Log Returns

Log Returns (also known as 'continuously compounded returns') are a method for calculating the rate of return on an investment over a period. Imagine an investment that grows not in daily or monthly steps, but constantly, every microsecond. Log returns tell you the total return you would have earned if your investment had compounded continuously. While this might sound abstract, it's a powerful tool that makes analyzing investment performance much simpler and more intuitive than the more common 'simple return' (calculated as (New Price - Old Price) / Old Price). For professional analysts and savvy investors, log returns are preferred for their neat mathematical properties, which are especially useful when dealing with long time series of data. They help in modeling stock prices and assessing risk management more accurately, turning messy multiplication problems into simple addition.

For most quick, back-of-the-envelope calculations, simple returns work fine. But when you start to analyze performance over time, log returns have two superpowers that make them the superior choice.

• The Magic of Addition: The standout feature of log returns is time-additivity. If you want to know your total return over a year, you can simply add up all the daily (or weekly, or monthly) log returns. This is impossible with simple returns, which require messy compounding calculations. For example, a 10% gain followed by another 10% gain isn't a 20% gain; it's a 21% gain. With log returns, you just add them up. This makes long-term performance analysis a breeze.
• Symmetry is Beautiful: Log returns make gains and losses more comparable. Let's say a stock jumps from $80 to $100 (a 25% gain) and later drops from $100 back to $80 (a 20% loss). The percentages are different and can be misleading. With log returns, the return from $80 to $100 is ln(100/80) = +0.223, and the return from $100 to $80 is ln(80/100) = -0.223. They are perfectly symmetrical. This symmetry makes statistical models, especially those for volatility, more reliable.

For a value investor, the game is about buying wonderful companies at fair prices, not obsessing over daily price wiggles. So, are log returns just a fancy toy for quants? Not at all. While a disciple of Warren Buffett focuses on a company's underlying business value, understanding performance and risk is still paramount. Log returns are a superior tool for measuring that risk. They are the bedrock of many quantitative analysis techniques used to calculate a stock's historical volatility. A clear understanding of a stock's true volatility can help you better assess its margin of safety. A highly volatile stock, even if it looks cheap, might carry hidden risks that justify demanding an even deeper discount. Think of it as another tool in your analytical toolkit to help you sleep well at night.

Don't let the name “logarithm” send you running for the hills. The calculation is straightforward and easy to do on any calculator or spreadsheet.

The formula to calculate the log return is: Log Return = ln(Ending Price / Beginning Price) Where 'ln' stands for the natural logarithm, a button you can find on most scientific calculators and in spreadsheet programs (e.g., `=LN()` in Excel).

Let's say you bought a share of “Capipedia Corp.” for $100, and a month later, it's trading at $110.

1. Simple Return: (110 - 100) / 100 = 0.10, or 10%
2. Log Return: ln(110 / 100) = ln(1.1) ≈ 0.0953, or 9.53%

As you can see, for small changes, the simple return and log return are very close. This is why for casual, single-period calculations, simple returns work just fine. However, the differences become more significant with larger price swings and over multiple periods, which is when the clean properties of log returns truly shine. For any serious performance analysis, they are the professional's choice.