Question

The historical returns on a portfolio had an average return of 19 percent and a standard deviation of 25 percent. Assume that returns on this portfolio follow a bell-shaped distribution.a. Approximately what percentage of returns were greater than 69 percent?b. Approximately what percentage of returns were below –56 percent?

Answer 1

Answer:

a) $P(X>69)=P(\frac{X-\mu}{\sigma}>\frac{69-\mu}{\sigma})=P(Z>\frac{69-19}{25})=P(Z>2)$

And we can find this probability using the complement rule and the z table or excel:

$P(Z>2)=1-P(Z$

b) $P(X$

And we can find this probability using the z table or excel:

$P(Z$

Step-by-step 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".  

Part a

Let X the random variable that represent the historical returns of a population, and for this case we know the distribution for X is given by:

$X \sim N(19,25)$  

Where $\mu=19$ and $\sigma=25$

We are interested on this probability

$P(X>69)$

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

$z=\frac{x-\mu}{\sigma}$

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

$P(X>69)=P(\frac{X-\mu}{\sigma}>\frac{69-\mu}{\sigma})=P(Z>\frac{69-19}{25})=P(Z>2)$

And we can find this probability using the complement rule and the z table or excel:

$P(Z>2)=1-P(Z$

Part b

We are interested on this probability

$P(X$

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

$z=\frac{x-\mu}{\sigma}$

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

$P(X$

And we can find this probability using the z table or excel:

$P(Z$