Final answer:
The arithmetic return for the stock over the last six years is 7.83%, while the geometric return is -0.35%. The arithmetic return is simply the average of the returns, whereas the geometric return takes account of the compounding effects over time.
Step-by-step explanation:
To calculate the arithmetic return and geometric return for the stock that had returns of 27%, 12%, 32%, -12%, 19%, and -31% over the last six years, we must first understand what each term means. The arithmetic return is the simple average of returns over the period, while the geometric return is the average rate of return that accounts for the effect of compounding.
To calculate the arithmetic return, you add up the annual returns and then divide by the number of years:
(27 + 12 + 32 - 12 + 19 - 31) / 6 = 7.83%
For the geometric return, we need to multiply the returns together as if they compunded over time and then take the sixth root (since there are six years) and subtract one. However, because we are dealing with percentages, we first need to convert them into their decimal form before multiplication:
((1 + 0.27) × (1 + 0.12) × (1 + 0.32) × (1 - 0.12) × (1 + 0.19) × (1 - 0.31))1/6 - 1 = -0.0035 or -0.35%
These calculations show that while the arithmetic average might seem positive, the geometric return gives a more accurate picture of performance over time, especially when there are negative years. It's important to note that when a sequence of returns includes negative numbers, the geometric return can actually be negative even when the arithmetic return is positive, as in this case.