Model explainer — theory, significance, and formulas.
Each model's background and principle in plain language, so newcomers to finance can follow along, followed in each section by how DIVA Quantizer implements it and how to use it.
Fama–French Five-Factor Model
The alpha the three factors left unexplained: was it really skill?
In 2015 Fama and French answered by adding two factors, profitability (RMW) and investment (CMA). DIVA can run the same portfolio through both the 3- and 5-factor models, so you can check directly whether the new factors explain the alpha.
Add two factors: profitability and investment.
In the 20-odd years after the 3-factor model, alphas it could not explain kept turning up. In particular, consistently profitable companies and companies that grow their assets conservatively earned more on average.
In 2015 the authors added these as profitability (RMW, high-margin minus low-margin firms) and investment (CMA, conservative minus aggressive investors) factors.
As with three factors, each beta is how strongly your return moved with that factor, not what you hold.
In DIVA, the five betas appear as coefficient cards at the top of the result page.
The same layout as the 3-factor result gains profitability beta and investment beta cards, each with its p-value and a significant / not-significant mark.
The portfolio personality (archetype) builds on the 3-factor reading and adds a modifier when profitability and investment tilts are clear.
The Small-Town Wanderer뚜벅이 소도시 여행자
Sticking to safe public transit, you still seek out hidden, hip little spots one by one.
In the example portfolio (AAPL 25% · MSFT 20% · SPY 25% · TLT 30%), the profitability beta is +0.30 (p 0.005), significant, and the investment beta +0.17 (p 0.19). The size beta that was clear under three factors turns neutral at −0.12, so the archetype changes from the 3-factor “The Luxury Director” to “The Small-Town Wanderer”.
A modifier (quality/junk × conservative/aggressive) is added only when both profitability and investment betas reach a magnitude of 0.3+ with p ≤ 0.10. Here the investment beta falls short, so there is no modifier.
Read style on five axes, and trace it back to holdings.
The signs of the profitability and investment betas reveal a portfolio’s style. Both positive means it moved with “quality plus conservative capital allocation” (often called the Buffett style); both negative, with “high growth plus aggressive investment” (the tech-growth style).
A portfolio beta is the weight-sum of holding betas, so you can trace which holdings created the style.
DIVA overlays the five betas on a single radar.
The Style Profile tab overlays your portfolio and the market (market beta 1, the rest 0) on five axes, showing at a glance which way you lean.
Because betas are weight-sums, the result page’s equalizer recomputes all five instantly as you change weights, without re-running the analysis.
About 90% of the example’s profitability beta of +0.30 comes from two holdings, MSFT (0.73) and AAPL (0.50). Moving with “highly profitable big tech” explains part of the performance.
Profitability beta by holding: MSFT 0.73 · AAPL 0.50 · SPY 0.09 · TLT 0.03; multiplying by the weights and summing gives the portfolio’s +0.30.
Read the same portfolio under three and five factors.
If an alpha seen under three factors shrinks under five, it was not skill but exposure to profitability or investment style.
Adding factors always nudges R² up, so fit is compared with the adjusted R², which accounts for the number of factors (k) and months observed (n).
In DIVA, you can compare the same portfolio under both models.
Run the same portfolio through three factors as well, and the 5-factor result’s equation card shows the R² change versus three factors.
Compare the alpha cards’ p-values across the two results to see straight away whether the new factors explain the 3-factor alpha.
| 3-factor | 5-factor | |
|---|---|---|
| Alpha (ann.) | −1.08% (p 0.74) | −1.59% (p 0.59) |
| Market beta | 0.83 | 0.83 |
| Size beta | −0.20 (p 0.02) | −0.12 (p 0.28) |
| Value beta | −0.26 (p <0.001) | −0.38 (p <0.001) |
| Profitability beta | — | +0.30 (p 0.005) |
| Investment beta | — | +0.17 (p 0.19) |
| R² | 79.8% | 83.5% |
| Adjusted R² | 78.9% | 82.2% |
and a significant profitability beta appears. Adjusted R² rises from 78.9% to 82.2%. The size beta, meanwhile, loses significance (see 06).
p-values in brackets; below 0.05 is significant.
From the formula to real data
The coefficients are the values that make the sum of squared gaps between actual excess return and the equation’s prediction (the residual ε) as small as possible (ordinary least squares, OLS).
The first value is alpha and the rest are the five betas. Each coefficient’s standard error gives a t and p value, telling whether it can be told apart from zero.
How DIVA fills in each part of the regression
- Rp − Rf
- Monthly returns over the last 6 years, in the window all holdings share, minus the factor dataset’s monthly risk-free rate (min. 12 months).
- MKT…CMA
- US holdings use the North American 5 factors from the Kenneth French Data Library (monthly); Korean holdings use DIVA’s own KRW 5 factors, computed from every KRX stock including delisted ones (from July 2016).
- KR + US
- Separate regressions per market, weight-averaged by amount; standard errors and p-values are combined with the delta method.
- p
- Coefficient cards are marked significant when p is below 0.05.
- Rf
- Expected return starts from the current 13-week US T-bill rate (^IRX), used site-wide (3.99% in the example).
- Display
- Alpha and expected return are annualized (×12).
Moments when the Fama–French 5-factor model helps
- 01Re-test a 3-factor alpha
See whether a 3-factor alpha is explained by profitability or investment exposure.
- 02Check what a “quality” ETF really does
The profitability beta shows whether a quality or dividend-growth ETF really moves like profitable companies.
- 03Understand a tech-heavy portfolio
Split “growth” into profitability and investment to see what kind of growth you hold.
- 04Find weights for five target betas
Set target betas on five axes in Factor Target and weights are re-split among your holdings.
For a simpler three-axis view, start with the 3-factor model. For efficiency per unit of risk see Markowitz, for beta and alpha against the market alone see CAPM, and for risk-based weights see HRP.
Reading the result with care
More factors can bring multicollinearity: factors strongly correlated with each other. Value and investment, for example, tend to move alike, which widens the standard errors of individual betas and makes their size and sign hard to trust. The usual check is the VIF.
DIVA does not display the VIF.
Instead, run the same portfolio through three and five factors and check whether a significant beta loses significance under five (the example’s size beta: p 0.02 → 0.28). That signals overlap with the new factors.
References
- E. F. Fama, K. R. French, “A five-factor asset pricing model,” Journal of Financial Economics, 2015.
- Kenneth R. French — Data Library · Source of the North American 5-factor data (Korean factors are DIVA’s own)
- Wikipedia — Fama–French five-factor model · RMW and CMA, and how the model differs from three factors
- Wikipedia — Variance Inflation Factor · The standard multicollinearity check
Now apply the theory to your own portfolio.
Enter your holdings and amounts to see the Fama–French 5-Factor analysis as real numbers.