Summary  
This chapter covers computing the standard deviation of time-series data, annualizing it for comparability, and applying rolling-window calculations to track variability over time.

General domain of usage  
Finance

Volatility is a central concept in finance, representing the degree of variation in asset returns over time. As a financial analyst, you use **volatility** to gauge the risk associated with an investment — assets with higher volatility are considered riskier because their prices fluctuate more unpredictably. Since investors often compare risk across assets or over different periods, it is important to standardize volatility. This is accomplished through **annualization**, which expresses volatility on an annual basis, and through **rolling windows**, which allow you to observe how risk changes over time.

# Create synthetic daily returns
set.seed(123)
daily_returns <- rnorm(252, mean = 0.0005, sd = 0.01)

# Calculate the standard deviation of daily returns
daily_volatility <- sd(daily_returns)

# Annualize the volatility (assuming 252 trading days per year)
annualized_volatility <- daily_volatility * sqrt(252)

# Print the annualized volatility
print(annualized_volatility)

Suppose your calculation produces an annualized volatility of `0.22`. This means that, based on historical daily returns, you expect the asset's return to fluctuate by about 22% over a typical year. The formula used is:  

$$
\text{Annualized Volatility} = \text{Standard Deviation of Daily Returns} * \sqrt{252}
$$
  
Here, $$252$$ is the typical number of trading days in a year. This adjustment allows you to compare the risk of assets regardless of the frequency of your return data.

# To analyze how risk changes over time, calculate rolling volatility
# Load the zoo package for rollapply
library(zoo)

# Calculate 21-day (approximately 1 month) rolling volatility
rolling_volatility <- rollapply(
  daily_returns,
  width = 21,
  FUN = sd,
  fill = NA,
  align = "right"
)

# Annualize the rolling volatility
rolling_annualized_volatility <- rolling_volatility * sqrt(252)

# Plot the rolling annualized volatility
plot(
  rolling_annualized_volatility,
  type = "l",
  main = "21-Day Rolling Annualized Volatility",
  xlab = "Time",
  ylab = "Annualized Volatility"
)

When you plot rolling annualized volatility, you see how risk fluctuates throughout the period. Peaks in the plot correspond to times of high uncertainty or market stress, while troughs indicate more stable periods. By interpreting these changes, you can identify when an asset was especially risky and when it was relatively calm, supporting better risk management and investment decisions.

What does volatility represent in the context of asset returns?

Learn to analyze financial data in R: work with time series, compute returns, assess risk, analyze portfolios, and visualize and forecast market data using practical, code-driven examples.

Represent, manipulate, and analyze financial time series data in R, focusing on price data and return calculations.

Quantify and interpret risk in financial data using volatility and Value at Risk (VaR) metrics in R.

Analyze multi-asset portfolios: compute returns, risk, and diversification metrics in R.

Measuring Volatility in Asset Returns

Measuring Volatility in Asset Returns