121834193 Elton Gruber 7e Solution Manual

Elton, Gruber, Brown, and Goetzmann 1-1 Modern Portfolio Theory and Investment Analysis, 7th Edition Solutions to Text Problems: Chapter 1

Elton, Gruber, Brown, and Goetzmann Modern Portfolio Theory and Investment Analysis, 7th Edition

Solutions to Text Problems: Chapter 1

Chapter 1: Problem 1 A. Opportunity Set With one dollar, you can buy 500 red hots and no rock candies (point A), or 100 rock candies and no red hots (point B), or any combination of red hots and rock candies (any point along the opportunity set line AB).

Algebraically, if X = quantity of red hots and Y = quantity of rock candies, then:

10012.0 =+ YX That is, the money spent on candies, where red hots sell for 0.2 cents a piece and rock candy sells for 1 cent a piece, cannot exceed 100 cents ($1.00). Solving the above equation for X gives:

YX 5500 = which is the equation of a straight line, with an intercept of 500 and a slope of 5.


B. Indifference Map Below is one indifference map. The indifference curves up and to the right indicate greater happiness, since these curves indicate more consumption from both candies. Each curve is negatively sloped, indicating a preference of more to less, and each curve is convex, indicating that the rate of exchange of red hots for rock candies decreases as more and more rock candies are consumed. Note that the exact slopes of the indifference curves in the indifference map will depend an the individuals utility function and may differ among students.

Chapter 1: Problem 2 A. Opportunity Set

The individual can consume everything at time 2 and nothing at time 1, which, assuming a riskless lending rate of 10%, gives the maximum time-2 consumption amount:

$20 + $20 (1 + 0.1) = $42.

Instead, the individual can consume everything at time 1 and nothing at time 2, which, assuming a riskless borrowing rate of 10%, gives the maximum time-1 consumption amount: $20 + $20 (1 + 0.1) = $38.18


The individual can also choose any consumption pattern along the line AB

(the opportunity set) below.

The opportunity set line can be determined as follows. Consumption at

time 2 is equal to the amount of money available in time 2, which is the income earned at time 2, $20, plus the amount earned at time 2 from any money invested at time 1, ($20 C1) (1 + 0.1): C2 = $20 + ($20 C1) (1.1) or C2 = $42 1.1C1 which is the equation of a straight line with an intercept of $42 and a slope of 1.1. B. Indifference Map We are given that the utility function of the individual is:

501),( 212121

CCCCCCU +++=

A particular indifference curve can be traced by setting U(C1,C2) equal to a constant and then varying C1 and C2. By changing the constant, we can trace out other indifference curves. For example, by setting U(C1,C2) equal to 50 we get:

5050

1 2121 =+++ CCCC or 24505050 2121 =++ CCCC


This indifference curve appears in the graph of the indifference map below as the curve labeled 50 (the lowest curve shown). By setting U(C1,C2) equal to 60, we get the curve labeled 60, etc.

C. Solution The optimum solution is where the opportunity set is tangent to the highest possible indifference curve (the point labeled E in the following graph).

This problem is meant to be solved graphically. Below, we show an analytical solution:

501),( 212121

CCCCCCU +++=

Substituting the equation of the opportunity set given in part A for C2 in the above equation gives:

501.142

1.1421),(2

111121

CCCCCCU

+++= To maximize the utility function, we take the derivative of U with respect to C1 and set it equal to zero:

0502.2

50421.11 1

1=+= C

dCdU

which gives C1 = $16.82. Substituting $16.82 for C1 in the equation of the opportunity set given in part A gives C2 = $23.50.


Chapter 1: Problem 3 If you consume nothing at time 1 and instead invest all of your time-1 income at a riskless rate of 10%, then at time 2 you will be able to consume all of your time-2 income plus the proceeds earned from your investment: $5,000 + $5,000 (1.1) = $10,500. If you consume nothing at time 2 and instead borrow at time 1 the present value of your time-2 income at a riskless rate of 10%, then at time 1 you will be able to consume all of the borrowed proceeds plus your time-1 income: $5,000 + $5,000 (1.1) = $9,545.45 All other possible optimal consumption patterns of time 1 and time 2 consumption appear on a straight line (the opportunity set) with an intercept of $10,500 and a slope of 1.1: C2 = $5,000 + ($5,000 C1) (1.1) = $10,500 1.1C1

Chapter 1: Problem 4 If you consume nothing at time 1 and instead invest all of your wealth plus your time-1 income at a riskless rate of 5%, then at time 2 you will be able to consume all of your time-2 income plus the proceeds earned from your investment: $20,000 + ($20,000 + $50,000)(1.05) = $93,500.


If you consume nothing at time 2 and instead borrow at time 1 the present value of your time-2 income at a riskless rate of 5%, then at time 1 you will be able to consume all of the borrowed proceeds plus your time-1 income and your wealth: $20,000 + $50,000 + $20,000 (1.05) = $89,047.62 All other possible optimal consumption patterns of time-1 and time-2 consumption appear on a straight line (the opportunity set) with an intercept of $93,500 and a slope of 1.05: C2 = $20,000 + ($20,000 + $50,000 C1) (1.05) = $93,500 1.05C1

Chapter 1: Problem 5 With Job 1 you can consume $30 + $50 (1.05) = $82.50 at time 2 and nothing at time 1, $50 + $30 (1.05) = $78.60 at time 1 and nothing at time 2, or any consumption pattern of time 1 and time 2 consumption shown along the line AB: C2 = $82.50 1.05C1. With Job 2 you can consume $40 + $40 (1.05) = $82.00 at time 2 and nothing at time 1, $40 + $40 (1.05) = $78.10 at time 1 and nothing at time 2, or any consumption pattern of time 1 and time 2 consumption shown along the line CD: C2 = $82.00 1.05C1.

The individual should select Job 1, since the opportunity set associated with it (line AB) dominates the opportunity set of Job 2 (line CD).


Chapter 1: Problem 6 With an interest rate of 10% and income at both time 1 and time 2 of $5,000, the opportunity set is given by the line AB: C2 = $5,000 + ($5,000 C1) (1.1) = $10,500 1.1C1 With an interest rate of 20% and income at both time 1 and time 2 of $5,000, the opportunity set is given by the line CD: C2 = $5,000 + ($5,000 C1) (1.2) = $11,000 1.2C1

Lines AB and CD intersect at point E (where C2 = time-2 income = $5,000 and C1 = time-1 income = $5,000). Along either line above point E, the individual is lending (consuming less at time 1 than the income earned at time 1); along either line below point E, the individual is borrowing (consuming more at time 1 than the income earned at time 1). Since the individual can only lend at 10% and must borrow at 20%, the individuals opportunity set is given by line segments AE and ED. Chapter 1: Problem 7

For P = 50, this is simply a plot of the function 1

12 1

50CC

C += .

For P = 100, this is simply a plot of the function 1

12 1

100C

CC +

= .


Chapter 1: Problem 8 This problem is analogous to Problem 2. We present the analytical solution below. The problem could be solved graphically, as in Problem 2. From Problem 3, the opportunity set is C2 = $10,500 1.1C1. Substituting this equation into the preference function P = C1 + C2 + C1 C2 yields:

21111 1.1500,10$1.1500,10$ CCCCP ++=

02.2500,10$1.11 11

=+= CdCdP

C1 = $4,772.68 C2 = $5,250.05

Chapter 1: Problem 9

Let X = the number of pizza slices, and let Y = the number of hamburgers. Then, if pizza slices are $2 each, hamburgers are $2.50 each, and you have $10, your opportunity set is given algebraically by $2X + $2.50Y = $10 Solving the above equation for X gives X = 5 1.25Y, which is the equation for a straight line with an intercept of 5 and a slope of 1.25. Graphically, the opportunity set appears as follows:

Assuming you like both pizza and hamburgers, your indifference curves will be negatively sloped, and you will be better off on an indifference curve to the right of another indifference curve. Assuming diminishing marginal rate of substitution between pizza slices and hamburgers (the lower the number of hamburgers you have, the more pizza slices you need to give up one more burger without changing your level of satisfaction), your indifference curves will also be convex.


A typical family of indifference curves appears below. Although you would rather be on an indifference curve as far to the right as possible, you are constrained by your $10 budget to be on an indifference curve that is on or to the left of the opportunity set. Therefore, your optimal choice is the combination of pizza slices and hamburgers that is represented by the point where your indifference curve is just tangent to the opportunity set (point A below).

Chapter 1: Problem 10 If you consume C1 at time 1 and invest (lend) the rest of your time-1 income at 5%, your time-2 consumption (C2) will be $50 from your time-2 income plus ($50 C1)(1.05) from your investment. Algebraically, the opportunity set is thus C2 = 50 + (50 - C1)(1.05) = 102.50 - 1.05C1 If C1 is 0 (no time-1 consumption), then from the above equation C2 will be $102.50. If C2 is 0, then C1 will be $97.62. Graphically, the opportunity set appears below, along with a typical family of indifference curves.


Chapter 1: Problem 11 If you consume C1 at time 1 and invest (lend) the rest of your time-1 income at 20%, your time-2 consumption (C2) will be $10,000 from your time-2 income plus $10,000 from your inheritance plus ($10,000 - C1)(1.20) from your investment. The opportunity set is thus C2 = $10,000 + $10,000 + ($10,000 - C1)(1.20) = $32,000 - 1.2C1 If C2 is 0 (no time-2 consumption), then you can borrow the present value of your time-2 income and your time-2 inheritance and spend that amount along with your time-1 income on time-1 consumption. Solving the above equation for C1 when C2 is 0 gives C1 = $26,666.67, which is the maximum that can be consumed at time 1. Similarly, if C1 is 0 (no time-1 consumption), then you can invest all of your time-1 income at 20% and spend the future value of your time-1 income plus your time-2 income and inheritance on time-2 consumption. From the above equation, C2 will be $32,000 when C1 is 0, which is the maximum that can be consumed at time 2. Chapter 1: Problem 12 If you consume nothing at time 2, then you can borrow the present value of your time-2 income for consumption at time 1. If the borrowing rate is 10% and your time-2 income is $100, then the present value (at time 1) of your time-2 income is $100/(1.1) = $90.91. You can borrow this amount and spend it along with your time- income of $100 on time-1 consumption. So the maximum you can consume at time 1 is $90.91 + $100 = $190.91. If you consume nothing at time 1 and instead invest all of your time-1 income of $100 at the lending rate of 5%, the future value (in period 2) of your period 1 income will be $100(1.05) = $105. You can then spend that amount along with your time 2 income of $100 on time-2 consumption. So the maximum you can consume at time 2 is $105 + $100 = $205. With two different interest rates, we have two separate equations for opportunity sets: one for borrowers and one for lenders. If you only consume some of your time 1 income at time 1 and invest the rest at 5%, you have the following opportunity set: C2 = 100 + (100 - C1)(1.05) = 205 - 1.05C1. If you only consume some of your time-2 income at time 2 and borrow the present value of the rest at 10% for consumption at time 1, your opportunity set is: C1 = 100 + (100 - C2)/(1.1) = 190.91 - C2/1.1,or, solving the equation for C2, C2 = 210 - 1.1C1


Graphically, the two lines appear as follows:

The lines intersect at point E (which represents your income endowment for times 1 and 2). Moving along the lines above point E represents lending (investing some time-1 income); moving along the lines below point E represents borrowing (spending more than your time-1 income on time-1 consumption). Since you can only lend at 5%, line segment AE represents your opportunity set if you choose to lend. Since you must borrow at 10%, line segment ED represents your opportunity set if you choose to borrow. So your total opportunity set is represented by AED.

Elton, Gruber, Brown, and Goetzmann 4-1 Modern Portfolio Theory and Investment Analysis, 7th Edition Solutions To Text Problems: Chapter 4

Elton, Gruber, Brown and Goetzmann Modern Portfolio Theory and Investment Analysis, 7th Edition


Chapter 4: Problem 1 A. Expected return is the sum of each outcome times its associated

probability.

Expected return of Asset 1 = =1R 16% 0.25 + 12% 0.5 + 8% 0.25 = 12% 2R = 6%; 3R = 14%; 4R = 12%

Standard deviation of return is the square root of the sum of the squares of each outcome minus the mean times the associated probability. Standard deviation of Asset 1 =

1 = [(16% 12%)2 0.25 + (12% 12%)2 0.5 + (8% 12%)2 0.25]1/2 = 81/2 = 2.83%

2 = 21/2 = 1.41%; 3 = 181/2 = 4.24%; 4 = 10.71/2 = 3.27% B. Covariance of return between Assets 1 and 2 = 12 = (16 12) (4 6) 0.25 + (12 12) (6 6) 0.5 + (8 12) (8 6) 0.25

= 4 The variance/covariance matrix for all pairs of assets is:

1 2 3 4 1 8 4 12 0 2 4 2 6 0 3 12 6 18 0 4 0 0 0 10.7

Correlation of return between Assets 1 and 2 = 141.183.2

412 =

= . The correlation matrix for all pairs of assets is:

1 2 3 4 1 1 1 1 0 2 1 1 1 0 3 1 1 1 0 4 0 0 0 1


C. Portfolio Expected Return A 1/2 12% + 1/2 6% = 9% B 13% C 12% D 10% E 13% F 1/3 12% + 1/3 6% + 1/3 14% = 10.67% G 10.67% H 12.67% I 1/4 12% + 1/4 6% + 1/4 14% + 1/4 12% = 11% Portfolio Variance A (1/2)2 8 + (1/2)2 2 + 2 1/2 1/2 ( 4) = 0.5 B 12.5 C 4.6 D 2 E 7 F (1/3)2 8 + (1/3)2 2 + (1/3)2 18 + 2 1/3 1/3 ( 4) + 2 1/3 1/3 12 + 2 1/3 1/3 ( 6) = 3.6 G 2 H 6.7 I (1/4)2 8 + (1/4)2 2 + (1/4)2 18 + (1/4)2 10.7

+ 2 1/4 1/4 ( 4) + 2 1/4 1/4 12 + 2 1/4 1/4 0 + 2 1/4 1/4 ( 6) + 2 1/4 1/4 0 + 2 1/4 1/4 0 = 2.7


Chapter 4: Problem 2 A. Monthly Returns

Month Security 1 2 3 4 5 6 A 3.7% 0.4% -6.5% 1.4% 6.2% 2.1% B 10.5% 0.5% 3.7% 1.0% 3.4% -1.4% C 1.4% 14.9% -1.4% 10.8% 4.9% 16.9%

B. Sample Average (Mean) Monthly Returns ( ) %22.1

6%1.2%2.6%4.1%5.6%4.0%7.3 =++++=AR

%95.2=BR

%92.7=CR C. Sample Standard Deviations of Monthly Returns

( ) ( ) ( ) ( ) ( ) ( )%92.334.15

6%22.1%1.2%22.1%2.6%22.1%4.1%22.1%5.6%22.1%4.0%22.1%7.3 222222

==

+++++=A

%8.342.14 ==B %78.602.46 ==C


D. Sample Covariances and Correlation Coefficients of Monthly Returns ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )17.2

6%95.2%4.1%22.1%1.2%95.2%4.3%22.1%2.6%95.2%0.1%22.1%4.1%95.2%7.3%22.1%5.6%95.2%5.0%22.1%4.0%95.2%5.10%22.1%7.3

=

+++++

=AB

24.7=AC ; 89.19=BC

15.08.392.3

17.2 ==AB 27.0=AC ; 77.0=BC E. Portfolio Returns and Standard Deviations Portfolio 1 (X1 = 1/2; X2= 1/2; X3= 0): %09.2%92.70%95.22/1%22.12/11 =++=PR

( ) ( ) ( )%92.253.8

89.1902/124.702/117.22/12/1202.46042.142/134.152/1 2221==

+++++=P

Portfolio 2 (X1 = 1/2; X2= 0; X3= 1/2): %57.42 =PR %35.496.182 ==P Portfolio 3 (X1 = 0; X2= 1/2; X3= 1/2): %44.53 =PR %27.217.53 ==P Portfolio 4 (X1 = 1/3; X2= 1/3; X3= 1/3): %03.44 =PR %47.209.64 ==P


Chapter 4: Problem 3 It is shown in the text below Table 4.8 that a formula for the variance of an equally weighted portfolio (where Xi = 1/N for i = 1, , N securities) is

( ) kjkj2j2P 1/N = + where 2j is the average variance across all securities, kj is the average covariance across all pairs of securities, and N is the number of securities. Using the above formula with 2j = 50 and kj = 10 we have: Portfolio Size (N) 2P 5 18 10 14 20 12 50 10.8 100 10.4 Chapter 4: Problem 4 As is shown in the text, as the number of securities (N) approaches infinity, an equally weighted portfolios variance (total risk) approaches a minimum equal to the average covariance of the pairs of securities in the portfolio, which in Problem 3 is given as 10. Therefore, 10% above the minimum risk level would result in the portfolios variance being equal to 11. Setting the formula shown in the above answer to Problem 3 equal to 11 and using 2j = 50 and kj = 10 we have: ( ) 11101050/12 =+= NP Solving the above equation for N gives N = 40 securities.


Chapter 4: Problem 5 As shown in the text, if the portfolio contains only one security, then the portfolios average variance is equal to the average variance across all securities, 2j . If instead an equally weighted portfolio contains a very large number of securities, then its variance will be approximately equal to the average covariance of the pairs of securities in the portfolio, kj . Therefore, the fraction of risk that of an individual security that can be eliminated by holding a large portfolio is expressed by the following ratio:

2

2

i

kji

From Table 4.9, the above ratio is equal to 0.6 (60%) for Italian securities and 0.8 (80%) for Belgian securities. Setting the above ratio equal to those values and solving for kj gives 24.0 ikj = for Italian securities and 22.0 ikj = for Belgian securities.

Thus, the ratio kj

kji

2

equals 5.14.0

4.02

22

=i

ii

for Italian securities and

42.0

2.02

22

=i

ii

for Belgian securities.

If the average variance of a single security, 2j , in each country equals 50, then

20504.04.0 2 === ikj for Italian securities and 10502.02.0 2 === ikj for Belgian securities. Using the formula shown in the preceding answer to Problem 3 with 2j = 50 and either kj = 20 for Italy or kj = 10 for Belgium we have: Portfolio Size (N securities) Italian 2P Belgian 2P 5 26 18 20 21.5 12 100 20.3 10.4


Chapter 4: Problem 6 The formula for an equally weighted portfolio's variance that appears below Table 4.8 in the text is

( ) kjkj2j2P 1/N = + where 2j is the average variance across all securities, kj is the average covariance across all securities, and N is the number of securities. The text below Table 4.8 states that the average variance for the securities in that table was 46.619 and that the average covariance was 7.058. Using the above equation with those two numbers, setting 2P equal to 8, and solving for N gives: 8 = 1/N (46.619 - 7.058) + 7.058 .942N = 39.561 N = 41.997. Since the portfolio's variance decreases as N increases, holding 42 securities will provide a variance less than 8, so 42 is the minimum number of securities that will provide a portfolio variance less than 8.

Elton, Gruber, Brown and Goetzmann 5-1 Modern Portfolio Theory and Investment Analysis, 7th Edition Solutions To Text Problems: Chapter 5



Chapter 5: Problem 1 From Problem 1 of Chapter 4, we know that: R 1 = 12% R 2 = 6% R 3 = 14% R 4 = 12% 21 = 8 22 = 2 23 = 18 24 = 10.7 1 = 2.83% 2 = 1.41% 3 = 4.24% 4 = 3.27% 12 = 4 13 = 12 14 = 0 23 = 6 24 = 0 34 = 0 12 = 1 13 = 1 14 = 0 23 = 1.0 24 = 0 34 = 0 In this problem, we will examine 2-asset portfolios consisting of the following pairs of securities: Pair Securities A 1 and 2 B 1 and 3 C 1 and 4 D 2 and 3 E 2 and 4 F 3 and 4 A. Short Selling Not Allowed (Note that the answers to part A.4 are integrated with the answers to parts A.1, A.2 and A.3 below.) A.1 We want to find the weights, the standard deviation and the expected return of the minimum-risk porfolio, also known as the global minimum variance (GMV) portfolio, of a pair of assets when short sales are not allowed.


We further know that the compostion of the GMV portfolio of any two assets i and j is:

ijji

ijjGMViX

222

2

+=

GMVi

GMVj XX = 1

Pair A (assets 1 and 2): Applying the above GMV weight formula to Pair A yields the following weights:

31

186

)4)(2(28)4(2

2 1222

21

1222

1 ==+=+

= GMVX (or 33.33%)

32

3111 12 === GMVGMV XX (or 66.67%)

This in turn gives the following for the GMV portfolio of Pair A:

%8%632%12

31 =+=GMVR

( ) ( ) ( ) ( ) 0432

3122

328

31 222 =

+

+

=GMV

0=GMV

Recalling that 12 = 1, the above result demonstrates the fact that, when two assets are perfectly negatively correlated, the minimum-risk portfolio of those two assets will have zero risk. Pair B (assets 1 and 3): Applying the above GMV weight formula to Pair B yields the following weights:

31 =GMVX (300%) and 23 =GMVX (200%)

This means that the GMV portfolio of assets 1 and 3 involves short selling asset 3. But if short sales are not allowed, as is the case in this part of Problem 1, then the GMV portfolio involves placing all of your funds in the lower risk security (asset 1) and none in the higher risk security (asset 3). This is obvious since, because the correlation between assets 1 and 3 is +1.0, portfolio risk is simply a linear combination of the risks of the two assets, and the lowest value that can be obtained is the risk of asset 1.


Thus, when short sales are not allowed, we have for Pair B:

11 =GMVX (100%) and 03 =GMVX (0%)

%121 == RRGMV ; 8212 == GMV ; %83.21 == GMV

For the GMV portfolios of the remaining pairs above we have: Pair GMViX

GMVjX GMVR GMV

C (i = 1, j = 4) 0.572 0.428 12% 2.14% D (i = 2, j = 3) 0.75 0.25 8% 0% E (i = 2, j = 4) 0.8425 0.1575 6.95% 1.3% F (i = 3, j = 4) 0.3728 0.6272 12.75% 2.59% A.2 and A.3 For each of the above pairs of securities, the graph of all possible combinations (portfolios) of the securities (the portfolio possibilties curves) and the efficient set of those portfolios appear as follows when short sales are not allowed: Pair A

The efficient set is the positively sloped line segment.


Pair B

The entire line is the efficient set. Pair C

Only the GMV portfolio is efficient.


Pair D

The efficient set is the positively sloped line segment. Pair E

The efficient set is the positively sloped part of the curve, starting at the GMV portfolio and ending at security 4.


Pair F

The efficient set is the positively sloped part of the curve, starting at the GMV portfolio and ending at security 3. B. Short Selling Allowed (Note that the answers to part B.4 are integrated with the answers to parts B.1, B.2 and B.3 below.) B.1 When short selling is allowed, all of the GMV portfolios shown in Part A.1 above are the same except the one for Pair B (assets 1 and 3). In the no-short-sales case in Part A.1, the GMV portfolio for Pair B was the lower risk asset 1 alone. However, applying the GMV weight formula to Pair B yielded the following weights:

31 =GMVX (300%) and 23 =GMVX (200%)

This means that the GMV portfolio of assets 1 and 3 involves short selling asset 3 in an amount equal to twice the investors original wealth and then placing the original wealth plus the proceeds from the short sale into asset 1. This yields the following for Pair B when short sales are allowed:

%8%142%123 ==GMVR ( ) ( ) ( ) ( ) ( )( )( )( ) 01223218283 222 =++=GMV

0=GMV Recalling that 13 = +1, this demonstrates the fact that, when two assets are perfectly positively correlated and short sales are allowed, the GMV portfolio of those two assets will have zero risk.


B.2 and B.3 When short selling is allowed, the portfolio possibilities graphs are extended. Pair A

The efficient set is the positively sloped line segment through security 1 and out toward infinity. Pair B

The entire line out toward infinity is the efficient set.


Pair C

Only the GMV portfolio is efficient. Pair D

The efficient set is the positively sloped line segment through security 3 and out toward infinity.


Pair E

The efficient set is the positively sloped part of the curve, starting at the GMV portfolio and extending past security 4 toward infinity. Pair F

The efficient set is the positively sloped part of the curve, starting at the GMV portfolio and extending past security 3 toward infinity.


C. Pair A (assets 1 and 2): Since the GMV portfolio of assets 1 and 2 has an expected return of 8% and a risk of 0%, then, if riskless borrowing and lending at 5% existed, one would borrow an infinite amount of money at 5% and place it in the GMV portfolio. This would be pure arbitrage (zero risk, zero net investment and positive return of 3%). With an 8% riskless lending and borrowing rate, one would hold the same portfolio one would hold without riskless lending and borrowing. (The particular portfolio held would be on the efficient frontier and would depend on the investors degree of risk aversion.) Pair B (assets 1 and 3): Since short sales are allowed in Part C and since we saw in Part B that when short sales are allowed the GMV portfolio of assets 1 and 3 has an expected return of 8% and a risk of 0%, the answer is the same as that above for Pair A. Pair C (assets 1 and 4): We have seen that, regardless of the availability of short sales, the efficient frontier for this pair of assets was a single point representing the GMV portfolio, with a return of 12%. With riskless lending and borrowing at either 5% or 8%, the new efficient frontier (efficient set) will be a straight line extending from the vertical axis at the riskless rate and through the GMV portfolio and out to infinity. The amount that is invested in the GMV portfolio and the amount that is borrowed or lent will depend on the investors degree of risk aversion. Pair D (assets 2 and 3): Since assets 2 and 3 are perfectly negatively correlated and have a GMV portfolio with an expected return of 8% and a risk of 0%, the answer is identical to that above for Pair A. Pair E (assets 2 and 4): We arrived at the following answer graphically; the analytical solution to this problem is presented in the subsequent chapter (Chapter 6). With a riskless rate of 5%, the new efficient frontier (efficient set) will be a straight line extending from the vertical axis at the riskless rate, passing through the portfolio where the line is tangent to the upper half of the original portfolio possibilities curve, and then out to infinity. The amount that is invested in the tangent portfolio and the amount that is borrowed or lent will depend on the investors degree of risk aversion. The tangent portfolio has an expected return of 9.4% and a standard deviation of 1.95%. With a riskless rate of 8%, the point of tangency occurs at infinity.


Pair F (assets 3 and 4): We arrived at the following answer graphically; the analytical solution to this problem is presented in the subsequent chapter (Chapter 6). With a riskless rate of 5%, the new efficient frontier (efficient set) will be a straight line extending from the vertical axis at the riskless rate, passing through the portfolio where the line is tangent to the upper half of the original portfolio possibilities curve, and then out to infinity. The amount that is invested in the tangent portfolio and the amount that is borrowed or lent will depend on the investors degree of risk aversion. The tangent (optimal) portfolio has an expected return of 12.87% and a standard deviation of 2.61%. With a riskless rate of 8%, the new efficient frontier will be a straight line extending from the vertical axis at the riskless rate, passing through the portfolio where the line is tangent to the upper half of the original portfolio possibilities curve, and then out to infinity. The tangent (optimal) portfolio has an expected return of 12.94% and a standard deviation of 2.64%. Chapter 5: Problem 2 From Problem 2 of Chapter 4, we know that: R A = 1.22% R B = 2.95% R C = 7.92% 2A = 15.34 2B = 14.42 2C = 46.02 A = 3.92% B = 3.8% C = 6.78% AB = 2.17 AC = 7.24 BC = 19.89 AB = 0.15 AC = 0.27 BC = 0.77 In this problem, we will examine 2-asset portfolios consisting of the following pairs of securities: Pair Securities 1 A and B 2 A and C 3 B and C


A. Short Selling Not Allowed (Note that the answers to part A.4 are integrated with the answers to parts A.1, A.2 and A.3 below.) A.1 We want to find the weights, the standard deviation and the expected return of the minimum-risk porfolio, also known as the global minimum variance (GMV) portfolio, of a pair of assets when short sales are not allowed. We further know that the compostion of the GMV portfolio of any two assets i and j is:

ijji

ijjGMViX

222

2

+=

GMVi

GMVj XX = 1

Pair 1 (assets A and B): Applying the above GMV weight formula to Pair 1 yields the following weights:

482.0)17.2)(2(42.1434.15

17.242.14222

2

=+=+

=ABBA

ABBGMVAX

(or 48.2%)

518.0482.011 === GMVAGMVB XX (or 51.8%)

This in turn gives the following for the GMV portfolio of Pair 1:

%12.2%95.2518.0%22.1482.0 =+=GMVR

( ) ( ) ( ) ( ) ( )( )( )( ) 52.817.2518.0482.0242.14518.034.15482.0 222 =++=GMV

%92.2=GMV


For the GMV portfolios of the remaining pairs above we have: Pair GMViX

GMVjX GMVR GMV

2 (i = A, j = C) 0.827 0.173 2.38% 3.73% 3 (i = B, j = C) 0.658 0.342 4.65% 1.63% A.2 and A.3 For each of the above pairs of securities, the graph of all possible combinations (portfolios) of the securities (the portfolio possibilties curves) and the efficient set of those portfolios appear as follows when short sales are not allowed: Pair 1

The efficient set is the positively sloped part of the curve, starting at the GMV portfolio and ending at security B.


Pair 2

The efficient set is the positively sloped part of the curve, starting at the GMV portfolio and ending at security C. Pair 3

The efficient set is the positively sloped part of the curve, starting at the GMV portfolio and ending at security C.


B. Short Selling Not Allowed (Note that the answers to part B.4 are integrated with the answers to parts B.1, B.2 and B.3 below.) B.1 When short selling is allowed, all of the GMV portfolios shown in Part A.1 above remain the same. B.2 and B.3 When short selling is allowed, the portfolio possibilities graphs are extended. Pair 1

The efficient set is the positively sloped part of the curve, starting at the GMV portfolio and extending past security B toward infinity.


Pair 2

The efficient set is the positively sloped part of the curve, starting at the GMV portfolio and extending past security C toward infinity. Pair 3

The efficient set is the positively sloped part of the curve, starting at the GMV portfolio and extending past security C toward infinity.


C. In all cases where the riskless rate of either 5% or 8% is higher than the returns on both of the individual securities, if short sales are not allowed, any rational investor would only invest in the riskless asset. Even if short selling is allowed, the point of tangency of a line connecting the riskless asset to the original portfolio possibilities curve occurs at infinity for all cases, since the original GMV portfolios return is lower than 5% in all cases. Chapter 5: Problem 3 The answers to this problem are given in the answers to part A.1 of Problem 2. Chapter 5: Problem 4 The locations, in expected return standard deviation space, of all portfolios composed entirely of two securities that are perfectly negatively correlated (say, security C and security S) are described by the equations for two straight lines, one with a positive slope and one with a negative slope. To derive those equations, start with the expressions for a two-asset portfolio's standard deviation when the two assets' correlation is 1 (the equations in (5.8) in the text), and solve for XC (the investment weight for security C). E.g., for the first equation:

( )

. + + = X

) + (X = + X + -X =

X X =

SC

SPC

SCCSP

SCSCCP

SCCCP

1

Now plug the above expression for XC into the expression for a two-asset portfolio's expected return and simplify:

( )

. + RR

+ RR + R=

+ RRR + R + R =

R + + + R +

+ =

RX + RX = R

PSC

SCS

SC

SCS

SC

SSSPCSCPS

SSC

SPC

SC

SP

SCCCP

+

1

1

The above equation is that of a straight line in expected return standard deviation space, with an intercept equal to the first term in brackets and a slope equal to the second term in brackets.


Solving for XC in the second equation in (5.8) gives:

( )( )

. +

= X

+ X= X + X=

X+ X=

SC

PSC

SCCSP

SCSCCP

SCCCP

1

Substitute the above expression for XC into the equation for expected return and simplify:

( )

. + RR

+ RR + R=

+ R + RRR + R =

R + + R +

=

RX + RX = R

PSC

CSS

SC

SCS

SC

SPSSCPCSS

SSC

PSC

SC

PS

SCCCP

+

1

1

The above equation is also that of a straight line in expected return standard deviation space, with an intercept equal to the first term in brackets and a slope equal to the second term in brackets. The intercept term for the above equation is identical to the intercept term for the first derived equation. The slope term is equal to 1 times the slope term of the first derived equation. So when one equation has a positive slope, the other equation has a negative slope (when the expected returns of the two assets are equal, the two lines are coincident), and both lines meet at the same intercept. Chapter 5: Problem 5 When equals 1, the least risky "combination" of securities 1 and 2 is security 2 held alone (assuming no short sales). This requires X1 = 0 and X2 = 1, where the X's are the investment weights. The standard deviation of this "combination" is equal to the standard deviation of security 2; P = 2 = 2. When equals -1, we saw in Chapter 5 that we can always find a combination of the two securities that will completely eliminate risk, and we saw that this combination can be found by solving X1 = 2/(1 + 2). So, X1 = 2/(5 + 2) = 2/7, and since the investment weights must sum to 1, X2 = 1 - X1 = 1 - 2/7 = 5/7. So a combination of 2/7 invested in security 1 and 5/7 invested in security 2 will completely eliminate risk when equals -1, and P will equal 0.


When equals 0, we saw in Chapter 5 that the minimum-risk combination of two assets can be found by solving X1 = 22/(12 + 22). So, X1 = 4/(25 + 4) = 4/29, and X2 = 1 - X1 = 1 - 4/29 = 25/29. When equals 0, the expression for the standard deviation of a two-asset portfolio is

( ) 22212121 1 XX = P +

Substituting 4/29 for X1 in the above equation, we have

%86.1841

2900841

2500841400

4292525

294 22

==

+=

+

=P

Chapter 5: Problem 6 If the riskless rate is 10%, then the risk-free asset dominates both risky assets in terms of risk and return, since it offers as much or higher expected return than either risky asset does, for zero risk. Assuming the investor prefers more to less and is risk averse, the optimal investment is the risk-free asset.



Solutions to Text Problems: Chapter 6 Chapter 6: Problem 1 The simultaneous equations necessary to solve this problem are: 5 = 16Z1 + 20Z2 + 40Z3 7 = 20Z1 + 100Z2 + 70Z3 13 = 40Z1 + 70Z2 + 196Z3 The solution to the above set of equations is: Z1 = 0.292831 Z2 = 0.009118 Z3 = 0.003309 This results in the following set of weights for the optimum (tangent) portfolio: X1 = .95929 (95.929%) X2 = .02987 (2.987%) X3 = .01084 (1.084% The optimum portfolio has a mean return of 10.146% and a standard deviation of 4.106%. Chapter 6: Problem 2 The simultaneous equations necessary to solve this problem are: 11 RF = 4Z1 + 10Z2 + 4Z3 14 RF = 10Z1 + 36Z2 + 30Z3

17 RF = 4Z1 + 30Z2 + 81Z3


The optimum portfolio solutions using Lintner short sales and the given values for RF are: RF = 6% RF = 8% RF = 10%

Z1 3.510067 1.852348 0.194631 Z2 1.043624 0.526845 0.010070 Z3 0.348993 0.214765 0.080537

X1 0.715950 0.714100 0.682350 X2 0.212870 0.203100 0.035290 X3 0.711800 0.082790 0.282350

Tangent (Optimum) Portfolio Mean Return 6.105% 6.419% 11.812% Tangent (Optimum) Portfolio Standard Deviation 0.737% 0.802% 2.971% Chapter 6: Problem 3 Since short sales are not allowed, this problem must be solved as a quadratic programming problem. The formulation of the problem is:

P

FP

X

RR=max

subject to:

11

==

N

iiX

0iX i


Chapter 6: Problem 4 This problem is most easily solved using The Investment Portfolio software that comes with the text, but, since all pairs of assets are assumed to have the same correlation coefficient of 0.5, the problem can also be solved manually using the constant correlation form of the Elton, Gruber and Padberg Simple Techniques described in a later chapter. To use the software, open up the Markowitz module, select file then new then group constant correlation to open up a constant correlation table. Enter the input data into the appropriate cells by first double clicking on the cell to make it active. Once the input data have been entered, click on optimizer and then run optimizer (or simply click on the optimizer icon). At that point, you can either select full Markowitz or simple method. If you select full Markowitz, you then select short sales allowed/riskless lending and borrowing and then enter 4 for both the lending and borrowing rate and click OK. A graph of the efficient frontier then appears. You may then hit the Tab key to jump to the tangent portfolio, then click on optimizer and then show portfolio (or simply click on the show portfolio icon) to view and print the composition (investment weights), mean return and standard deviation of the tangent (optimum) portfolio. If instead you select simple method, you then select short sales allowed with riskless asset and enter 4 for the riskless rate and click OK. A table showing the investment weights of the tangent portfolio then appears. Regardless of the method used, the resulting investment weights for the optimum portfolio are as follows:

Asset i Xi

1 5.999% 2 17.966% 3 21.676% 4 0.478% 5 29.585% 6 12.693% 7 59.170% 8 14.793% 9 3.442%

10 189.224%


Given the above weights, the optimum (tangent) portfolio has a mean return of 18.907% and a standard deviation of 3.297%. The efficient frontier is a positively sloped straight line starting at the riskless rate of 4% and extending through the tangent portfolio (T) and out to infinity:

Chapter 6: Problem 5 Since the given portfolios, A and B, are on the efficient frontier, the GMV portfolio can be obtained by finding the minimum-risk combination of the two portfolios:

31

20216362016

2222

=+=+

=ABBA

ABBGMVAX

3111 == GMVAGMVB XX

This gives %33.7=GMVR and %83.3=GMV Also, since the two portfolios are on the efficient frontier, the entire efficient frontier can then be traced by using various combinations of the two portfolios, starting with the GMV portfolio and moving up along the efficient frontier (increasing the weight in portfolio A and decreasing the weight in portfolio B). Since XB = 1 XA the efficient frontier equations are:

( ) ( )AABAAAP XXRXRXR +=+= 18101

( ) ( )( ) ( )AAAA

ABAABAAAP

XXXX

XXXX

++=++=

14011636

12122

2222


Since short sales are allowed, the efficient frontier will extend beyond portfolio A and out toward infinity. The efficient frontier appears as follows:



Solutions to Text Problems: Chapter 7 Chapter 7: Problem 1 We will illustrate the answers for stock A and the market portfolio (S&P 500); the answers for stocks B and C are found in an identical manner. The sample mean monthly return on stock A is:

%946.212

94.048.775.1207.118.197.879.216.357.112.427.1505.1212

12

1

=

++++++=

==t

At

A

RR

The sample mean monthly return on the market portfolio (the answer to part 1.E) is:

%005.312

15.147.216.646.311.277.643.441.448.441.299.528.1212

12

1

=

++++++++=

==t

mt

m

RR

Using data given in the problem and the above two sample mean monthly returns, we have the following: Month t AAt RR ( )2AAt RR mmt RR ( )2mmt RR ( )( )mmtAAt RRRR

1 9.104 82.883 9.275 86.026 84.44 2 12.324 151.881 2.985 8.910 36.79 3 -7.066 49.928 -0.595 0.354 4.2 4 -1.376 1.893 1.475 2.176 -2.03 5 0.214 0.046 1.405 1.974 0.3 6 -5.736 32.902 1.425 2.031 -8.17 7 -11.916 141.991 -9.775 95.551 116.48 8 -4.126 17.024 -5.115 26.163 21.1 9 -1.876 3.519 0.455 0.207 -0.85

10 9.804 96.118 3.155 9.954 30.93 11 4.534 20.557 -0.535 0.286 -2.43 12 -3.886 15.101 -4.155 17.264 16.15

Sum 0.00 613.84 0.00 250.90 296.91


The sample variance and standard deviation of the stock As monthly return are:

( )

15.5112

84.61312

12

1

2

2 ==

==t

AAt

A

RR

%15.715.51 ==A The sample variance (the answer to part 1.F) and standard deviation of the market portfolios monthly return are:

( )

91.2012

90.25012

12

1

2

2 ==

==t

mmt

m

RR

%57.491.20 ==m The sample covariance of the returns on stock A and the market portfolio is:

( )( )[ ]

74.2412

91.29612

12

1 ==

==t

mmtAAt

Am

RRRR

The sample correlation coefficient of the returns on stock A and the market portfolio (the answer to part 1.D) is:

757.057.415.7

74.24 === mAAm

Am

The sample beta of stock A (the answer to part 1.B) is:

183.191.2074.24

2 ===m

AmA

The sample alpha of stock A (the answer to part 1.A) is: %609.0%005.3183.1%946.2 === mAAA RR Each months sample residual is security As actual return that month minus the return that month predicted by the regression. The regressions predicted monthly return is: mtAAedictedtA RR =Pr,,


The sample residual for each month t is then: edictedtAAtAt RR Pr,,= So we have the following: Month t AtR edictedtAR Pr,, At 2At

1 12.05 13.92 -1.87 3.5 2 15.27 6.48 8.79 77.26 3 -4.12 2.24 -6.36 40.45 4 1.57 4.69 -3.12 9.73 5 3.16 4.61 -1.45 2.1 6 -2.79 4.63 -7.42 55.06 7 -8.97 -8.62 -0.35 0.12 8 -1.18 -3.11 1.93 3.72 9 1.07 3.48 -2.41 5.81

10 12.75 6.68 6.07 36.84 11 7.48 2.31 5.17 26.73 12 -0.94 -1.97 1.02 1.04

Sum: 0.00 262.36 Since the sample residuals sum to 0 (because of the way the sample alpha and beta are calculated), the sample mean of the sample residuals also equals 0 and the sample variance and standard deviation of the sample residuals (the answer to part 1.C) are:

( )

863.2112

36.2621212

12

1

12

12 ===

=== t

Att

AAt

A

%676.4863.21 ==A


Repeating the above analysis for all the stocks in the problem yields: Stock A Stock B Stock C alpha 0.609% 2.964% 3.422% beta 1.183 1.021 2.322 correlation with market 0.757 0.684 0.652 standard deviation of sample residuals* 4.676% 4.983% 12.341% with %005.3=mR and 91.202 =m . *Note that most regression programs use N 2 for the denominator in the sample residual variance formula and use N 1 for the denominator in the other variance formulas (where N is the number of time series observations). As is explained in the text, we have instead used N for the denominator in all the variance formulas. To convert the variance from a regression program to our results, simply multiply the

variance by either N

N 2 or N

N 1 . Chapter 7: Problem 2 A. A.1 The Sharpe single-index model's formula for a security's mean return is

R + = R miii Using the alpha and beta for stock A along with the mean return on the market portfolio from Problem 1 we have: %946.2005.3183.1609.0 =+=AR Similarly: %032.6=BR ; %556.3=CR


The Sharpe single-index model's formula for a security's variance of return is: 2222 imii += Using the beta and residual standard deviation for stock A along with the variance of return on the market portfolio from Problem 1 we have: 14.51676.491.20183.1 222 =+=A Similarly:

62.462 =b ; 0.2652 =c A.2 From Problem 1 we have: %946.2=AR ; %031.6=BR ; %554.3=CR 15.512 =A ; 61.462 =B ; 0.2652 =C B. B.1 According to the Sharpe single-index model, the covariance between the returns on a pair of assets is: 2mjiijSIM = Using the betas for stocks A and B along with the variance of the market portfolio from Problem 1 we have: 254.2591.20021.1183.1 ==ABSIM Similarly: 433.57=ACSIM ; 568.49=BCSIM


B.2 The formula for sample covariance from the historical time series of 12 pairs of returns on security i and security j is:

( )( )

12

12

1=

= t

jjtiit

ij

RRRR

Applying the above formula to the monthly data given in Problem 1 for securities A, B and C gives: 462.18=AB ; 618.61=AC ; 085.54=BC C. C.1 Using the earlier results from the Sharpe single-index model, the mean monthly return and standard deviation of an equally weighted portfolio of stocks A, B and C are:

%18.4%556.331%032.6

31%946.2

31 =++=PR

%348.8

57.493143.57

3125.25

3120.265

3162.46

3115.51

31 222222

=

+

+

+

+

+

=P

C.2 Using the earlier results from the historical data, the mean monthly return and standard deviation of an equally weighted portfolio of stocks A, B and C are:

%18.4%554.331%031.6

31%946.2

31 =++=PR

%374.8

08.543162.61

3146.18

3120.265

3162.46

3115.51

31 222222

=

+

+

+

+

+

=P


D. The slight differences between the answers to parts A.1 and A.2 are simply due to rounding errors. The results for sample mean return and variance from either the Sharpe single-index model formulas or the sample-statistics formulas are in fact identical. The answers to parts B.1 and B.2 differ for sample covariance because the Sharpe single-index model assumes the covariance between the residual returns of securities i and j is 0 (cov(i j ) = 0), and so the single-index form of sample covariance of total returns is calculated by setting the sample covariance of the sample residuals equal to 0. The sample-statistics form of sample covariance of total returns incorporates the actual sample covariance of the sample residuals. The answers in parts C.1 and C.2 for mean returns on an equally weighted portfolio of stocks A, B and C are identical because the Sharpe single-index model formula for the mean return on an individual stock yields a result identical to that of the sample-statistics formula for the mean return on the stock. The answers in parts C.1 and C.2 for standard deviations of return on an equally weighted portfolio of stocks A, B and C are different because the Sharpe single-index model formula for the sample covariance of returns on a pair of stocks yields a result different from that of the sample-statistics formula for the sample covariance of returns on a pair of stocks. Chapter 7: Problem 3 Recall from the text that the Vasicek techniques forecast of security is beta ( 2i ) is:

121

21

21

121

21

21

2 iii

ii

+++=

where 1 is the average beta across all sample securities in the historical period (in this problem referred to as the market beta), 1i is the beta of security i in the historical period, 21 is the variance of all the sample securities betas in the historical period and 21i is the square of the standard error of the estimate of beta for security i in the historical period. If the standard errors of the estimates of all the betas of the sample securities in the historical period are the same, then, for each security i, we have: ai =21 where a is a constant across all the sample securities.


Therefore, we have for any security i:

( ) 11121

21

121

2 1 iii XXaaa

+=+++=

This shows that, under the assumption that the standard errors of all historical betas are the same, the forecasted beta for any security using the Vasicek technique is a simple weighted average (proportional weighting) of 1 (the market beta) and 1i (the securitys historical beta), where the weights are the same for each security. Chapter 7: Problem 4 Letting the historical period of the year of monthly returns given in Problem 1 equal 1 (t = 1), then the forecast period equals 2 and the Blume forecast equation is: 12 60.041.0 ii += Using the earlier answer to Problem 1 for the estimate of beta from the historical period for stock A along with the above equation we obtain the stocks forecasted beta: 120.1183.160.041.060.041.0 12 =+=+= AA Similarly: 023.12 =B ; 803.12 =C Chapter 7: Problem 5 A. The single-index model's formula for security i's mean return is

R + = R miii Since Rm equals 8%, then, e.g., for security A we have:

%=+ =

x + =R + = R mAAA

14122

85.12


Similarly:

%4.13=BR ; %4.7=CR ; %2.11=DR B. The single-index model's formula for security i's own variance is:

. + = 2e2m

2i

2i i

Since m = 5, then, e.g., for security A we have:

( ) ( ) ( )25.65

355.1 222

= =

+ = 2e2m

2A

2A A

+

Similarly: 2B = 43.25; 2C = 20; 2D = 36.25 C. The single-index model's formula for the covariance of security i with security j is

2mjijiij = = Since 2m = 25, then, e.g., for securities A and B we have:

75.48

253.15.1== = 2mBAAB

Similarly: AC = 30; AD = 33.75; BC = 26; BD = 29.25; CD = 18 Chapter 7: Problem 6 A. Recall that the formula for a portfolio's beta is:

iiN

1 = iP X =

The weight for each asset (Xi) in an equally weighted portfolio is simply 1/N, where N is the number of assets in the portfolio.


Since there are four assets in Problem 5, N = 4 and Xi equals 1/4 for each asset in an equally weighted portfolio of those assets. So:

( )

125.1

5.441

9.08.03.15.141

41

41

41

41

=

=

=

= DCBAP

+++

+++

B. Recall that the definition of a portfolio's alpha is:

iiN

1 = iP X =

Using 1/4 as the weight for each asset, we have:

( )

5.2

1041

413241

41

41

41

41

=

=

=

= DCBAP

+++

+++

C. Recall that a formula for a portfolios variance using the single-index model is:

=

+=N

ieimPP iX

1

22222 Using 1/4 as the weight for each asset, we have:

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

52.33

1641916125125.1

4412

411

413

415125.1

2

22

22

22

22

222

=++++=

+

+

+

+=P


D. Using the single-index models formula for a portfolios mean return we have:

%5.11

8125.15.2=

+=+= mPPP RR

Chapter 7: Problem 7 Using 12 677.0343.0 ii += and the historical betas given in Problem 5 we can forecast, e.g., the beta for security A:

3585.10155.1343.0

5.1677.0343.0677.0343.0 12

=+=

+=+= AA

Similarly: 2231.12 =B ; 8846.02 =C ; 9523.02 =D Chapter 7: Problem 8 Using the historical betas given in Problem 5 and Vasiceks formula, we can forecast, e.g., the beta of security A:

( )( ) ( )

( )( ) ( )

2932.18795.04137.0

5.15863.014137.0

5.10441.00625.0

0625.010441.00625.0

0441.0

5.121.025.0

25.0121.025.0

21.022

2

22

2

121

21

21

121

21

21

2

=+=

+=+++=

+++=

+++= AAAA

A

Similarly: 1137.12 =B ; 8683.02 =C ; 9390.02 =D



Solutions to Text Problems: Chapter 8 Chapter 8: Problem 1 Given the correlation coefficient of the returns on a pair of securities i and j, the securities covariance can be expressed as the securities correlation coefficient times the product of their standard deviations:

jiijij = But if we assume that all pairs of securities have the same constant correlation, * , then the constant-correlation expression for covariance is:

jiijCC *= Given the assumptions of the Sharpe single-index model, the single-index models expression for the covariance between the returns on a pair of securities is:

jijmim

mm

mjjm

m

miim

mm

jm

m

im

mjiijSIM

==

==

222

222

2

If the assumptions of both the constant correlation and single-index model hold, then we have ijij SIMCC = :

jijmimji =* or jmim =* This must hold for all pairs of securities, including i and j, i and k and j and k. So we have:

jmim =* kmim =* kmjm =*

The only solution to the above set of equations is: * === kmjmim

Therefore, for any security i we have:

imm

iim

m

miim

m

imi

====

*

22

In other words, given that all pairs of securities have the same correlation coefficient and that the Sharpe single-index model holds, each securitys beta is proportional to its standard deviation, where the proportion is a constant across all

securities equal to m *

.


Chapter 8: Problem 2 Start with a general 3-index model of the form:

iiiiii cIbIbIbaR ++++= *3*3*2*2*1*1* (1)

Set 1*1 II = and define an index I2 which is orthogonal to I1 as follows:

tdII ++= 110*2 or ( )110*22 IIdI t +==

which gives:

2110*2 III ++=

Substituting the above expression into equation (1) and rearranging we get: ( ) ( ) iiiiiiii cIbIbIbbbaR ++++++= *3*32*211*2*10*2* The first term in the above equation is a constant, which we can define as 1a . The coefficient in the second term of the above equation is also a constant, which we can define as 1ib . We can then rewrite the above equation as:

iiiiii cIbIbIbaR ++++= *3*32*211 (2) Now define an index I3 which is orthogonal to I1 and I2 as follows:

teIII +++= 22110*3 or ( )22110*33 IIIeI t ++== which gives:

322110*3 IIII +++=

Substituting the above expression into equation (2) and rearranging we get: ( ) iiiiiiiii cIbIbbIbbbaR ++++ ++ += 3*322*3*2113103 In the above equation, the first term and all the coefficients of the new orthogonal indices are constants, so we can rewrite the equation as:

iiiiii cIbIbIbaR ++++= 332211


Chapter 8: Problem 3 Recall from the earlier chapter on the single-index model that an expression for the covariance of returns on two securities i and j is: ( ) ( )[ ] ( )[ ] [ ]jimmjimmijmmjiij eeRReRReRR E E E E 2 +++ = The first term contains the variance of the market portfolio, the second two terms contain the covariance of the market portfolio with the residuals and the last term is the covariance of the residuals. Given that one of the models assumptions is that the covariance of the market portfolio with the residuals is zero and that, from the problem, the covariance of the residuals equals a constant K, the derived covariance between the two securities is:

Kmjiij += 2 One expression for the variance of a portfolio is:

= ==

+=N

j

N

jkk

jkkj

N

iiiP XXX

1 11

222

Recalling that the single-index models expression for the variance of a security is

2222eimii += and substituting that expression and the derived expression for

covariance into the above equation and rearranging gives:

++=

++

=

++=

+++=

= ==

= ====

= === =

= = ====

N

j

N

jkk

kj

N

ieiimP

N

j

N

jkk

kj

N

ieiim

N

iii

N

iii

N

j

N

jkk

kj

N

ieii

N

i

N

jmjiji

N

j

N

j

N

jkk

kj

N

jkk

mkjkj

N

ieii

N

imiiP

XXKX

KXXXXX

KXXXXX

KXXXXXX

1 11

2222

1 11

222

11

1 11

22

1 1

2

1 1 11

2

1

22

1

2222


Chapter 8: Problem 4 Using the result from Problem 2, we have:

iiiiii cIbIbIbaR ++++= 332211 Since the residual ci always has a mean of zero (by construction if necessary), we have the following expression for expected return:

332211 IbIbIbaR iiiii +++= The variance formula is: ( )( )

( ) ( ) ( )( ) +++=

+++++++=2

333222111

2332211332211

2

E

E

iiii

iiiiiiiiii

cIIbIIbIIb

IbIbIbacIbIbIba

Carrying out the squaring, noting that the indices are all orthogonal with each other and making the usual assumption that the residual is uncorrelated with any index gives us:

223

23

22

22

21

21

2ciIiIiIii bbb +++=

The covariance formula is: ( )( )( )( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ]jjjjiiiijjjjjjjjj

iiiiiiiiii

cIIbIIbIIbcIIbIIbIIb

IbIbIbacIbIbIba

IbIbIbacIbIbIba

++++++=

++++++++++++++=

333222111333222111

332211332211

3322113322112

E

E

Carrying out the multiplication, noting that the indices are all orthogonal with each other, making the usual assumption that the residuals are uncorrelated with any index and assuming that the residuals are uncorrelated with each other gives us:

2333

2222

2111 IjiIjiIjiij bbbbbb ++=


Chapter 8: Problem 5 The formula for a security's expected return using a general two-index model is:

2211 IbIbaR iiii ++=

Using the above formula and data given in the problem, the expected return for, e.g., security A is:

%1249.088.02

2211

=++=

++= IbIbaR AAAA

Similarly:

%17=BR ; %6.12=CR The two-index models formula for a securitys own variance is:

222

22

21

21

2ciIiIii bb ++=

Using the above formula, the variance for, e.g., security A is:

( ) ( ) ( ) ( ) ( )6225.1140625.556.2

25.29.028.0 22222

222

22

21

21

2

=++=++=

+= cAIAIAA bb

Similarly, 2B = 16.4025, and 2C = 13.0525. C. The two-index model's formula for the covariance of security i with security j is:

2222

2111 IjiIjiijj bbbb +=

Using the above formula, the covariance of, e.g., security A with security B is:

( )( )( ) ( )( )( )8325.103125.752.3

5.23.19.021.18.0 222222

2111

=+=+=

+= IBAIBAAB bbbb

Similarly, AC = 9.0675, and BC = 12.8975.


Chapter 8: Problem 6 For an industry-index model, the text gives two formulas for the covariance between securities i and k. If firms i and k are both in industry j, the covariance between their securities' returns is given by:

22Ijkjijmkmimik bbbb +=

Otherwise, if the firms are in different industries, the covariance of their securities' returns is given by:

2mkmimik bb =

If only firms A and B are in the same industry, then:

( )( )( ) ( )( )( )8325.103125.752.3

5.23.19.021.18.0 222222

2

=+=+=+= IBAmBmAmAB bbbb

The second formula should be used for the other pairs of firms:

( )( )( ) 88.229.08.0 22

=== mCmAmAC bb

( )( )( ) 96.329.01.1 22

=== mCmBmBC bb

Chapter 8: Problem 7

The answers for this problem are found in the same way as the answers for problem 6, except that now only firms B and C are in the same industry. So for firms B and C, the covariance between their securities' returns is:

( )( )( ) ( )( )( )8975.129375.896.3

5.21.13.129.01.1 22

2222

2

=+=+=+= ICBmCmBmBC bbbb


The other formula should be used for the other pairs of firms:

( )( )( ) 52.321.18.0 22

=== mBmAmAB bb

( )( )( ) 88.229.08.0 22

=== mCmAmAC bb

Chapter 8: Problem 8 To answer this problem, use the procedure described in Appendix A of the text. First, I1 is defined as being equal to I*1 , then I*2 is regressed on I1 to obtain the given regression equation. Since dt is uncorrelated with I1 by the techniques of regression analysis, dt is an orthogonal index to I1. So, define I2 = dt. Then express the given regression equation as: I*2 = 1 + 1.3 I1 + I2. Now, substitute the above equation for I*2 into the given multi-index model and simplify: iR = 2 + 1.1 I*1 + 1.2 I*2 + ci = 2 + 1.1 I1 + 1.2 (1 + 1.3 I1 + I2) + ci = 2 + 1.1 I1 + 1.2 + 1.56 I1 + 1.2 I2 + ci = 3.2 + 2.66 I1 + 1.2 I2 + ci The two-index model has now been transformed into one with orthogonal indices I1 and I2, where I1 = I*1, and I2 = dt = I*2 - 1 - 1.3 I1.




Chapter 9: Problem 1 In the table below, given that the riskless rate equals 5%, the securities are ranked in descending order by their excess return over beta.

Security Rank i Fi RR i

Fi RR ( )2

ei

iFi RR

22

ei

i

( )

=

i

j ej

jFj RR

12

=

i

j ej

j

12

2

Ci

1 1 10 10.0000 0.3333 0.0333 0.3333 0.0333 2.5000 6 2 9 6.0000 1.3500 0.2250 1.6833 0.2583 4.6980 2 3 7 4.6667 0.5250 0.1125 2.2083 0.3708 4.6910 5 4 4 4.0000 0.2000 0.0500 2.4083 0.4208 4.6242 4 5 3 3.7500 0.2400 0.0640 2.6483 0.4848 4.5286 3 6 6 3.0000 0.3000 0.1000 2.9483 0.5848 4.3053

The numbers in the column above labeled Ci were obtained by recalling from the text that, if the Sharpe single-index model holds: ( )

+

=

=

=

i

j ej

jm

i

j ej

jFjm

i

RR

C

12

22

12

2

1

Thus, given that 2m = 10: 500.2

333.1333.3

0333.01013333.010

1 ==+=C

698.4583.3833.16

2583.01016833.110

2 ==+=C

etc.

With no short sales, we only include those securities for which ii

Fi CRR > . Thus, only securities 1 and 6 (the highest and second highest ranked securities in the above table) are in the optimal (tangent) portfolio. We could have stopped our

calculations after the first time we found a ranked security for which ii

Fi CRR


Since security 6 (the second highest ranked security, where i = 2) is the last ranked

security in descending order for which ii

Fi CRR > , we set C* = C2 = 4.698 and solve for the optimum portfolios weights using the following formulas:

= *2 C

RRZi

Fi

ei

ii

=

= 21i

i

ii

Z

ZX

This gives us:

( ) 1767.0698.410301

1 =

=Z

( ) 1953.0698.4610

5.12 =

=Z

3720.01953.01767.021 =+=+ ZZ

475.03720.01767.0

1 ==X

525.03720.01953.0

2 ==X Since i = 1 for security 1 and i = 2 for security 6, the optimum (tangent) portfolio when short sales are not allowed consists of 47.5% invested in security 1 and 52.5% invested in security 6. Chapter 9: Problem 2 This problem uses the same input data as Problem 1. When short sales are allowed, all securities are included and C* is equal to the value of Ci for the lowest ranked security. Referring back to the table given in the answer to Problem 1, we see that the lowest ranked security is security 3, where i = 6. Therefore, we have C* = C6 = 4.3053.


To solve for the optimum portfolios weights, we use the following formulas:

= *2 C

RRZi

Fi

ei

ii

and

=

= 61i

i

ii

Z

ZX (for the standard definition of short sales)

or

=

= 61i

i

ii

Z

ZX (for the Lintner definition of short sales)

So we have:

( ) 1898.03053.410301

1 =

=Z

( ) 2542.03053.4610

5.12 =

=Z

( ) 0271.03053.4667.420

5.13 =

=Z

( ) 0153.03053.44201

4 =

=Z

( ) 0444.03053.475.310

8.05 =

=Z

( ) 0653.03053.4340

0.26 =

=Z

3461.00653.00444.00153.00271.02542.01898.06

1=++=

=iiZ

5961.00653.00444.00153.00271.02542.01898.06

1=+++++=

=iiZ


This gives us the following weights (by rank order) for the optimum portfolios under either the standard definition of short sales or the Lintner definition of short sales: Standard Definition Lintner Definition

Security 1 (i = 1) 5484.03461.01898.0

1 ==X 3184.05961.01898.0

1 ==X

Security 6 (i = 2) 7345.03461.02542.0

2 ==X 4264.05961.02542.0

2 ==X

Security 2 (i = 3) 0783.03461.00271.0

3 ==X 0455.05961.00271.0

3 ==X

Security 5 (i = 4) 0442.03461.00153.0

4 ==X 0257.05961.00153.0

4 ==X

Security 4 (i = 5) 1283.03461.00444.0

5 ==X 0745.05961.00444.0

5 ==X

Security 3 (i = 6) 1887.03461.00653.0

6 ==X 1095.05961.00653.0

6 ==X Chapter 9: Problem 3 With short sales allowed but no riskless lending or borrowing, the optimum portfolio depends on the investors utility function and will be found at a point along the upper half of the minimum-variance frontier of risky assets, which is the efficient frontier when riskless lending and borrowing do not exist. As is described in the text, the entire efficient frontier of risky assets can be delineated with various combinations of any two efficient portfolios on the frontier. One such efficient portfolio was found in Problem 2. By simply solving Problem 2 using a different value for RF , another portfolio on the efficient frontier can be found and then the entire efficient frontier can be traced using combinations of those two efficient portfolios.


Chapter 9: Problem 4 In the table below, given that the riskless rate equals 5%, the securities are ranked in descending order by their excess return over standard deviation.

Security Rank i Fi RR i

Fi RR ( )

=

i

j j

Fj RR

1

i+1 Ci

1 1 10 1.00 1.00 0.5000 0.5000 2 2 15 1.00 2.00 0.3333 0.6667 5 3 5 1.00 3.00 0.2500 0.7500 6 4 9 0.90 3.90 0.2000 0.7800 4 5 7 0.70 4.60 0.1667 0.7668 3 6 13 0.65 5.25 0.1429 0.7502 7 7 11 0.55 5.80 0.1250 0.7250

The numbers in the column above labeled Ci were obtained by recalling from the text that, if the constant-correlation model holds: ( )

+= =i

j j

Fji

RRi

C11

Thus, given that = 0.5 for all pairs of securities: 5000.00.15.01 ==C 6667.00.23333.02 ==C etc.

With no short sales, we only include those securities for which ii

Fi CRR > . Thus, only securities 1, 2, 5 and 6 (the four highest ranked securities in the above table) are in the optimal (tangent) portfolio. We could have stopped our calculations

after the first time we found a ranked security for which ii

Fi CRR


Since security 6 (the fourth highest ranked security, where i = 4) is the last ranked

security in descending order for which ii

Fi CRR > , we set C* = C4 = 0.78 and solve for the optimum portfolios weights using the following formulas:

( )

=*

11 CRRZ

i

Fi

ii

=

= 41i

i

ii

Z

ZX

This gives us:

( )( ) ( ) 0440.078.01105.01

1 =

=Z

( )( ) ( ) 0293.078.01155.01

2 =

=Z

( )( ) ( ) 0880.078.0155.01

3 =

=Z

( )( ) ( ) 0240.078.09.0105.01

4 =

=Z

1853.00240.00880.00293.00440.04321 =+++=+++ ZZZZ

2375.01853.00440.0

1 ==X

1581.01853.00293.0

2 ==X

4749.01853.00880.0

3 ==X

1295.01853.00240.0

4 ==X Since i = 1 for security 1, i = 2 for security 2, i = 3 for security 5 and i = 4 for security 6, the optimum (tangent) portfolio when short sales are not allowed consists of 23.75% invested in security 1, 15.81% % invested in security 2, 47.49% % invested in security 5 and 12.95% invested in security 6.


Chapter 9: Problem 5 This problem uses the same input data as Problem 4. When short sales are allowed, all securities are included and C* is equal to the value of Ci for the lowest ranked security. Referring back to the table given in the answer to Problem 4, we see that the lowest ranked security is security 7, where i = 7. Therefore, we have C* = C7 = 0.725. To solve for the optimum portfolios weights, we use the following formulas:

( )

=*

11 CRRZ

i

Fi

ii

and

=

= 71i

i

ii

Z

ZX (for the standard definition of short sales)

or

=

= 71i

i

ii

Z

ZX (for the Lintner definition of short sales)

So we have:

( )( ) ( ) 0550.0725.01105.01

1 =

=Z

( )( ) ( ) 0367.0725.01155.01

2 =

=Z

( )( ) ( ) 1100.0725.0155.01

3 =

=Z

( )( ) ( ) 0350.0725.09.0105.01

4 =

=Z

( )( ) ( ) 0050.0725.07.0105.01

5 =

=Z

( )( ) ( ) 0075.0725.065.0205.01

6 =

=Z

( )( ) ( ) 0175.0725.055.0205.01

7 =

=Z

2067.00175.00075.00050.00350.01100.00367.00550.07

1=+++=

=iiZ

2667.00175.00075.00050.00350.01100.00367.00550.07

1=++++++=

=iiZ


This gives us the following weights (by rank order) for the optimum portfolios under either the standard definition of short sales or the Lintner definition of short sales: Standard Definition Lintner Definition

Security 1 (i = 1) 2661.02067.00550.0

1 ==X 2062.02667.00550.0

1 ==X

Security 2 (i = 2) 1776.02067.00367.0

2 ==X 1376.02667.00367.0

2 ==X

Security 5 (i = 3) 5322.02067.01100.0

3 ==X 4124.02667.01100.0

3 ==X

Security 6 (i = 4) 1703.02067.00350.0

4 ==X 1312.02667.00350.0

4 ==X

Security 4 (i = 5) 0242.02067.00050.0

5 ==X 0187.02667.00050.0

5 ==X

Security 3 (i = 6) 0363.02067.00075.0

6 ==X 0281.02667.00075.0

6 ==X

Security 7 (i = 7) 0847.02067.00175.0

7 ==X 0656.02667.00175.0

7 ==X Chapter 9: Problem 6 With short sales allowed but no riskless lending or borrowing, the optimum portfolio depends on the investors utility function and will be found at a point along the upper half of the minimum-variance frontier of risky assets, which is the efficient frontier when riskless lending and borrowing do not exist. As is described in the text, the entire efficient frontier of risky assets can be delineated with various combinations of any two efficient portfolios on the frontier. One such efficient portfolio was found in Problem 5. By simply solving Problem 5 using a different value for RF , another portfolio on the efficient frontier can be found and then the entire efficient frontier can be traced using combinations of those two efficient portfolios.




Chapter 11: Problem 1 Expected utility of investment A = 1/3 7.5 + 1/3 12.5 + 1/3 31.5 = 17.0 Expected utility of investment B = 1/4 4.0 + 1/2 17.5 + 1/4 40.0 = 19.75 Expected utility of investment C = 1/5 0.5 + 3/5 31.5 + 1/5 144.0 = 47.8 Investment A is preferred because it has the highest level of expected utility. Chapter 11: Problem 2 Expected utility of investment A = 1/3 0.45 + 1/3 0.41 + 1/3 0.33 = 0.40 Expected utility of investment B = 1/4 0.50 + 1/2 0.38 + 1/4 0.32 = 0.39 Expected utility of investment C = 1/5 1 + 3/5 0.33 + 1/5 0.24 = 0.45 Investment B is preferred because it has the highest level of expected utility. Chapter 11: Problem 3 Expected utility of investment A = 2/5 8.96 + 1/5 14 + 2/5 21.84 = 15.12 Expected utility of investment B = 1/2 6 + 1/4 17.76 + 1/4 36 = 16.44 Investment A is preferred because it has the highest level of expected utility. Chapter 11: Problem 4 For an investor to be indifferent, the expected utility of investment B must be set equal to that of investment A. Referring back to Problem 3, we see that the given probabilities are for the first two of the three outcomes in investment B. So we need to solve for the probabilities of those two outcomes that make investment Bs expected utility level equal to that of As. Since the last outcome in investment B has a probability of 1/4, the first two probabilities must sum to 3/4. Therefore we have: X 6 + (3/4 X) 17.76 + 1/4 36 = 15.12


Solving for X: X = 0.61 Therefore, the first outcomes probability of 0.5 would have to be increased by 0.11 to 0.61, and the second outcomes probability of 0.25 would have to be reduced by 0.11 to 0.14. Chapter 11: Problem 5 The investor will prefer the investment that maximizes expected utility of terminal wealth. Recall that the formula for expected utility of wealth (E[U(W)]) is: ( )[ ] ( ) ( ) =

WWPWUWUE

where each P(W) is the probability associated with each particular outcome of wealth (W). Since ( ) 205.0 WWWU = , we have: Investment A:

( )[ ] ( ) ( ) ( )525.4

3.055.055.42.075.33.01005.0105.0705.072.0505.05E 222

=++=

++=WU

Investment B:

( )[ ] ( ) ( ) ( )635.4

1.095.46.08.43.02.41.0905.098.0805.083.0605.06E 222

=++=

++=WU

Investment B is preferred over investment A since B provides higher expected utility. Chapter 11: Problem 6 To solve this problem, set the expected utility of investment A in Problem 5 equal to 4.635 (the expected utility of investment B) and solve for the value of the first outcome in investment A: ( ) ( ) ( ) 635.43.01005.0105.0705.072.005.0 222 =++ XX 635.45.1275.201.2.0 2 =++ XX 086202 =+ XX


The equation above is a quadratic equation with two roots. Using the quadratic formula, the roots are found to be 6.26 and 13.74. So, the minimum amount that the first outcome of investment A would have to change by for the investor to be indifferent between investments A and B would be $6.26 $5 = $1.26 (an increase), since both investments would then provide the same level of expected utility. Chapter 11: Problem 7 Roys safety-first criterion is to minimize Prob(RP < RL). If RL = 5%, then (assuming an initial investment of $100) for the outcomes in Problem 1 we have: Prob(RA < 5%) = 0.0 Prob(RB < 5%) = 0.25 Prob(RC < 5%) = 0.20 Thus, using Roys safety-first criterion, investment in A is preferred over investments in B and C, and investment in C is preferred over investment in B. Chapter 11: Problem 8 Kataoka's safety-first criterion is to maximize RL subject to Prob(RP < RL) . If = 10%, then (assuming an initial investment of $100) for the outcomes in Problem 1 the maximum RL is: 4.99% for A 3.99% for B 0.99% for C Thus, A is preferred to B and C, and B is preferred to C. Chapter 11: Problem 9 Employing Telser's criterion, we see that (assuming an initial investment of $100) Projects A, B and C in Problem 1 do not satisfy the constraint Prob(Rp 5%) 10%. So, investments A, B, and C are indistinguishable using Telsers criterion with RL = 5%.

Elton, Gruber, Brown, and Goetzmann 11-4 Modern Portfolio Theory and Investment Analysis, 7th Edition Solutions To Text Probl

121834193 Elton Gruber 7e Solution Manual

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