Niles is making an investment with an expected return of 12 percent. If the standard deviation of the return is 4.5 percent, and if Niles is investing $100,000, then what dollar amount is Niles 90 percent sure that he will have at the end of the year?
A) $100,000.00
B) $104,597.50
C) $116,500.00
D) $119,402.50
To calculate the dollar amount that Niles is 90 percent sure he will have at the end of the year, we need to use the concept of confidence intervals.
The expected return of 12 percent is the average return that Niles can expect to earn on his investment. The standard deviation of 4.5 percent measures the amount of variability or risk associated with the investment.
To calculate the confidence interval, we multiply the standard deviation by the appropriate number of standard deviations for a 90 percent confidence level. For a normal distribution, this value is approximately 1.645.
First, let's calculate the amount of return Niles can expect to earn:
Expected return = 12% of $100,000 = $12,000
Next, let's calculate the amount of variability or risk associated with the investment:
Standard deviation = 4.5% of $100,000 = $4,500
Now, let's calculate the margin of error:
Margin of error = 1.645 * $4,500 = $7,392.25
To find the lower bound of the confidence interval, we subtract the margin of error from the expected return:
Lower bound = $12,000 - $7,392.25 = $4,607.75
To find the upper bound of the confidence interval, we add the margin of error to the expected return:
Upper bound = $12,000 + $7,392.25 = $19,392.25
Therefore, Niles can be 90 percent sure that he will have between $4,607.75 and $19,392.25 at the end of the year.
Based on the given options, the closest dollar amount to the upper bound of the confidence interval is $19,402.50. Therefore, the correct answer is D) $119,402.50.