Returns and the Bell Curve An investment has an expected return of 11 percent per year with a standard deviation of 26 percent. Assuming that the returns on this investment are at least roughly normally distributed, how often do you expect to earn less than -15 percent?

And we can find this probability using the normal standard distribution table or excel and we got:

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the expected return, and for this case we know the distribution for X is given by:

Where and

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability using the normal standard distribution table or excel and we got:

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