# Using Arithmetic and Logarithmic Returns to Compare Investments

Editors Note: Comparing different investments with different dollar amounts and time periods is arithmetically more challenging than it first appears. When I do my side-by-side comparisons (such as IBM, Decker, or Boston Beer), I always do the same dollar amount and the same time period. Simon does a really good job of explaining how to do this if they are not the same dollar and time period.

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Ralph and Jackie are both keen investors in the stock market. In this article I’m going to explain how they can use two different methods of calculating a rate of return in order to compare the performance of their investments.

Ralph buys \$1000 worth of stock in AlphaCorp. He holds it for exactly 2 years and then sells it for \$1200. Jackie buys \$3000 worth of stock in BetaCorp. She holds her stock for 1 year, and sells it for \$3300. Just to keep the example simple, we’ll assume that neither Ralph nor Jackie receive any dividend payments from their stock.

Ralph and Jackie now want to compare their investments. They know that there are two main methods they could use: the Arithmetic Return and the Logarithmic Return (often shortened to Log Return).

Ralph’s total profit is \$200, and Jackie’s is \$300. So this tells them that Jackie has made more money overall. But she also invested more. Investing more usually means she took a greater risk (if the stock went down she would lose more money). To take this into account, they want to know the profit as a percentage of the amount invested. This is exactly what the Arithmetic Return gives them.

For Ralph’s investment in AlphaCorp the arithmetic return is 20%. For Jackie’s investment in BetaCorp the arithmetic return is 10%. So based on the arithmetic return, it looks like Ralph has made the better investment, as his gained 20% in value compared to 10% for Jackie’s investment.

But notice that Jackie sold her stock after 1 year, while Ralph held his for 2 years. The arithmetic return doesn’t include the duration of the investment, so these values cannot really be compared meaningfully. So now Ralph and Jackie compare their investments using the Logarithmic Return, which does take this into account to give an annual rate of return for each investment.

For Ralph’s investment in AlphaCorp the log return is 9.12%. For Jackie’s investment in BetaCorp the log return is 9.53%. Both of these are annual rates, so they can be directly compared.

So who made the better investment?

Based on the logarithmic return, Jackie’s investment was slightly better than Ralph’s. However, there is a slight caveat to mention, which is that Jackie sold her investment after 1 year, while Ralph held his for 2 years. If she really wants to do better than Ralph over 2 years, she will need to find another investment to make for the second year that does at least as well as Ralph’s 9.12%.

Both the Arithmetic Return and the Log Return are useful ways to compare the return on investments. The log return is normally the best choice for investments that were held for different lengths of time because it gives you an annual rate that you can compare between investments.

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