AIM 1: Define, calculate and interpret the expected value.

1、An investor is considering purchasing ACQ. There is a 30% probability that ACQ will be acquired in the next two months. If ACQ is acquired, there is a 40% probability of earning a 30% return on the investment and a 60% probability of earning 25%. If ACQ is not acquired, the expected return is 12%. What is the expected return on this investment?

A) 18.3%.

B) 16.5%.

C) 12.3%.

D) 17.4%.

The correct answer is B

E(r) = (0.70 × 0.12) + (0.30 × 0.40 × 0.30) + (0.30 × 0.60 × 0.25) = 0.165.

2、An analyst has knowledge of the beginning-of-period expected returns, standard deviations of return, and market value weights for the assets that comprise a portfolio. The analyst does not require the covariances of returns between asset pairs to calculate the:

A) variance of the return on the portfolio.

B) correlations between asset pairs.

C) expected return on the portfolio.

D) reduction in risk due to diversification.

The correct answer is C

All that is required to calculate the expected return for the portfolio is the portfolio weights and individual asset expected returns. All other items are functions of the covariance.

AIM 2: List and discuss the properties of expected value.

1、There is a 30% chance that the economy will be good and a 70% chance that it will be bad. If the economy is good, your returns will be 20% and if the economy is bad, your returns will be 10%. What is your expected return?

A) 15%.

B) 17%.

C) 13%.

D) 18%.

The correct answer is

Expected value is the probability weighted average of the possible outcomes of the random variable. The expected return is: ((0.3) × (0.2)) + ((0.7) × (0.1)) = (0.06) + (0.07) = 0.13.

2、Tully Advisers, Inc., has determined four possible economic scenarios and has projected the portfolio returns for two portfolios for their client under each scenario. Tully’s economist has estimated the probability of each scenario as shown in the table below. Given this information, what is the expected return on portfolio A?

A)    9.25%.

B)    11.55%.

C)   10.75%.

D)   12.95%.

The correct answer is B

The expected return is equal to the sum of the products of the probabilities of the scenarios and their respective returns: = (0.15)(0.17) + (0.20)(0.14) + (0.25)(0.12) + (0.40)(0.08) = 0.1155 or 11.55%.

3、

The expected value of X is closest to:

A)    1.6

B)    1.5

C)   1.2

D)   1.8

The correct answer is A

p(X = 1) = 0.05 + 0.15 + 0.20 = 0.40, and p(X = 2) = 0.15 + 0.15 + 0.30 = 0.60, so E(X) = 0.40(1) + 0.60(2) = 1.6.

If you know that X is equal to 1, the probability that Y is equal to 2 is closest to:

A) 0.50

B) 0.15

C) 0.38

D) 0.30

The correct answer is C

p(Y = 2|X = 1) = 0.15/(0.05 + 0.15 + 0.20) = 0.375.

The variance of Y is closest to:

A) 2.27

B) 0.61

C) 1.51

D) 0.76

The correct answer is D

p(Y = 1) = 0.05 + 0.15 = 0.20, p(Y = 2) = 0.15 + 0.15 = 0.30, and p(Y = 3) = 0.20 + 0.30 = 0.50. Thus, the mean of Y is equal to μ = 0.20(1) + 0.30(2) + 0.50(3) = 2.3, and the variance is calculated as 0.20(1 – 2.3)2 + 0.30(2 – 2.3)2 + 0.50(3 – 2.3)2 = 0.61.

4、The characteristic function of the product of independent random variables is equal to the:

A) square root of the product of the individual characteristic functions.

B) exponential root of the product of the individual characteristic functions.

C) product of the individual characteristic functions.

D) sum of the individual characteristic functions.

The correct answer is C

The characteristic function of the sum of independent random variables is equal to the product of the individual characteristic functions. E(XY) = E(X) × E(Y).

AIM 5: List and discuss the properties of variance.

1、Use the following probability distribution to calculate the standard deviation for the portfolio.

A)  6.0%.

B)  6.5%.

C)  7.0%.

D)  5.5%.

The correct answer is D

[0.30 × (0.15 ? 0.066)2 + 0.70 × (0.03 ? 0.066)2]1/2 = 5.5%.

2、Use the following joint probability distribution to answer the questions below.

	Y=1	Y=2	Y=3
X=1	0.05	0.05	0.10
X=2	0.05	0.10	0.15
X=3	0.15	0.15	0.20

The expected value of Y is closest to:

A) 2.2.

B) 0.2.

C) 1.0.

D) 2.3.

The correct answer is A

p(Y=1)=0.05+0.05+0.15=0.25, p(Y=2)=0.05+0.10+0.15=0.30, and p(Y=3)=0.10+0.15+0.20=0.45, so E(Y)=0.25(1)+0.30(2)+0.45(3)=2.20.

A) 0.05.

B) 0.25.

C) 0.17.

D) 0.20.

The correct answer is C

p(X=1|Y=2)=0.05/(0.05+0.10+0.15)=0.17.

A) 0.54.

B) 0.67.

C) 0.61.

D) 0.74.

The correct answer is C

p(X=1)=0.05+0.05+0.10=0.20, p(X=2)=0.05+0.10+0.15=0.30, and p(X=3)=0.15+0.15+0.20=.50. Thus, the mean of X is equal to μ=0.20(1)+0.30(2)+0.50(3)=2.3, and the variance is calculated as 0.20(1-2.3)2+0.30(2-2.3)2+0.50(3-2.3)2=0.61.

3、The variance of the sum of two independent random variables is equal to the sum of their variances:

A) plus a positive covariance term.

B) plus zero.

C) minus a positive covariance term.

D) plus a non-zero covariance term.

The correct answer is B

Independent random variables have a covariance of zero. Hence, the variance of the sum of two variables will be the sum of the variables’ variances.