The Capital Asset Pricing Model (CAPM) is a financial model that assumes returns on a portfolio are normally distributed. Suppose a portfolio has an average annual return of 17.9% (i. e., an average gain of 17.9%) with a standard deviation of 34%. A return of 0% means the value of the portfolio doesn't change, a negative return means that the portfolio loses money, and a positive return means that the portfolio gains money. (Round your answers to two decimal places.)a.) What percent of years does this portfolio lose money, i. e. have a return less than 0%? b.) What is the cutoff for the highest 15% of annual returns with this portfolio?

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 and standard deviation , the zscore of a measure X is given by:

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:

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

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.

The cutoff for the highest 15% of annual returns with this portfolio is a return of 53.16%.