Presentation is loading. Please wait.

Presentation is loading. Please wait.

1 1 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole.

Similar presentations


Presentation on theme: "1 1 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole."— Presentation transcript:

1 1 1 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. SLIDES BY SLIDES BY John Loucks St. Edward’s Univ.

2 2 2 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 12 Advanced Optimization Applications n Data Envelopment Analysis n Revenue Management n Portfolio Models and Asset Allocation n Nonlinear Optimization n Constructing an Index Fund

3 3 3 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n Data envelopment analysis (DEA) is an LP application used to determine the relative operating efficiency of units with the same goals and objectives. n DEA creates a fictitious composite unit made up of an optimal weighted average ( W 1, W 2,…) of existing units. n An individual unit, k, can be compared by determining E, the fraction of unit k ’s input resources required by the optimal composite unit. n If E < 1, unit k is less efficient than the composite unit and be deemed relatively inefficient. n If E = 1, there is no evidence that unit k is inefficient, but one cannot conclude that k is absolutely efficient.

4 4 4 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n The DEA Model Min E s.t.Weighted outputs > Unit k ’s output (for each measured output) Weighted inputs < E [Unit k ’s input] (for each measured input) Sum of weights = 1 E, weights > 0

5 5 5 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis The Langley County School District is trying to determine the relative efficiency of its three high schools. In particular, it wants to evaluate Roosevelt High. The district is evaluating performances on SAT scores, the number of seniors finishing high school, and the number of students who enter college as a function of the number of teachers teaching senior classes, the prorated budget for senior instruction, and the number of students in the senior class.

6 6 6 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n n Input Roosevelt Lincoln Washington Roosevelt Lincoln Washington Senior Faculty Budget ($100,000's) Senior Enrollments

7 7 7 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n n Output Roosevelt Lincoln Washington Roosevelt Lincoln Washington Average SAT Score High School Graduates High School Graduates College Admissions College Admissions

8 8 8 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n Define the Decision Variables E = Fraction of Roosevelt's input resources required by the composite high school E = Fraction of Roosevelt's input resources required by the composite high school w 1 = Weight applied to Roosevelt's input/output resources by the composite high school w 2 = Weight applied to Lincoln’s input/output resources by the composite high school w 3 = Weight applied to Washington's input/output resources by the composite high school

9 9 9 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n Define the Objective Function Minimize the fraction of Roosevelt High School's input resources required by the composite high school: Min E Min E

10 10 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n Define the Constraints Sum of the Weights is 1: (1) w 1 + w 2 + w 3 = 1 (1) w 1 + w 2 + w 3 = 1 Output Constraints: Output Constraints: Since w 1 = 1 is possible, each output of the composite school must be at least as great as that of Roosevelt: Since w 1 = 1 is possible, each output of the composite school must be at least as great as that of Roosevelt: (2) 800 w w w 3 > 800 (SAT Scores) (2) 800 w w w 3 > 800 (SAT Scores) (3) 450 w w w 3 > 450 (Graduates) (3) 450 w w w 3 > 450 (Graduates) (4) 140 w w w 3 > 140 (College Admissions) (4) 140 w w w 3 > 140 (College Admissions)

11 11 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n Define the Constraints (continued) Input Constraints: The input resources available to the composite school is a fractional multiple, E, of the resources available to Roosevelt. Since the composite high school cannot use more input than that available to it, the input constraints are: The input resources available to the composite school is a fractional multiple, E, of the resources available to Roosevelt. Since the composite high school cannot use more input than that available to it, the input constraints are: (5) 37 w w w 3 < 37 E (Faculty) (6) 6.4 w w w 3 < 6.4 E (Budget) (6) 6.4 w w w 3 < 6.4 E (Budget) (7) 850 w w w 3 < 850 E (Seniors) (7) 850 w w w 3 < 850 E (Seniors) Nonnegativity of variables: Nonnegativity of variables: E, w 1, w 2, w 3 > 0 E, w 1, w 2, w 3 > 0

12 12 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n Computer Solution Objective Function Value = Variable Value Reduced Cost Variable Value Reduced Cost E W W W W W W

13 13 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Data Envelopment Analysis n Conclusion The output shows that the composite school is made up of equal weights of Lincoln and Washington. Roosevelt is 76.5% efficient compared to this composite school when measured by college admissions (because of the 0 slack on this constraint #4). It is less than 76.5% efficient when using measures of SAT scores and high school graduates (there is positive slack in constraints 2 and 3.)

14 14 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management n Another LP application is revenue management. n Revenue management involves managing the short- term demand for a fixed perishable inventory in order to maximize revenue potential. n The methodology was first used to determine how many airline seats to sell at an early-reservation discount fare and many to sell at a full fare. n Application areas now include hotels, apartment rentals, car rentals, cruise lines, and golf courses.

15 15 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management LeapFrog Airways provides passenger service for LeapFrog Airways provides passenger service for Indianapolis, Baltimore, Memphis, Austin, and Tampa. LeapFrog has two WB828 airplanes, one based in Indianapolis and the other in Baltimore. Each morning the Indianapolis based plane flies to Austin with a stopover in Memphis. The Baltimore based plane flies to Tampa with a stopover in Memphis. Both planes have a coach section with a 120-seat capacity.

16 16 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. LeapFrog uses two fare classes: a discount fare D LeapFrog uses two fare classes: a discount fare D class and a full fare F class. Leapfrog’s products, each referred to as an origin destination itinerary fare (ODIF), are listed on the next slide with their fares and forecasted demand. LeapFrog wants to determine how many seats it should allocate to each ODIF. LeapFrog wants to determine how many seats it should allocate to each ODIF. Revenue Management

17 17 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. ODIF OriginIndianapolisIndianapolisIndianapolisIndianapolisIndianapolisIndianapolisBaltimoreBaltimoreBaltimoreBaltimoreBaltimoreBaltimoreMemphisMemphisMemphisMemphisDestinationMemphisAustinTampaMemphisAustinTampaMemphisAustinTampaMemphisAustinTampaAustin Tampa Austin Tampa FareClassDDDFFFDDDFFFDDFFODIFCodeIMDIADITDIMFIAFITFBMDBADBTDBMFBAFBTFMADMTDMAFMTF Fare Demand Revenue Management

18 18 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management n Define the Decision Variables There are 16 variables, one for each ODIF: IMD = number of seats allocated to Indianapolis-Memphis- Discount class Discount class IAD = number of seats allocated to Indianapolis-Austin- Discount class ITD = number of seats allocated to Indianapolis-Tampa- Discount class IMF = number of seats allocated to Indianapolis-Memphis- Full Fare class IAF = number of seats allocated to Indianapolis-Austin-Full Fare class

19 19 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management n Define the Decision Variables (continued) ITF = number of seats allocated to Indianapolis-Tampa- Full Fare class Full Fare class BMD = number of seats allocated to Baltimore-Memphis- Discount class Discount class BAD = number of seats allocated to Baltimore-Austin- Discount class Discount class BTD = number of seats allocated to Baltimore-Tampa- Discount class Discount class BMF = number of seats allocated to Baltimore-Memphis- Full Fare class Full Fare class BAF = number of seats allocated to Baltimore-Austin- Full Fare class Full Fare class

20 20 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management n Define the Decision Variables (continued) BTF = number of seats allocated to Baltimore-Tampa- Full Fare class Full Fare class MAD = number of seats allocated to Memphis-Austin- Discount class Discount class MTD = number of seats allocated to Memphis-Tampa- Discount class Discount class MAF = number of seats allocated to Memphis-Austin- Full Fare class Full Fare class MTF = number of seats allocated to Memphis-Tampa- Full Fare class Full Fare class

21 21 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management n Define the Objective Function Maximize total revenue: Max (fare per seat for each ODIF) Max (fare per seat for each ODIF) x (number of seats allocated to the ODIF) x (number of seats allocated to the ODIF) Max 175IMD + 275IAD + 285ITD + 395IMF + 425IAF + 475ITF + 185BMD + 315BAD + 425IAF + 475ITF + 185BMD + 315BAD + 290BTD + 385BMF + 525BAF + 490BTF + 290BTD + 385BMF + 525BAF + 490BTF + 190MAD + 180MTD + 310MAF + 295MTF + 190MAD + 180MTD + 310MAF + 295MTF

22 22 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management n Define the Constraints There are 4 capacity constraints, one for each flight leg: Indianapolis-Memphis leg Indianapolis-Memphis leg (1) IMD + IAD + ITD + IMF + IAF + ITF < 120 (1) IMD + IAD + ITD + IMF + IAF + ITF < 120 Baltimore-Memphis leg Baltimore-Memphis leg (2) BMD + BAD + BTD + BMF + BAF + BTF < 120 (2) BMD + BAD + BTD + BMF + BAF + BTF < 120 Memphis-Austin leg Memphis-Austin leg (3) IAD + IAF + BAD + BAF + MAD + MAF < 120 (3) IAD + IAF + BAD + BAF + MAD + MAF < 120 Memphis-Tampa leg Memphis-Tampa leg (4) ITD + ITF + BTD + BTF + MTD + MTF < 120 (4) ITD + ITF + BTD + BTF + MTD + MTF < 120

23 23 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management n Define the Constraints (continued) There are 16 demand constraints, one for each ODIF: (5) IMD < 44(11) BMD < 26(17) MAD < 5 (5) IMD < 44(11) BMD < 26(17) MAD < 5 (6) IAD < 25(12) BAD < 50(18) MTD < 48 (6) IAD < 25(12) BAD < 50(18) MTD < 48 (7) ITD < 40(13) BTD < 42(19) MAF < 14 (7) ITD < 40(13) BTD < 42(19) MAF < 14 (8) IMF < 15(14) BMF < 12(20) MTF < 11 (8) IMF < 15(14) BMF < 12(20) MTF < 11 (9) IAF < 10(15) BAF < 16 (9) IAF < 10(15) BAF < 16 (10) ITF < 8(16) BTF < 9 (10) ITF < 8(16) BTF < 9

24 24 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Revenue Management n Computer Solution Objective Function Value = Variable Value Reduced Cost IMD IMD IAD IAD ITD ITD IMF IMF IAF IAF ITF ITF BMD BMD BAD BAD

25 25 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Portfolio Models and Asset Management n Asset allocation involves determining how to allocate investment funds across a variety of asset classes such as stocks, bonds, mutual funds, real estate. n Portfolio models are used to determine percentage of funds that should be made in each asset class. n The goal is to create a portfolio that provides the best balance between risk and return.

26 26 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. John Sweeney is an investment advisor who is John Sweeney is an investment advisor who is attempting to construct an "optimal portfolio" for a client who has $400,000 cash to invest. There are ten different investments, falling into four broad categories that John and his client have identified as potential candidate for this portfolio. The investments and their important characteristics are listed in the table on the next slide. Note that Unidyde Corp. under Equities and Unidyde Corp. under Debt are two separate investments, whereas First General REIT is a single investment that is considered both an equities and a real estate investment. The investments and their important characteristics are listed in the table on the next slide. Note that Unidyde Corp. under Equities and Unidyde Corp. under Debt are two separate investments, whereas First General REIT is a single investment that is considered both an equities and a real estate investment. Portfolio Model

27 27 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Portfolio Model Exp. Annual Exp. Annual After Tax Liquidity Risk After Tax Liquidity Risk Category Investment Return Factor Factor Equities Unidyde Corp. 15.0% (Stocks)CC’s Restaurants 17.0% First General REIT 17.5% First General REIT 17.5% Debt Metropolis Electric 11.8% (Bonds) Unidyde Corp. 12.2% Lewisville Transit 12.0% Lewisville Transit 12.0% Real Estate Realty Partners 22.0% 0 50 First General REIT ( --- See above --- ) First General REIT ( --- See above --- ) Money T-Bill Account 9.6% 80 0 Money Mkt. Fund 10.5% Money Mkt. Fund 10.5% Saver's Certificate 12.6% 0 0 Saver's Certificate 12.6% 0 0

28 28 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Portfolio Model Formulate a linear programming problem to Formulate a linear programming problem to accomplish John's objective as an investment advisor which is to construct a portfolio that maximizes his client's total expected after-tax return over the next year, subject to the limitations placed upon him by the client for the portfolio. (Limitations listed on next two slides.)

29 29 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Portfolio Model Portfolio Limitations Portfolio Limitations 1. The weighted average liquidity factor for the portfolio 1. The weighted average liquidity factor for the portfolio must to be at least 65. must to be at least The weighted average risk factor for the portfolio must 2. The weighted average risk factor for the portfolio must be no greater than 55. be no greater than No more than $60,000 is to be invested in Unidyde 3. No more than $60,000 is to be invested in Unidyde stocks or bonds. stocks or bonds. 4. No more than 40% of the investment can be in any one 4. No more than 40% of the investment can be in any one category except the money category. category except the money category. 5. No more than 20% of the total investment can be in 5. No more than 20% of the total investment can be in any one investment except the money market fund. any one investment except the money market fund.continued

30 30 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Portfolio Model Portfolio Limitations (continued) Portfolio Limitations (continued) 6. At least $1,000 must be invested in the Money Market 6. At least $1,000 must be invested in the Money Market fund. fund. 7. The maximum investment in Saver's Certificates is 7. The maximum investment in Saver's Certificates is $15,000. $15, The minimum investment desired for debt is $90, The minimum investment desired for debt is $90, At least $10,000 must be placed in a T-Bill account. 9. At least $10,000 must be placed in a T-Bill account.

31 31 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Portfolio Model n Define the Decision Variables X1 = $ amount invested in Unidyde Corp. (Equities) X2 = $ amount invested in CC’s Restaurants X3 = $ amount invested in First General REIT X4 = $ amount invested in Metropolis Electric X5 = $ amount invested in Unidyde Corp. (Debt) X6 = $ amount invested in Lewisville Transit X7 = $ amount invested in Realty Partners X8 = $ amount invested in T-Bill Account X9 = $ amount invested in Money Mkt. Fund X10 = $ amount invested in Saver's Certificate X10 = $ amount invested in Saver's Certificate

32 32 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Portfolio Model n Define the Objective Function Maximize the total expected after-tax return over the next year: Max.15X1 +.17X X X X5 Max.15X1 +.17X X X X5 +.12X6 +.22X X X X X6 +.22X X X X10

33 33 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Portfolio Model Total funds invested must not exceed $400,000: (1) X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 = 400,000 Weighted average liquidity factor must to be at least 65: (2)100X X X3 + 95X4 + 92X5 + 79X6 + 80X X9 > 65(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) 65(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) Weighted average risk factor must be no greater than 55: (3)60X1 + 70X2 + 75X3 + 20X4 + 30X5 + 22X6 + 50X7 + 10X9 < 55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) 55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) No more than $60,000 to be invested in Unidyde Corp: (4)X1 + X5 < 60,000 n Define the Constraints

34 34 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Portfolio Model n Define the Constraints (continued) No more than 40% of the $400,000 investment can be in any one category except the money category: (5) X1 + X2 + X3 < 160,000 (6) X4 + X5 + X6 < 160,000 (7)X3 + X7 < 160,000 No more than 20% of the $400,000 investment can be in any one investment except the money market fund: (8) X2 < 80,000(12) X7 < 80,000 (9) X3 < 80,000(13) X8 < 80,000 (10) X4 < 80,000(14) X10 < 80,000 (11) X6 < 80,000

35 35 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Portfolio Model n Define the Constraints (continued) At least $1,000 must be invested in the Money Market fund: (15) X9 > 1,000 The maximum investment in Saver's Certificates is $15,000: (16) X10 < 15,000 The minimum investment the Debt category is $90,000: (17) X4 + X5 + X6 > 90,000 At least $10,000 must be placed in a T-Bill account: (18) X8 > 10,000 Non-negativity of variables: Xj > 0 j = 1,..., 10 Xj > 0 j = 1,..., 10

36 36 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Portfolio Model n Solution Summary Total Expected After-Tax Return = $64,355 X1 = $0 invested in Unidyde Corp. (Equities) X1 = $0 invested in Unidyde Corp. (Equities) X2 = $80,000 invested in CC’s Restaurants X2 = $80,000 invested in CC’s Restaurants X3 = $80,000 invested in First General REIT X3 = $80,000 invested in First General REIT X4 = $0 invested in Metropolis Electric X4 = $0 invested in Metropolis Electric X5 = $60,000 invested in Unidyde Corp. (Debt) X5 = $60,000 invested in Unidyde Corp. (Debt) X6 = $74,000 invested in Lewisville Transit X6 = $74,000 invested in Lewisville Transit X7 = $80,000 invested in Realty Partners X7 = $80,000 invested in Realty Partners X8 = $10,000 invested in T-Bill Account X8 = $10,000 invested in T-Bill Account X9 = $1,000 invested in Money Mkt. Fund X9 = $1,000 invested in Money Mkt. Fund X10 = $15,000 invested in Saver's Certificate X10 = $15,000 invested in Saver's Certificate

37 37 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. A portfolio manager has been asked to develop a portfolio for the firm’s conservative clients who express a strong aversion to risk. The manager’s task is to determine the proportion of the portfolio to invest in each of six mutual funds so that the portfolio provides the best return possible with a minimum risk. A portfolio manager has been asked to develop a portfolio for the firm’s conservative clients who express a strong aversion to risk. The manager’s task is to determine the proportion of the portfolio to invest in each of six mutual funds so that the portfolio provides the best return possible with a minimum risk. The annual return (%) for five 1-year periods for the six mutual funds are shown on the next slide. The portfolio manager thinks that the returns for the five years shown in the table are scenarios that can be used to represent the possibilities for the next year. The annual return (%) for five 1-year periods for the six mutual funds are shown on the next slide. The portfolio manager thinks that the returns for the five years shown in the table are scenarios that can be used to represent the possibilities for the next year. Conservative Portfolio

38 38 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Conservative Portfolio Planning Scenarios Mutual Fund Year 1 Year 2 Year 3 Year 4 Year 5 Foreign Stock Intermediate-Term Bond Large-Cap Growth Large-Cap Value Small-Cap Growth Small-Cap Value S&P 500 Return

39 39 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Conservative Portfolio n Define the Decision Variables FS = proportion invested in foreign stock mutual fund FS = proportion invested in foreign stock mutual fund IB = proportion invested in intermediate-term bond fund IB = proportion invested in intermediate-term bond fund LG = proportion invested in large-cap growth fund LG = proportion invested in large-cap growth fund LV = proportion invested in large-cap value fund LV = proportion invested in large-cap value fund SG = proportion invested in small-cap growth fund SG = proportion invested in small-cap growth fund SV = proportion invested in small-cap value fund SV = proportion invested in small-cap value fund

40 40 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Conservative Portfolio Minimum returns for five scenarios: Minimum returns for five scenarios: – M FS IB LG LV SG SV ≥ 0 – M FS IB LG LV SG SV ≥ 0 – M FS IB LG LV SG – 6.70 SV ≥ 0 – M FS – 1.33 IB LG LV SG SV ≥ 0 – M – FS IB – LG – 5.37 LV – 9.02 SG SV ≥ 0 Sum of the proportions must equal 1: Sum of the proportions must equal 1: FS + IB + LG + LV + SG + SV = 1 Non-negativity Non-negativity M, FS, IB, LG, LV, SG, SV ≥ 0 M, FS, IB, LG, LV, SG, SV ≥ 0 n Constraints n Objective Function Maximize the minimum return for the portfolio: Maximize the minimum return for the portfolio: Max M Max M

41 41 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Conservative Portfolio n Optimal Solution The optimal value of the objective function is (The optimal portfolio will earn 6.445% in the worst- case scenario.) 55.4% of the portfolio should be invested in the 55.4% of the portfolio should be invested in the intermediate-term bond fund. intermediate-term bond fund. 13.2% of the portfolio should be invested in the 13.2% of the portfolio should be invested in the large-cap growth fund. large-cap growth fund. 31.4% of the portfolio should be invested in the 31.4% of the portfolio should be invested in the small-cap value fund. small-cap value fund.

42 42 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Moderate Portfolio A portfolio manager would like to construct a portfolio for clients who are willing to accept a moderate amount of risk in order to attempt to achieve better returns. Suppose that clients in this risk category are willing to accept some risk, but do not want the annual return for the portfolio to drop below 2%. A portfolio manager would like to construct a portfolio for clients who are willing to accept a moderate amount of risk in order to attempt to achieve better returns. Suppose that clients in this risk category are willing to accept some risk, but do not want the annual return for the portfolio to drop below 2%. The annual return (%) for five 1-year periods for the six mutual funds are shown on the next slide. The portfolio manager thinks that the returns for the five years shown in the table are scenarios that can be used to represent the possibilities for the next year. The annual return (%) for five 1-year periods for the six mutual funds are shown on the next slide. The portfolio manager thinks that the returns for the five years shown in the table are scenarios that can be used to represent the possibilities for the next year.

43 43 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Moderate Portfolio Minimum returns for five scenarios: Minimum returns for five scenarios: – M FS IB LG LV SG SV ≥ 2 – M FS IB LG LV SG SV ≥ 2 – M FS IB LG LV SG – 6.70 SV ≥ 2 – M FS – 1.33 IB LG LV SG SV ≥ 2 – M – FS IB – LG – 5.37 LV – 9.02 SG SV ≥ 2 Sum of the proportions must equal 1: Sum of the proportions must equal 1: FS + IB + LG + LV + SG + SV = 1 Non-negativity Non-negativity M, FS, IB, LG, LV, SG, SV ≥ 0 M, FS, IB, LG, LV, SG, SV ≥ 0 n Constraints

44 44 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Moderate Portfolio n Objective Function Maximize the minimum return for the portfolio: Maximize the minimum return for the portfolio: Max 12.03FS IB LG LV SG SV The coefficient of FS in the objective function is given by: 0.2(10.06) + 0.2(13.12) + 0.2(13.47) + 0.2(45.42) + 0.2( – 21.93) (10.06) + 0.2(13.12) + 0.2(13.47) + 0.2(45.42) + 0.2( – 21.93) The coefficient of IB is given by: 0.2(17.64) + 0.2(3.25) + 0.2(7.51) = 0.2( – 1.33) = 0.2(7.36) = (17.64) + 0.2(3.25) + 0.2(7.51) = 0.2( – 1.33) = 0.2(7.36) = 6.89 … and so on. Thus, the objective function is:

45 45 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Moderate Portfolio n Optimal Solution Invest 10.8% of the portfolio in a large-cap growth Invest 10.8% of the portfolio in a large-cap growth mutual fund. mutual fund. Invest 41.5% in a small-cap growth mutual fund. Invest 41.5% in a small-cap growth mutual fund. Invest 47.7% in a small-cap value mutual fund. Invest 47.7% in a small-cap value mutual fund. This allocation provides a maximum expected return of 17.33%. The portfolio return will only be 2% if scenarios 3 or 5 occur (constraints 3 and 5 are binding). The portfolio return will be % if scenario 1 occurs, % if scenario 2 occurs, and % if scenario 4 occurs.

46 46 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Nonlinear Optimization n Many business processes behave in a nonlinear manner. The price of a bond is a nonlinear function of interest rates. The price of a bond is a nonlinear function of interest rates. The price of a stock option is a nonlinear function of the price of the underlying stock. The price of a stock option is a nonlinear function of the price of the underlying stock. The marginal cost of production often decreases with the quantity produced. The marginal cost of production often decreases with the quantity produced. The quantity demanded for a product is often a nonlinear function of the price. The quantity demanded for a product is often a nonlinear function of the price.

47 47 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Nonlinear Optimization n A nonlinear optimization problem is any optimization problem in which at least one term in the objective function or a constraint is nonlinear. n Nonlinear terms include n The nonlinear optimization problems presented on the upcoming slides can be solved using computer software such as LINGO and Excel Solver.

48 48 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Armstrong Bike Co. produces two new lightweight bicycle frames, the Flyer and the Razor, that are made from special aluminum and steel alloys. The cost to produce a Flyer frame is $100, and the cost to produce a Razor frame is $120. We can not assume that Armstrong will sell all the frames it can produce. As the selling price of each frame model – Flyer and Razor - increases, the quantity demanded for each model goes down. n Armstrong Bike Co. Nonlinear Optimization

49 49 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Assume that the demand for Flyer frames F and the demand for Razor frames R are given by: F = 750 – 5 P F F = 750 – 5 P F R = 400 – 2 P R R = 400 – 2 P R where P F = the price of a Flyer frame P R = the price of a Razor frame. P R = the price of a Razor frame. n The profit contributions (revenue – cost) are: P F F  100 F for Flyer frames P F F  100 F for Flyer frames P R R  120 R for Razor frames P R R  120 R for Razor frames Nonlinear Optimization

50 50 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Profit Contribution as a Function of Demand Solving F = 750  5 P F for P F we get:Solving F = 750  5 P F for P F we get: P F = 150   F P F = 150   F Substituting 150   F for P F in P F F  100 F we get: Substituting 150   F for P F in P F F  100 F we get: P F F  100 F = F (150   F )  100 F = 50 F   F 2 P F F  100 F = F (150   F )  100 F = 50 F   F 2 Solving R = 400  2 P R for P R we get:Solving R = 400  2 P R for P R we get: P R = 200   R P R = 200   R Substituting 200   R for P R in P R R  120 R we get: Substituting 200   R for P R in P R R  120 R we get: P R R  120 R = R (200   R )  120 R = 80 R   R 2 P R R  120 R = R (200   R )  120 R = 80 R   R 2 Nonlinear Optimization

51 51 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Total Profit Contribution Total Profit Contribution = 50 F –  F R –  R 2 Total Profit Contribution = 50 F –  F R –  R 2 This function is an example of a quadratic function This function is an example of a quadratic function because the nonlinear terms have a power of 2. because the nonlinear terms have a power of 2. Nonlinear Optimization

52 52 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. A supplier can deliver a maximum of 500 A supplier can deliver a maximum of 500 pounds of the aluminum alloy and 420 pounds of the steel alloy weekly. The number of pounds of each alloy needed per frame is summarized below. Aluminum Alloy Steel Alloy Aluminum Alloy Steel Alloy Flyer 2 3 Flyer 2 3 Razor 4 2 Razor 4 2 How many Flyer and Razor frames should Armstrong produce each week? Nonlinear Optimization

53 53 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Max 50 F –  F R –  R 2 (Total Weekly Profit) s.t. 2 F + 4 R < 500 (Aluminum Available) 3 F + 2 R < 420 (Steel Available) 3 F + 2 R < 420 (Steel Available) F, R > 0 (Non-negativity) F, R > 0 (Non-negativity) n Problem Formulation Nonlinear Optimization

54 54 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Total Profit Contribution First, we will solve the unconstrained version of First, we will solve the unconstrained version of this nonlinear program to find the values of F and R that maximize the above total profit contribution function (with the production constraints ignored). Nonlinear Optimization

55 55 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Optimal Solution for Unconstrained Problem 3 F + 2 R < 420 2F + 4 R < 500 UnconstrainedOptimum (125, 80) Profit = $6, x1x1x1x1 x2x2x2x FeasibleRegion An Unconstrained Problem

56 56 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Total Profit Contribution Now we will solve the constrained version of this Now we will solve the constrained version of this nonlinear program to find the values of F and R that maximize the total profit contribution function with the production constraints enforced. A Constrained Problem

57 57 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Objective Function Contour Lines $6, x2x2x2x $6,200.00Contour $6,075.47Contour $5,500.00Contour x1x1x1x1 A Constrained Problem

58 58 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Optimal Solution for Constrained Problem x1x1x1x1 x2x2x2x $6,075.47Contour ConstrainedOptimum (92.45, 71.32) Profit = $6, A Constrained Problem

59 59 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Optimal Solution Produce Flyer frames per week.Produce Flyer frames per week. Produce Razor frames per week.Produce Razor frames per week. Profit per week is $6, Profit per week is $6, Use pounds of aluminum alloy per week (of the 500 pounds available per week).Use pounds of aluminum alloy per week (of the 500 pounds available per week). Use the entire 420 pounds of steel alloy available per week.Use the entire 420 pounds of steel alloy available per week. A Constrained Problem

60 60 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Local and Global Optima n A feasible solution is a local optimum if there are no other feasible solutions with a better objective function value in the immediate neighborhood. For a maximization problem the local optimum corresponds to a local maximum. For a maximization problem the local optimum corresponds to a local maximum. For a minimization problem the local optimum corresponds to a local minimum. For a minimization problem the local optimum corresponds to a local minimum.  A feasible solution is a global optimum if there are no other feasible points with a better objective function value in the feasible region.  Obviously, a global optimum is also a local optimum.

61 61 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Multiple Local Optima n Nonlinear optimization problems can have multiple local optimal solutions, in which case we want to find the best local optimum. n Nonlinear problems with multiple local optima are difficult to solve and pose a serious challenge for optimization software. n In these cases, the software can get “stuck” and terminate at a local optimum. n There can be a severe penalty for finding a local optimum that is not a global optimum. n Developing algorithms capable of finding the global optimum is currently a very active research area.

62 62 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Multiple Local Optima n Consider the function n The shape of this function is shown on the next slide. n The hills and valleys in the graph show that this function has several local maximums and local minimums. n There are two local minimums, one of which is the the global minimum. n There are three local maximums, one of which is the global maximum.

63 63 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Multiple Local Optima

64 64 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Single Local Optimum n Consider the function n The shape of this function is shown on the next slide. n A function that is bowl-shaped down is called a concave function. n The maximum value for this particular function is 0 and the point (0, 0) gives the optimal value of 0. n Functions such as this one have a single local maximum that is also a global maximum. n This type of nonlinear problem is relatively easy to maximize.

65 65 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Single Local Optimum

66 66 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Single Local Optimum n Consider the function n The shape of this function is shown on the next slide. n A function that is bowl-shaped up is called a convex function. n The minimum value for this particular function is 0 and the point (0, 0) gives the optimal value of 0. n Functions such as this one have a single local minimum that is also a global minimum. n This type of nonlinear problem is relatively easy to minimize.

67 67 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Single Local Optimum Y Y X X Z Z

68 68 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Constructing an Index Fund n Index funds are a very popular investment vehicle in the mutual fund industry. n Vanguard 500 Index Fund is the largest mutual fund in the U.S. with over $70 billion in net assets in n An index fund is an example of passive asset management. n The key idea behind an index fund is to construct a portfolio of stocks, mutual funds, or other securities that closely matches the performance of a broad market index such as the S&P 500. n Behind the popularity of index funds is research that basically says “you can’t beat the market.”

69 69 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Constructing an Index Fund n Lymann Brothers Investments Lymann Brothers has a substantial number of clients who wish to own a mutual fund portfolio that closely matches the performance of the S&P 500 stock index. A manager at Lymann Brothers has selected five mutual funds (shown on the next slide) that will be considered for inclusion in the portfolio. The manager must decide what percentage of the portfolio should be invested in each mutual fund.

70 70 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Constructing an Index Fund Annual Returns (Planning Scenarios) Annual Returns (Planning Scenarios) Mutual Fund Year 1Year 2Year 3Year 4 International Stock Large-Cap Blend Mid-Cap Blend Small-Cap Blend Intermediate Bond S&P n Mutual Fund Performance in 4 Selected Years

71 71 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Constructing an Index Fund n Define the 9 Decision Variables IS = proportion of portfolio invested in international stock LC = proportion of portfolio invested in large-cap blend MC = proportion of portfolio invested in mid-cap blend SC = proportion of portfolio invested in small-cap blend IB = proportion of portfolio invested in intermediate bond R 1 = portfolio return for scenario 1 (year 1) R 2 = portfolio return for scenario 2 (year 2) R 3 = portfolio return for scenario 3 (year 3) R 4 = portfolio return for scenario 4 (year 4)

72 72 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Constructing an Index Fund n Define the Objective Function Min ( R 1 – 13) 2 + ( R 2 – 12) 2 + ( R 3 – 7) 2 + ( R 4 – 2) 2 n Define the 6 Constraints (including non-negativity) IS LC MC SC IB = R IS LC MC SC IB = R IS LC MC  1.92 SC IB = R IS LC MC  1.92 SC IB = R 3  3.13 IS LC  1.04 MC SC IB = R 4  3.13 IS LC  1.04 MC SC IB = R 4 IS + LC + MC + SC + IB = 1 IS, LC, MC, SC, IB > 0

73 73 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Constructing an Index Fund n Optimal Solution for Lymann Brothers Example R 1 = 12.51(12.51% portfolio return for scenario 1) R 1 = 12.51(12.51% portfolio return for scenario 1) R 2 = (12.90% portfolio return for scenario 2) R 2 = (12.90% portfolio return for scenario 2) R 3 = 7.13 ( 7.13% portfolio return for scenario 3) R 3 = 7.13 ( 7.13% portfolio return for scenario 3) R 4 = 2.51 ( 2.51% portfolio return for scenario 4) R 4 = 2.51 ( 2.51% portfolio return for scenario 4) IS = 0 ( 0.0% of portfolio in international stock) IS = 0 ( 0.0% of portfolio in international stock) LC = 0 ( 0.0% of portfolio in large-cap blend) LC = 0 ( 0.0% of portfolio in large-cap blend) MC =.332 (33.2% of portfolio in mid-cap blend) MC =.332 (33.2% of portfolio in mid-cap blend) SC =.161 (16.1% of portfolio in small-cap blend) SC =.161 (16.1% of portfolio in small-cap blend) IB =.507 (50.7% of portfolio in intermediate bond) IB =.507 (50.7% of portfolio in intermediate bond) 100.0% of portfolio

74 74 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Constructing an Index Fund n Lymann Brothers Portfolio Return vs. S&P 500 Return Scenario Portfolio ReturnS&P 500 Return

75 75 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 12


Download ppt "1 1 Slide © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole."

Similar presentations


Ads by Google