Question

The historical returns on a balanced portfolio have had an average return of 12% and a standard deviation of 20%. Assume that returns on this portfolio follow a normal distribution. [Use Excel commands instead of the z table.]

a. What percentage of returns were greater than 52%? (Round your answer to 2 decimal places.)

b. What percentage of returns were below −48%? (Round your answer to 2 decimal places.)

Answer 1

X:Return on a balanced portfolio

X follows a Normal distribution with mean 12% and standard deviation of 20%

a. What percentage of returns were greater than 52%

Probability that returns were greater than 52% = P(X>52)

P(X>52) = 1 - P(X52)

Excel function:

NORM.DIST function

Returns the normal distribution for the specified mean and standard deviation. This function has a very wide range of applications in statistics, including hypothesis testing.

Syntax

NORM.DIST(x,mean,standard_dev,cumulative)

The NORM.DIST function syntax has the following arguments:

• X     Required. The value for which you want the distribution.

• Mean     Required. The arithmetic mean of the distribution.

• Standard_dev     Required. The standard deviation of the distribution.

• Cumulative     Required. A logical value that determines the form of the function. If cumulative is TRUE, NORM.DIST returns the cumulative distribution function; if FALSE, it returns the probability density function.

P(X52) = NORM.DIST(52,12,20,TRUE) = 0.9772

P(X>52) = 1 - P(X52) = 1 -0.9772=0.0228

Percentage of returns were greater than 52% =100 x Probability that returns were greater than 52% = 100 x 0.0228 = 2.28%

b. What percentage of returns were below −48%?

Probability that returns were below -48% = P(X<-48%)

P(X<-48%) =NORM.DIST(-48,12,20,TRUE) = 0.0013

percentage of returns were below −48% = 100 x Probability that returns were below -48% =100 x 0.0013 =0.13%

percentage of returns were below −48% = 0.13%