Question

Portfolio returns. 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 11.1% (i.e. an average gain of 11.1%) with a standard deviation of 40%. 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 all answers to 4 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 13% of annual returns with this portfolio?

Answer 1

The percent of years does the portfolio lose money. That is, find the probability P(X<0)

Let X be the random variable defined by returns on a portfolio follows normal distribution with mean(μ) 11.1 % and standard deviation(σ) 40%.

The probability P(X<0) is,

P(X<0)=P(x-11.1/40<0-11.1/40)

=P(z<-0.2775)

=0.3907 ============>> Use ''=NORMSDIST(-0.2775)'' in Excl.

The percent of the year’s portfolio lose money, that is have a return less than 0% is 39.07%

b.

The cutoff for the highest 13% of annual returns with this portfolio is obtained below:

P(X>x)=0.13

P(X<x)=1-0.13

=0.87

From the “standard Normal table”, the area covered for value of 0.87 is obtained at . z=1.1264

The cutoff for the highest 13% of annual returns with this portfolio is,

Z=(x-mean)/standard deviation.

1.1264=(x-11.1)/40

x=11.1+1.1264*40

=56.16

The cutoff for the highest 13% of annual returns with this portfolio is 56.16%