What Beta Means: Measuring a Stock’s Risk

Finance

Historical betas of BIST30 stocks.

Uraz Akgül
2022-10-03

The volatility of a stock in relation to the overall market is measured by beta (\(\beta\)). The market, such as the BIST100 Index, has a beta of 1.0 by definition, and individual stocks are ranked based on how much they deviate from the market.

\(\beta = \frac{cov(R_e,R_m)}{var(R_m)}\)

\(\beta:\) Beta coefficient

\(R_e:\) The return on an individual stock

\(R_m:\) The return on the overall market

\(cov(R_e,R_m):\) How changes in a stock’s returns are related to changes in the market’s returns.

\(var(R_m):\) How far the market’s data points spread out from their average value

Regression analysis is used to calculate beta. The beta coefficient can be interpreted as follows:

The data you can access by downloading the post35.xlsx file from here is from Reuters.

library(tidyverse)

bist30 <- readxl::read_excel("data.xlsx")

Consider a stock and a market index. To build a regression model, we’ll use stock returns as the dependent variable and market index returns as the independent variable.

Let’s start with the daily returns.

master <- bist30 %>% 
  mutate_at(
    vars(-DATE), function(x) lag(log(lead(x) / x))
  ) %>% 
  slice(-1) %>% 
  pivot_longer(!c(DATE,XU100), names_to = "TICKER", values_to = "Return") %>% 
  arrange(TICKER) %>% 
  mutate(
    "YEAR" = lubridate::year(DATE)
  )

We can now build the regression models.

betas <- data.frame(
  ticker = rep(unique(master$TICKER), each = length(unique(master$YEAR))),
  year = rep(unique(master$YEAR), length(unique(master$TICKER))),
  beta = NA
)

for(i in 1:nrow(betas)) {
  
  ticker <- betas$ticker[i]
  year <- betas$year[i]
  
  df_model <- master %>%
    filter(TICKER == ticker & YEAR == year) %>%
    na.omit()
  
  model <- lm(df_model$Return ~ df_model$XU100)
  beta <- model$coefficients[2]
  
  betas$beta[i] <- beta
  
}

I’d like to show the year-based betas of the stocks we are studying on.

Let’s rank the stocks’ betas for 2022.

The codes for the last two data visualizations are provided below.

betas %>% 
  ggplot(aes(x = year, y = beta)) +
  geom_line() +
  geom_point() +
  geom_hline(yintercept = 1, linetype = "dashed") +
  facet_wrap(~ticker) +
  ggthemes::theme_fivethirtyeight() +
  theme(strip.text = element_text(size = 15),
        plot.title = element_text(size = 20, hjust = 0.5),
        axis.text.y = element_text(size = 15),
        axis.text.x = element_blank(),
        panel.grid.major = element_blank()) +
  scale_y_continuous(breaks = c(0,1)) +
  labs(title = "Historical Betas of BIST30 Stocks, 2013-2022") +
  annotate(
    geom = "rect",
    xmin = -Inf,
    xmax = Inf,
    ymin = 1,
    ymax = Inf,
    fill = "red",
    alpha = .2
  )

betas %>%
  filter(year == 2022) %>%
  ggplot(aes(x = reorder(ticker, -beta), y = beta)) +
  geom_point(size = 5, alpha = .5) +
  ggrepel::geom_text_repel(aes(label = ticker), size = 7) +
  geom_hline(yintercept = 1, linetype = "dashed") +
  ggthemes::theme_fivethirtyeight() +
  theme(
    axis.text.x = element_blank(),
    panel.grid.major = element_blank(),
    axis.text.y = element_text(size = 15)
  ) +
  annotate(
    geom = "rect",
    xmin = -Inf,
    xmax = Inf,
    ymin = 1,
    ymax = Inf,
    fill = "red",
    alpha = .2
  )