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You are considering the risk-return profile of two mutual funds for investment. The relatively risky fund promises an expected return of 7% with a standard deviation of of 13%. The relatively less risky fund promises an expected return and standard deviation of 4.5% and 5.2%, respectively. Assume that the returns are approximately normally distributed.

a. Which mutual fund will you pick if your objective is to minimize the probability of earning a negative return?

b. Which mutual fund will you pick if your objective is to maximize the probability of earning a return above 9%?

Respuesta :

Answer:

a) The relatively less risky fund should be chosen.

b) The relatively risky fund should be chosen.

Step-by-step explanation:

Problems of normally distributed samples are solved using 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.

a. Which mutual fund will you pick if your objective is to minimize the probability of earning a negative return?

We should pick whichever fund has the lower probability of earning a negative return.

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

Relatively risky:

[tex]\mu = 7, \sigma = 13[/tex]

Then

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{0 - 7}{13}[/tex]

[tex]Z = -0.54[/tex]

[tex]Z = -0.54[/tex] has a pvalue of 0.2946.

29.46% probability of earning a negative return.

Less risky:

[tex]\mu = 4.5, \sigma = 5.2[/tex]

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{0 - 4.5}{5.2}[/tex]

[tex]Z = -0.87[/tex]

[tex]Z = -0.87[/tex] has a pvalue of 0.1922

19.22% probability of earning a negative return.

The relatively less risky fund should be chosen.

b. Which mutual fund will you pick if your objective is to maximize the probability of earning a return above 9%?

We should pick whichever fund has a higher probability of earning a return above 9%.

This probability is 1 subtracted by the pvalue of Z when X = 9.

Relatively risky:

[tex]\mu = 7, \sigma = 13[/tex]

Then

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{9 - 7}{13}[/tex]

[tex]Z = 0.15[/tex]

[tex]Z = 0.15[/tex] has a pvalue of 0.5596

1 - 0.5596 = 0.4404

44.04% probability of earning a return above 9%.

Less risky:

[tex]\mu = 4.5, \sigma = 5.2[/tex]

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{9 - 4.5}{5.2}[/tex]

[tex]Z = 0.87[/tex]

[tex]Z = 0.87[/tex] has a pvalue of 0.8078

1 - 0.8078 = 0.1922

19.22% probability of earning a return above 9%.

The riskier fund has the higher probability, so it should be chosen.