Question

Diversified Investment annual return follows a normal distribution. If 99% of the distribution lies between 2.5% and 10.3%, determine the expected mean return if the standard deviation is 1.3%

A) 8.2% B) 5.1% C)12.8% D) 6.9

I do not understand the concepts to this question, can someone please explain exactly what this question is asking and what the steps are to answering it?

Answer 1

Normal distribution is a type of continuous probability distribution for a real-valued random variable.

In simple words, Normal distribution is a type of set of possible outcome values of a variable whose graph looks like a bell curve shape as shown below:

99.7% 95% - - 68% 4- 32- 10 1 2

This figure implies that if a variable follows a normal distribution, then 68% of the time its value is likely to be within +-1 standard deviation from the mean. Similarly, for 99.7% of the trials (i.e. with 99% confidence) its value will be within the range of -3* SD to +3* SD around the mean (total range 6* Standard deviation)

So in the given question, 99% of the time the return value is falling within 2.5% & 10.3%

So, mean + 3 * 1.3% should be approximately equal to 10.3%

and mean - 3 * 1.3% should be approximately equal to 2.5%

thus mean = 10.3% - 3*1.3% = 6.4%

2.5% + 3*1.3% = 6.4%

The closest to this in the options is 6.9% So the correct answer is option D)

Note that we used the approximate method here using the common confidence interval of 99.7% confidence level. If you want to calculate exactly you can look up the readymade normal distribution tables to find the exact confidence interval (range of standard deviations from the mean) corresponding to 99% probability and then use the same approach as above.