# Load required packages
library(tidyquant)
library(tidyverse)
library(broom)
library(DT)
library(ggplot2)
library(ggthemes)
library(highcharter)
library(frenchdata) # install.packages("frenchdata")R Note 3 - Factor Model Estimation
Course Information
Course Name: Investments | 000351051
Semester: Spring 2025
Introduction
In this document, we estimate factor models for a stock’s returns. A factor model decomposes a stock’s return into systematic components driven by common risk factors and a residual idiosyncratic component. We begin with the simplest case — the market model (one factor: the market return) — and progressively extend it to the Fama-French three-factor model (Fama and French, 1993) and the Carhart four-factor model (Carhart, 1996), which adds a momentum factor.
The general \(K\)-factor model is:
\[R_{i,t} - R_{f,t} = \alpha_i + \sum_{k=1}^{K} \beta_{i,k} \, F_{k,t} + \varepsilon_{i,t}\]
where \(R_{f,t}\) is the risk-free rate, \(F_{k,t}\) are factor returns (e.g., market, size, value, momentum), and \(\beta_{i,k}\) are the corresponding factor loadings estimated by OLS.
1. Fetching Stock Data
We collect daily adjusted price data for NVIDIA (NVDA) using tidyquant. The estimation window spans 2020–2022, a three-year horizon commonly used in empirical finance.1
stock_ticker <- "NVDA"
market_ticker <- "^GSPC"
start_date <- "2020-01-01"
end_date <- "2022-12-31"
stock_data <- tq_get(stock_ticker, from = start_date, to = end_date)
market_data <- tq_get(market_ticker, from = start_date, to = end_date)Preview Data
datatable(stock_data) |>
formatRound(columns = c("open","high","low","close","adjusted"), digits = 2)2. Data Preparation
We calculate daily returns for both the stock and the market index using tq_transmute(), then merge them.
stock_returns <- stock_data |>
tq_transmute(select = adjusted,
mutate_fun = periodReturn,
period = "daily",
col_rename = "stock_return")
market_returns <- market_data |>
tq_transmute(select = adjusted,
mutate_fun = periodReturn,
period = "daily",
col_rename = "market_return")
returns_data <- left_join(stock_returns, market_returns, by = "date") |>
na.omit()
datatable(returns_data) |>
formatRound(columns = c("stock_return", "market_return"), digits = 4)Visualize Stock Prices
highchart() |>
hc_title(text = "NVIDIA Inc. (NVDA) Stock Prices") |>
hc_xAxis(type = "datetime", title = list(text = "Date")) |>
hc_yAxis(title = list(text = "Adjusted Closing Price (USD)")) |>
hc_add_series(stock_data, type = "line",
hcaes(x = date, y = adjusted),
name = "Adjusted Price") |>
hc_tooltip(pointFormat = "Price: <b>${point.y:.2f}</b>") |>
hc_add_theme(hc_theme_smpl())3. The Market Model (One-Factor Model)
The market model is the simplest factor model. It regresses a stock’s excess return on the market excess return:2
\[R_{i,t} - R_{f,t} = \alpha_i + \beta_i \,(R_{m,t} - R_{f,t}) + \varepsilon_{i,t}\]
where:
- \(\alpha_i\) is the stock’s abnormal return independent of the market.
- \(\beta_i\) is the market beta — the sensitivity of the stock’s excess return to the market excess return. This is the measure of systematic risk.
- \(\varepsilon_{i,t}\) is the idiosyncratic component unexplained by the market.
Scatterplot: NVDA vs. S&P 500
Code
fit_market <- lm(stock_return ~ market_return, data = returns_data)
returns_data$fit <- predict(fit_market)
highchart() |>
hc_title(text = "Linear Relation: NVDA vs. S&P 500") |>
hc_xAxis(title = list(text = "Market Return (S&P 500)")) |>
hc_yAxis(title = list(text = "Stock Return (NVDA)")) |>
hc_add_series(data = returns_data,
type = "scatter",
hcaes(x = market_return, y = stock_return),
name = "Daily Returns") |>
hc_add_series(data = returns_data[order(returns_data$market_return), ],
type = "line",
hcaes(x = market_return, y = fit),
name = "Fitted Line") |>
hc_tooltip(pointFormat = "Market: {point.x:.4f}<br>Stock: {point.y:.4f}") |>
hc_legend(align = "center", verticalAlign = "bottom") |>
hc_add_theme(hc_theme_smpl())Regression Output
model_1f <- lm(stock_return ~ market_return, data = returns_data)
summary(model_1f)
Call:
lm(formula = stock_return ~ market_return, data = returns_data)
Residuals:
Min 1Q Median 3Q Max
-0.084558 -0.014089 -0.001877 0.013956 0.112267
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0012297 0.0008435 1.458 0.145
market_return 1.6562127 0.0526407 31.463 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.02319 on 754 degrees of freedom
Multiple R-squared: 0.5676, Adjusted R-squared: 0.5671
F-statistic: 989.9 on 1 and 754 DF, p-value: < 2.2e-16Extracting \(\hat{\beta}\) and Idiosyncratic Volatility
beta_hat <- tidy(model_1f) |>
filter(term == "market_return") |>
pull(estimate)
idio_vol <- sd(resid(model_1f))
cat("Estimated market beta for", stock_ticker, ":", round(beta_hat, 4), "\n")Estimated market beta for NVDA : 1.6562 cat("Idiosyncratic volatility (residual SD):", round(idio_vol, 6), "\n")Idiosyncratic volatility (residual SD): 0.023171 Variance Decomposition
The market model implies a clean decomposition of total variance:
\[\underbrace{\sigma_i^2}_{\text{total}} = \underbrace{\beta_i^2 \,\sigma_m^2}_{\text{systematic}} + \underbrace{\sigma_{\varepsilon,i}^2}_{\text{idiosyncratic}}\]
sigma_total <- sd(returns_data$stock_return)
sigma_mkt <- sd(returns_data$market_return)
sys_part <- beta_hat^2 * sigma_mkt^2
idio_part <- idio_vol^2
tibble(
Component = c("Total variance", "Systematic", "Idiosyncratic"),
Value = c(sigma_total^2, sys_part, idio_part),
Fraction = c(1, sys_part / sigma_total^2, idio_part / sigma_total^2)
) |>
datatable() |>
formatRound(columns = c("Value", "Fraction"), digits = 6)4. Retrieving Factor Data from Kenneth French’s Data Library
The most authoritative source of factor return data for empirical asset pricing is Kenneth French’s Data Library, hosted at Dartmouth. The frenchdata package provides a convenient R interface to download these datasets directly.
# Install once
install.packages("frenchdata")library(frenchdata)4.1 Browsing Available Datasets
datasets <- get_french_data_list()
# Show the first 20 available datasets
head(datasets, 20)$info
[1] "Information collected from: https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html on Fri Jun 26 23:51:46 2026"
$files_list
# A tibble: 297 × 3
name file_url details_url
<chr> <chr> <chr>
1 Fama/French 3 Factors ftp/F-F_Rese… Data_Libra…
2 Fama/French 3 Factors [Weekly] ftp/F-F_Rese… Data_Libra…
3 Fama/French 3 Factors [Daily] ftp/F-F_Rese… Data_Libra…
4 Fama/French 5 Factors (2x3) ftp/F-F_Rese… Data_Libra…
5 Fama/French 5 Factors (2x3) [Daily] ftp/F-F_Rese… Data_Libra…
6 Portfolios Formed on Size ftp/Portfoli… Data_Libra…
7 Portfolios Formed on Size [ex.Dividends] ftp/Portfoli… Data_Libra…
8 Portfolios Formed on Size [Daily] ftp/Portfoli… Data_Libra…
9 Portfolios Formed on Book-to-Market ftp/Portfoli… Data_Libra…
10 Portfolios Formed on Book-to-Market [ex. Dividends] ftp/Portfoli… Data_Libra…
# ℹ 287 more rows4.2 Downloading the Fama-French Three Factors (Daily)
The dataset "Fama/French 3 Factors [Daily]" contains four daily series: the market excess return (Mkt-RF), the size factor (SMB), the value factor (HML), and the risk-free rate (RF).
ff3_raw <- download_french_data("Fama/French 3 Factors [Daily]")New names:
• `` -> `...1`# The data is stored in the $subsets list; the first element contains daily data
ff3_daily <- ff3_raw$subsets$data[[1]] |>
mutate(date = ymd(date)) |> # parse date from YYYYMMDD integer
rename(
MktRF = `Mkt-RF`,
RF = RF,
SMB = SMB,
HML = HML
) |>
mutate(across(c(MktRF, RF, SMB, HML), ~ . / 100)) # convert from % to decimal
head(ff3_daily)# A tibble: 6 × 5
date MktRF SMB HML RF
<date> <dbl> <dbl> <dbl> <dbl>
1 1926-07-01 0.0009 -0.0025 -0.0027 0.0001
2 1926-07-02 0.0045 -0.0033 -0.0006 0.0001
3 1926-07-06 0.0017 0.003 -0.0039 0.0001
4 1926-07-07 0.0009 -0.0058 0.0002 0.0001
5 1926-07-08 0.0022 -0.0038 0.0019 0.0001
6 1926-07-09 -0.0071 0.0043 0.0057 0.0001Note on units: French’s data are published as percentages (e.g.,
0.35means 0.35%). Dividing by 100 converts them to decimals consistent with return calculations in R.
5. The Fama-French Three-Factor Model
Fama and French (1993) argue that the market alone cannot fully explain the cross-section of expected returns. Two additional factors capture systematic risks related to firm size and book-to-market ratio:
\[R_{i,t} - R_{f,t} = \alpha_i + \beta_i^{MKT}\,\text{MktRF}_t + \beta_i^{SMB}\,\text{SMB}_t + \beta_i^{HML}\,\text{HML}_t + \varepsilon_{i,t}\]
- MktRF (Market Risk Premium): \(R_m - R_f\). The market factor; same as the CAPM.
- SMB (Small Minus Big): average return of small-cap portfolios minus large-cap portfolios. Captures the size premium.
- HML (High Minus Low): average return of high book-to-market (value) portfolios minus low book-to-market (growth) portfolios. Captures the value premium.
5.1 Merging Stock Returns with French Factors
# Restrict factors to the estimation window
ff3_est <- ff3_daily |>
filter(date >= start_date, date <= end_date)
# Compute stock excess return: stock_return - RF
ff3_merged <- returns_data |>
left_join(ff3_est |> select(date, MktRF, SMB, HML, RF), by = "date") |>
mutate(excess_return = stock_return - RF) |>
na.omit()
datatable(ff3_merged |> select(date, excess_return, MktRF, SMB, HML)) |>
formatRound(columns = c("excess_return","MktRF","SMB","HML"), digits = 5)5.2 Regression: Fama-French Three-Factor Model
model_ff3 <- lm(excess_return ~ MktRF + SMB + HML, data = ff3_merged)
summary(model_ff3)
Call:
lm(formula = excess_return ~ MktRF + SMB + HML, data = ff3_merged)
Residuals:
Min 1Q Median 3Q Max
-0.079407 -0.011536 -0.000834 0.009757 0.100355
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.001331 0.000686 1.940 0.0528 .
MktRF 1.616106 0.042881 37.688 <2e-16 ***
SMB 0.108462 0.086459 1.254 0.2101
HML -0.902552 0.050859 -17.746 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.01885 on 752 degrees of freedom
Multiple R-squared: 0.7149, Adjusted R-squared: 0.7138
F-statistic: 628.5 on 3 and 752 DF, p-value: < 2.2e-165.3 Interpreting the Factor Loadings
tidy(model_ff3) |>
datatable() |>
formatRound(columns = c("estimate","std.error","statistic","p.value"), digits = 4)The key takeaways:
- \(\hat\beta^{MKT}\): a large positive value means the stock amplifies market moves — i.e., high systematic risk.
- \(\hat\beta^{SMB}\): a negative loading is typical for large-cap stocks like NVDA; it loads negatively on the small-cap premium.
- \(\hat\beta^{HML}\): a negative loading indicates a growth stock (low book-to-market); NVDA is a canonical growth firm.
5.4 Comparing Model Fit: One-Factor vs. Three-Factor
glance(model_1f) |> select(r.squared, adj.r.squared, sigma, nobs)# A tibble: 1 × 4
r.squared adj.r.squared sigma nobs
<dbl> <dbl> <dbl> <int>
1 0.568 0.567 0.0232 756glance(model_ff3) |> select(r.squared, adj.r.squared, sigma, nobs)# A tibble: 1 × 4
r.squared adj.r.squared sigma nobs
<dbl> <dbl> <dbl> <int>
1 0.715 0.714 0.0189 756The three-factor model typically explains more of the variance in individual stock returns (higher \(R^2\)) compared to the single-factor market model.
6. The Momentum Factor and the Carhart Four-Factor Model
Jegadeesh and Titman (1993) document a striking empirical regularity: stocks that performed well (poorly) over the past 6–12 months tend to continue performing well (poorly) over the next 3–12 months. This momentum anomaly is pervasive across markets and time periods and is not captured by the Fama-French three factors.
Carhart (1996) incorporates momentum into the factor framework by adding a fourth factor:
\[R_{i,t} - R_{f,t} = \alpha_i + \beta_i^{MKT}\,\text{MktRF}_t + \beta_i^{SMB}\,\text{SMB}_t + \beta_i^{HML}\,\text{HML}_t + \beta_i^{MOM}\,\text{MOM}_t + \varepsilon_{i,t}\]
- MOM (Winners Minus Losers, or UMD — Up Minus Down): the average return of high prior-return (“winner”) portfolios minus low prior-return (“loser”) portfolios, constructed using returns from months \(t-12\) to \(t-2\) (skipping the most recent month to avoid short-term reversal contamination).
The Carhart model is widely used in mutual fund performance evaluation. A fund’s \(\alpha_i\) (“Carhart alpha”) after controlling for all four factors is a cleaner measure of manager skill than raw excess return.
6.1 Downloading the Momentum Factor (Daily)
The momentum factor is available as a separate dataset on French’s website.
mom_raw <- download_french_data("Momentum Factor (Mom) [Daily]")New names:
• `` -> `...1`mom_daily <- mom_raw$subsets$data[[1]] |>
mutate(date = ymd(date),
MOM = Mom / 100) |> # convert % to decimal
select(date, MOM)
head(mom_daily)# A tibble: 6 × 2
date MOM
<date> <dbl>
1 1926-11-03 0.0054
2 1926-11-04 -0.0051
3 1926-11-05 0.0117
4 1926-11-06 -0.0003
5 1926-11-08 -0.0002
6 1926-11-09 0.00146.2 Visualizing the Momentum Factor
Code
mom_plot <- mom_daily |>
filter(date >= "2000-01-01") |>
mutate(MOM_cum = cumprod(1 + MOM) - 1)
highchart(type = "stock") |>
hc_title(text = "Momentum Factor (MOM) — Cumulative Return (2000–)") |>
hc_yAxis(title = list(text = "Cumulative Return"),
labels = list(format = "{value:.0%}")) |>
hc_add_series(mom_plot, type = "line",
hcaes(x = date, y = MOM_cum),
name = "MOM (Winners − Losers)") |>
hc_tooltip(valueDecimals = 4) |>
hc_legend(enabled = FALSE) |>
hc_add_theme(hc_theme_smpl())Momentum crashes: note the dramatic drawdown around 2009. After the financial crisis, past losers rebounded violently while past winners lagged, producing a catastrophic reversal for momentum strategies. This pattern — momentum working well in normal times but crashing in market rebounds — is a well-known feature of the factor and an active area of research (e.g., Daniel and Moskowitz, 2016).
6.3 All Four Factors Together
Code
all_factors <- ff3_daily |>
inner_join(mom_daily, by = "date") |>
filter(date >= "2000-01-01") |>
mutate(
MktRF_cum = cumprod(1 + MktRF) - 1,
SMB_cum = cumprod(1 + SMB) - 1,
HML_cum = cumprod(1 + HML) - 1,
MOM_cum = cumprod(1 + MOM) - 1
)
highchart(type = "stock") |>
hc_title(text = "Carhart Four Factors — Cumulative Returns (2000–)") |>
hc_yAxis(title = list(text = "Cumulative Return"),
labels = list(format = "{value:.0%}")) |>
hc_add_series(all_factors, type = "line",
hcaes(x = date, y = MktRF_cum), name = "Mkt-RF") |>
hc_add_series(all_factors, type = "line",
hcaes(x = date, y = SMB_cum), name = "SMB") |>
hc_add_series(all_factors, type = "line",
hcaes(x = date, y = HML_cum), name = "HML") |>
hc_add_series(all_factors, type = "line",
hcaes(x = date, y = MOM_cum), name = "MOM") |>
hc_tooltip(valueDecimals = 4, shared = TRUE) |>
hc_legend(align = "center", verticalAlign = "bottom") |>
hc_add_theme(hc_theme_smpl())6.4 Regression: Carhart Four-Factor Model
# Merge all four factors with the stock excess return
carhart_data <- ff3_merged |>
left_join(mom_daily, by = "date") |>
na.omit()
model_carhart <- lm(excess_return ~ MktRF + SMB + HML + MOM,
data = carhart_data)
summary(model_carhart)
Call:
lm(formula = excess_return ~ MktRF + SMB + HML + MOM, data = carhart_data)
Residuals:
Min 1Q Median 3Q Max
-0.079100 -0.011629 -0.001042 0.009681 0.101144
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.001311 0.000685 1.913 0.0561 .
MktRF 1.629793 0.043474 37.489 <2e-16 ***
SMB 0.136476 0.087693 1.556 0.1201
HML -0.871629 0.053559 -16.274 <2e-16 ***
MOM 0.084716 0.046635 1.817 0.0697 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.01883 on 751 degrees of freedom
Multiple R-squared: 0.7161, Adjusted R-squared: 0.7146
F-statistic: 473.7 on 4 and 751 DF, p-value: < 2.2e-166.5 Extracting and Interpreting All Factor Loadings
tidy(model_carhart) |>
datatable() |>
formatRound(columns = c("estimate","std.error","statistic","p.value"), digits = 4)For a high-momentum stock like NVDA:
- A positive \(\hat\beta^{MOM}\) means NVDA tends to ride momentum — it has behaved like a “winner” stock that continues to win.
- After controlling for momentum, the alpha \(\hat\alpha\) represents abnormal return that cannot be attributed to any of the four systematic risk factors.
6.6 Model Comparison: One-Factor → Two-Factor → Four-Factor
bind_rows(
glance(model_1f) |> mutate(Model = "Market (1F)"),
glance(model_ff3) |> mutate(Model = "Fama-French (3F)"),
glance(model_carhart) |> mutate(Model = "Carhart (4F)")
) |>
select(Model, r.squared, adj.r.squared, sigma, AIC, BIC) |>
datatable() |>
formatRound(columns = c("r.squared","adj.r.squared","sigma","AIC","BIC"), digits = 4)Moving from the one-factor market model to the four-factor Carhart model generally increases \(R^2\) and lowers \(\sigma\) (residual standard error), reflecting a more complete account of the systematic sources of risk that drive the stock’s returns.
7. Summary
This note built factor model estimation step by step:
| Model | Factors | Key insight |
|---|---|---|
| Market model | MktRF | \(\beta\) captures co-movement with the market; idiosyncratic risk is the rest. |
| Fama-French (3F) | MktRF, SMB, HML | Size and value are additional priced risk sources beyond the market. |
| Carhart (4F) | MktRF, SMB, HML, MOM | Momentum (past winners/losers) adds further explanatory power; widely used in fund evaluation. |
The regression mechanics are the same throughout — only the right-hand-side variables change. Factor data from Kenneth French’s library makes replication straightforward via the frenchdata package.
Exercises
Change the stock. Repeat the full four-factor estimation for Apple (AAPL) and Tesla (TSLA) over 2020–2022. Compare the factor loadings. Which stock loads more heavily on momentum? Which has a larger idiosyncratic volatility?
Interpretation. For NVDA’s Carhart regression, construct a table showing: (a) the estimated loading, (b) its standard error, (c) its t-statistic, and (d) a one-sentence economic interpretation of each coefficient.
Extended window. Re-estimate the Carhart model for NVDA over 2015–2022 (eight years). Do the loadings change substantially? What does stability (or instability) of loadings imply for the assumption that beta is constant?
Variance decomposition. Using the Carhart model residuals for NVDA, compute: total variance, systematic variance attributable to each factor (\(\hat\beta_k^2 \,\text{Var}(F_k)\)), and idiosyncratic variance. How much of total variance is explained by the market alone vs. all four factors combined?
Alpha hunting. Download data for a mutual fund ETF of your choice (e.g., ARKK). Estimate its Carhart alpha over 2019–2023. Is the alpha statistically significant? What does this suggest about the fund’s performance after adjusting for factor exposures?
References
- Carhart, M. M. (1996). On persistence in mutual fund performance. Journal of Finance, 52(1), 57–82.
- Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1), 3–56.
- Jegadeesh, N., & Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock market efficiency. Journal of Finance, 48(1), 65–91.
- French, K. R. Data Library. https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
Happy coding! 🚀
Footnotes
There is no unique answer to how to set the estimation window. Three years of daily data (approximately 756 observations) is a widely used convention. The implicit assumption is that the estimated loadings from the historical period reflect the stock’s risk exposures in recent periods.↩︎
For simplicity, some practitioners omit the risk-free rate and regress raw returns on raw market returns. Including the risk-free rate on both sides produces slightly different estimates but the difference is usually minor at daily frequency.↩︎