the bar must rise (they suggest about 3.0). 8. RMW (robust minus weak operating profitability) and CMA (conservative minus aggressive asset growth, that is, investment). 9. A factor can drive co-movement (variance) without carrying a premium, as industry factors do; only priced factors add to expected returns. 10. (a) Brinson attribution. (b) A factor regression (alpha against factors). (c) The cost of capital, usually CAPM based (mba/mba-capm-wacc-card). ## Revision history | Version | Date | Author | Change summary | |---|---|---|---| | 0.1 | YYYY-MM-DD | [Your name] | First pass through the card | | 0.2 | YYYY-MM-DD | [Your name] | *Example:* Replaced Example data with downloaded factor files and a 60-month fund series | ## Standard and licence note Based on the factor models in the primary works listed above (Sharpe 1963, Ross 1976, Fama and French 1993 and 2015, Carhart 1997) and the related empirical literature. The text, examples, numbers and questions are original; no textbook text is copied or paraphrased. Fund I, Fund J, the Tallow Creek Medical Center foundation and Northfield Outdoor Co. are fictional, and every return series is invented *Example* data generated for teaching, not real factor returns. The Kenneth R. French Data Library is named only as a data source. The template is independent and not affiliated with or endorsed by any business school, publisher or data provider. Check current data before you rely on a figure. Template licence: MIT.