Quiz 1

The Returns and Standard Deviations for stocks 1, 2 and 3 are the following --------------Stock 1 Stock 2 Stock 3 Mean Return (R) 10% 8% 12% Standard Deviation 2% 1% 3% The correlation coefficients for the returns between stocks 1, 2 and 3 are the following Correlations Stock 1 Stock 2 Stock 3 Stock 1 1 .75 .25 Stock 2 .75 1 0 Stock 3 .25 0 1 A portfolio is created where 1/3 of the portfolio weight is in Stock 1, 1/3 is in Stock 2 and 1/3 is in Stock 3. The mean return of the portfolio is:

10% Rp = X1 R1 + X2 R2 + X3 R3 = .1/3 (.10) + .1/3 (.08) + 1/3 (.12) = 10%

Assume that market portfolio M has an expected annual return of 15% and standard deviation of 5%. An individual then leverages this market portfolio M by borrowing an amount equal to his/her own equity, at an interest rate of 4%, thereby investing twice as much as his/her equity in market portfolio M. The standard deviation for this leveraged investment is: a. 5% b. 10% c. 15% d. 6%

10% σp = wM σM = 2.00 (5%) = 10% What is shown below is not necessary for the answer, but if this portfolio was not leveraged, nor loaned, then Rp = wrf Rrf + wM RM = 0 (4%) + 1.00 (15%) = 15% σp = wM σM = 1.00 (5%) = 5%

Stocks X and Y have the following expected returns and standard deviations of their return distributions. Assume that a portfolio is constructed consisting of 25% weight in Stock X and 75% weight in Stock Y. Stock X Stock Y Expected return (mean) 8% 12% Standard deviation 3% 5% In addition, the correlation between these two stock's returns distributions is -0.20. The standard deviation for this portfolio is: a. 3.67% b. 1.35% c. 4% d. 4.50%

3.67% σP = [ X2X σ2X +X2Y σ2Y + 2 XX XY σX σY ρXY ]0.50 = [ 0.252 (.03)2 +0.752 (.05)2 + (2) (0.25) (0.75) (.03) (.05) (-0.20) ]0.50 = [ 0.0625 (.0009) +0.5625 (.0025) + (2) (0.25) (0.75) (.03) (.05) (-0.20) ]0.50 = [ .00005625 + .00140625 - .0001125 ] 0.50 = [0.00135 ] 0.50 = 0.036742346 = 3.6742346%

Two companies Amber and Bolt are manufacturers of glass. The securities of the companies are listed and traded in the New York Stock Exchange. An investor's portfolio consists of these two securities in the proportion of 5/6 and 1/6 respectively. Amber's security has an expected return of 20% and a standard deviation of 8%. Bolt has an expected return of 15% and a standard deviation of 5%. The correlation coefficient between the two securities is 0.6. Calculate the expected return and the standard deviation of the investor's portfolio. a. Expected Return = 19.17%; Standard Deviation = 7.20% b. Expected Return = 20.19%; Standard Deviation = 8.20% c. Expected Return = 17%; Standard Deviation = 7.0% d. Expected Return = 18.17%; Standard Deviation = 8.0%

Expected Return = 19.17%; Standard Deviation = 7.20%

A fully invested portfolio consisting of stocks A and B exists, where XA is the weigh in Stock A. The symbol sp is the risk (standard deviation) of the portfolio, sA the risk (standard deviation) of stock A, and sB the risk (standard deviation) of stock B. When the correlation coefficient between the return distributions of stock A and stock B is zero, i.e. rA,B is -1.0, then sp is always zero. a. True b. False c. Uncertain

False

The following diagram shows the investment opportunity set for portfolios containing stocks A and B. You need to know that: -Point A on the graph represents a portfolio with 100% in stock A -Point B represents a portfolio with 100% in stock B *A C graph starting with A at the tip then going Z, Y, X (at center of line), W, B Under the assumptions in part (e), would it be wise for an investor to invest all of his or her money in stock A? Why or why not?

No. When the investor has the ability to borrow or lend at the risk-free rate, only the portfolios on the tangency line are efficient. Note in the graph above that by borrowing at the risk-free rate and investing everything in the optimal risky portfolio (y, in this case), the investor can create portfolios that that dominate A.

Assume the following regarding the expected return and standard deviation of stocks A and B. Expected Return Standard Deviation Stock A 15% 5% Stock B 10% 3% Show in equation form the RETURN AS A FUNCTION OF RISK for a portfolio fully invested in stocks A and B if the correlation coefficient between A and B,rA,B, is +1.0 (plus one), the weight in Stock A equal XA and the weight in Stock B to equal (1 - XA).(12 points)

Rp = (XA) (RA) + (XB) (RB) = (XA) (RA) + (1 -XA) (RB) = (XA) (15%) + (1 -XA) (10%) = .15 XA + .10 - .10 XA Rp = .10 + .05 XA (1) sp = [ (XA)2 (sA)2 + (XB)2 (sB)2 + 2 (XA) (XB) (sA) (sB) (rAB) ]0.5 sp = [ (XA)2 (sA)2 + (1 - XA)2 (sB)2 + 2 (XA) (1 - XA) (sA) (sB) (rAB) ]0.5 and since rA,B is +1.0 sp = [ (XA)2 (sA)2 + (1 - XA)2 (sB)2 + 2 (XA) (1 - XA) (sA) (sB) ]0.5 by factoring obtain sp = [ (XA) (sA) + (1 - XA) (sB) ] sp = [ (XA) (.05) + (1 - XA) (.03) ] sp = [ (XA) (.05) + (.03 - .03 XA) ] sp = [ .05 XA + .03 - .03 XA ] sp = [ .02 XA + .03 ] Now, solve for XA, and obtain, XA = (sp - .03) / .02 (2) Thus, since equation (1) is Rp = .10 + .05 XA And equation (2) is XA = (sp - .03) / .02 Substitute (2) into (1), and obtain the relationship between return and risk, Rp = .10 + .05 [(sp - .03) / .02] Rp = .10 + .05 sp / .02 - .05 (.03) / .02 Rp = .10 + 2.5 sp - .075 Rp = .025 + 2.5 sp

Assume the following information for Stocks A and B. -------------Mean Return (R) 14% 10% Standard Deviation 4% 3% If a fully invested portfolio is formed with Stocks A and B where the weight in Stock A is 60%, i.e. XA is 60%, and the correlation between stocks A and B is + 1.0, then the risk (standard deviation) of this portfolio is: a. between 0% and 1% b. between 1% and 2% c. between 2% and 3% d. between 3% and 4%

between 3% and 4% sp = (XA) (sA) + (1 - XA) (sB) = .60 (4%) + (1-.60) (3%) = 2.4% + 1.2% = 3.6%

Stocks A and B have the following expected returns and standard deviations of their return distributions. -------------------------Stock A Stock B Expected return (mean) 6% 12% Standard deviation 1% 2% If the covariance between the returns of stocks A and B (ρA,B) is 0.0001, then the correlation coefficient between the returns for stocks A and B is: a. This depends on the weights for stocks A and B in the portfolio b. +1.0 c. +0.50 d. 0.0

+0.50 rAB = σA,B / sA sB = σA,B / (0.01) (0.02) = 0.0001 / 0.0002 = + 0.50

Stocks A and B have the following distribution of expected returns for years 2013, 2014 and 2015. (Note that the standard deviation information provided below is not adjusted for degrees of freedom.) Year Stock A Return Stock B Return 2013 4% 22% 2014 8% 14% 2015 12% 6% Expected return 8% 14% Standard deviation 3.266% 6.532% The covariance between the return distributions for Stock A and Stock B is (do not adjust for degrees of freedom): a. -0.006 b. -0.002 c. -0.004 d. zero

-0.002 The covariance between the return distributions of Stock A and B is σAB = -0.00213, that is σAB = (.04 - .08) (.22 - .14) + (.08 - .08)(.14 - .14) + (.12 - .08)(.06 - .14) = -0.00213

Stock A has the following distribution of expected returns for selected market conditions. Market Condition Return Probability Good 4% 0.25 Average 6% 0.50 Poor 8% 0.2 Expected return ? Standard deviation ? Stock A's standard deviation is: a. 0.02% b. 2% c. 1.41% d. 0.66

1.41% [.25 (.04 - .06)2 + .50 (.06 - .06)2 + .25 (.08 - .06)2] 0.50 = [.0002] 0.50 = .014142 = 1.4142%

Assume that market portfolio M has an expected annual return of 15% and standard deviation of 5%. An individual then leverages this market portfolio M by borrowing an amount equal to his/her own equity, at an interest rate of 4%, thereby investing twice as much as his/her equity in market portfolio M. a. The expected return for this leveraged investment is: a. 26% b. 15% c. 11% d. 30%

26% Rp = wrf Rrf + wM RM = -1.00 (4%) + 2.00 (15%) = 26%

In the graph shown in question 16 - reproduced below - consider adding a risk-free security earning a risk-free return that is then combined with an efficient portfolio and its return. Show the risk-return relationship in this case, and label that relationship (what is the resulting line called?) *A graph looking like a C with Stock A at the top and Stock B at the bottom*

Capital market line *Graph with the same C with a upward slope line connecting at the back point of the C*

A fully invested portfolio consisting of stocks A and B exists, where XA is the weigh in Stock A. The symbol sp is the risk (standard deviation) of the portfolio, sA the risk (standard deviation) of stock A, and sB the risk (standard deviation) of stock B. When the correlation coefficient between the return distributions of stock A and stock B is perfectly negative, i.e. rA,B is -1.0, then sp = [ - (XA) (sA) - (1 - XA) (sB) ]. a. True b. False c. Uncertain

False

A fully invested portfolio consisting of stocks A and B exists, where XA is the weigh in Stock A. The symbol sp is the risk (standard deviation) of the portfolio, sA the risk (standard deviation) of stock A, and sB the risk (standard deviation) of stock B. When the correlation coefficient between the returns of stock A and stock B is perfectly positive, i.e. rA,B is +1.0, then sp = [ (XA) (sA) + (1 - XA) (sB) ]^2 a. True b. False c. Uncertain

False

Assume a portfolio exists consisting of three stocks, namely stocks 1, 2 and 3, with the weights of each stock in the portfolio designated as X1, X2 and X3, respectively. The risk (standard deviation) of the portfolio (σp), using the conventional designations for standard deviation and correlations, can be generically written as follows: σp = [X1 σ1 + X2 σ2 + X3 σ3 + 2 X1 X2 σ1 σ2 ρ12+ 2 X1 X3 σ1 σ3 ρ13 + 2 X2 X3 σ2 σ3 ρ23] a. True b. False c. Uncertain

False The risk (standard deviation) of the portfolio (σp) is: σp = [X21 σ21 +X22 σ22 +X23 σ23 +2 X1 X2 σ1 σ2 ρ12 +2 X1 X3 σ1 σ3 ρ13 +2 X2 X3 σ2 σ3 ρ23]0.5

The following diagram shows the investment opportunity set for portfolios containing stocks A and B. You need to know that: -Point A on the graph represents a portfolio with 100% in stock A -Point B represents a portfolio with 100% in stock B *A C graph starting with A at the tip then going Z, Y, X (at center of line), W, B Is the correlation between A and B greater than, equal to, or less than 1. How do you know?

Less than 1. Correlation can't be greater than 1, and if correlation equaled 1 (meaning that A and B were perfectly positively correlated), then the IOS between A and B would be a straight line.

Consider the following graph depicting the return and risk for a portfolio *A graph looking like a C with Stock A at the top and Stock B at the bottom* What is point "x" called? If all possible investments in a portfolio are considered, then what is the line from point "x" to point "y" called,?

Minimum risk portfolio Efficient frontier

Which of the following is correct of how the returns on assets move together? a. Positive and negative deviations between assets at similar times give a negative covariance. b. Positive and negative deviations between assets at dissimilar times give a negative covariance. c. Positive and negative deviations between assets give a zero covariance. d. Positive and negative deviations between assets at dissimilar times give a positive covariance.

Positive and negative deviations between assets at dissimilar times give a negative covariance.

The following diagram shows the investment opportunity set for portfolios containing stocks A and B. You need to know that: -Point A on the graph represents a portfolio with 100% in stock A -Point B represents a portfolio with 100% in stock B *A C graph starting with A at the tip then going Z, Y, X (at center of line), W, B Which labeled point on the graph represents the minimum variance portfolio?

X

The following diagram shows the investment opportunity set for portfolios containing stocks A and B. You need to know that: -Point A on the graph represents a portfolio with 100% in stock A -Point B represents a portfolio with 100% in stock B *A C graph starting with A at the tip then going Z, Y, X (at center of line), W, B If A and B are the only investments available to an investor, which of the labeled portfolios are efficient?

a. x, y, z, and A

The following diagram shows the investment opportunity set for portfolios containing stocks A and B. You need to know that: -Point A on the graph represents a portfolio with 100% in stock A -Point B represents a portfolio with 100% in stock B *A C graph starting with A at the tip then going Z, Y, X (at center of line), W, B Suppose a risk-free asset exists, allowing an investor to invest or borrow at the risk-free rate of 3%. If the above graph is drawn perfectly to scale, which labeled point represents the optimal risky portfolio.

a. y. Note on the graph that the tangency line from the risk-free asset intercepts the IOS at y.

Assume the following information for Stocks A and B. -------------Mean Return (R) 14% 10% Standard Deviation 4% 3% If a fully invested portfolio is formed with Stocks A and B where the weight in Stock A is 60%, i.e. XA is 60%, and the correlation between stocks A and B is - 1.0, then the risk (standard deviation) of this portfolio is: a. between 0.5% and 1.5% b. between 1.5% and 2.5% c. between 2.5% and 3.5% d. 0% e. less than zero

between 0.5% and 1.5% sp = (XA) (sA) - (1 - XA) (sB) = .60 (4%) - (1-.60) (3%) = 2.4% - 1.3% = 1.1% or sp = - (XA) (sA) + (1 - XA) (sB) = - .60 (4%) + (1-.60) (3%) = -2.4% + 1.3% = - 1.1%

The Returns and Standard Deviations for stocks 1, 2 and 3 are the following --------------Stock 1 Stock 2 Stock 3 Mean Return (R) 10% 8% 12% Standard Deviation 2% 1% 3% The correlation coefficients for the returns between stocks 1, 2 and 3 are the following Correlations Stock 1 Stock 2 Stock 3 Stock 1 1 .75 .25 Stock 2 .75 1 0 Stock 3 .25 0 1 A portfolio is created where 1/3 of the portfolio weight is in Stock 1, 1/3 is in Stock 2 and 1/3 is in Stock 3. The standard deviation of the portfolio is: a. between 0% and 1% b. between 1% and 2% c. between 2% and 3% d. 2%

between 1% and 2% σp = [X21 σ21 + X22 σ22 + X23 σ23 + 2 X1 X2 σ1 σ2 ρ12+ 2 X1 X3 σ1 σ3 ρ13 + 2 X2 X3 σ2 σ3 ρ23] 0.50 = [(1/3)2 (.02)2 + (1/3)2 (.01)2 + (1/3)2 (.03)2 + 2 (1/3) (1/3) (.02) (.01) (.75) + 2 (1/3) (1/3) (.02) (.03) (.25) + 2 (1/3)(1/3) (.01) (.03) (0)] 0.50 = [0.11111 (.0004) + 0.11111 (.0001) + 0.11111 (.0009) + 2 (0.11111) (0.00015) + 2 (0.11111) (0.00015) + (0)] 0.50 = [0.11111 (.0004) + 0.11111 (.0001) + 0.11111 (.0009) + 2 (0.11111) (0.00015) + 2 (0.11111) (0.00015) + (0)] 0.50 = [.000155554 for first 3 terms + .000066666 for second 3 terms] 0.50 = [0.00022222] 0.50= 1.4907045%

Stocks A and B have the following distribution of expected returns for years 2013, 2014 and 2015. (Note that the standard deviation information provided below is not adjusted for degrees of freedom.) Year Stock A Return Stock B Return 2013 4% 22% 2014 8% 14% 2015 12% 6% Expected return 8% 14% Standard deviation 3.266% 6.532% The correlation coefficient between the return distributions for Stock A and Stock B is (do not adjust for degrees of freedom): a. plus one b. minus one c. 0.50 d. zero

minus one The correlation coefficient between the return distributions of Stock A and B is -1.0. This calculation is obtained as follows: ρAB = σAB / σA σB = -0.00213 / (.03266) (.06532) = -1 where σAB = -0.00213

The following diagram shows the investment opportunity set for portfolios containing stocks A and B. You need to know that: -Point A on the graph represents a portfolio with 100% in stock A -Point B represents a portfolio with 100% in stock B *A C graph starting with A at the tip then going Z, Y, X (at center of line), W, B Which labeled point on the graph represents a portfolio with 88% invested in stock A and the rest in B?

z. This should be obvious, since a portfolio with 88% in A will be much closer to A than B on the curve. You can also confirm mathematically by noting from the graph that E(rA) ≈ 8.5% and E(rB) ≈ 4.5%. Thus, a portfolio with 87% in A will have E(rP) ≈ 0.88(.085) + 0.12(.045) = 0.0802, which is approximately the expected return of portfolio z in the graph.

The distribution of returns for assets A and B, as shown below, depends on the state of the market. State of Market Probability Asset A Return Asset B Return Good .25 5% 8% Average .5 7% 11% Poor .25 9% 14% Mean 7% 11% Standard Deviation 1.2142% 2.1213% The covariance between the return distributions of assets A and B (σA,B) is:

σA,B = [.25 (.05 - .07) (.08 - .11) + .50 (.07 - .07) (.11 - .11) + .25 (.09 - .07) (.14 - .11)] = [.25 ( - .02) ( - .03) + .25 (.02) (.03)] = [.25 ( .0006) + .25 (.0006)] = [.00015) + .00015)] = .0003 In integers, but answer should be in decimals. σA,B =[.25 (5 - 7) (8 - 11) + .50 (7 - 7) (11 - 11) + .25 (9 - 7) (14 - 11)] = [.25 ( - 2) ( - 3) + .25 (2) (3)] = [.25 (6) + .25 (6)] = [1.5 + 1.5] = 3