Sensitivity Analysis with Data Tables. 10% annual interest now =$110 one year later. 10% annual interest now =$121 one year later
|
|
- Clifton Garrison
- 5 years ago
- Views:
Transcription
1 Sensitivity Analysis with Data Tables Time Value of Money: A Special kind of Trade-Off: 10% annual interest now =$110 one year later 10% annual interest now =$121 one year later 10% annual interest now =$121 two years later General Formulation: PV(X) = Present Value of dollar amount X R = interest rate per period (e.g. R = 10% = 0.10) FV n (X) = Future Value of X after n periods n FV ( X ) = (1 + R) PV ( X ) n PV ( X ) = FV n ( X ) (1 + R) n Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 1
2 Note difference between: 10% Annual Interest Rate and 10% Annual Interest Rate compounded monthly 1. Annual Interest = $110 dollars after one year 2. 10% Annual Interest compounded monthly over 12 periods = *$100 = $ *100 Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 2
3 Stream of Cash Flow: Interest Rate 10.00% At the End of Year Future Value Factor Present Value 0 -$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $38.55 NPV $ Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 3
4 Constant Interest Rate Formulation: X k = cost/revenue at the end of period k R% = interest rate per period NPV ( X, X, L X ) 0 1 n = n k = 0 X k ( 1+ R) k Varying Interest Rate Formulation: X k = cost/revenue at the end of period k R k % = interest rate during period k NPV ( X, X, L X ) 0 1 n = n k k k = 0 ( 1+ R j ) j= 1 X Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 4
5 Note: cost X k negative ; revenue X k positive Comparison of Two Streams of Cash Flows Cash Flow for Two Projects $ $ $50.00 $0.00 -$ $ $ Project 1 Project 2 Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 5
6 Cash Flow Characteristics Project 1: Small Start-Up Cost and Increasing Profits. Cash Flow Characteristics Project 2: Large Start-Up Costs and Decreasing Profits. WHICH ONE WOULD YOU PREFER? ANSWER: DEPENDS ON THE INTEREST RATE Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 6
7 Interest 10.00% Rate Time Project 1 Project 2 0 -$ $ $10.00 $ $20.00 $ $30.00 $ $40.00 $ $50.00 $ $60.00 $ $70.00 $ $80.00 $ $90.00 $ $ $5.00 NPV $ $ Is there an Interest Rate at which Project 2 would be preferred? Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 7
8 R NPV 1 NPV 2 NPV1 - NPV 2 PREFERRED $ $ $ % $ $ $ NPV % $ $ $ NPV % $ $ $90.04 NPV % $ $ $75.38 NPV % $ $ $62.23 NPV % $ $ $50.45 NPV % $ $ $39.88 NPV % $ $ $30.41 NPV % $ $ $21.93 NPV % $ $ $14.33 NPV % $ $ $7.53 NPV % $ $ $1.45 NPV % $ $ $3.98 NPV % $ $ $8.83 NPV % $ $ $13.15 NPV % $ $ $16.99 NPV % $ $ $20.41 NPV % $ $ $23.43 NPV % $ $ $26.11 NPV % $ $ $28.46 NPV 2 Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 8
9 NPV $ $ $50.00 Difference between NPV 1 and NPV 2 as a function of Interest Rate Project 1 preferred Project 2 preferred $0.00 -$ % 3.00% 5.00% 7.00% 9.00% 11.00% 13.00% Interest Rate 15.00% 17.00% 19.00% NPV1 - NPV 2 It appears that NPV 1 and NPV 2 break even between 12% and 13 %. Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 9
10 The Break Even Point for the interest rate can be calculated exactly using the GOALSEEK function in Excel. GOALSEEK allows to search for root of the equation where F(x) is a continuous function. F(x)=0, GOALSEEK Method is similar to Bisection Method or Newton-Raphson Method Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 10
11 BISECTION METHOD: Starting at interval [a 1,b 1 ] established such that F(a 1 )*F(b 1 )<0 EXAMPLE BISECTION METHOD a b 2 b 4 b 2 =b 3 b 1 a 1 = a 2 a 3 a b 1 Stop when a + k b F k 2 < δ or b k a k < δ Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 11
12 NEWTON-RAPHSON METHOD: Requires Best-Guess and being able to calculate first order derivative EXAMPLE NEWTON RAPHSON METHOD F(a 1 ) a k+ 1 = a k F( ak ) d F( a dx k ) a 1 a 2 a 3 a 4 d F( a1) ( x a1) + F( a1) dx Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 12
13 Stop when a + k b F k 2 < δ QUESTION: DOES NEWTON-RAPHSON ALWAYS WORK? ANSWER: NO! EXAMPLE NEWTON RAPHSON METHOD F(a 1 ) a k+ 1 = a k F( ak ) d F( ak ) dx a 4 a 3 a 1 a 2 d dx F( a1) ( x a1) + F( a1) Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 13
14 Newton-Raphson may not converge at all or may converge to another solution of the equation F(x)=0. GOALSEEK FUNCTION of EXCEL is similar to the Newton-Raphson method. If a solution is not found, does not necessarily mean that none exists. Try different starting values! Be careful when using this method!!! Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 14
15 Interest 12.26% Rate Time Project 1 Project 2 NPV1 - NPV 2 0 -$ $ $ $10.00 $ $20.00 $ $30.00 $ $40.00 $ $50.00 $ $60.00 $ $70.00 $ $80.00 $ $90.00 $ $ $5.00 NPV $ $ Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 15
16 HOME WORK 1: 1. Consider the cash flows of Project 1 and Project 2. Assume that at the beginning of the project the interest rate equals 10%, and that over the duration of the project (10 years), the interest rate increases each year by 0.5%. Which Project is preferred based on Net Present Value? 2. Determine the rate of increase in interest at which you would be indifferent between Project 1 and Project 2 based on Net Present Value. Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 16
17 Two Way Data Tables Description Case Study: A product costs $3.00 per unit to make. Demand for the product is determined by two factors: The price of the product, i.e. the lower the price, the higher the demand. The advertising budget, i.e. the more you advertise the higher the demand. It is estimated (e.g. using market research) that Demand D (in thousands of units) is: 3 D = A 7P + A* 1 P where A = advertising budget (in $000 s) and P = Price per Unit. Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 17
18 Profit can be calculated as a function of A and P as well: 1. Profit=Total Revenue Total Cost Advertising Budget : A $ Price per Unit : P $ Total Revenue = 1000*D*P Cost per Unit : C $ Total Cost = Production Cost + Advertising Cost 4. Production Cost = 1000*D*$3 5. Advertising Cost = 1000*A Demand (in 1000's) Revenue $356, Production Cost $106, Advertising Cost $120, Total Cost $226, Profit $129, Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 18
19 Using Data Tables in Excel we can produce a graph of the DEMAND as a function of A and P. Demand Advertising Budget Price Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 19
20 Demand Price Advertisin g Budget Note that: Demand decreases when Price Increases Demand increases when you spend more money on advertising Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 20
21 Using Data Tables we can produce a graph of the Profit as a function of A and P. Profit $129, $20.00 $40.00 $60.00 $80.00 $ $ $ $ $ $ $156, $180, $203, $225, $247, $269, $291, $313, $335, $ $82, $105, $127, $149, $171, $192, $213, $234, $256, $ $20, $40, $60, $80, $100, $120, $140, $160, $180, $ 4.00 $31, $15, $1, $19, $37, $55, $73, $91, $110, $ 5.00 $71, $59, $46, $31, $16, $ $15, $31, $47, $ 6.00 $98, $92, $83, $72, $60, $47, $34, $20, $5, $ 7.00 $113, $113, $109, $102, $93, $83, $73, $62, $50, $ 8.00 $115, $122, $123, $120, $116, $109, $102, $94, $85, $ 9.00 $105, $119, $126, $128, $127, $125, $121, $116, $110, $ $82, $104, $116, $124, $127, $129, $129, $127, $124, $ $47, $77, $95, $108, $116, $122, $126, $128, $129, $ $1, $37, $62, $80, $94, $104, $112, $118, $122, $ $63, $15, $17, $41, $59, $74, $86, $96, $105, Max Profit $115, $122, $126, $128, $127, $129, $129, $128, $129, Note that: For a fixed advertising budget there is a price at which profit is optimal For a fixed price there is an advertising budget at which profit is optimal (less clear from the figure) Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 21
22 Is there a combination of Advertising Budget and Price at which profit is maximized? $1.00 $5.00 Price $9.00 $13.00 $20.00 $60.00 $ $ $ $150,000 $100,000 $50,000 $0 -$50,000 -$100,000 -$150,000 -$200,000 -$250,000 -$300,000 -$350,000 Profit Advertising Budget Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 22
23 Using Solver to Maximize Profit General Formulation of Optimization Problems: Max (or Min): F(x 1, x n ) Subject to : G 1 (x 1, x n ) 0 G 2 (x 1, x n ) 0 M G m (x 1, x n ) 0 Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 23
24 Function F(x 1, x n ) is called the objective function with variables x 1, x n. Functions G j (x 1, x n ) j=1,,m are called the constrained functions (or constraints). The set S = n {( x1, L, xn ) G1( x1, L, xn ) 0, L, Gm ( x1, L, x ) 0} is called the feasible set (informally, the set of allowable solutions). IMPORTANT: The feasible set is bounded Optimization problem has a solution Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 24
25 (i.e. there is a feasible solution that globally maximizes the objective function) Objective Function Local Maxima Global Maximum x a 0 b x 0 Constraint Function Constraint Function Feasible Set is Bounded Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 25
26 Linear Optimization Problems: Both the objective function and the constrained functions are of the form a 1 x 1 + a 2 x 2 + L+ a n x n Non-Linear Optimization Problems: Objective function or constrained functions are non-linear. An optimization method solving a Non-Linear Optimization problem may get trapped in a local maximum. QUESTION? Under what conditions on the objective function and the constraint functions is a typical maximization method guaranteed to find a global maximum? Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 26
27 ANSWER: When the objective function is CONCAVE and constraint functions are CONVEX A function is concave on [a,b] when xy, [ ab, ] : f ( λ x + (1 λ) y) λ f ( x) + (1 λ) f ( y), λ [0,1] f ( λ x + (1 λ) y) f ( y) f (x) λ f ( x) + (1 λ) f ( y) x y Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 27
28 A function is convex on [x,y] when xy, [ ab, ] : f ( λ x + (1 λ) y) λ f ( x) + (1 λ) f ( y), λ [0,1] λ f ( x) + (1 λ) f ( y) f ( y) f (x) f ( λ x + (1 λ) y) x y Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 28
29 QUESTION? Under what conditions on the objective function and the constraint functions is a typical minimization method guaranteed to find a global minimum? ANSWER: When the objective function is CONVEX and constraint functions are CONVEX An optimization problem is convex if: 1. The constraint functions G j (x 1, x n ) are convex functions. 2. If the optimization problem is a minimization problem the objective function F(x 1, x n ) is convex. Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 29
30 3. If the optimization problem is a maximization problem the objective function F(x 1, x n ) is concave. Practical Implications: If you can show that the optimization problem is convex and your optimization algorithm finds an optimal solution, your solution is a global optimum. If you cannot show that the optimization problem is convex and your optimization algorithm finds an optimal solution, your solution is a local optimum and you can never guarantee global optimality. Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 30
31 Graph 3D-Convex Maximization Problem. F(x,y) y x Constraint 1 Constraint 3 Feasible Set Constraint 2 Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 31
32 Basic approach to solving non-linear maximization problem: 1. Start at a feasible solution, choose a feasible search direction of ascent and follow that direction until you go downhill. 2. Choose a new search direction of ascent and follow that until you go downhill. 3. Stop when you cannot find a feasible search direction of ascent. (E.g. Steepest Ascent Method, Conjugate Gradient Method). Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 32
33 Graph 3D- Linear Maximization Problem. F(x,y) Constraint 5 y x Constraint 1 Global Optimum Constraint 4 Constraint 2 Feasible Set Constraint 3 Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 33
34 Note that: Global Optimum of a Linear Optimization Problem is attained at a corner point of the feasible set Corner points of a feasible set in a Linear Optimization Problem are called Extreme Points (or Vertices). Each Linear Optimization problem has a finite number of extreme points. Basic approach to solving linear optimization problem: Enumerate extreme points, evaluate the objective function at extreme points and stop when you cannot improve. (E.g. Simplex Method). Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 34
35 QUESTION? WHY CAN YOU NOT USE A LINEAR OPTIMIZATION METHOD TO SOLVE A NON-LINEAR OPTIMIZATION PROBLEM? ANSWER: LINEAR-OPTIMIZATION METHODS ARE TYPICALLY LIMITED TO ENUMERATING EXTREME POINTS AND A GLOBAL OPTIMUM OF A NON- LINEAR OPTIMIZATION PROBLEM DOES NOT HAVE TO BE ATTAINED IN AN EXTREME POINT. Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 35
36 QUESTION? CAN YOU USE A NON-LINEAR OPTIMIZATION METHOD TO SOLVE A LINEAR OPTIMIZATION PROBLEM? ANSWER: YES, BUT THIS MAY OR MAY NOT BE COMPUTATIONALLY EFFICIENT. EXAMPLE: INTERIOR POINT METHOD OF KARMAKAR HAS BETTER THEORETICAL COMPUTATIONAL COMLEXITY THAN SIMPLEX METHOD Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 36
37 Going back to the optimization problem of our case study: Maximize : Profit = (1000*P 3000)*D 1000*A Subject to : P 13 Where: 3 D = A 7P + A* 1 P IS THIS OPTIMIZATION PROBLEM LINEAR? IS THIS OPTIMIZATION PROBLEM CONVEX? Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 37
38 Using SOLVER in EXCEL and the GRG Optimization method 1, we find the following optimal solution: Advertising Budget : A $ Price per Unit : P $10.09 Cost per Unit : C $3.00 Demand (in 1000's) Revenue $371, Production Cost $110, Advertising Cost $131, Total Cost $241, Profit $129, GRG Optimization method is the Generalized Reduced Gradient optimization method, a non-linear optimization approach. Details of this approach are beyond the scope of this course. Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 38
39 HOMEWORK 2: PROVE OR PROVIDE COUNTER EXAMPLE IS THE SUM OF TWO CONCAVE FUNCTIONS CONCAVE? IS THE PRODUCT OF TWO CONCAVE FUNCTIONS CONCAVE? Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 39
40 Two-way Data Tables and Decision Making Uncertainty Description Case Study: Eli Daisy must decide on monthly storage capacity for a new drug. The drug will sell for a period of 10 years at a price of $7 per unit. The production cost per drug unit is $4. One additional unit of storage capacity will cost $75 to build per drug unit. Storage capacity cost $1 annually per drug unit for maintenance. Production can only occur at the beginning of the month (Set-up Cost for production are exorbitant). You always produce the full storage capacity as FDA has a monthly expiration date on the drug. Clearly, the most ideal (profitable scenario) would be to produce the same amount as the monthly demand for the drug. The monthly demand for the drug will be constant over this 10 year period but is uncertain. Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 40
41 Probability (0.1) (0.2) (0.3) Monthly Demand 100, , ,000 Note that: (0.2) (0.1) (0.1) 400, , ,000 You would prefer not to produce more on a monthly basis than your monthly demand as demand is constant. You do not want excess storage capacity as even unused storage space cost $1 annually in maintenance cost. Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 41
42 Profit Calculation given fixed Storage Capacity and a known Monthly Demand for 10 years: Profit = Total Revenue Total Cost Total Revenue = 10*(Units Sold per Year)*Price Unit Sold Per Year = 12* Min(Storage Capacity, Monthly Demand) Total Cost = Building Cost + 10 *Annual Maintenance Cost + 10 *Annual Production Cost Building Cost = Storage Capacity*$15 Annual Maintenance Cost = Storage Capacity*$1 Annual Production Cost = 12*Storage Capacity*$4. Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 42
43 Capacity Planning (10 Year Period) Variable Value Unit Production Cost 4 Sales price 7 Building cost per Unit Storage Capacity 75 Annual maintenance cost per unit of Storage Capacity 1 Storage Capacity Level 300,000 Monthly demand 100,000 Planning Horizon 10 Revenue over Planning Horizon Unit Monthly sales 100,000 Total Revenue $ 84,000,000 Total Costs over Planning Horizon Building Cost (Fixed) $ 22,500,000 Maintenance Cost (Variable) $ 3,000,000 Production Costs (Variable) $ 144,000,000 Total costs $ 169,500,000 Profit -$85,500,000 Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 43
44 Possible Capacity Levels are 100, ,000. What amount of storage capacity should Eli build for this 10 year period? Solve Decision Problem using Expected Monetary Value (EMV) Expected Value of discrete random variable Y: E Y [ Y ] = n i= 1 y i Pr( Y = y i ) = n i= 1 y i p i Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 44
45 EMV= $4.5 Trade Ticket -$1 Keep Ticket $0 EMV= $4 EMV= $4.5 Max Profit Win (0.20) $24 $25 Lose (0.80) -$1 $0 Win (0.45) $10 $10 Lose (0.55) $0 $0 y Pr(Y=y) y*pr(y=y) $ $4.80 -$ $0.80 $4.00 =EMV y Pr(Y=y) y*pr(y=y) $ $4.50 $ $0.00 $4.50 =EMV Interpretation: Playing the lottery a lot of times will result in an average payoff equal to the EMV. Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 45
46 Returning to our Case Study: probabilities profit Monthtly Demand $ (85,500,000.00) Mean profit $27,500, $27,500, $27,500, $27,500, $27,500, $27,500, $ 27,500, $29,000, $55,000, $55,000, $55,000, $55,000, $55,000, $ 46,600,000 Annual $85,500, $1,500, $82,500, $82,500, $82,500, $82,500, $ 48,900,000 Capacity $142,000, $58,000, $26,000, $110,000, $110,000, $110,000, $ 26,000, $198,500, $114,500, $30,500, $53,500, $137,500, $137,500, $ (13,700,000) $255,000, $171,000, $87,000, $3,000, $81,000, $165,000, $ (61,800,000) Probability Monthly Demand Probability Monthly Demand EMV = $27,500K (0.1) (0.2) (0.3) 100, , ,000 EMV = -$61,800K (0.1) (0.2) (0.3) 100, , ,000 (0.2) (0.1) (0.1) 400, , ,000 (0.2) (0.1) (0.1) 400, , ,000 FIRST ROW LAST ROW Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 46
47 10 Year Capacity 100, , , , , ,000 EMV 27,500,000 EMV 46,600,000 EMV 48,900,000 EMV 26,000,000 EMV -13,700,000 EMV -61,800,000 CONCLUSION: SET STORAGE CAPACITY AT 300,000 Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 47
48 Variance : VARIANCE AND STANDARD DEVIATION OF Y: Var Y E Y E Y 2 ( ) = σ Y = [ [ ] ] ( ) 2 n = i i i= 1 p *( y E[ Y]) 2 Standard Deviation : σ 2 Y = σ Y Informal Interpretation: Standard deviation is the best guess distance from the mean for an arbritrary outcome Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 48
49 Squared deviation Variance Std dev E E E E E E E+00 $ E E E E E E E+14 $25,200, E E E E E E E+15 $55,719, E E E E E E E+15 $84,000, E E E E E E E+16 $104,915, E E E E E E E+16 $119,090, CONCLUSION: A Storage Capacity of 300,000 yields the highest expected profit of $48,900,000 over 10 years with a standard deviation of $55,719,296. IS THIS A GOOD INVESTMENT OPPORTUNITY? ANSWER DEPENDS ON DECISION MAKER S RISK AVERNESS! Pr( EMV < 0 Storage Capacity=100,000)=0% Pr( EMV < 0 Storage Capacity=200,000)=10% Pr(EMV < 0 Storage Capacity=300,000)=30% Lecture Notes by: Dr. J. Rene van Dorp Chapter 1 - Page 49
Continuing Education Course #287 Engineering Methods in Microsoft Excel Part 2: Applied Optimization
1 of 6 Continuing Education Course #287 Engineering Methods in Microsoft Excel Part 2: Applied Optimization 1. Which of the following is NOT an element of an optimization formulation? a. Objective function
More informationOptimization Models one variable optimization and multivariable optimization
Georg-August-Universität Göttingen Optimization Models one variable optimization and multivariable optimization Wenzhong Li lwz@nju.edu.cn Feb 2011 Mathematical Optimization Problems in optimization are
More informationReview of Expected Operations
Economic Risk and Decision Analysis for Oil and Gas Industry CE81.98 School of Engineering and Technology Asian Institute of Technology January Semester Presented by Dr. Thitisak Boonpramote Department
More informationCSCI 1951-G Optimization Methods in Finance Part 00: Course Logistics Introduction to Finance Optimization Problems
CSCI 1951-G Optimization Methods in Finance Part 00: Course Logistics Introduction to Finance Optimization Problems January 26, 2018 1 / 24 Basic information All information is available in the syllabus
More informationMathematics for Management Science Notes 07 prepared by Professor Jenny Baglivo
Mathematics for Management Science Notes 07 prepared by Professor Jenny Baglivo Jenny A. Baglivo 2002. All rights reserved. Calculus and nonlinear programming (NLP): In nonlinear programming (NLP), either
More information1 Overview. 2 The Gradient Descent Algorithm. AM 221: Advanced Optimization Spring 2016
AM 22: Advanced Optimization Spring 206 Prof. Yaron Singer Lecture 9 February 24th Overview In the previous lecture we reviewed results from multivariate calculus in preparation for our journey into convex
More informationMicroeconomics of Banking: Lecture 2
Microeconomics of Banking: Lecture 2 Prof. Ronaldo CARPIO September 25, 2015 A Brief Look at General Equilibrium Asset Pricing Last week, we saw a general equilibrium model in which banks were irrelevant.
More informationCS 3331 Numerical Methods Lecture 2: Functions of One Variable. Cherung Lee
CS 3331 Numerical Methods Lecture 2: Functions of One Variable Cherung Lee Outline Introduction Solving nonlinear equations: find x such that f(x ) = 0. Binary search methods: (Bisection, regula falsi)
More informationGame Theory: Normal Form Games
Game Theory: Normal Form Games Michael Levet June 23, 2016 1 Introduction Game Theory is a mathematical field that studies how rational agents make decisions in both competitive and cooperative situations.
More informationOptimization for Chemical Engineers, 4G3. Written midterm, 23 February 2015
Optimization for Chemical Engineers, 4G3 Written midterm, 23 February 2015 Kevin Dunn, kevin.dunn@mcmaster.ca McMaster University Note: No papers, other than this test and the answer booklet are allowed
More informationAnswer Key for M. A. Economics Entrance Examination 2017 (Main version)
Answer Key for M. A. Economics Entrance Examination 2017 (Main version) July 4, 2017 1. Person A lexicographically prefers good x to good y, i.e., when comparing two bundles of x and y, she strictly prefers
More informationCharacterization of the Optimum
ECO 317 Economics of Uncertainty Fall Term 2009 Notes for lectures 5. Portfolio Allocation with One Riskless, One Risky Asset Characterization of the Optimum Consider a risk-averse, expected-utility-maximizing
More informationTrust Region Methods for Unconstrained Optimisation
Trust Region Methods for Unconstrained Optimisation Lecture 9, Numerical Linear Algebra and Optimisation Oxford University Computing Laboratory, MT 2007 Dr Raphael Hauser (hauser@comlab.ox.ac.uk) The Trust
More informationLecture Quantitative Finance Spring Term 2015
implied Lecture Quantitative Finance Spring Term 2015 : May 7, 2015 1 / 28 implied 1 implied 2 / 28 Motivation and setup implied the goal of this chapter is to treat the implied which requires an algorithm
More informationLecture 7: Bayesian approach to MAB - Gittins index
Advanced Topics in Machine Learning and Algorithmic Game Theory Lecture 7: Bayesian approach to MAB - Gittins index Lecturer: Yishay Mansour Scribe: Mariano Schain 7.1 Introduction In the Bayesian approach
More informationSOLVING ROBUST SUPPLY CHAIN PROBLEMS
SOLVING ROBUST SUPPLY CHAIN PROBLEMS Daniel Bienstock Nuri Sercan Özbay Columbia University, New York November 13, 2005 Project with Lucent Technologies Optimize the inventory buffer levels in a complicated
More informationApplications of Linear Programming
Applications of Linear Programming lecturer: András London University of Szeged Institute of Informatics Department of Computational Optimization Lecture 8 The portfolio selection problem The portfolio
More informationSupport Vector Machines: Training with Stochastic Gradient Descent
Support Vector Machines: Training with Stochastic Gradient Descent Machine Learning Spring 2018 The slides are mainly from Vivek Srikumar 1 Support vector machines Training by maximizing margin The SVM
More informationGolden-Section Search for Optimization in One Dimension
Golden-Section Search for Optimization in One Dimension Golden-section search for maximization (or minimization) is similar to the bisection method for root finding. That is, it does not use the derivatives
More informationThe homework is due on Wednesday, September 7. Each questions is worth 0.8 points. No partial credits.
Homework : Econ500 Fall, 0 The homework is due on Wednesday, September 7. Each questions is worth 0. points. No partial credits. For the graphic arguments, use the graphing paper that is attached. Clearly
More informationEcon 172A, W2002: Final Examination, Solutions
Econ 172A, W2002: Final Examination, Solutions Comments. Naturally, the answers to the first question were perfect. I was impressed. On the second question, people did well on the first part, but had trouble
More informationPart 3: Trust-region methods for unconstrained optimization. Nick Gould (RAL)
Part 3: Trust-region methods for unconstrained optimization Nick Gould (RAL) minimize x IR n f(x) MSc course on nonlinear optimization UNCONSTRAINED MINIMIZATION minimize x IR n f(x) where the objective
More informationMS-E2114 Investment Science Exercise 4/2016, Solutions
Capital budgeting problems can be solved based on, for example, the benet-cost ratio (that is, present value of benets per present value of the costs) or the net present value (the present value of benets
More informationRisk-Return Optimization of the Bank Portfolio
Risk-Return Optimization of the Bank Portfolio Ursula Theiler Risk Training, Carl-Zeiss-Str. 11, D-83052 Bruckmuehl, Germany, mailto:theiler@risk-training.org. Abstract In an intensifying competition banks
More informationHandout 8: Introduction to Stochastic Dynamic Programming. 2 Examples of Stochastic Dynamic Programming Problems
SEEM 3470: Dynamic Optimization and Applications 2013 14 Second Term Handout 8: Introduction to Stochastic Dynamic Programming Instructor: Shiqian Ma March 10, 2014 Suggested Reading: Chapter 1 of Bertsekas,
More informationFinancial Optimization ISE 347/447. Lecture 15. Dr. Ted Ralphs
Financial Optimization ISE 347/447 Lecture 15 Dr. Ted Ralphs ISE 347/447 Lecture 15 1 Reading for This Lecture C&T Chapter 12 ISE 347/447 Lecture 15 2 Stock Market Indices A stock market index is a statistic
More informationStock Repurchase with an Adaptive Reservation Price: A Study of the Greedy Policy
Stock Repurchase with an Adaptive Reservation Price: A Study of the Greedy Policy Ye Lu Asuman Ozdaglar David Simchi-Levi November 8, 200 Abstract. We consider the problem of stock repurchase over a finite
More informationWhat can we do with numerical optimization?
Optimization motivation and background Eddie Wadbro Introduction to PDE Constrained Optimization, 2016 February 15 16, 2016 Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15 16, 2016
More informationCS360 Homework 14 Solution
CS360 Homework 14 Solution Markov Decision Processes 1) Invent a simple Markov decision process (MDP) with the following properties: a) it has a goal state, b) its immediate action costs are all positive,
More informationEconomics 2010c: Lecture 4 Precautionary Savings and Liquidity Constraints
Economics 2010c: Lecture 4 Precautionary Savings and Liquidity Constraints David Laibson 9/11/2014 Outline: 1. Precautionary savings motives 2. Liquidity constraints 3. Application: Numerical solution
More informationCSCI 1951-G Optimization Methods in Finance Part 07: Portfolio Optimization
CSCI 1951-G Optimization Methods in Finance Part 07: Portfolio Optimization March 9 16, 2018 1 / 19 The portfolio optimization problem How to best allocate our money to n risky assets S 1,..., S n with
More informationChapter 6 Analyzing Accumulated Change: Integrals in Action
Chapter 6 Analyzing Accumulated Change: Integrals in Action 6. Streams in Business and Biology You will find Excel very helpful when dealing with streams that are accumulated over finite intervals. Finding
More informationPenalty Functions. The Premise Quadratic Loss Problems and Solutions
Penalty Functions The Premise Quadratic Loss Problems and Solutions The Premise You may have noticed that the addition of constraints to an optimization problem has the effect of making it much more difficult.
More informationPORTFOLIO OPTIMIZATION AND EXPECTED SHORTFALL MINIMIZATION FROM HISTORICAL DATA
PORTFOLIO OPTIMIZATION AND EXPECTED SHORTFALL MINIMIZATION FROM HISTORICAL DATA We begin by describing the problem at hand which motivates our results. Suppose that we have n financial instruments at hand,
More informationConsumption, Investment and the Fisher Separation Principle
Consumption, Investment and the Fisher Separation Principle Consumption with a Perfect Capital Market Consider a simple two-period world in which a single consumer must decide between consumption c 0 today
More informationINTERNATIONAL UNIVERSITY OF JAPAN Public Management and Policy Analysis Program Graduate School of International Relations
Hun Myoung Park (4/18/2018) LP Interpretation: 1 INTERNATIONAL UNIVERSITY OF JAPAN Public Management and Policy Analysis Program Graduate School of International Relations DCC5350 (2 Credits) Public Policy
More informationLecture 2: Fundamentals of meanvariance
Lecture 2: Fundamentals of meanvariance analysis Prof. Massimo Guidolin Portfolio Management Second Term 2018 Outline and objectives Mean-variance and efficient frontiers: logical meaning o Guidolin-Pedio,
More informationLecture 1: The market and consumer theory. Intermediate microeconomics Jonas Vlachos Stockholms universitet
Lecture 1: The market and consumer theory Intermediate microeconomics Jonas Vlachos Stockholms universitet 1 The market Demand Supply Equilibrium Comparative statics Elasticities 2 Demand Demand function.
More informationFINA 695 Assignment 1 Simon Foucher
Answer the following questions. Show your work. Due in the class on March 29. (postponed 1 week) You are expected to do the assignment on your own. Please do not take help from others. 1. (a) The current
More informationEconomic Risk and Decision Analysis for Oil and Gas Industry CE School of Engineering and Technology Asian Institute of Technology
Economic Risk and Decision Analysis for Oil and Gas Industry CE81.98 School of Engineering and Technology Asian Institute of Technology January Semester Presented by Dr. Thitisak Boonpramote Department
More informationAdvanced Operations Research Prof. G. Srinivasan Dept of Management Studies Indian Institute of Technology, Madras
Advanced Operations Research Prof. G. Srinivasan Dept of Management Studies Indian Institute of Technology, Madras Lecture 23 Minimum Cost Flow Problem In this lecture, we will discuss the minimum cost
More informationIntroduction to Numerical Methods (Algorithm)
Introduction to Numerical Methods (Algorithm) 1 2 Example: Find the internal rate of return (IRR) Consider an investor who pays CF 0 to buy a bond that will pay coupon interest CF 1 after one year and
More informationDecomposition Methods
Decomposition Methods separable problems, complicating variables primal decomposition dual decomposition complicating constraints general decomposition structures Prof. S. Boyd, EE364b, Stanford University
More informationLog-Robust Portfolio Management
Log-Robust Portfolio Management Dr. Aurélie Thiele Lehigh University Joint work with Elcin Cetinkaya and Ban Kawas Research partially supported by the National Science Foundation Grant CMMI-0757983 Dr.
More informationChapter 2 Linear programming... 2 Chapter 3 Simplex... 4 Chapter 4 Sensitivity Analysis and duality... 5 Chapter 5 Network... 8 Chapter 6 Integer
目录 Chapter 2 Linear programming... 2 Chapter 3 Simplex... 4 Chapter 4 Sensitivity Analysis and duality... 5 Chapter 5 Network... 8 Chapter 6 Integer Programming... 10 Chapter 7 Nonlinear Programming...
More informationMS-E2114 Investment Science Lecture 5: Mean-variance portfolio theory
MS-E2114 Investment Science Lecture 5: Mean-variance portfolio theory A. Salo, T. Seeve Systems Analysis Laboratory Department of System Analysis and Mathematics Aalto University, School of Science Overview
More informationLecture 17: More on Markov Decision Processes. Reinforcement learning
Lecture 17: More on Markov Decision Processes. Reinforcement learning Learning a model: maximum likelihood Learning a value function directly Monte Carlo Temporal-difference (TD) learning COMP-424, Lecture
More informationAP CALCULUS AB CHAPTER 4 PRACTICE PROBLEMS. Find the location of the indicated absolute extremum for the function. 1) Maximum 1)
AP CALCULUS AB CHAPTER 4 PRACTICE PROBLEMS Find the location of the indicated absolute extremum for the function. 1) Maximum 1) A) No maximum B) x = 0 C) x = 2 D) x = -1 Find the extreme values of the
More informationFebruary 24, 2005
15.053 February 24, 2005 Sensitivity Analysis and shadow prices Suggestion: Please try to complete at least 2/3 of the homework set by next Thursday 1 Goals of today s lecture on Sensitivity Analysis Changes
More informationMidterm 1, Financial Economics February 15, 2010
Midterm 1, Financial Economics February 15, 2010 Name: Email: @illinois.edu All questions must be answered on this test form. Question 1: Let S={s1,,s11} be the set of states. Suppose that at t=0 the state
More informationGraphs Details Math Examples Using data Tax example. Decision. Intermediate Micro. Lecture 5. Chapter 5 of Varian
Decision Intermediate Micro Lecture 5 Chapter 5 of Varian Decision-making Now have tools to model decision-making Set of options At-least-as-good sets Mathematical tools to calculate exact answer Problem
More informationMulti-period mean variance asset allocation: Is it bad to win the lottery?
Multi-period mean variance asset allocation: Is it bad to win the lottery? Peter Forsyth 1 D.M. Dang 1 1 Cheriton School of Computer Science University of Waterloo Guangzhou, July 28, 2014 1 / 29 The Basic
More informationThe Irrevocable Multi-Armed Bandit Problem
The Irrevocable Multi-Armed Bandit Problem Ritesh Madan Qualcomm-Flarion Technologies May 27, 2009 Joint work with Vivek Farias (MIT) 2 Multi-Armed Bandit Problem n arms, where each arm i is a Markov Decision
More informationLINEAR PROGRAMMING. Homework 7
LINEAR PROGRAMMING Homework 7 Fall 2014 Csci 628 Megan Rose Bryant 1. Your friend is taking a Linear Programming course at another university and for homework she is asked to solve the following LP: Primal:
More informationBudget Management In GSP (2018)
Budget Management In GSP (2018) Yahoo! March 18, 2018 Miguel March 18, 2018 1 / 26 Today s Presentation: Budget Management Strategies in Repeated auctions, Balseiro, Kim, and Mahdian, WWW2017 Learning
More informationt g(t) h(t) k(t)
Problem 1. Determine whether g(t), h(t), and k(t) could correspond to a linear function or an exponential function, or neither. If it is linear or exponential find the formula for the function, and then
More informationUNIVERSITY OF KWAZULU-NATAL
UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: June 006 Subject, course and code: Mathematics 34 (MATH34P Duration: 3 hours Total Marks: 00 INTERNAL EXAMINERS: Mrs. A. Campbell, Mr. P. Horton, Dr. M. Banda
More informationRobust Optimization Applied to a Currency Portfolio
Robust Optimization Applied to a Currency Portfolio R. Fonseca, S. Zymler, W. Wiesemann, B. Rustem Workshop on Numerical Methods and Optimization in Finance June, 2009 OUTLINE Introduction Motivation &
More informationx x x1
Mathematics for Management Science Notes 08 prepared by Professor Jenny Baglivo Graphical representations As an introduction to the calculus of two-variable functions (f(x ;x 2 )), consider two graphical
More informationChapter 3. A Consumer s Constrained Choice
Chapter 3 A Consumer s Constrained Choice If this is coffee, please bring me some tea; but if this is tea, please bring me some coffee. Abraham Lincoln Chapter 3 Outline 3.1 Preferences 3.2 Utility 3.3
More informationBudget Constrained Choice with Two Commodities
1 Budget Constrained Choice with Two Commodities Joseph Tao-yi Wang 2013/9/25 (Lecture 5, Micro Theory I) The Consumer Problem 2 We have some powerful tools: Constrained Maximization (Shadow Prices) Envelope
More informationDefinition 4.1. In a stochastic process T is called a stopping time if you can tell when it happens.
102 OPTIMAL STOPPING TIME 4. Optimal Stopping Time 4.1. Definitions. On the first day I explained the basic problem using one example in the book. On the second day I explained how the solution to the
More informationECON 6022B Problem Set 2 Suggested Solutions Fall 2011
ECON 60B Problem Set Suggested Solutions Fall 0 September 7, 0 Optimal Consumption with A Linear Utility Function (Optional) Similar to the example in Lecture 3, the household lives for two periods and
More informationIntroduction to Computational Finance and Financial Econometrics Introduction to Portfolio Theory
You can t see this text! Introduction to Computational Finance and Financial Econometrics Introduction to Portfolio Theory Eric Zivot Spring 2015 Eric Zivot (Copyright 2015) Introduction to Portfolio Theory
More informationLECTURE 1 : THE INFINITE HORIZON REPRESENTATIVE AGENT. In the IS-LM model consumption is assumed to be a
LECTURE 1 : THE INFINITE HORIZON REPRESENTATIVE AGENT MODEL In the IS-LM model consumption is assumed to be a static function of current income. It is assumed that consumption is greater than income at
More information1 Economical Applications
WEEK 4 Reading [SB], 3.6, pp. 58-69 1 Economical Applications 1.1 Production Function A production function y f(q) assigns to amount q of input the corresponding output y. Usually f is - increasing, that
More informationChapter 7: Random Variables
Chapter 7: Random Variables 7.1 Discrete and Continuous Random Variables 7.2 Means and Variances of Random Variables 1 Introduction A random variable is a function that associates a unique numerical value
More information6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2
6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2 Daron Acemoglu and Asu Ozdaglar MIT October 14, 2009 1 Introduction Outline Review Examples of Pure Strategy Nash Equilibria Mixed Strategies
More informationIntroduction to Operations Research
Introduction to Operations Research Unit 1: Linear Programming Terminology and formulations LP through an example Terminology Additional Example 1 Additional example 2 A shop can make two types of sweets
More informationChapter 3: Model of Consumer Behavior
CHAPTER 3 CONSUMER THEORY Chapter 3: Model of Consumer Behavior Premises of the model: 1.Individual tastes or preferences determine the amount of pleasure people derive from the goods and services they
More informationContents Critique 26. portfolio optimization 32
Contents Preface vii 1 Financial problems and numerical methods 3 1.1 MATLAB environment 4 1.1.1 Why MATLAB? 5 1.2 Fixed-income securities: analysis and portfolio immunization 6 1.2.1 Basic valuation of
More informationChapter 7 One-Dimensional Search Methods
Chapter 7 One-Dimensional Search Methods An Introduction to Optimization Spring, 2014 1 Wei-Ta Chu Golden Section Search! Determine the minimizer of a function over a closed interval, say. The only assumption
More informationMarket Risk Analysis Volume I
Market Risk Analysis Volume I Quantitative Methods in Finance Carol Alexander John Wiley & Sons, Ltd List of Figures List of Tables List of Examples Foreword Preface to Volume I xiii xvi xvii xix xxiii
More informationThe method of Maximum Likelihood.
Maximum Likelihood The method of Maximum Likelihood. In developing the least squares estimator - no mention of probabilities. Minimize the distance between the predicted linear regression and the observed
More informationOptimization in Financial Engineering in the Post-Boom Market
Optimization in Financial Engineering in the Post-Boom Market John R. Birge Northwestern University www.iems.northwestern.edu/~jrbirge SIAM Optimization Toronto May 2002 1 Introduction History of financial
More informationProbability and Stochastics for finance-ii Prof. Joydeep Dutta Department of Humanities and Social Sciences Indian Institute of Technology, Kanpur
Probability and Stochastics for finance-ii Prof. Joydeep Dutta Department of Humanities and Social Sciences Indian Institute of Technology, Kanpur Lecture - 07 Mean-Variance Portfolio Optimization (Part-II)
More informationSolution Guide to Exercises for Chapter 4 Decision making under uncertainty
THE ECONOMICS OF FINANCIAL MARKETS R. E. BAILEY Solution Guide to Exercises for Chapter 4 Decision making under uncertainty 1. Consider an investor who makes decisions according to a mean-variance objective.
More informationThe method of false position is also an Enclosure or bracketing method. For this method we will be able to remedy some of the minuses of bisection.
Section 2.2 The Method of False Position Features of BISECTION: Plusses: Easy to implement Almost idiot proof o If f(x) is continuous & changes sign on [a, b], then it is GUARANTEED to converge. Requires
More informationLinear Programming: Simplex Method
Mathematical Modeling (STAT 420/620) Spring 2015 Lecture 10 February 19, 2015 Linear Programming: Simplex Method Lecture Plan 1. Linear Programming and Simplex Method a. Family Farm Problem b. Simplex
More informationTI-83 Plus Workshop. Al Maturo,
Solving Equations with one variable. Enter the equation into: Y 1 = x x 6 Y = x + 5x + 3 Y 3 = x 3 5x + 1 TI-83 Plus Workshop Al Maturo, AMATURO@las.ch We shall refer to this in print as f(x). We shall
More informationStochastic Programming and Financial Analysis IE447. Midterm Review. Dr. Ted Ralphs
Stochastic Programming and Financial Analysis IE447 Midterm Review Dr. Ted Ralphs IE447 Midterm Review 1 Forming a Mathematical Programming Model The general form of a mathematical programming model is:
More informationPROBABILITY AND STATISTICS
Monday, January 12, 2015 1 PROBABILITY AND STATISTICS Zhenyu Ye January 12, 2015 Monday, January 12, 2015 2 References Ch10 of Experiments in Modern Physics by Melissinos. Particle Physics Data Group Review
More informationRisk Management for Chemical Supply Chain Planning under Uncertainty
for Chemical Supply Chain Planning under Uncertainty Fengqi You and Ignacio E. Grossmann Dept. of Chemical Engineering, Carnegie Mellon University John M. Wassick The Dow Chemical Company Introduction
More informationSOLUTION QUANTITATIVE TOOLS IN MANAGEMENT MAY (x) 5000 ( ) ( )
QUESTION 1 a) Annual Gross Income Less than 6000 6000 and less than 8000 8000 and less than 10000 10000 and less than 14000 14000 and less than 20000 20000 and less than 32000 32000 and above The mean:
More informationHomework #2 Graphical LP s.
UNIVERSITY OF MASSACHUSETTS Isenberg School of Management Department of Finance and Operations Management FOMGT 353-Introduction to Management Science Homework #2 Graphical LP s. Show your work completely
More informationAn Empirical Examination of the Electric Utilities Industry. December 19, Regulatory Induced Risk Aversion in. Contracting Behavior
An Empirical Examination of the Electric Utilities Industry December 19, 2011 The Puzzle Why do price-regulated firms purchase input coal through both contract Figure and 1(a): spot Contract transactions,
More informationF A S C I C U L I M A T H E M A T I C I
F A S C I C U L I M A T H E M A T I C I Nr 38 27 Piotr P luciennik A MODIFIED CORRADO-MILLER IMPLIED VOLATILITY ESTIMATOR Abstract. The implied volatility, i.e. volatility calculated on the basis of option
More informationGame Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India October 2012
Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India October 22 COOPERATIVE GAME THEORY Correlated Strategies and Correlated
More informationCEC login. Student Details Name SOLUTIONS
Student Details Name SOLUTIONS CEC login Instructions You have roughly 1 minute per point, so schedule your time accordingly. There is only one correct answer per question. Good luck! Question 1. Searching
More informationConsumer Theory. The consumer s problem: budget set, interior and corner solutions.
Consumer Theory The consumer s problem: budget set, interior and corner solutions. 1 The consumer s problem The consumer chooses the consumption bundle that maximizes his welfare (that is, his utility)
More informationEco504 Spring 2010 C. Sims FINAL EXAM. β t 1 2 φτ2 t subject to (1)
Eco54 Spring 21 C. Sims FINAL EXAM There are three questions that will be equally weighted in grading. Since you may find some questions take longer to answer than others, and partial credit will be given
More informationthat internalizes the constraint by solving to remove the y variable. 1. Using the substitution method, determine the utility function U( x)
For the next two questions, the consumer s utility U( x, y) 3x y 4xy depends on the consumption of two goods x and y. Assume the consumer selects x and y to maximize utility subject to the budget constraint
More informationEE/AA 578 Univ. of Washington, Fall Homework 8
EE/AA 578 Univ. of Washington, Fall 2016 Homework 8 1. Multi-label SVM. The basic Support Vector Machine (SVM) described in the lecture (and textbook) is used for classification of data with two labels.
More informationOptimization Models in Financial Engineering and Modeling Challenges
Optimization Models in Financial Engineering and Modeling Challenges John Birge University of Chicago Booth School of Business JRBirge UIUC, 25 Mar 2009 1 Introduction History of financial engineering
More informationSCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT. BF360 Operations Research
SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT BF360 Operations Research Unit 3 Moses Mwale e-mail: moses.mwale@ictar.ac.zm BF360 Operations Research Contents Unit 3: Sensitivity and Duality 3 3.1 Sensitivity
More informationMA200.2 Game Theory II, LSE
MA200.2 Game Theory II, LSE Answers to Problem Set [] In part (i), proceed as follows. Suppose that we are doing 2 s best response to. Let p be probability that player plays U. Now if player 2 chooses
More information4. Introduction to Prescriptive Analytics. BIA 674 Supply Chain Analytics
4. Introduction to Prescriptive Analytics BIA 674 Supply Chain Analytics Why is Decision Making difficult? The biggest sources of difficulty for decision making: Uncertainty Complexity of Environment or
More informationAdvanced Financial Economics Homework 2 Due on April 14th before class
Advanced Financial Economics Homework 2 Due on April 14th before class March 30, 2015 1. (20 points) An agent has Y 0 = 1 to invest. On the market two financial assets exist. The first one is riskless.
More informationPortfolio selection with multiple risk measures
Portfolio selection with multiple risk measures Garud Iyengar Columbia University Industrial Engineering and Operations Research Joint work with Carlos Abad Outline Portfolio selection and risk measures
More informationOptimization Methods in Management Science
Problem Set Rules: Optimization Methods in Management Science MIT 15.053, Spring 2013 Problem Set 6, Due: Thursday April 11th, 2013 1. Each student should hand in an individual problem set. 2. Discussing
More information