# Advanced Statistics Problems

1. Suppose you have invested only in two stocks, A and B. The returns on the two stocks depend on the following three states of the economy, which are equally likely to happen:

State of Return on Return on

Economy Stock A (%) Stock B (%)

Bear 6.30 _ 3.70

Normal 10.50 6.40

Bull 15.60 25.30

a. Calculate the expected return on each stock.

b. Calculate the standard deviation of returns on each stock.

c. Calculate the covariance and correlation between the returns on the two stocks.

2. Security F has an expected return of 12 percent and a standard deviation of 9 percent per year. Security G has an expected return of 18 percent and a standard deviation of 25 percent per year.

a. What is the expected return on a portfolio composed of 30 percent of security F and

70 percent of security G?

b. If the correlation between the returns of security F and security G is 0.2, what is the standard deviation of the portfolio described in part (a)?

3. There are three securities in the market. The following chart shows their possible payoffs.

Probability Return on Return on Return on

State of Outcome Security 1 (%) Security 2 (%) Security 3 (%)

1 0.1 0.25 0.25 0.10

2 0.4 0.20 0.15 0.15

3 0.4 0.15 0.20 0.20

4 0.1 0.10 0.10 0.25

a. What is the expected return and standard deviation of each security?

b. What are the covariances and correlations between the pairs of securities?

c. What is the expected return and standard deviation of a portfolio with half of its funds invested in security 1 and half in security 2?

d. What is the expected return and standard deviation of a portfolio with half of its funds?

Invested in security 1 and half in security 3?

e. What is the expected return and standard deviation of a portfolio with half of its funds invested in security 2 and half in security 3?

f. What do your answers in parts (a), (c), (d), and (e) imply about diversification?

4. Assume there are N securities in the market. The expected return on every security is

10 percent. All securities also have the same variance of 0.0144.The covariance between any pair of securities is 0.0064.

a. What is the expected return and variance of an equally weighted portfolio containing all N securities? Note: the weight of each security in the portfolio is 1/N.

b. What will happen to the variance of the portfolio as N approaches infinity?

c. What characteristics of a security are most important in the determination of the variance of a well-diversified portfolio?

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1. Suppose you have invested only in two stocks, A and B. The returns on the two stocks depend on the following three states of the economy, which are equally likely to happen:

State of Return on Return on

Economy Stock A (%) Stock B (%)

Bear 6.30 _ 3.70

Normal 10.50 6.40

Bull 15.60 25.30

a. Calculate the expected return on each stock.

A: The expected return is: (1/3) * (6.3 + 10.50 + 15.60) = 10.8% for Stock A

(1/3) * (-3.7 + 6.4 + 25.30) ≈ 9.33% for Stock B

We divide by three because each state of the economy is equally likely.

b. Calculate the standard deviation of returns on each stock.

The standard deviation is simply the square root of the variance.

The variance for Stock A is: [(6.30 - 10.8)2 + (10.50 - 10.8)2 + (15.60 - 10.8)2] / 3 = 21.69

The variance for Stock B is: [(-3.70 - 9.33)2 + (6.40 - 9.33)2 + (25.30 - 9.33)2] / 3 ≈ 216.70

Thus the standard deviation for Stock A is SD(A) = 4.66 and for stock B is SD(B) = 14.72.

c. Calculate the covariance and correlation between the returns on the two stocks.

One way to find the covariance is by the following formula:

Cov (A, B) = E(AB) - E(A)*E(B)

where A is Stock A, B is Stock B, and E stands for the expectation or mean (see part a for an example). So first we have to find E(AB):

State of Return on Return on A*B

Economy Stock A (%) Stock B (%)

(A) (B)

Bear 6.30 _ 3.70 -23.31

Normal 10.50 6.40 67.2

Bull 15.60 25.30 394.68

So E(AB) = (1/3) * (-23.31 + 67.2 + 394.68) = 146.19

As a result the covariance is: 146.19 - 10.8 * 9.33 = 45.43

The correlation is simply cov(A,B) / [SD(A)*SD(B)] = 45.43 / (4.66 * 14.72) = 0.66.

Remember that a correlation of zero means the two stocks are not correlated, while a correlation of 1 means that they are perfectly correlated. A correlation of -1 means that they are perfectly inversely related (as one stock moves in one direction, the other moves in the exact opposite direction). Correlation is always between -1 and 1.

2. Security F has an expected return of 12 percent and ...

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