Investors remember 1987 as the year stocks lost 20% of their value in a single day. for 1987 as a whole, the mean return of all common stocks on the new york stock exchange was µ = -4.8%. (that is, these stocks lost an average of -4.8%. of their value in 1987.) the standard deviation of returns was about s = 25%.

(a) what are the mean and the standard deviation of the distribution of 7-stock portfolios in 1987.
µ =-4.8 %
s =9.45 %

(b) assuming that the population distribution of returns on individual common stocks is normal, what is the probability that a randomly chosen stock showed a return of at least 7% in 1987?

(c) assuming that the population distribution of returns on individual common stocks is normal, what is the probability that a randomly chosen portfolio of 3 stocks showed a return of at least 7% in 1987?

(d) what percentage of 3-stock portfolios lost money in 1987?

a)

b) 31.92% probability that a randomly chosen stock showed a return of at least 7% in 1987.

c) 20.61% probability that a randomly chosen portfolio of 3 stocks showed a return of at least 7% in 1987.

d) 62.93% of 3-stock portfolios lost money in 1987.

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a random variable X, with mean and standard deviation , a large sample size can be approximated to a normal distribution with mean and standard deviation .

Normal probability distribution:

Problems of normally distributed samples can be solved using 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 problem, we have that:

(a) What are the mean and the standard deviation of the distribution of 7-stock portfolios in 1987.

By the Central Limit Theorem, we have that:

(b) Assuming that the population distribution of returns on individual common stocks is normal, what is the probability that a randomly chosen stock showed a return of at least 7% in 1987?

This is 1 subtracted by the pvalue of Z when X = 7. So

has a pvalue of 0.6808.

So 1-0.6808 = 0.3192 = 31.92% probability that a randomly chosen stock showed a return of at least 7% in 1987.

(c) Assuming that the population distribution of returns on individual common stocks is normal, what is the probability that a randomly chosen portfolio of 3 stocks showed a return of at least 7% in 1987?

By the central Limit theorem, we use

This is 1 subtracted by the pvalue of Z when X = 7. So

has a pvalue of 0.7939.

So 1-0.7939 = 0.2061 = 20.61% probability that a randomly chosen portfolio of 3 stocks showed a return of at least 7% in 1987.

(d) What percentage of 3-stock portfolios lost money in 1987?

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

has a pvalue of 0.6293.

So 62.93% of 3-stock portfolios lost money in 1987.

step-by-step explanation:

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