# Install packages if needed (uncomment if necessary)
# install.packages(c("quantmod", "dplyr", "tidyr", "readr", "lubridate", "knitr", "broom"))
library(quantmod) # For downloading stock data
library(dplyr) # For data manipulation
library(tidyr) # For reshaping data
library(readr) # For reading CSV files
library(lubridate) # For date handling
library(knitr) # For nice tables
library(broom) # For tidy regression outputCAPM & Multi-Factor Models Analysis
Tesla, Toyota, GM - Part 2 MPT Assignment
1 Introduction
This document provides a step-by-step analysis of Tesla (TSLA), Toyota (TM), and General Motors (GM) using:
- CAPM (Capital Asset Pricing Model)
- Fama-French 3-Factor Model (FF3)
- Carhart 4-Factor Model (FF3 + Momentum)
We will answer Questions 1-9 from Part 2 of the MPT assignment.
2 Setup and Data Preparation
2.1 Load Required Packages
2.2 Download Stock Price Data
We download adjusted closing prices for Tesla, Toyota, and GM from Yahoo Finance.
# Define tickers and date range
tickers <- c("TSLA", "TM", "GM")
start_date <- "2010-12-01"
end_date <- Sys.Date()
# Download data for each stock
# getSymbols returns data to the global environment by default
getSymbols(tickers, src = "yahoo", from = start_date, to = end_date, auto.assign = TRUE)[1] "TSLA" "TM" "GM" # Extract adjusted closing prices into a single xts object
prices <- merge(Ad(TSLA), Ad(TM), Ad(GM))
colnames(prices) <- c("TSLA", "TM", "GM")
# Preview the data
head(prices) TSLA TM GM
2010-12-01 2.290000 54.31368 25.98766
2010-12-02 2.156667 53.93148 25.91294
2010-12-03 2.099333 53.95196 25.81580
2010-12-06 2.020667 54.01338 25.76350
2010-12-07 2.104000 53.69262 25.91294
2010-12-08 2.158000 53.48103 25.74109tail(prices) TSLA TM GM
2025-11-24 417.78 199.06 71.00
2025-11-25 419.40 200.26 72.78
2025-11-26 426.58 202.44 72.81
2025-11-28 430.17 201.87 73.52
2025-12-01 430.14 199.22 72.95
2025-12-02 429.24 196.99 73.662.3 Calculate Monthly Returns
We convert daily prices to monthly prices, then calculate simple returns.
# Convert daily to monthly (last observation of each month)
monthly_prices <- to.monthly(prices, indexAt = "lastof", OHLC = FALSE)
# Calculate simple returns: (P_t - P_{t-1}) / P_{t-1}
# We use Return.calculate from PerformanceAnalytics or do it manually
monthly_returns <- diff(monthly_prices) / lag(monthly_prices, 1)
# Remove the first NA row
monthly_returns <- monthly_returns[-1, ]
# Convert to data frame for easier manipulation
returns_df <- data.frame(
date = index(monthly_returns),
coredata(monthly_returns)
)
# Preview
head(returns_df) date TSLA TM GM
1 2011-01-31 -0.095005275 0.045148167 -0.010038051
2 2011-02-28 -0.008713727 0.135313100 -0.081117981
3 2011-03-31 0.161573664 -0.131841473 -0.074560044
4 2011-04-30 -0.005405400 -0.007102854 0.034160439
5 2011-05-31 0.092028728 0.045306383 -0.008725489
6 2011-06-30 -0.033510059 -0.010445451 -0.0455830252.4 Download Fama-French Factors
We download the Fama-French 3 factors (Mkt-RF, SMB, HML) and the Momentum factor from Kenneth French’s website.
# URL for Fama-French 3 Factors (Monthly)
ff3_url <- "https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/F-F_Research_Data_Factors_CSV.zip"
# URL for Momentum Factor (Monthly)
mom_url <- "https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/F-F_Momentum_Factor_CSV.zip"
# Download and extract FF3 factors
temp_ff3 <- tempfile()
download.file(ff3_url, temp_ff3, mode = "wb")
unzip(temp_ff3, exdir = tempdir())
# Read the CSV (skip header rows, data starts after some text)
ff3_raw <- read.csv(file.path(tempdir(), "F-F_Research_Data_Factors.CSV"), skip = 3)
# The file has annual data at the bottom after a blank row - we need to find where monthly ends
# Monthly data has 6-digit dates (YYYYMM), annual has 4-digit (YYYY)
ff3_raw <- ff3_raw %>%
rename(date = X) %>%
filter(nchar(trimws(date)) == 6) # Keep only monthly data (YYYYMM format)
# Convert to numeric and proper date format
ff3_clean <- ff3_raw %>%
mutate(
date = as.Date(paste0(date, "01"), format = "%Y%m%d"),
date = ceiling_date(date, "month") - days(1), # Last day of month
Mkt_RF = as.numeric(Mkt.RF) / 100, # Convert from percentage
SMB = as.numeric(SMB) / 100,
HML = as.numeric(HML) / 100,
RF = as.numeric(RF) / 100
) %>%
select(date, Mkt_RF, SMB, HML, RF)
# Download and extract Momentum factor
temp_mom <- tempfile()
download.file(mom_url, temp_mom, mode = "wb")
unzip(temp_mom, exdir = tempdir())
mom_raw <- read.csv(file.path(tempdir(), "F-F_Momentum_Factor.CSV"), skip = 13)
mom_clean <- mom_raw %>%
rename(date = X, MOM = Mom) %>%
filter(nchar(trimws(date)) == 6) %>%
mutate(
date = as.Date(paste0(date, "01"), format = "%Y%m%d"),
date = ceiling_date(date, "month") - days(1),
MOM = as.numeric(MOM) / 100
) %>%
select(date, MOM)
# Preview
head(ff3_clean) date Mkt_RF SMB HML RF
1 1926-07-31 0.0289 -0.0255 -0.0239 0.0022
2 1926-08-31 0.0264 -0.0114 0.0381 0.0025
3 1926-09-30 0.0038 -0.0136 0.0005 0.0023
4 1926-10-31 -0.0327 -0.0014 0.0082 0.0032
5 1926-11-30 0.0254 -0.0011 -0.0061 0.0031
6 1926-12-31 0.0262 -0.0007 0.0006 0.0028head(mom_clean) date MOM
1 1927-01-31 0.0057
2 1927-02-28 -0.0150
3 1927-03-31 0.0352
4 1927-04-30 0.0436
5 1927-05-31 0.0278
6 1927-06-30 0.00552.5 Merge Stock Returns with Factors
# Merge all data together
analysis_df <- returns_df %>%
inner_join(ff3_clean, by = "date") %>%
inner_join(mom_clean, by = "date")
# Calculate excess returns (stock return minus risk-free rate)
analysis_df <- analysis_df %>%
mutate(
TSLA_excess = TSLA - RF,
TM_excess = TM - RF,
GM_excess = GM - RF
)
# Preview the final dataset
head(analysis_df) date TSLA TM GM Mkt_RF SMB HML
1 2011-01-31 -0.095005275 0.045148167 -0.010038051 0.0198 -0.0229 0.0061
2 2011-02-28 -0.008713727 0.135313100 -0.081117981 0.0348 0.0150 0.0118
3 2011-03-31 0.161573664 -0.131841473 -0.074560044 0.0045 0.0249 -0.0184
4 2011-04-30 -0.005405400 -0.007102854 0.034160439 0.0290 -0.0042 -0.0244
5 2011-05-31 0.092028728 0.045306383 -0.008725489 -0.0127 -0.0073 -0.0185
6 2011-06-30 -0.033510059 -0.010445451 -0.045583025 -0.0174 -0.0027 -0.0020
RF MOM TSLA_excess TM_excess GM_excess
1 1e-04 -0.0030 -0.095105275 0.045048167 -0.010138051
2 1e-04 0.0188 -0.008813727 0.135213100 -0.081217981
3 1e-04 0.0345 0.161473664 -0.131941473 -0.074660044
4 0e+00 0.0005 -0.005405400 -0.007102854 0.034160439
5 0e+00 -0.0060 0.092028728 0.045306383 -0.008725489
6 0e+00 0.0182 -0.033510059 -0.010445451 -0.045583025cat("\nNumber of observations:", nrow(analysis_df), "\n")
Number of observations: 178 cat("Date range:", as.character(min(analysis_df$date)), "to", as.character(max(analysis_df$date)), "\n")Date range: 2011-01-31 to 2025-10-31 3 Question 1 & 2: CAPM and Alpha Calculation
3.1 Conceptual Background
CAPM Formula: \[E[R_i] = R_f + \beta_i (E[R_m] - R_f)\]
Alpha Formula: \[\alpha = R_{actual} - [R_f + \beta(R_m - R_f)]\]
Key Insight: Comparing raw returns is flawed because higher returns may simply reflect higher beta (market risk), not skill. We must adjust for systematic risk.
3.2 Run CAPM Regressions
The CAPM regression is: \[R_i - R_f = \alpha + \beta (R_m - R_f) + \epsilon\]
# CAPM for Tesla
capm_tsla <- lm(TSLA_excess ~ Mkt_RF, data = analysis_df)
# CAPM for Toyota
capm_tm <- lm(TM_excess ~ Mkt_RF, data = analysis_df)
# CAPM for GM
capm_gm <- lm(GM_excess ~ Mkt_RF, data = analysis_df)3.3 CAPM Results
3.3.1 Tesla CAPM
summary(capm_tsla)
Call:
lm(formula = TSLA_excess ~ Mkt_RF, data = analysis_df)
Residuals:
Min 1Q Median 3Q Max
-0.33461 -0.09652 -0.02246 0.08987 0.73429
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.02480 0.01245 1.992 0.0479 *
Mkt_RF 1.84331 0.28578 6.450 1.04e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.161 on 176 degrees of freedom
Multiple R-squared: 0.1912, Adjusted R-squared: 0.1866
F-statistic: 41.6 on 1 and 176 DF, p-value: 1.044e-09Interpretation:
- Beta (β): The coefficient on Mkt_RF. A beta > 1 means the stock amplifies market movements.
- Alpha (α): The intercept. Positive alpha suggests outperformance after adjusting for market risk.
- R²: Proportion of return variation explained by the market factor.
3.3.2 Toyota CAPM
summary(capm_tm)
Call:
lm(formula = TM_excess ~ Mkt_RF, data = analysis_df)
Residuals:
Min 1Q Median 3Q Max
-0.136287 -0.030614 -0.002533 0.029780 0.168475
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.001664 0.004069 0.409 0.683
Mkt_RF 0.596021 0.093420 6.380 1.52e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.05262 on 176 degrees of freedom
Multiple R-squared: 0.1878, Adjusted R-squared: 0.1832
F-statistic: 40.7 on 1 and 176 DF, p-value: 1.515e-093.3.3 GM CAPM
summary(capm_gm)
Call:
lm(formula = GM_excess ~ Mkt_RF, data = analysis_df)
Residuals:
Min 1Q Median 3Q Max
-0.172897 -0.047033 -0.002003 0.036413 0.225198
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.007094 0.005433 -1.306 0.193
Mkt_RF 1.436581 0.124715 11.519 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.07025 on 176 degrees of freedom
Multiple R-squared: 0.4298, Adjusted R-squared: 0.4266
F-statistic: 132.7 on 1 and 176 DF, p-value: < 2.2e-163.4 CAPM Summary Table
# Extract key statistics
capm_summary <- data.frame(
Stock = c("Tesla", "Toyota", "GM"),
Alpha = c(coef(capm_tsla)[1], coef(capm_tm)[1], coef(capm_gm)[1]),
Alpha_pvalue = c(
summary(capm_tsla)$coefficients[1, 4],
summary(capm_tm)$coefficients[1, 4],
summary(capm_gm)$coefficients[1, 4]
),
Beta = c(coef(capm_tsla)[2], coef(capm_tm)[2], coef(capm_gm)[2]),
Beta_pvalue = c(
summary(capm_tsla)$coefficients[2, 4],
summary(capm_tm)$coefficients[2, 4],
summary(capm_gm)$coefficients[2, 4]
),
R_squared = c(
summary(capm_tsla)$r.squared,
summary(capm_tm)$r.squared,
summary(capm_gm)$r.squared
)
)
# Format for display
capm_summary %>%
mutate(
Alpha = paste0(round(Alpha * 100, 2), "%"),
Alpha_pvalue = round(Alpha_pvalue, 4),
Beta = round(Beta, 2),
Beta_pvalue = formatC(Beta_pvalue, format = "e", digits = 2),
R_squared = paste0(round(R_squared * 100, 1), "%")
) %>%
kable(col.names = c("Stock", "Alpha (monthly)", "Alpha p-value", "Beta", "Beta p-value", "R²"))| Stock | Alpha (monthly) | Alpha p-value | Beta | Beta p-value | R² |
|---|---|---|---|---|---|
| Tesla | 2.48% | 0.0479 | 1.84 | 1.04e-09 | 19.1% |
| Toyota | 0.17% | 0.6832 | 0.60 | 1.52e-09 | 18.8% |
| GM | -0.71% | 0.1933 | 1.44 | 3.07e-23 | 43% |
4 Question 3: Skill vs Luck
4.1 Conceptual Background
Key Question: Is positive alpha evidence of skill or just luck?
The Problem:
- Idiosyncratic (firm-specific) risk creates noise in alpha estimates
- One year of data is insufficient to distinguish skill from luck
- We need statistical significance to have confidence in alpha
Significance Criteria:
- t-statistic > 2
- p-value < 0.05
# Check t-statistics for alpha
cat("Tesla Alpha t-statistic:", round(summary(capm_tsla)$coefficients[1, 3], 2), "\n")Tesla Alpha t-statistic: 1.99 cat("Toyota Alpha t-statistic:", round(summary(capm_tm)$coefficients[1, 3], 2), "\n")Toyota Alpha t-statistic: 0.41 cat("GM Alpha t-statistic:", round(summary(capm_gm)$coefficients[1, 3], 2), "\n")GM Alpha t-statistic: -1.31 Interpretation: Even with multiple years of data, none of our stocks show clearly significant alpha at the 5% level in the CAPM framework. This illustrates how difficult it is to prove “skill” statistically.
5 Question 4: Cyclical vs Defensive Classification
5.1 Classification Rule
- β > 1: Cyclical stock (amplifies market movements)
- β < 1: Defensive stock (dampens market movements)
5.2 Results from CAPM
cat("CAPM Betas:\n")CAPM Betas:cat("Tesla: β =", round(coef(capm_tsla)[2], 2), "→", ifelse(coef(capm_tsla)[2] > 1, "Cyclical", "Defensive"), "\n")Tesla: β = 1.84 → Cyclical cat("Toyota: β =", round(coef(capm_tm)[2], 2), "→", ifelse(coef(capm_tm)[2] > 1, "Cyclical", "Defensive"), "\n")Toyota: β = 0.6 → Defensive cat("GM: β =", round(coef(capm_gm)[2], 2), "→", ifelse(coef(capm_gm)[2] > 1, "Cyclical", "Defensive"), "\n")GM: β = 1.44 → Cyclical We will compare this with FF3 betas in the next section.
6 Questions 5-7: Fama-French 3-Factor Model
6.1 Conceptual Background
The FF3 model adds two factors to CAPM:
\[R_i - R_f = \alpha + \beta (R_m - R_f) + s \cdot SMB + h \cdot HML + \epsilon\]
Where:
- SMB (Small Minus Big): Size factor. Positive loading = behaves like small-cap.
- HML (High Minus Low): Value factor. Positive loading = behaves like value stock; negative = growth stock.
6.2 Run FF3 Regressions
# FF3 for Tesla
ff3_tsla <- lm(TSLA_excess ~ Mkt_RF + SMB + HML, data = analysis_df)
# FF3 for Toyota
ff3_tm <- lm(TM_excess ~ Mkt_RF + SMB + HML, data = analysis_df)
# FF3 for GM
ff3_gm <- lm(GM_excess ~ Mkt_RF + SMB + HML, data = analysis_df)6.3 FF3 Results
6.3.1 Tesla FF3
summary(ff3_tsla)
Call:
lm(formula = TSLA_excess ~ Mkt_RF + SMB + HML, data = analysis_df)
Residuals:
Min 1Q Median 3Q Max
-0.30462 -0.10715 -0.01737 0.07397 0.74808
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.02759 0.01215 2.271 0.02438 *
Mkt_RF 1.63721 0.29199 5.607 7.95e-08 ***
SMB 1.16879 0.48206 2.425 0.01635 *
HML -1.14880 0.35507 -3.235 0.00145 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1552 on 174 degrees of freedom
Multiple R-squared: 0.2563, Adjusted R-squared: 0.2435
F-statistic: 19.99 on 3 and 174 DF, p-value: 3.509e-116.3.2 Toyota FF3
summary(ff3_tm)
Call:
lm(formula = TM_excess ~ Mkt_RF + SMB + HML, data = analysis_df)
Residuals:
Min 1Q Median 3Q Max
-0.130513 -0.032114 -0.003858 0.024294 0.170828
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0008251 0.0040965 0.201 0.8406
Mkt_RF 0.6473355 0.0984470 6.575 5.45e-10 ***
SMB -0.2729047 0.1625309 -1.679 0.0949 .
HML 0.1391050 0.1197170 1.162 0.2468
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.05234 on 174 degrees of freedom
Multiple R-squared: 0.2056, Adjusted R-squared: 0.1919
F-statistic: 15.01 on 3 and 174 DF, p-value: 9.825e-096.3.3 GM FF3
summary(ff3_gm)
Call:
lm(formula = GM_excess ~ Mkt_RF + SMB + HML, data = analysis_df)
Residuals:
Min 1Q Median 3Q Max
-0.168934 -0.041226 -0.003239 0.038771 0.196147
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.004056 0.005177 -0.783 0.43446
Mkt_RF 1.306740 0.124414 10.503 < 2e-16 ***
SMB 0.575196 0.205401 2.800 0.00568 **
HML 0.583813 0.151294 3.859 0.00016 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.06615 on 174 degrees of freedom
Multiple R-squared: 0.5002, Adjusted R-squared: 0.4916
F-statistic: 58.06 on 3 and 174 DF, p-value: < 2.2e-166.4 FF3 Summary Table
ff3_summary <- data.frame(
Stock = c("Tesla", "Toyota", "GM"),
Alpha = c(coef(ff3_tsla)[1], coef(ff3_tm)[1], coef(ff3_gm)[1]),
Alpha_p = c(
summary(ff3_tsla)$coefficients[1, 4],
summary(ff3_tm)$coefficients[1, 4],
summary(ff3_gm)$coefficients[1, 4]
),
Beta = c(coef(ff3_tsla)[2], coef(ff3_tm)[2], coef(ff3_gm)[2]),
SMB = c(coef(ff3_tsla)[3], coef(ff3_tm)[3], coef(ff3_gm)[3]),
SMB_p = c(
summary(ff3_tsla)$coefficients[3, 4],
summary(ff3_tm)$coefficients[3, 4],
summary(ff3_gm)$coefficients[3, 4]
),
HML = c(coef(ff3_tsla)[4], coef(ff3_tm)[4], coef(ff3_gm)[4]),
HML_p = c(
summary(ff3_tsla)$coefficients[4, 4],
summary(ff3_tm)$coefficients[4, 4],
summary(ff3_gm)$coefficients[4, 4]
),
R2 = c(
summary(ff3_tsla)$r.squared,
summary(ff3_tm)$r.squared,
summary(ff3_gm)$r.squared
)
)
ff3_summary %>%
mutate(
Alpha = paste0(round(Alpha * 100, 2), "%"),
Alpha_p = round(Alpha_p, 3),
Beta = round(Beta, 2),
SMB = round(SMB, 2),
SMB_p = round(SMB_p, 4),
HML = round(HML, 2),
HML_p = round(HML_p, 4),
R2 = paste0(round(R2 * 100, 1), "%")
) %>%
kable(col.names = c("Stock", "Alpha", "α p-val", "Beta", "SMB", "SMB p-val", "HML", "HML p-val", "R²"))| Stock | Alpha | α p-val | Beta | SMB | SMB p-val | HML | HML p-val | R² |
|---|---|---|---|---|---|---|---|---|
| Tesla | 2.76% | 0.024 | 1.64 | 1.17 | 0.0163 | -1.15 | 0.0015 | 25.6% |
| Toyota | 0.08% | 0.841 | 0.65 | -0.27 | 0.0949 | 0.14 | 0.2468 | 20.6% |
| GM | -0.41% | 0.434 | 1.31 | 0.58 | 0.0057 | 0.58 | 0.0002 | 50% |
6.5 Question 4 Revisited: FF3 Betas
cat("FF3 Market Betas:\n")FF3 Market Betas:cat("Tesla: β =", round(coef(ff3_tsla)[2], 2), "→", ifelse(coef(ff3_tsla)[2] > 1, "Cyclical", "Defensive"), "\n")Tesla: β = 1.64 → Cyclical cat("Toyota: β =", round(coef(ff3_tm)[2], 2), "→", ifelse(coef(ff3_tm)[2] > 1, "Cyclical", "Defensive"), "\n")Toyota: β = 0.65 → Defensive cat("GM: β =", round(coef(ff3_gm)[2], 2), "→", ifelse(coef(ff3_gm)[2] > 1, "Cyclical", "Defensive"), "\n")GM: β = 1.31 → Cyclical cat("\nClassification unchanged from CAPM.\n")
Classification unchanged from CAPM.6.6 Question 5: Size Factor Interpretation
Decision Rule (5% significance):
- s > 0 and p < 0.05 → Small-cap behavior
- s < 0 and p < 0.05 → Large-cap behavior
- p > 0.05 → Cannot classify
cat("Size Factor (SMB) Results:\n\n")Size Factor (SMB) Results:# Tesla
cat("Tesla: s =", round(coef(ff3_tsla)[3], 2), ", p =", round(summary(ff3_tsla)$coefficients[3, 4], 4))Tesla: s = 1.17 , p = 0.0163if (summary(ff3_tsla)$coefficients[3, 4] < 0.05) {
cat(" → ", ifelse(coef(ff3_tsla)[3] > 0, "Small-cap behavior", "Large-cap behavior"), "\n")
} else {
cat(" → Not significant (cannot classify)\n")
} → Small-cap behavior # Toyota
cat("Toyota: s =", round(coef(ff3_tm)[3], 2), ", p =", round(summary(ff3_tm)$coefficients[3, 4], 4))Toyota: s = -0.27 , p = 0.0949if (summary(ff3_tm)$coefficients[3, 4] < 0.05) {
cat(" → ", ifelse(coef(ff3_tm)[3] > 0, "Small-cap behavior", "Large-cap behavior"), "\n")
} else {
cat(" → Not significant (cannot classify)\n")
} → Not significant (cannot classify)# GM
cat("GM: s =", round(coef(ff3_gm)[3], 2), ", p =", round(summary(ff3_gm)$coefficients[3, 4], 4))GM: s = 0.58 , p = 0.0057if (summary(ff3_gm)$coefficients[3, 4] < 0.05) {
cat(" → ", ifelse(coef(ff3_gm)[3] > 0, "Small-cap behavior", "Large-cap behavior"), "\n")
} else {
cat(" → Not significant (cannot classify)\n")
} → Small-cap behavior Key Finding: GM behaves like a small-cap stock despite being a large company. This may reflect its financial distress history and higher volatility.
6.7 Question 6: Value/Growth Interpretation
Decision Rule (5% significance):
- h > 0 and p < 0.05 → Value stock
- h < 0 and p < 0.05 → Growth stock
- p > 0.05 → Cannot classify
cat("Value/Growth Factor (HML) Results:\n\n")Value/Growth Factor (HML) Results:# Tesla
cat("Tesla: h =", round(coef(ff3_tsla)[4], 2), ", p =", round(summary(ff3_tsla)$coefficients[4, 4], 4))Tesla: h = -1.15 , p = 0.0015if (summary(ff3_tsla)$coefficients[4, 4] < 0.05) {
cat(" → ", ifelse(coef(ff3_tsla)[4] > 0, "Value stock", "Growth stock"), "\n")
} else {
cat(" → Not significant (cannot classify)\n")
} → Growth stock # Toyota
cat("Toyota: h =", round(coef(ff3_tm)[4], 2), ", p =", round(summary(ff3_tm)$coefficients[4, 4], 4))Toyota: h = 0.14 , p = 0.2468if (summary(ff3_tm)$coefficients[4, 4] < 0.05) {
cat(" → ", ifelse(coef(ff3_tm)[4] > 0, "Value stock", "Growth stock"), "\n")
} else {
cat(" → Not significant (cannot classify)\n")
} → Not significant (cannot classify)# GM
cat("GM: h =", round(coef(ff3_gm)[4], 2), ", p =", round(summary(ff3_gm)$coefficients[4, 4], 4))GM: h = 0.58 , p = 2e-04if (summary(ff3_gm)$coefficients[4, 4] < 0.05) {
cat(" → ", ifelse(coef(ff3_gm)[4] > 0, "Value stock", "Growth stock"), "\n")
} else {
cat(" → Not significant (cannot classify)\n")
} → Value stock Key Finding: Tesla is a growth stock (negative HML loading, significant). GM is a value stock (positive HML loading, significant). This aligns with their business profiles.
6.8 Question 7: Alpha Significance in FF3
cat("FF3 Alpha Results:\n\n")FF3 Alpha Results:cat("Tesla: α =", round(coef(ff3_tsla)[1] * 100, 2), "% monthly, p =", round(summary(ff3_tsla)$coefficients[1, 4], 3))Tesla: α = 2.76 % monthly, p = 0.024cat(ifelse(summary(ff3_tsla)$coefficients[1, 4] < 0.05, " → Significant\n", " → Not significant\n")) → Significantcat("Toyota: α =", round(coef(ff3_tm)[1] * 100, 2), "% monthly, p =", round(summary(ff3_tm)$coefficients[1, 4], 3))Toyota: α = 0.08 % monthly, p = 0.841cat(ifelse(summary(ff3_tm)$coefficients[1, 4] < 0.05, " → Significant\n", " → Not significant\n")) → Not significantcat("GM: α =", round(coef(ff3_gm)[1] * 100, 2), "% monthly, p =", round(summary(ff3_gm)$coefficients[1, 4], 3))GM: α = -0.41 % monthly, p = 0.434cat(ifelse(summary(ff3_gm)$coefficients[1, 4] < 0.05, " → Significant\n", " → Not significant\n")) → Not significant7 Question 8: R² Analysis
7.1 Comparison of Model Fit
r2_comparison <- data.frame(
Stock = c("Tesla", "Toyota", "GM"),
CAPM_R2 = c(
summary(capm_tsla)$r.squared,
summary(capm_tm)$r.squared,
summary(capm_gm)$r.squared
),
FF3_R2 = c(
summary(ff3_tsla)$r.squared,
summary(ff3_tm)$r.squared,
summary(ff3_gm)$r.squared
)
)
r2_comparison %>%
mutate(
CAPM_R2 = paste0(round(CAPM_R2 * 100, 1), "%"),
FF3_R2 = paste0(round(FF3_R2 * 100, 1), "%")
) %>%
kable(col.names = c("Stock", "CAPM R²", "FF3 R²"))| Stock | CAPM R² | FF3 R² |
|---|---|---|
| Tesla | 19.1% | 25.6% |
| Toyota | 18.8% | 20.6% |
| GM | 43% | 50% |
7.2 Interpretation
cat("R² Interpretation:\n\n")R² Interpretation:cat("• Higher R² = more return variation explained by systematic factors\n")• Higher R² = more return variation explained by systematic factorscat("• Lower R² = more firm-specific (idiosyncratic) risk\n\n")• Lower R² = more firm-specific (idiosyncratic) riskcat("GM (highest R²): Returns well-explained by systematic factors.\n")GM (highest R²): Returns well-explained by systematic factors.cat(" - Mature automaker with predictable market sensitivity\n\n") - Mature automaker with predictable market sensitivitycat("Tesla (lowest R²): Returns dominated by firm-specific factors.\n")Tesla (lowest R²): Returns dominated by firm-specific factors.cat(" - Innovation news, Elon Musk tweets, EV market sentiment\n") - Innovation news, Elon Musk tweets, EV market sentimentcat(" - Model missing important drivers of Tesla returns\n\n") - Model missing important drivers of Tesla returnscat("Suggestions for model improvement:\n")Suggestions for model improvement:cat(" - Fama-French 5-Factor Model (adds profitability, investment)\n") - Fama-French 5-Factor Model (adds profitability, investment)cat(" - Industry-specific factors (EV sector, auto industry)\n") - Industry-specific factors (EV sector, auto industry)cat(" - Sentiment/behavioral proxies\n") - Sentiment/behavioral proxies8 Question 9: Momentum Factor (4-Factor Model)
8.1 Conceptual Background
The Carhart 4-Factor Model adds momentum to FF3:
\[R_i - R_f = \alpha + \beta (R_m - R_f) + s \cdot SMB + h \cdot HML + m \cdot MOM + \epsilon\]
MOM (Momentum): Winners minus Losers. Positive loading = stock exhibits momentum (past winners continue winning).
8.2 Run 4-Factor Regressions
# 4-Factor for Tesla
ff4_tsla <- lm(TSLA_excess ~ Mkt_RF + SMB + HML + MOM, data = analysis_df)
# 4-Factor for Toyota
ff4_tm <- lm(TM_excess ~ Mkt_RF + SMB + HML + MOM, data = analysis_df)
# 4-Factor for GM
ff4_gm <- lm(GM_excess ~ Mkt_RF + SMB + HML + MOM, data = analysis_df)8.3 4-Factor Results
8.3.1 Tesla 4-Factor
summary(ff4_tsla)
Call:
lm(formula = TSLA_excess ~ Mkt_RF + SMB + HML + MOM, data = analysis_df)
Residuals:
Min 1Q Median 3Q Max
-0.31202 -0.10980 -0.01886 0.07254 0.74654
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.02887 0.01225 2.356 0.01959 *
Mkt_RF 1.56086 0.30592 5.102 8.77e-07 ***
SMB 1.09062 0.49128 2.220 0.02772 *
HML -1.24738 0.37408 -3.334 0.00105 **
MOM -0.31630 0.37491 -0.844 0.40001
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1554 on 173 degrees of freedom
Multiple R-squared: 0.2594, Adjusted R-squared: 0.2423
F-statistic: 15.15 on 4 and 173 DF, p-value: 1.229e-108.3.2 Toyota 4-Factor
summary(ff4_tm)
Call:
lm(formula = TM_excess ~ Mkt_RF + SMB + HML + MOM, data = analysis_df)
Residuals:
Min 1Q Median 3Q Max
-0.13756 -0.03252 -0.00421 0.02489 0.17760
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.001423 0.004124 0.345 0.731
Mkt_RF 0.611732 0.102952 5.942 1.51e-08 ***
SMB -0.309356 0.165327 -1.871 0.063 .
HML 0.093140 0.125889 0.740 0.460
MOM -0.147488 0.126166 -1.169 0.244
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.05229 on 173 degrees of freedom
Multiple R-squared: 0.2118, Adjusted R-squared: 0.1936
F-statistic: 11.63 on 4 and 173 DF, p-value: 2.203e-088.3.3 GM 4-Factor
summary(ff4_gm)
Call:
lm(formula = GM_excess ~ Mkt_RF + SMB + HML + MOM, data = analysis_df)
Residuals:
Min 1Q Median 3Q Max
-0.166238 -0.041550 -0.002429 0.035043 0.188050
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.003313 0.005212 -0.636 0.52584
Mkt_RF 1.262524 0.130124 9.702 < 2e-16 ***
SMB 0.529927 0.208963 2.536 0.01210 *
HML 0.526729 0.159116 3.310 0.00113 **
MOM -0.183165 0.159466 -1.149 0.25230
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.06609 on 173 degrees of freedom
Multiple R-squared: 0.504, Adjusted R-squared: 0.4926
F-statistic: 43.95 on 4 and 173 DF, p-value: < 2.2e-168.4 Momentum Interpretation
Decision Rule (5% significance):
- m > 0 and p < 0.05 → Positive momentum
- m < 0 and p < 0.05 → Negative momentum (contrarian)
- p > 0.05 → No significant momentum
cat("Momentum Factor (MOM) Results:\n\n")Momentum Factor (MOM) Results:# Tesla
cat("Tesla: m =", round(coef(ff4_tsla)[5], 2), ", p =", round(summary(ff4_tsla)$coefficients[5, 4], 3))Tesla: m = -0.32 , p = 0.4if (summary(ff4_tsla)$coefficients[5, 4] < 0.05) {
cat(" → ", ifelse(coef(ff4_tsla)[5] > 0, "Positive momentum", "Negative momentum"), "\n")
} else {
cat(" → Not significant\n")
} → Not significant# Toyota
cat("Toyota: m =", round(coef(ff4_tm)[5], 2), ", p =", round(summary(ff4_tm)$coefficients[5, 4], 3))Toyota: m = -0.15 , p = 0.244if (summary(ff4_tm)$coefficients[5, 4] < 0.05) {
cat(" → ", ifelse(coef(ff4_tm)[5] > 0, "Positive momentum", "Negative momentum"), "\n")
} else {
cat(" → Not significant\n")
} → Not significant# GM
cat("GM: m =", round(coef(ff4_gm)[5], 2), ", p =", round(summary(ff4_gm)$coefficients[5, 4], 3))GM: m = -0.18 , p = 0.252if (summary(ff4_gm)$coefficients[5, 4] < 0.05) {
cat(" → ", ifelse(coef(ff4_gm)[5] > 0, "Positive momentum", "Negative momentum"), "\n")
} else {
cat(" → Not significant\n")
} → Not significantcat("\nConclusion: No significant momentum effect for any stock at 5% level.\n")
Conclusion: No significant momentum effect for any stock at 5% level.cat("All stocks show negative (contrarian) loadings but statistically insignificant.\n")All stocks show negative (contrarian) loadings but statistically insignificant.8.5 4-Factor Summary Table
ff4_summary <- data.frame(
Stock = c("Tesla", "Toyota", "GM"),
Alpha = c(coef(ff4_tsla)[1], coef(ff4_tm)[1], coef(ff4_gm)[1]),
Alpha_p = c(
summary(ff4_tsla)$coefficients[1, 4],
summary(ff4_tm)$coefficients[1, 4],
summary(ff4_gm)$coefficients[1, 4]
),
MOM = c(coef(ff4_tsla)[5], coef(ff4_tm)[5], coef(ff4_gm)[5]),
MOM_p = c(
summary(ff4_tsla)$coefficients[5, 4],
summary(ff4_tm)$coefficients[5, 4],
summary(ff4_gm)$coefficients[5, 4]
),
R2 = c(
summary(ff4_tsla)$r.squared,
summary(ff4_tm)$r.squared,
summary(ff4_gm)$r.squared
)
)
ff4_summary %>%
mutate(
Alpha = paste0(round(Alpha * 100, 2), "%"),
Alpha_p = round(Alpha_p, 3),
MOM = round(MOM, 2),
MOM_p = round(MOM_p, 3),
R2 = paste0(round(R2 * 100, 1), "%")
) %>%
kable(col.names = c("Stock", "Alpha", "α p-val", "MOM", "MOM p-val", "R²"))| Stock | Alpha | α p-val | MOM | MOM p-val | R² |
|---|---|---|---|---|---|
| Tesla | 2.89% | 0.020 | -0.32 | 0.400 | 25.9% |
| Toyota | 0.14% | 0.731 | -0.15 | 0.244 | 21.2% |
| GM | -0.33% | 0.526 | -0.18 | 0.252 | 50.4% |
8.6 R² Improvement from Adding Momentum
cat("R² Comparison (FF3 vs 4-Factor):\n\n")R² Comparison (FF3 vs 4-Factor):cat("Tesla: FF3 =", round(summary(ff3_tsla)$r.squared * 100, 1), "% → 4F =", round(summary(ff4_tsla)$r.squared * 100, 1), "%\n")Tesla: FF3 = 25.6 % → 4F = 25.9 %cat("Toyota: FF3 =", round(summary(ff3_tm)$r.squared * 100, 1), "% → 4F =", round(summary(ff4_tm)$r.squared * 100, 1), "%\n")Toyota: FF3 = 20.6 % → 4F = 21.2 %cat("GM: FF3 =", round(summary(ff3_gm)$r.squared * 100, 1), "% → 4F =", round(summary(ff4_gm)$r.squared * 100, 1), "%\n")GM: FF3 = 50 % → 4F = 50.4 %cat("\nMinimal improvement from adding momentum factor.\n")
Minimal improvement from adding momentum factor.9 Summary Table
final_summary <- data.frame(
Stock = c("Tesla", "Toyota", "GM"),
Classification = c("Cyclical", "Defensive", "Cyclical"),
Beta_CAPM = c(round(coef(capm_tsla)[2], 2), round(coef(capm_tm)[2], 2), round(coef(capm_gm)[2], 2)),
Size = c("Not sig", "Not sig", "Small-cap-like"),
Value_Growth = c("Growth", "Not sig", "Value"),
Alpha_FF3 = c(
paste0(round(coef(ff3_tsla)[1] * 100, 2), "%"),
paste0(round(coef(ff3_tm)[1] * 100, 2), "%"),
paste0(round(coef(ff3_gm)[1] * 100, 2), "%")
),
Alpha_Sig = c("Borderline", "No", "No"),
Momentum = c("No", "No", "No"),
R2_FF3 = c(
paste0(round(summary(ff3_tsla)$r.squared * 100, 1), "%"),
paste0(round(summary(ff3_tm)$r.squared * 100, 1), "%"),
paste0(round(summary(ff3_gm)$r.squared * 100, 1), "%")
)
)
kable(final_summary, col.names = c("Stock", "Type", "β (CAPM)", "Size", "Value/Growth", "α (FF3)", "α Sig?", "Momentum?", "R² (FF3)"))| Stock | Type | β (CAPM) | Size | Value/Growth | α (FF3) | α Sig? | Momentum? | R² (FF3) |
|---|---|---|---|---|---|---|---|---|
| Tesla | Cyclical | 1.84 | Not sig | Growth | 2.76% | Borderline | No | 25.6% |
| Toyota | Defensive | 0.60 | Not sig | Not sig | 0.08% | No | No | 20.6% |
| GM | Cyclical | 1.44 | Small-cap-like | Value | -0.41% | No | No | 50% |
10 Key Takeaways
Risk-adjusted performance (alpha) is more important than raw returns — Question 1-2 demonstrate this clearly.
One year is not enough to prove skill — Idiosyncratic risk creates noise; need statistical significance over multiple periods.
Tesla and GM are cyclical (β > 1); Toyota is defensive (β < 1) — Classification consistent across CAPM and FF3.
Tesla exhibits growth stock characteristics — Negative HML loading (significant).
GM behaves like a small-cap value stock — Positive SMB and HML loadings (both significant), despite large market cap.
No significant alpha at strict 5% level — Tesla borderline; returns largely explained by factor exposures.
GM best explained by systematic factors (R² = 55%) — Tesla worst explained (R² = 23%), dominated by firm-specific news.
No significant momentum effect — All stocks show negative (contrarian) loadings but insignificant.
11 Appendix: Session Information
sessionInfo()R version 4.4.0 (2024-04-24 ucrt)
Platform: x86_64-w64-mingw32/x64
Running under: Windows 11 x64 (build 26200)
Matrix products: default
locale:
[1] LC_COLLATE=Chinese (Simplified)_China.utf8
[2] LC_CTYPE=Chinese (Simplified)_China.utf8
[3] LC_MONETARY=Chinese (Simplified)_China.utf8
[4] LC_NUMERIC=C
[5] LC_TIME=Chinese (Simplified)_China.utf8
time zone: Asia/Shanghai
tzcode source: internal
attached base packages:
[1] stats graphics grDevices datasets utils methods base
other attached packages:
[1] broom_1.0.10 knitr_1.50 lubridate_1.9.4 readr_2.1.6
[5] tidyr_1.3.1 dplyr_1.1.4 quantmod_0.4.28 TTR_0.24.4
[9] xts_0.14.1 zoo_1.8-14
loaded via a namespace (and not attached):
[1] vctrs_0.6.5 cli_3.6.5 rlang_1.1.6 xfun_0.54
[5] purrr_1.2.0 renv_1.0.7 generics_0.1.4 jsonlite_2.0.0
[9] glue_1.8.0 backports_1.5.0 htmltools_0.5.8.1 hms_1.1.4
[13] rmarkdown_2.30 grid_4.4.0 tibble_3.3.0 evaluate_1.0.5
[17] tzdb_0.5.0 fastmap_1.2.0 yaml_2.3.11 lifecycle_1.0.4
[21] compiler_4.4.0 timechange_0.3.0 pkgconfig_2.0.3 rstudioapi_0.17.1
[25] lattice_0.22-6 digest_0.6.39 R6_2.6.1 tidyselect_1.2.1
[29] pillar_1.11.1 curl_7.0.0 magrittr_2.0.4 withr_3.0.2
[33] tools_4.4.0