Question

Suppose that an investor opens an account by investing $1,000. At the beginning of each of the next four years, he deposits an additional $1,000 each year, and he then liquidates the account at the end of the total five-year period. Suppose that the yearly returns in this account, beginning in year 1, are as follows: −9 percent, 17 percent, 9 percent, 14 percent, and −4 percent. Calculate the arithmetic and geometric average returns for this investment.

Answer 1

The arithmetic average return of the investment is calculated by summing up the yearly returns and dividing by the number of years, resulting in 5.4%. The geometric average return is found by multiplying all terms presented as a factor of (1 + return), taking the fifth root, and then subtracting 1, which accounts for the effect of compounding.

Step-by-step explanation:

To calculate the arithmetic average return for the investment mentioned in the question, you sum up all the yearly returns and divide by the number of years. The total sum of the yearly returns (-9% + 17% + 9% + 14% - 4%) is 27%. Dividing by 5 years, the arithmetic average return is 5.4%.

To find the geometric average return, you multiply all the returns presented as a factor (1 + return), take the fifth root (since there are five years), and subtract 1. This will be calculated as follows: [(1 - 0.09) * (1 + 0.17) * (1 + 0.09) * (1 + 0.14) * (1 - 0.04)]^(1/5) - 1. After performing the calculations, you find the geometric average return.

The arithmetic average return gives a simple average without considering the effects of compounding, while the geometric average return accounts for the compounding effect over the periods.

Answer 2

The arithmetic average returns for this investment= 5.4%

The geometric average returns for this investment= 4.89%

Step-by-step explanation:

Arithmetic average return is simple the addition of all the returns by divided the years which is 5

=(-9)+17+9+14+(-4)/5 =5.4%.

while the geometric mean will require you to first add 1 to each return(this will help you deal with the negative numbers.

{(0.91)*1.17*1.09*1.14*(0.96)^(1/5)}-1=0.0489, to change it to percentage simply multiply it by 100 and the final answer will be 4.89%