# Using arithmetic and logarithmic returns to compare investments

Ralph and Jackie are avid stock market investors. In this article, I will explain how they can use two different methods to calculate the rate of return to compare the performance of their investments.

Ralph buys \$ 1,000 worth of shares in AlphaCorp. He held it for exactly 2 years and then sold it for \$ 1,200. Jackie buys shares worth \$ 3,000 at BetaCorp. She keeps her stock for 1 year and sells it for \$ 3,300. To facilitate the example, we will assume that neither Ralph nor Jackie receive dividends from their shares.

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

Ralph’s total winnings are \$ 200 and Jackie’s \$ 300. This tells them that Jackie has made more money overall. But it is also investing more. Investing more often means that she has taken a greater risk (if stocks fall, she will 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 seems that Ralph made the better investment, as he earned 20% in value compared to 10% for Jackie’s investment.

But notice that Jackie sold his shares after 1 year, while Ralph held his for 2 years. The arithmetic return does not include the duration of the investment, so these values ​​really cannot be compared in terms of meaning. So now Ralph and Jackie are comparing their investments, using the logarithmic return, which takes this into account to give an annual rate of return for each investment.

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

So who made the better investment?

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