Use of arithmetic and logarithmic returns to compare investments

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Ralph and Jackie are two investors interested in the stock market. In this article I will explain how they can use two different methods to calculate a rate of return in order to compare the return on their investments.

Ralph buys $ 1,000 worth of shares from AlphaCorp. He keeps it for exactly 2 years and then sells it for $ 1200. Jackie buys $ 3,000 worth from BetaCorp. He holds his shares for a year and sells them for $ 3,300. Just to simplify the example, let’s assume that neither Ralph nor Jackie receive any dividend payments from their shares.

Ralph and Jackie now want to compare their investments. They know that there are two main methods they could use: arithmetic return and logarithmic return (often reduced to record return).

Ralph’s total profit is $ 200 and Jackie’s is $ 300. So this tells them that Jackie has made more money overall. But he also invested more. Investing more usually means you took a bigger risk (if stocks went down, you 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 arithmetic return provides 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 best investment, as his investment earned 20% compared to Jackie’s 10% investment.

But notice that Jackie sold his shares after a year, while Ralph kept his for 2 years. The arithmetic return does not include the duration of the investment, so these values ​​cannot be compared significantly. So now Ralph and Jackie compare their investments using the logarithmic return, which takes it into account to give an annual rate of return for each investment.

For Ralph’s investment in AlphaCorp, the return on record is 9.12%. For Jackie’s investment in BetaCorp, the return on record is 9.53%. Both are annual fees, so they can be compared directly.

So who made the best 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 a year, while Ralph kept her investment for 2 years. If you really want to do better than Ralph for two years, you will need to find another investment to make for the second year that makes at least 9.12% Ralph.

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