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### Chapter 11

School: MIT
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Word Count: 355

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SHORT A B C D E F G H I J K L NO SALES c Means 8% 9% 10% 11% 0.05 0.03 0.2 0.9 Ctrl+A works the VBA program which calculates efficient portfolios for no-short sales. This program iteratively substitutes a constant ranging from -3.5% 'till 16% (1/2% jumps) and calculates the optimal portfolio. 16.0% &lt;-- This is the constant Optimal portfolio proportions x1 0.0000 x2 0.0000 x3 0.0000 x4...

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SHORT A B C D E F G H I J K L NO SALES c Means 8% 9% 10% 11% 0.05 0.03 0.2 0.9 Ctrl+A works the VBA program which calculates efficient portfolios for no-short sales. This program iteratively substitutes a constant ranging from -3.5% 'till 16% (1/2% jumps) and calculates the optimal portfolio. 16.0% <-- This is the constant Optimal portfolio proportions x1 0.0000 x2 0.0000 x3 0.0000 x4 1.0000 Total 1 ### Portfolio mean Portfolio sigma Theta 11.00% <-- {=MMULT(TRANSPOSE(C12:C15),G4:G7)} 94.87% <-- {=SQRT(MMULT(TRANSPOSE(C12:C15),MMULT(B4:E7,C12:C15)))} -5.27% <-- =(B18-C9)/B19 Note: Because the formulas in cells B18 and B19 use the TRANSPOSE function, they must be entered as arrays, with [Ctrl]-[Shift]-[Enter] . The curly brackets are entered automatically by Excel. See section 9.4 of Chapter 9 for more details. To compute the values given in pages 201-204: Substitute in the appropriate value for c, and then bring up the Excel Solver: Tools|Solver. In most cases this is enough; however, you may have to load the Solver: Tools|Add-ins, then click the box for the Solver Add-in. Variance-covariance matrix 0.1 0.03 -0.08 0.03 0.2 0.02 -0.08 0.02 0.3 0.05 0.03 0.2 c M N Sigma 20.24% 20.25% 20.25% 20.25% 20.25% 20.26% 20.26% 20.27% 20.27% 20.28% 20.29% 20.30% 20.31% 20.32% 20.34% 20.37% 20.41% 20.46% 20.54% 20.67% 20.90% 21.36% 23.27% 31.91% 45.25% 60.76% 74.30% 94.87% 94.87% 94.87% 94.87% 94.87% 94.87% 94.87% 94.87% 94.87% 94.87% 94.87% 94.87% 94.87% Means minus c 0.0% 1.0% 2.0% 3.0% Means 8% 9% 10% 11% 0.05 0.03 0.2 0.9 c -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14 0.145 0.15 0.155 0.16 8.0% z 1.4705 0.1701 0.8110 -0.1454 mean variance covariance O P Q R S x1 0.605 0.604 0.603 0.603 0.602 0.601 0.599 0.598 0.597 0.595 0.593 0.591 0.589 0.586 0.582 0.578 0.573 0.566 0.557 0.545 0.528 0.499 0.427 0.200 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 x2 0.089 0.089 0.089 0.089 0.090 0.090 0.091 0.091 0.092 0.093 0.093 0.094 0.095 0.097 0.098 0.100 0.102 0.105 0.108 0.113 0.120 0.132 0.163 0.259 0.251 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 x3 0.307 0.307 0.307 0.308 0.309 0.309 0.310 0.311 0.311 0.312 0.313 0.315 0.316 0.318 0.320 0.322 0.325 0.329 0.334 0.342 0.352 0.368 0.386 0.422 0.488 0.556 0.300 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 x4 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.025 0.119 0.260 0.444 0.700 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 RESULTS Variance-covariance matrix 0.1 0.03 -0.08 0.03 0.2 0.02 -0.08 0.02 0.3 0.05 0.03 0.2 x 63.76% 7.38% 35.17% -6.30% z 0.0287 0.0371 0.0605 0.0171 8.59% 3.72% y 20.04% 25.87% 42.19% 11.90% transpose y 20.04% 25.87% 42.19% 11.90% sigma 23.51% 98.98% 94.77% 90.56% 86.37% 82.18% 78.01% 73.85% 69.71% 65.59% 61.50% 57.43% 53.40% 49.41% 45.49% 41.63% 31.91% 28.60% 25.58% 22.97% 20.93% 19.64% 19.23% x 63.76% 7.38% 35.17% -6.30% transpose19.78% 21.20% 23.34% 26.02% 29.08% 32.43% 35.98% mean 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 12.07% 9.46% 10.18% 4.10% prop. x mean variance sigma 0.5 12.07% 5.53% 23.51% start step -3 0.18 Efficient Frontier, Short Sales Allowed 9.5% 9.0% Mea n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 8.5% 8.0% 7.5% 18% 19% 20% 21% Sigma 22% 23% 24% -3 -2.82 -2.64 -2.46 -2.28 -2.1 -1.92 -1.74 -1.56 -1.38 -1.2 -1.02 -0.84 -0.66 -0.48 0 0.18 0.36 0.54 0.72 0.9 1.08 1.26 1.44 1.62 1.8 1.98 2.16 2.34 Mean 8.70% 8.70% 8.70% 8.71% 8.71% 8.71% 8.71% 8.71% 8.71% 8.72% 8.72% 8.72% 8.73% 8.73% 8.74% 8.74% 8.75% 8.76% 8.78% 8.80% 8.82% 8.87% 9.01% 9.46% 10.01% 10.44% 10.70% 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% 11.00% Efficient Frontier N o Short Sales Allowed 12% 11% 11% Mean 10% 10% 9% 9% 8% 18% 28% 38% 4 8% 58% Sigma 68% 78% 88% 98% 108% Comparing Two Efficient Frontiers For low sigmas the two frontiers coincide. For higher sigmas, no restrictions on short sales gives higher returns. 12.0% 11.0% Me a n 10.0% 9.0% 8.0% 7.0% 6.0% 19% 29% 39% 49% 59% 69% Sigma No short sales Sh ort sales allowed 79% 89% 99% Note: To get the two data series (with and without short sales) to chart on the same set of axes, we copy all the sigmas in one column and then copy the means in two adjacent columns. The graph is produced with an XY graph format. down to here Sigma No short sales Short sales allowed 20.24% 8.70% 20.25% 8.70% 20.25% 8.70% 20.25% 8.71% 20.25% 8.71% 20.26% 8.71% 20.26% 8.71% 20.27% 8.71% 20.27% 8.71% 20.28% 8.72% 20.29% 8.72% 20.30% 8.72% 20.31% 8.73% 20.32% 8.73% 20.34% 8.74% 20.37% 8.74% 20.41% 8.75% 20.46% 8.76% 20.54% 8.78% 20.67% 8.80% 20.90% 8.82% 21.36% 8.87% 23.27% 9.01% 31.91% 9.46% 45.25% 10.01% 60.76% 10.44% 74.30% 10.70% 94.87% 11.00% 94.87% 11.00% 94.87% 11.00% 94.87% 11.00% 94.87% 11.00% 94.87% 11.00% 94.87% 11.00% 94.87% 11.00% 94.87% 11.00% 94.87% 11.00% 94.87% 11.00% 94.87% 11.00% 94.87% 11.00% 98.98% 12.07% 94.77% 12.07% 90.56% 12.07% 86.37% 12.07% 82.18% 12.07% 78.01% 12.07% 73.85% 12.07% 69.71% 12.07% 65.59% 12.07% 61.50% 12.07% 57.43% 12.07% 53.40% 12.07% 49.41% 12.07% 45.49% 12.07% 41.63% 12.07% 31.91% 12.07% 28.60% 12.07% 25.58% 12.07% 22.97% 12.07% 20.93% 12.07% 19.64% 12.07% 19.23% 12.07% 19.78% 12.07% 21.20% 12.07% 23.34% 12.07% 26.02% 12.07% 29.08% 12.07% 32.43% 12.07% 35.98% 12.07%
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CSE 21: Homework 3October 12, 2009Problem 1In how many ways can 6 people be assigned to 4 nonempty teams?Problem 2An urn contains 5 red marbles and 6 white marbles.(a) How many ways can 4 marbles be drawn?(b) What if we must have 2 red marbles and
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Name:Student ID:CSE 21Midterm #2 SolutionsNovember 17, 2009Each problem is worth 20 points. Show your work, especially on decision trees. Also, makesure you write legibly so that I have a chance of being able to read your solutions! Additionalscrat
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CSE21 FA11Homework #2 (10/3/11)In this homework, we will consider ordinary decks of playing cards whichhave 52 cards, with 13 of each of the four suits (Hearts, Spades, Diamondsand Clubs) with each suit have the 13 ranks (Ace, 2, 3, . . . , Jack, Quee
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CSE21 FA11Homework #3 (10/10/11)3.1.Prove by induction thatnk=1k2 =n(n+1)(2n+1).63.2. It is desired to form three committees from a group of 3 men and 3women.(a) In how many ways can this be done?(b) Suppose that each of the committees must c
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CSE21 FA11Homework #3 Solutions (10/15/11)3.1. Prove by induction thatnk2 =k=1n(n + 1)(2n + 1)6Sketch of proof. Step 1. We need to verify the base case for n = 1, i.e.1k=1=k=11 (1 + 1) (2 + 1).6Step 2. Proof by induction. We assume that n=1
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CSE21 FA11Homework #4 (10/17/11)4.1.Prove by induction thatnk=1k3 = (nk=1k )2 .(Hint: First get an explicit form form the second sum. Then use induction.)4.2. (Note: This problem is fairly tough! Ill be impressed by anyone whocan get this one!
UCSD - CSE - 21
CSE21 FA11Homework #4 (10/17/11)4.1.Sketch of proof. Recall that we showed in class that n=1 k = n(n2+1) for allk22n 1. We will prove that by induction n=1 k 3 = n (n4+1) . We assume thatknn2 (n+1)23for some xed n 1. Note thatk=1 k =4n+1n
UCSD - CSE - 21
CSE21 FA11Homework #5 (10/24/11)5.1. (a) 10 cards are drawn at random one at a time with replacement froman ordinary deck of cards.What is the sample space? What is the probabilitythat no Ace appears on any of the draws? What is the probability that a
UCSD - CSE - 21
CSE21 FA11Homework #5 (10/17/11)5.1Solution. (a) The probability that no Ace appears on any of the drawsis ( 48 )10 . The probability that at least one King appears in 10 draws is521 ( 48 )10 . The probability that at least 2 Queens appear in the 10
UCSD - CSE - 21
CSE21 FA11Homework #6 (10/31/11)6.1. Five boys and three girls are throwing Frisbees. Each boy has oneFrisbee and throws it to a random girl. What is the probability that each ofthe girls gets at least one of thrown Frisbees?6.2. A box contains 20 li
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CSE21 FA11Homework #6 (10/31/11)6.1. Five boys and three girls are throwing Frisbees. Each boy has oneFrisbee and throws it to a random girl. What is the probability that each ofthe girls gets at least one of thrown Frisbees?Answer: S (5, 3) 3!35=
UCSD - CSE - 21
CSE21 FA11Homework #7 (11/7/11)7.1. A bin contains 4 red balls, 5 white balls and 6 blue balls. A randomsubset S of 4 balls is removed (without replacement). Consider the following3 events:(1) E1 : S has exactly 2 red balls.;(2) E2 : S has balls of
UCSD - CSE - 21
CSE21 FA11Homework #7 Solutions7.1. A bin contains 4 red balls, 5 white balls and 6 blue balls. A randomsubset S of 4 balls is removed (without replacement). Consider the following3 events:(1) E1 : S has exactly 2 red balls.;(2) E2 : S has balls of
UCSD - CSE - 21
CSE21 FA11Homework #8 (11/14/11)8.1. An urn contains r red marbles and b blue marbles. A random marbleM1 is drawn out. If M1 is red, then it is put back into the urn along withc more blue marbles. On the other hand, if M1 is blue, then it is put back
UCSD - CSE - 21
CSE21 FA11Homework #8 (11/17/11)8.1Solution. Draw the decision tree. Then compute:(a)P r(M2 = red) =r+drbr+.r+b+cr+b r+b+dr+b(b)P r(M2 = red|M1 = red) =P r(M1 = blue|M2 = blue) =r.r+b+cbbr+b r+d+brb+cbb+ r+b r+d+br+d r+b+c8.2S
UCSD - CSE - 21
CSE21 FA11Homework #9 (11/22/11)9.1Does every connected graph has a spanning tree? Give either a proof or acounterexample.9.2(a) Select a spanning tree of minimum total weight from the following weightedgraph.(b) Find all spanning trees (list thei