Business Statistics problem on my homework

Question

A young investment manager tells his client that the probability of making a positive return with his suggested portfolio is 84%. What is the risk (standard deviation) that this investment manager has assumed in his calculation if it is known that returns are normally distributed with a mean of 4.1%?

Standard deviation? I tried looking on the Z table for .84 and I plugged in the corresponding number, but I didn't get the right answer. Isn't 84% and 4.1% both means? so how do I find the standard deviation with two means?

Answer 1

0.84 is not a mean. $\mu = 4.1$ and you want to find out $\sigma$ from knowing the fact that $P(X \geq 0) = 0.84.$

The calculation that you need to do is,

$P(X \geq 0) = P(\frac{X - \mu}{\sigma} \geq \frac{0-\mu}{\sigma}) = P(Z \geq \frac{-4.1}{\sigma}) = 0.84 $

That is, find the z value from the table$(P(Z \geq z) = 0.84)$ and then solve for $\sigma$ from $\frac{-4.1}{\sigma} = z.$

Answer 2

$.84 \ne \mu$; in fact, they are very different units. $\mu = 4.1\%$ is the average return on a portfolio (not a probability), whereas $P(X \geq 0\%) =.84$ is a probability.

Note that the choice of $.84$ may not be arbitrary -- for the normal distribution, $.841$ is the culmulative probability from the left tail to the $+1\sigma$, or equivalently, from the right tail to $-1\sigma$. That should give you some insight to the z-score. (Recall the formula for standardizing a normally distributed variable: $z=(x-\mu)/\sigma$.)