Answer :
Answer:
a) This portfolio loses money in 29.81% of the years.
b) The cutoff for the highest 15% of annual returns with this portfolio is a return of 53.16%.
Step-by-step explanation:
When the distribution is normal, we use the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 question:
[tex]\mu = 17.9, \sigma = 34[/tex]
a.) What percent of years does this portfolio lose money, i.e. have a return less than 0%?
We have to find the pvalue of Z when X = 0. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{0 - 17.9}{34}[/tex]
[tex]Z = -0.53[/tex]
[tex]Z = -0.53[/tex] has a pvalue of 0.2981
This portfolio loses money in 29.81% of the years.
b.) What is the cutoff for the highest 15% of annual returns with this portfolio?
This is the 100 - 15 = 85th percentile, which is X when Z has a pvalue of 0.85. So X when Z = 1.037.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.037 = \frac{X - 17.9}{34}[/tex]
[tex]X - 17.9 = 1.037*34[/tex]
[tex]X = 53.16[/tex]
The cutoff for the highest 15% of annual returns with this portfolio is a return of 53.16%.