Question

"the Capital Asset Pricing Model is a financial model that assumes returns on a portfolio are normally distributed. Suppose a portfolio has an average annual return of 14.7% (i.e. an average gain of 14.7%) with a standard deviation of 33%. 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.

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"

Answer 1

a) In 32.82% this portfolio lose money, i.e. have a return less than 0%

b) The cutoff for the highest 15% of annual returns with this portfolio is an annual return of 48.86%.

Explanation:

Problems of normally distributed samples are 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 = 0.147 \sigma = 0.33`

a.) What percent of years does this portfolio lose money, i.e. have a return less than 0%

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

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

`Z = (0 - 0.147)/(0.33)`

`Z = -0.445`

`Z = -0.445` has a pvalue of 0.3282

In 32.82% 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"

This is X when Z has a pvalue of 1-0.15 = 0.85. So it is X when Z = 1.035.

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

`1.035 = (X -  0.147)/(0.33)`

`X - 0.147 = 0.33*1.035`

`X = 0.4886`

The cutoff for the highest 15% of annual returns with this portfolio is an annual return of 48.86%.