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 Three-Factor Model
If your portfolio beat the market, was that skill, or the reward for carrying some risk?
Fama and French explained return by how much it moved with three risk factors: market, size and value. DIVA runs this regression on your holdings and shows where the return came from, along with what is left unexplained (alpha).
Split return into market, size and value.
The earlier standard, CAPM, explained all return with a single market risk. In reality, at the same market beta, small companies tended to beat large ones and cheap value stocks tended to beat expensive growth stocks.
Fama and French read this as those stocks carrying risks separate from the market, and widened risk from one kind to three: market, size and value.
A beta does not mean you hold such stocks; it is how strongly your return moved with that factor.
In DIVA, the regression shows up as coefficient cards at the top of the result page.
Alpha and the three betas each get a card, and every card carries its t and p values with a significant / not-significant mark, so you can judge whether a value could be chance.
Hover a card and the matching term lights up in the result page’s regression equation, linking each number to its place in the formula.
The size and direction of the three betas set the portfolio personality (archetype).
The Luxury Director명품 디렉터
Holding the classic identity of luxury intact, you read the flow of trends with a flexible, tasteful eye.
The example portfolio (AAPL 25% · MSFT 20% · SPY 25% · TLT 30%) has a market beta of 0.83, size beta −0.20 and value beta −0.26: swinging less than the market, it moved with large caps and growth stocks. These three betas set the archetype “The Luxury Director”.
Rules: a market beta of 0.85–1.15 is market-neutral; size and value betas get a direction (large/small, growth/value) only when their magnitude is 0.15+ and p is below 0.10.
A beta is co-movement strength, and it changes over time.
The market beta is how many percent your portfolio moved, on average, when the market moved 1%. 1.0 means market-like strength; 0.5 means you followed half of it.
A positive size beta means you rose when small caps led; negative, when large caps led. A positive value beta means you rose when value led; negative, when growth led.
All of these are patterns of co-movement, not holdings. A large-cap ETF alone can show a positive size beta if it moved with small caps over the window, and the same portfolio’s betas change when the window changes.
DIVA shows betas two ways: as one number and over time.
The Style Profile tab overlays the full-period betas on the market (market beta 1, the rest 0) to show which way you lean.
The Beta Stability tab chains betas from rolling 12-month regressions, showing change a single full-period beta cannot.
A portfolio beta is the weight-average of holding betas, so the result page’s equalizer recomputes betas instantly as you change weights, without re-running the analysis.
The example’s size beta went from −0.21 (2021-11) to −1.03 (2026-08), and its value beta from −0.53 to +0.07: change the full-period betas (size −0.20, value −0.26) cannot show.
A 12-month window is noisy. Watch for changes in direction rather than any single value.
Alpha may be skill, or a risk the model cannot see.
Alpha (α) is the return left over once the three factors are accounted for. A significantly positive alpha might be stock-picking skill, or a reward for a risk the three factors do not include.
t and p values separate it from luck (typically p < 0.05). R², read alongside, is the share of the portfolio’s movement the three factors explain.
DIVA leaves alpha out of expected return.
Nothing guarantees a past alpha will persist, so expected return is computed only as risk-free + Σ(beta × the factor’s average premium over the window).
The Expected Return tab splits that value into the risk-free rate and each factor’s contribution, showing which exposures the forward estimate leans on.
* Statistical estimate, not a guarantee of realized return. R² = 80% — the 3-factor model explains most of the return variance. Alpha (α -1.08%), the part factors can't explain, is a non-persistent residual rather than a model-justified return, so it is excluded from the expected return.
The example’s expected return of 13.73% is the risk-free 3.99% plus market +11.35%p, size +0.53%p and value −2.15%p; alpha (−1.08% a year, p 0.74) is left out. Value is negative because the value premium averaged +8.42% a year over the window while the portfolio moved with growth (value beta −0.26).
Regress the bond ETF TLT alone and alpha comes out at −14.99% a year with an R² of 28.8%. Interest-rate risk is in none of the three factors, so bond-price moves land in “alpha”; a large alpha is a cue to suspect a missing risk before skill.
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 three 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·SMB·HML
- US holdings use the US 3 factors from the Kenneth French Data Library (monthly); Korean holdings use DIVA’s own KRW 3 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 3-factor model helps
- 01Check your portfolio’s style
The three betas show whether you moved with large or small caps, value or growth.
- 02Check whether excess return is skill
Alpha’s p-value tells whether the excess return stands out from luck.
- 03Diagnose portfolios with Korean stocks
DIVA’s own Korean factors let mixed Korean and US portfolios be read the same way.
- 04Find weights that hit the betas you want
Set target betas in Factor Target (e.g. market beta 0.7) and weights are re-split among your holdings.
To split out profitability and investment style too, see the 5-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
Effects the three factors could not explain, such as profitability, conservative investment and momentum, were found later, and the same authors extended the model to five factors. Non-equity assets such as bonds and gold are also poorly explained by equity factors, so a heavy weight in them lowers R².
DIVA lets you run the same portfolio through five factors.
You can compare directly with the 5-factor result to see whether the 3-factor alpha is really exposure to profitability or investment style.
R² appears on the result page’s reliability card, so a low fit caused by heavy non-equity weights is easy to spot.
References
- E. F. Fama, K. R. French, “Common risk factors in the returns on stocks and bonds,” Journal of Financial Economics, 1993.
- Kenneth R. French — Data Library · Source of the US factor data (Korean factors are DIVA’s own)
- Wikipedia — Fama–French three-factor model · Definition and history of the model
- Nobel Prize — Eugene F. Fama (2013) · Official Nobel citation
Now apply the theory to your own portfolio.
Enter your holdings and amounts to see the Fama–French 3-Factor analysis as real numbers.