Question

An investment has an expected return of 12 percent per year with a standard deviation of 6 percent. Assuming that the returns on this investment are at least roughly normally distributed, what percentage of the time do you expect to lose money

Answer 1

You expect to lose money 2.28% of the time.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 12, \sigma = 6`

What percentage of the time do you expect to lose money?

This is the pvalue of Z when X = 0. So

`Z = (X - \mu)/(\sigma)`

`Z = (0 - 12)/(6)`

`Z = -2`

`Z = -2` has a pvalue of 0.0228.

So you expect to lose money 2.28% of the time.