...BUS 431a: Investment and Portfolio Management SPRING 2013 Problem Set #5 (team work) 1. (15 points) Suppose the economy can be .in one of the following two states: (i) Boom or “good” state and (ii) Recession or “bad” state. Each state can occur with an equal opportunity (50%). The annual return on the market and a certain security X in the two states of the economy are as follows: • Market: at the end of the year, the market is expected to yield a return of 30% in the good state and a return of (-10%) in the bad state; • Security X: at the end of the year, the security is expected to yield a return of 40% in the good state and a return of (-35%) in the bad state; Furthermore, assume that annual risk-free rate of return is 5%. A. (5 points) Calculate the beta of security X relative to the market. B. (5 points) Calculate the alpha of security X. C. (5 points) Draw the security market line (SML). Please, label the axes and all points (including the market portfolio, the risk-free security, and security X) in the graph clearly. Identify alpha in the graph. Solution: 1. (5 points) Beta of security X is: Market portfolio: (1/2) probability of getting 30% and (1/2) probability of getting -10%. So, expected/mean return and the standard deviation of the market are: [pic] [pic] Security X: (1/2) probability of getting 40% and (1/2) probability of getting -35%. So, expected/mean return and the standard deviation of security X are: ...
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...MATH 55 SOLUTION SET—SOLUTION SET #5 Note. Any typos or errors in this solution set should be reported to the GSI at isammis@math.berkeley.edu 4.1.8. How many different three-letter initials with none of the letters repeated can people have. Solution. One has 26 choices for the first initial, 25 for the second, and 24 for the third, for a total of (26)(25)(24) possible initials. 4.1.18. How many positive integers less than 1000 (a) are divisible by 7? (b) are divisible by 7 but not by 11? (c) are divisible by both 7 and 11? (d) are divisible by either 7 or 11? (e) are divisible by exactly one of 7 or 11? (f ) are divisible by neither 7 nor 11? (g) have distinct digits? (h) have distinct digits and are even? Solution. (a) Every 7th number is divisible by 7. Since 1000 = (7)(142) + 6, there are 142 multiples of seven less than 1000. (b) Every 77th number is divisible by 77. Since 1000 = (77)(12) + 76, there are 12 multiples of 77 less than 1000. We don’t want to count these, so there are 142 − 12 = 130 multiples of 7 but not 11 less than 1000. (c) We just figured this out to get (b)—there are 12. (d) Since 1000 = (11)(90) + 10, there are 90 multiples of 11 less than 1000. Now, if we add the 142 multiples of 7 to this, we get 232, but in doing this we’ve counted each multiple of 77 twice. We can correct for this by subtracting off the 12 items that we’ve counted twice. Thus, there are 232-12=220 positive integers less than 1000 divisible by 7 or 11. (e) If we want to exclude the multiples...
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...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). then: Algebraically, if X = quantity of red hots and Y = quantity of rock candies, 0.2 X + 1Y = 100 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: X = 500 − 5Y which is the equation of a straight line, with an intercept of 500 and a slope of −5. Elton, Gruber, Brown, and Goetzmann Modern Portfolio Theory and Investment Analysis, 7th Edition Solutions to Text Problems: Chapter 1 1-1 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 individual’s utility function and may differ among students. Chapter 1: Problem 2 A. Opportunity Set The individual...
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...Local Search Heuristics for the Vehicle Routing Problem Chris Gro¨r1 e Oak Ridge National Laboratory, 1 Bethel Valley Rd, Oak Ridge, TN 37831 cgroer@gmail.com Bruce Golden R.H. Smith School of Business, University of Maryland, College Park, MD 20742, USA, bgolden@rhsmith.umd.edu Edward Wasil2 Kogod School of Business, American University, Washington, DC 20016, USA, ewasil@american.edu The vehicle routing problem (VRP) is a difficult and well-studied combinatorial optimization problem. Real-world instances of the VRP can contain hundreds and even thousands of customer locations and can involve many complicating constraints, necessitating the use of heuristic methods. We present a software library of local search heuristics that allow one to quickly generate solutions to VRP instances. The code has a logical, object-oriented design and uses efficient data structures to store and modify solutions. The core of the library is the implementation of seven local search operators that share a similar interface and are designed to be extended to handle additional options with minimal code change. The code is well-documented, straightforward to compile, and is freely available online. The code contains several applications that can be used to generate solutions to the capacitated VRP. Computational results indicate that these applications are able to generate solutions that are within about one percent of the best-known solution on benchmark problems. Key words: vehicle routing; optimization; heuristics;...
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...Database Modeling and Design – 4th Edition (2006) Toby Teorey, Sam Lightstone, Tom Nadeau Exercises with Solutions – Solutions Manual ER and UML Conceptual Data Modeling Problem 2-1 Draw a detailed ER diagram for an car rental agency database (e.g. Hertz), keeping track of current rental location of each car, its current condition and history of repairs, and customer information for a local office, expected return date, return location, car status (ready, being-repaired, currently-rented, being-cleaned). Select attributes from your intuition about the situation, and list them separately from the diagram, but associated with a particular entity or relationship in the ER model. Solution to 2-1 Problem 2-2 Given the following assertions for a relational database that represents the current term enrollment at a large university, draw an ER diagram for this schema that takes into account all the assertions given. There are 2000 instructors, 4000 courses, and 30,000 students. Use as many ER constructs as you can to represent the true semantics of the problem. Assertions: An instructor may teach one or more courses in a given term (average is 2.0 courses). An instructor must direct the research of at least one student (average = 2.5 students). A course may have none, one, or two prerequisites (average = 1.5 prerequisites). A course may exist even if no students are currently enrolled. All courses are taught by exactly one instructor. The...
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...A Comparison of Solution Procedures for the Flow Shop Scheduling Problem with Late Work Criterion Jacek Błażewicz *) Erwin Pesch 1) Małgorzata Sterna 2) Frank Werner 3) *) Institute of Computing Science, Poznań University of Technology Piotrowo 3A, 60-965 Poznań, Poland phone: +48 (61) 8790 790 fax: +48 (61) 8771 525 blazewic@sol.put.poznan.pl Institute of Information Systems, FB 5 - Faculty of Economics, University of Siegen Hölderlinstrasse 3, 57068 Siegen, Germany pesch@fb5.uni-siegen.de Institute of Computing Science, Poznań University of Technology Piotrowo 3A, 60-965 Poznań, Poland Malgorzata.Sterna@cs.put.poznan.pl Faculty of Mathematics, Otto-von-Guericke-University PSF 4120, 39016 Magdeburg, Germany Frank.Werner@mathematik.uni-magdeburg.de 1) 2) 3) 1 A Comparison of Solution Procedures for the Flow Shop Scheduling Problem with Late Work Criterion Abstract In this paper, we analyze different solution procedures for the two-machine flow shop scheduling problem with a common due date and the weighted late work criterion, i.e. for problem F2 | dj = d | Yw, which is known to be binary NP-hard. In computational experiments, we compare the practical efficiency of a dynamic programming approach, an enumerative method and a heuristic list scheduling procedure. Test results show that each solution method has its advantages and none of them can be rejected from the consideration a priori. Keywords: flow shop, late work, dynamic programming, enumerative...
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...simple linear programming problems in terms of an objective function to be maximized or minimized subject to a set of constraints. • find feasible solutions for maximization and minimization linear programming problems using the graphical method of solution. • solve maximization linear programming problems using the simplex method. • construct the Dual of a linear programming problem. • solve minimization linear programming problems by maximizing their Dual. 0.1.2 Introduction One of the major applications of linear algebra involving systems of linear equations is in finding the maximum or minimum of some quantity, such as profit or cost. In mathematics the process of finding an extreme value (maximum or minimum) of a quantity (normally called a function) is known as optimization . Linear programming (LP) is a branch of Mathematics which deals with modeling a decision problem and subsequently solving it by mathematical techniques. The problem is presented in a form of a linear function which is to be optimized (i.e maximized or minimized) subject to a set of linear constraints. The function to be optimized is known as the objective function . Linear programming finds many uses in the business and industry, where a decision maker may want to utilize limited available resources in the best possible manner. The limited resources may include material, money, manpower, space and time. Linear Programming provides various methods of solving such problems. In this unit, we present...
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...UNIVERSITY EM 516 – LOGISTICS ASSIGNMENT 4 P-MEDIAN PROBLEM 1892876 - Nurşin ATAK Instructor: Sinan GÜREL December 26th, 2013 ANKARA 1. ABOUT P-MEDIAN PROBLEM: The problem is to locate one or more facilities (sources), such as warehouses, to serve a number of demand points (sinks) of known locations, volumes, and transportation rates. Fixed costs of a candidate set of facilities may also be known. The candidate set of facilities is selected from the demand points. The objective is to find the best P locations from the M candidate sites where P is less than or equal to M. 2. PROBLEM INSTANCE: The problem instance given in PMED01.dat file is considered for comparing Logware method and Linear Programming method. Problem data is taken from PMED01.dat file and demonstrated in Table 1. There are 12 markets being served from up to 5 candidate warehouse locations. The product is shipped over a road network. The annual volumes of the markets, the transportation rates, and the candidate sites with their fixed costs and X&Y Coordinates are shown in Table 1. Point X Y Volume, Transport Rate, No Coordinate Coordinate cwt $/cwt/mi M1 200 100 3000 0,002 M2 500 200 5000 0,0015 M3 900 100 17000 0,002 M4 700 400 12000 0,0013 M5 1000 500 10000 0,0012 M6 200 500 9000 0,0015 M7 200 700 24000 0,002 M8 400 700 14000 0,0014 M9 500 800 23000 0,0024 M10 800 900 30000 0,0011 M11 1200 300 4000 0,0001 M12 200 2000 2000 0,0002 Table 1. Problem Data of 12 Markets Fixed Candidate Cost, $ Sites 200...
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... Contents Preface 4 Mathematics of Finance 1. Simple Interest . . . . . . . . . . . . . . . . . . . . . . . 2. Discrete and Continuous Compound Interest . . . . . . 3. Ordinay Annuity, Future Value and Sinking Fund . . . 4. Present Value of an Ordinay Annuity and Amortization . . . . Matrices and Systems of Linear Equations 5. Solving Linear Systems Using Augmented Matrices . . . . 6. Gauss-Jordan Elimination . . . . . . . . . . . . . . . . . . 7. The Algebra of Matrices . . . . . . . . . . . . . . . . . . 8. Inverse Matrices and their Applications to Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . Linear Programming 9. Solving Systems of Linear Inequalities . . . . . . . . . . . . . . 10. Geometric Method for Solving Linear Programming Problems 11. Simplex Method for Solving Linear Programming Problems . 12. The Dual Problem: Minimization with ≥ Constraints . . . . . Counting Principles, Permuations, and 13. Sets . . . . . . . . . . . . . . . . . . 14. Counting Principles . . . . . . . . . 15. Permutations and Combinations . . . . . . 5 5 12 19 26 . . . . 34 34 42 53 62 . . . . 69 69 77 86 97 Combinations 106 . . . . . . . . . . . . . . . 106 . . . . . . . . . . . . . . . 117 . . . . . . . . . . . . . . . 123 Probability 129 16. Sample Spaces, Events, and Probability . . . . . . . . . . . . . 129 17. Probability of Unions and Intersections;...
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...many more applications is the assumption of divisibility (see Sec. 3.3), which requires that noninteger values be permissible for decision variables. In many practical problems, the decision variables actually make sense only if they have integer values. For example, it is often necessary to assign people, machines, and vehicles to activities in integer quantities. If requiring integer values is the only way in which a problem deviates from a linear programming formulation, then it is an integer programming (IP) problem. (The more complete name is integer linear programming, but the adjective linear normally is dropped except when this problem is contrasted with the more esoteric integer nonlinear programming problem, which is beyond the scope of this book.) The mathematical model for integer programming is the linear programming model (see Sec. 3.2) with the one additional restriction that the variables must have integer values. If only some of the variables are required to have integer values (so the divisibility assumption holds for the rest), this model is referred to as mixed integer programming (MIP). When distinguishing the all-integer problem from this mixed case, we call the former pure integer programming. For example, the Wyndor Glass Co. problem presented in Sec. 3.1 actually would have been an IP problem if the two decision variables x1 and x2 had represented the total number of units to be produced of products 1 and 2, respectively, instead of the production rates....
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...2 2.1 Introduction to Equations 2.2 Linear Equations 2.3 Introduction to Problem Solving 2.4 Formulas 2.5 Linear Inequalities Linear Equations and Inequalities Education is not the filling of a pail, but the lighting of a fire. — WILLIAM BUTLER YEATS M athematics is a unique subject that is essential for describing, or modeling, events in the real world. For example, ultraviolet light from the sun is responsible for both tanning and burning exposed skin. Mathematics lets us use numbers to describe the intensity of ultraviolet light. The table shows the maximum ultraviolet intensity measured in milliwatts per square meter for various latitudes and dates. Latitude 0 10 20 30 40 50 Mar. 21 325 311 249 179 99 57 June 21 254 275 292 248 199 143 Sept. 21 325 280 256 182 127 75 Dec. 21 272 220 143 80 34 13 ISBN 1-256-49082-2 Source: J. Williams, The USA Today Weather Almanac. If a student from Chicago, located at a latitude of 42°, spends spring break in Hawaii with a latitude of 20°, the sun’s ultraviolet rays in Hawaii will be approximately 249 2.5 99 times as intense as they are in Chicago. Equations can be used to describe, or model, the intensity of the sun at various latitudes. In this chapter we will focus on linear equations and the related concept of linear inequalities. 89 Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by...
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...The Critical Section Problem Problem Description Informally, a critical section is a code segment that accesses shared variables and has to be executed as an atomic action. The critical section problem refers to the problem of how to ensure that at most one process is executing its critical section at a given time. Important: Critical sections in different threads are not necessarily the same code segment! Concurrent Software Systems 2 1 Problem Description Formally, the following requirements should be satisfied: Mutual exclusion: When a thread is executing in its critical section, no other threads can be executing in their critical sections. Progress: If no thread is executing in its critical section, and if there are some threads that wish to enter their critical sections, then one of these threads will get into the critical section. Bounded waiting: After a thread makes a request to enter its critical section, there is a bound on the number of times that other threads are allowed to enter their critical sections, before the request is granted. Concurrent Software Systems 3 Problem Description In discussion of the critical section problem, we often assume that each thread is executing the following code. It is also assumed that (1) after a thread enters a critical section, it will eventually exit the critical section; (2) a thread may terminate in the non-critical section. while (true) { entry section critical section exit...
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...2 2.1 Introduction to Equations 2.2 Linear Equations 2.3 Introduction to Problem Solving 2.4 Formulas 2.5 Linear Inequalities Linear Equations and Inequalities Education is not the filling of a pail, but the lighting of a fire. — WILLIAM BUTLER YEATS M athematics is a unique subject that is essential for describing, or modeling, events in the real world. For example, ultraviolet light from the sun is responsible for both tanning and burning exposed skin. Mathematics lets us use numbers to describe the intensity of ultraviolet light. The table shows the maximum ultraviolet intensity measured in milliwatts per square meter for various latitudes and dates. Latitude 0 10 20 30 40 50 Mar. 21 325 311 249 179 99 57 June 21 254 275 292 248 199 143 Sept. 21 325 280 256 182 127 75 Dec. 21 272 220 143 80 34 13 ISBN 1-256-49082-2 Source: J. Williams, The USA Today Weather Almanac. If a student from Chicago, located at a latitude of 42°, spends spring break in Hawaii with a latitude of 20°, the sun’s ultraviolet rays in Hawaii will be approximately 249 2.5 99 times as intense as they are in Chicago. Equations can be used to describe, or model, the intensity of the sun at various latitudes. In this chapter we will focus on linear equations and the related concept of linear inequalities. 89 Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by...
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...Abstract: In this paper we first see how NP problems can be categorized and look at some examples. We then analyze the subset sum problem and prove its NP completeness. We look at various algorithms used to solve the problem and the efficiency of each algorithm. Hence we prove that subset sum problem is NP complete. Keywords: NP complete, NP hard, complexity, subset sum, exponential I. Introduction In the study of computation complexity of problems, we find out the resources required during computation like time, memory and space required. We check if the problem can be solved in polynomial time by some algorithm. An algorithm is said to be solved in polynomial time if its worst case efficiency belongs to O (p (n)) where p (n) is the polynomial of the problems input size. Problems that can be solved in polynomial time are called tractable and those that cannot be solved in polynomial time are intractable. The formal definition of NP problem is as follows: A problem is said to be Non-deterministically Polynomial (NP) if we can find a nondeterministic Turing machine that can solve the problem in a polynomial number of nondeterministic moves. Equivalent definition of NP problem: A problem is said to be NP if 1. its solution comes from a finite set of possibilities, and 2. It takes polynomial time to verify the correctness of a possible solution. NP-complete problems NP-complete problems are a subset of class NP. An problem p is NP-complete, if 1. p ∈ NP (you can solve...
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...please feel free to Google or YouTube the concepts. Within the directions of each problem, you will see key words that you should be using in your research. Show all work that is required to be performed in order to solve the problem! Partial credit will be awarded for every correct step given even though the final answer might be incorrect. Points will be deducted if your work is messy and illegible. All answers must be exact and simplified unless the problem asks for an approximation! 1) Evaluate the expression. 5xy 6 y if x 3 & y 4 2) Evaluate the expression. 3x 2 2 xy if x 3 & y 4 3) Evaluate the expression. x 3 y y x if x 6 & y 2 4) Evaluate the expression. xy xx y if x 6 & y 2 5) Combine the like terms. 11x 5 y 7 y 4 x 6) Combine the like terms. 13x 2 7 x 8x 2 x Ocean Township HS Mathematics Department (Algebra 2 Summer Assignment) 1 7) Combine the like terms. 7 xy x 2 y 9 x 2 y 4 xy 8) Combine the like terms. 8x 2 12 x 3 3x 2 5x 3 9) Solve the linear equation. 2 x 9 25 10) Solve the linear equation. 11 20 3x 11) Solve the linear equation. x 7 11 3 12) Solve the linear equation. 2 x 6 x 24 13) Solve the linear equation. x 4 x 6 30 14) Solve the linear equation. 2x 4 11 15) Solve the linear equation. 53 x 25...
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