Log Returns are a sophisticated way to measure percentage changes in stock prices that addresses key limitations of simple percentage calculations. Unlike regular returns, log returns provide a more accurate representation of compound growth over time.

Why use log returns instead of regular percentage changes?

• Makes math easier when dealing with multiple time periods
• They help us analyze risk and volatility better
• They follow a normal distribution (bell curve) more closely
• Banks and financial analysts use them every day

Example: NVDA stock moving from $400 to $440 has the same log return magnitude as moving from $440 to $400.

The formula for log return is simple:

\[r_t = \ln\left(\frac{P_t}{P_{t-1}}\right) = \ln(P_t) - \ln(P_{t-1})\]

Where:

• \(r_t\) = log return at time \(t\)
• \(P_t\) = stock price at time \(t\)
  
• \(P_{t-1}\) = stock price at previous time
• \(\ln\) = natural logarithm

Key Property: Log returns can be added together \[\text{Total Return} = r_1 + r_2 + r_3 + ... + r_n\]

# ----------------------------------------------------------------
# For simplicity, I decided to not import any stock tracking API
# ----------------------------------------------------------------

# NVDA simulation
n_days <- 252
initial_price <- 400

# Generate daily log returns
daily_returns <- rnorm(n_days, mean = 0.002, sd = 0.035)

# Convert to stock prices
stock_prices <- numeric(n_days + 1)
stock_prices[1] <- initial_price

for(i in 2:(n_days + 1)) {
  stock_prices[i] <- stock_prices[i-1] * exp(daily_returns[i-1])
}

# Create data frame
stock_data <- data.frame(day = 1:(n_days + 1),
                        price = stock_prices,
                        returns = c(NA, daily_returns))