library("tidyverse")
library("quantmod")
theme_set(theme_bw())
# ------------------------------------------------------------------------------
# Get data
# ------------------------------------------------------------------------------
# Define date range
start_date <- "2000-01-01"
end_date <- Sys.Date()
# Download Equity data from Yahoo Finance
getSymbols("AAPL", from = start_date, to = end_date, auto.assign = TRUE)
getSymbols("SPY", from = start_date, to = end_date, auto.assign = TRUE)
getSymbols("^IRX", from = start_date, to = end_date, auto.assign = TRUE) # 13-Week T-Bill Yield Index
# Calculate weekly returns for equity tickers
aapl_weekly <- weeklyReturn(AAPL, type = "arithmetic")
spy_weekly <- weeklyReturn(SPY, type = "arithmetic")
# Process the T-Bill (^IRX) data
# Extract the closing yield values matching our equity weekly date index
# Using Ad() gets the adjusted close, or Cl() for regular close
t_bill_weekly_yield <- Cl(IRX)[index(spy_weekly)]
# Forward-fill any missing data gap days (like market holidays)
t_bill_weekly_yield <- na.locf(t_bill_weekly_yield, na.rm = TRUE)
# Convert annualized yield percentage to weekly decimal: (Yield / 100) / 52
t_bill_weekly_return <- (t_bill_weekly_yield / 100) / 52
# Merge all series into one object and rename columns
merged_returns <- merge(aapl_weekly, spy_weekly, t_bill_weekly_return)
colnames(merged_returns) <- c(
"AAPL_Return",
"SPY_Return",
"TBill_Weekly_Yield"
)
# Convert the xts object to a standard data.frame
d <- data.frame(Date = index(merged_returns), coredata(merged_returns)) |>
na.omit() |>
remove_rownames() |>
mutate(
AAPL_Excess_Return = AAPL_Return - TBill_Weekly_Yield,
SPY_Excess_Return = SPY_Return - TBill_Weekly_Yield
)
# ------------------------------------------------------------------------------
# Exploratory Data Analysis (EDA)
# ------------------------------------------------------------------------------
scatter_plot <- ggplot(d, aes(x = SPY_Excess_Return, y = AAPL_Excess_Return)) +
geom_point(alpha = 0.5) +
labs(
title = "Scatter Plot of AAPL vs SPY Weekly Returns",
x = "SPY Weekly Excess Return",
y = "AAPL Weekly Excess Return"
)
scatter_plot
# Hexbin plot
hexbin_plot <- ggplot(d, aes(x = SPY_Excess_Return, y = AAPL_Excess_Return)) +
geom_hex(bins = 30) +
labs(
title = "Hexbin Plot of AAPL vs SPY Weekly Returns",
x = "SPY Weekly Excess Return",
y = "AAPL Weekly Excess Return"
) +
scale_fill_gradient(low = "lightblue", high = "darkblue")
hexbin_plot
hexbin_smooth_plot <- hexbin_plot + geom_smooth(method = "lm", formula = y ~ x)
hexbin_smooth_plot
# ------------------------------------------------------------------------------
# Fit the CAPM model using simple linear regression
# ------------------------------------------------------------------------------
# Fit the linear regression model: AAPL excess return ~ SPY excess return
capm_model <- lm(AAPL_Excess_Return ~ SPY_Excess_Return, data = d)
# View the summary of the CAPM model
summary(capm_model)
coef(capm_model)
confint(capm_model)
summary(capm_model)$r.squared5 Capital Asset Pricing Model
TipLearning objectives
After this lecture you should be able to
- derive the CAPM regression equation from the CAPM formula,
- interpret the intercept and slope of the fitted regression in financial terms, i.e. excess return and beta,
- fit the CAPM model in R using weekly returns downloaded from Yahoo Finance, and
- evaluate a fitted CAPM model by comparing its \(R^2\) and estimated beta to those of another asset.
The Capital Asset Pricing Model (CAPM) is a widely used model in finance that describes the relationship between systematic risk and expected return for assets, particularly stocks. The CAPM formula is given by: \[E[R] = R_f + \beta (E[R_m] - R_f)\] where
- \(R\) is the return of the asset,
- \(R_f\) is the risk-free rate, and
- \(R_m\) is the return of the market portfolio.
The return of the asset is calculated based on the current and previous price of the asset. If the price at time \(t\) is \(P_t\) and the price at time \(t-1\) is \(P_{t-1}\), then the return of the asset at time \(t\) is given by: \[R_t = \frac{P_t - P_{t-1}}{P_{t-1}}.\]
To fit a simple linear regression model to the CAPM, we can rewrite the equation in the form of a linear regression model: \[\underbrace{R_t - R_{t,f}}_{\text{response}} = \beta_0 + \beta_1 \underbrace{(R_{t,m} - R_{t,f})}_{\text{explanatory}} + \epsilon_t\] where, at time \(t\),
- \(R_t\) is the asset return,
- \(R_{t,f}\) is the risk-free return, and
- \(R_{t,m}\) is the market return.
Taking expectations of both sides and substituting the CAPM formula for \(E[R_t]\) and \(E[R_{t,m}]\),
\[E[R_t - R_{t,f}] = \beta_1 (E[R_{t,m}] - R_{t,f}),\]
which matches the regression equation above only if \(\beta_0 = 0\). CAPM is therefore making a testable prediction about the intercept, not just the slope.
The coefficient interpretations in this model are
- \(\beta_0\) is the asset’s abnormal return, or alpha — the part of its excess return not explained by market exposure. CAPM implies \(\beta_0 = 0\), so an estimate far from zero is evidence against the model, and
- \(\beta_1\) is the asset’s sensitivity to market movements, also known as the asset’s beta.
5.1 Example: AAPL vs SPY
We fit the CAPM using Apple (AAPL) as the asset, the S&P 500 ETF (SPY) as the market portfolio, and the 13-week T-bill yield (^IRX) as the risk-free rate.
NoteCode for this analysis
Call:
lm(formula = AAPL_Excess_Return ~ SPY_Excess_Return, data = d)
Residuals:
Min 1Q Median 3Q Max
-0.49795 -0.02051 -0.00147 0.02132 0.25639
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.003956 0.001136 3.483 0.000512 ***
SPY_Excess_Return 1.094719 0.046198 23.696 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.04237 on 1392 degrees of freedom
Multiple R-squared: 0.2874, Adjusted R-squared: 0.2869
F-statistic: 561.5 on 1 and 1392 DF, p-value: < 2.2e-16 (Intercept) SPY_Excess_Return
0.003956031 1.094718953 2.5 % 97.5 %
(Intercept) 0.001727829 0.006184234
SPY_Excess_Return 1.004093509 1.185344397[1] 0.28743615.2 Class activity
Find a stock, or an index, whose CAPM fit against SPY has
- a higher \(R^2\) than AAPL’s, and
- a more extreme beta, i.e. farther from 1 in either direction, than AAPL’s.
Adapt the code above by substituting your chosen ticker for AAPL (getSymbols() accepts any Yahoo Finance ticker), re-fit the model, and report the resulting \(R^2\) and \(\hat\beta_1\) alongside AAPL’s for comparison.