Logarithmic Return

Logarithmic Return (also known as log return or continuously compounded return) is a method for calculating an investment's rate of return. While most investors are familiar with the Simple Return (calculating the percentage change in price), the logarithmic return offers a few mathematical superpowers that make it a favorite among financial analysts and quants. It's calculated using the natural logarithm of the ratio of the final price to the initial price: ln(Final Price / Initial Price). Think of it this way: while a simple return tells you the direct path from Point A to Point B, a log return breaks the journey into an infinite number of tiny, continuous steps. This “continuously compounded” nature means that gains and losses over different periods can simply be added together, a property called time-additivity. This makes it incredibly useful for analyzing historical Volatility and is a cornerstone of many sophisticated Risk Management models. For the everyday investor, understanding this concept provides a deeper insight into how professionals model and manage investment risk.

For small price changes, log returns and simple returns are nearly identical. So why the extra math? The primary advantage of log returns lies in their elegant behavior over multiple time periods.

The most celebrated feature of log returns is that they are additive over time. Simple returns are not; they must be multiplied (or “geometrically linked”), which can be cumbersome. This property makes analyzing performance over time much cleaner. Let's say you buy a share of “Capipedia Corp.” for $100.

• Day 1: The price jumps to $120.
  • The simple return is ($120 - $100) / $100 = +20%.
  • The log return is ln($120 / $100) = +18.23%.
• Day 2: The price falls from $120 back to $100.
  • The simple return is ($100 - $120) / $120 = -16.67%.
  • The log return is ln($100 / $120) = -18.23%.

Now, let's look at the total return over the two days. Your stock is back where it started, so the true total return is 0%.

• Adding the simple returns gives: 20% + (-16.67%) = 3.33%. This is wrong! To get the correct answer with simple returns, you have to link them: (1 + 0.20) x (1 - 0.1667) - 1 = 0%.
• Adding the log returns gives: 18.23% + (-18.23%) = 0%. Perfect! The sum of the log returns directly and accurately reflects the total performance.

A dedicated Value Investing practitioner might scoff, “I care about a company's Fundamental Analysis and Margin of Safety, not fancy math for day traders!” And they'd have a point. A value investor's focus is on the long-term intrinsic value of a business, not fleeting price wiggles. However, understanding log returns is still valuable for two key reasons:

• Understanding Risk: Log returns are the standard input for measuring historical volatility. Because their distribution often resembles a Normal Distribution (the classic “bell curve”), they form the bedrock of many financial models, including the famous Black-Scholes Model used for pricing options. For a value investor, this isn't about predicting prices but about understanding the character of a stock's past price movements. A stock with highly volatile log returns might require a larger margin of safety to compensate for the higher risk and uncertainty.
• Portfolio Context: When analyzing a portfolio over many years, the additive nature of log returns makes it far simpler to calculate and compare long-term performance and risk metrics across different assets and timeframes. It provides a more robust framework for understanding how your collection of wonderful businesses has truly performed.

So, while you won't be calculating log returns to decide if a company is cheap, knowing what they are and why they matter gives you a more complete toolkit for assessing the risk side of the investment equation.