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A hypothetical investment in a single stock initially costs €100. One year later, the stock is trading at €200. At the end of the second year, the stock price falls back to the original purchase price of €100. No dividends are paid during the two-year period. Calculate the arithmetic and geometric mean annual returns.

First, we need to find the Year 1 and Year 2 annual returns with Equation 1.

Return in Year 1 = 200/100 – 1 = 100%

Return in Year 2 = 100/200 – 1 = –50%

The arithmetic mean of the annual returns is (100% − 50%)/2 = 25%.

Before we find the geometric mean, we must convert the percentage rates of return to (1 + *R _{t}*). After this adjustment, the geometric mean from Equation 6 is √2.0×0.502.0×0.50 – 1 = 0 percent.

The geometric mean return of 0 percent accurately reflects that the ending value of the investment in Year 2 equals the starting value in Year 1. The compound rate of return on the investment is 0 percent. The arithmetic mean return reflects the average of the one-year returns.

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