Screening Prospects Dominance Transparencies for chapter 4.

Slides:



Advertisements
Similar presentations
Economics of Information (ECON3016)
Advertisements

Chapter 7 Hypothesis Testing
The simplex algorithm The simplex algorithm is the classical method for solving linear programs. Its running time is not polynomial in the worst case.
Auction Theory Class 5 – single-parameter implementation and risk aversion 1.
Chapter 5 Finance Application (i) Simple Cash Balance Problem (ii) Optimal Equity Financing of a corporation (iii) Stochastic Application Cash Balance.
The securities market economy -- theory Abstracting again to the two- period analysis - - but to different states of payoff.
Utility Theory.
15 THEORY OF GAMES CHAPTER.
Risk Analysis November 3, Risk analysis is used when one or more of the numbers going into our analysis is a random variable.
1 Decision Making and Utility Introduction –The expected value criterion may not be appropriate if the decision is a one-time opportunity with substantial.
Problems With Decision Criteria Transparencies for chapter 2.
Managerial Decision Modeling with Spreadsheets
Lecture 4 Environmental Cost - Benefit - Analysis under risk and uncertainty.
1 Utility Theory. 2 Option 1: bet that pays $5,000,000 if a coin flipped comes up tails you get $0 if the coin comes up heads. Option 2: get $2,000,000.
P.V. VISWANATH FOR A FIRST COURSE IN INVESTMENTS.
Lecture 4 on Individual Optimization Risk Aversion

Three Approaches to Value There are three general approaches that we use to value any asset. –Discounted Cash Flow Valuation –Relative Valuation –Contingent.
Chapter 7 Appendix Stochastic Dominance
U.C. Berkeley© M. Spiegel and R. Stanton, BA203 Present Value Fundamentals Richard Stanton Class 2, September 1, 2000.
L1: Risk and Risk Measurement1 Lecture 1: Risk and Risk Measurement We cover the following topics in this part –Risk –Risk Aversion Absolute risk aversion.
Uncertainty and Consumer Behavior
Game Theory Here we study a method for thinking about oligopoly situations. As we consider some terminology, we will see the simultaneous move, one shot.
Complexity 19-1 Complexity Andrei Bulatov More Probabilistic Algorithms.
Drake DRAKE UNIVERSITY Fin 288 Valuing Options Using Binomial Trees.
Decision Trees and Utility Theory
CHAPTER SIX THE PORTFOLIO SELECTION PROBLEM. INTRODUCTION n THE BASIC PROBLEM: given uncertain outcomes, what risky securities should an investor own?
Visual Recognition Tutorial
Risk Aversion and Capital Allocation to Risky Assets
Chapter 6 An Introduction to Portfolio Management.
Inferences About Process Quality
INVESTMENTS | BODIE, KANE, MARCUS ©2011 The McGraw-Hill Companies CHAPTER 6 Risk Aversion and Capital Allocation to Risky Assets.
Class 2 September 16, Derivative Securities latest correction: none yet Lecture.
Probability Distributions W&W Chapter 4. Continuous Distributions Many variables we wish to study in Political Science are continuous, rather than discrete.
Chapter 9 Numerical Integration Numerical Integration Application: Normal Distributions Copyright © The McGraw-Hill Companies, Inc. Permission required.
Investment Analysis and Portfolio Management
Game Theory.
Cost of Capital MF 807 Corporate Finance Professor Thomas Chemmanur.
Decision Trees and Influence Diagrams Dr. Ayham Jaaron.
STOCHASTIC DOMINANCE APPROACH TO PORTFOLIO OPTIMIZATION Nesrin Alptekin Anadolu University, TURKEY.
Some Background Assumptions Markowitz Portfolio Theory
Investment Analysis and Portfolio Management Chapter 7.
Chapter 3: The Decision Usefulness Approach to Financial Reporting
INVESTMENTS | BODIE, KANE, MARCUS Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin CHAPTER 18 Option Valuation.
Decision Making Under Uncertainty and Risk 1 By Isuru Manawadu B.Sc in Accounting Sp. (USJP), ACA, AFM
Transformations of Risk Aversion and Meyer’s Location Scale Lecture IV.
1 Chapter 7 Applying Simulation to Decision Problems.
DISCRETE PROBABILITY DISTRIBUTIONS
Investment Analysis and Portfolio Management First Canadian Edition By Reilly, Brown, Hedges, Chang 6.
Dominance Since Player I is maximizing her security level, she prefers “large” payoffs. If one row is smaller (element- wise) than another,
Agresti/Franklin Statistics, 1 of 87  Section 7.2 How Can We Construct a Confidence Interval to Estimate a Population Proportion?
Consumer Choice With Uncertainty Part II: Expected Utility & Jensen’s Inequality Agenda: 1.From Expected Value to Expected Utility: The VNM 2.Jensen’s.
Slide Slide 1 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Overview.
Risk and Return: Portfolio Theory and Assets Pricing Models
Decision theory under uncertainty
1 Estimating Return and Risk Chapter 7 Jones, Investments: Analysis and Management.
Models for Strategic Marketing Decision Making. Market Entry Decisions To enter first or to wait Sources of First-Mover Advantages –Technological leadership.
1 Chapter 8 Revising Judgments in the Light of New Information.
Chapter 11 – Introduction to Risk Analysis u Why do individuals, companies, and stockholders take risks?
King Faisal University [ ] 1 Business School Management Department Finance Pre-MBA Dr Abdeldjelil Ferhat BOUDAH 1.
Money and Banking Lecture 11. Review of the Previous Lecture Application of Present Value Concept Internal Rate of Return Bond Pricing Real Vs Nominal.
Theory of Computational Complexity Probability and Computing Ryosuke Sasanuma Iwama and Ito lab M1.
Risk Efficiency Criteria Lecture XV. Expected Utility Versus Risk Efficiency In this course, we started with the precept that individual’s choose between.
1 Civil Systems Planning Benefit/Cost Analysis Scott Matthews Courses: / / Lecture 16.
© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-1 American Options The value of the option if it is left “alive” (i.e.,
Mean-Swap Variance,Portfolio Theory and Asset Pricing
Chapter 7 Appendix Stochastic Dominance
Further Topics on Random Variables: Derived Distributions
Further Topics on Random Variables: Derived Distributions
Further Topics on Random Variables: Derived Distributions
Presentation transcript:

Screening Prospects Dominance Transparencies for chapter 4

Introduction w In chapter 2 two types of dominance are introduced. Outcome dominance Mean-variance dominance Ask the students to recall them. What role dominance plays?

Outcome Dominance w Outcome dominance w Option a j dominates ak if and only if w y ij  y ik for all I w and y ij > y ik for at least one  i w This criterion is useful for eliminating options that are inferior. It reduces number of options and problem complexity.

Example for Outcome Dominance w Consider the following payoff matrix w a 1 a 2 a 3 w  w  w  w a 1 dominates a 2

Mean -Variance Dominance w Option a j dominates option a i iff w E(a j )  E(a i ) and V(a j ) ≤ V(a i ) with one of them an inequality. Option j E(a j ) V(a j ) Option 3 dominates options 2, option 1 dominates option 4. The efficient set: ES = { Options 1, 3 and 5}

Mean -Variance Dominance w Option a j dominates option a i iff w E(a j )  E(a i ) and V(a j ) ≤ V(a i ) with one of them an inequality. Option j E(a j ) V(a j ) Option 3 dominates options 2, option 1 dominates option 4. The efficient set: ES = { Options 1, 3 and 5}

Assumption For Dominance Validity w The Decision Maker is a Utility Maximazer w It is assumed if f j (y) dominates f i (y) then, In other words U(a j ) ≥ U(a i )

First-Degree Stochastic Dominance (FSD) w This is a generalization of payoff dominance to deal with a set of payoff distributions. w Assumptions: U(y) increasing, the decision maker prefers more. U(y) is smooth and differentiable. Discuss why this assumptions needed.

FSD Continued w a j dominates a i in FSD sense iff F i (y)  F i (y)  y Where F(y) is the cumulative probability distribution of the payoff. In other words option j provides a higher chance of obtaining higher pay.

FSD Example 1 w Check FSD for the following two course of actions p(  ) a 1 a

FSD Example 1 w Let us form pay off distributions w a 1 Y f 1 (y) F 1 (y) a 2 Y f 2 (y) F 2 (y)

Checking FSD Example1 w Interval F 1 (y) Sign F 2 (y) (- , 8) 0 = 0 [8, 9) 0.4 > 0 [9, 10) 0.4 > 0.3 [10, 11) 1.0 > 0.7 [11,  ) 1.0 = 1.0 F 1 (y)  F 2 (y), Therefore a 1 dominates a 2 in FSD The above can be done graphically also.

FSD Example 2 w Let a 1 provides a payoff Y 1 distributed U(-1, 1). w a 2 provides a payoff vector Y 2 distributed U(-2,2). w Check for FSD between a 1 and a 2.

Structure The Problem w f 1 (y) = ½  -1  y  1 = 0 otherwise f 2 (y) = ¼  -2  y  2 = 0  y < -1 = 0 if y < -1 F 1 (y) = ½ y + ½  -1  y  1 = 1  y > 1

FSD Example 2 = 0  y < -1 F 2 (y) = ¼ y + ½  -2  y  2 = 1  y > 2 It can be seen clearly that the two functions intersect at y=0 Therefore FSD fails or inconclusive.

Second-degree Stochastic Dominance (SSD) w Assumptions: All FSD assumptions Decision maker risk averse U(y) concave. w Let us define w a j dominates a i in SSD sense iff D i (z) ≥ D j (z)  z

SSD w Other characterization for SSD is given as: In other words the area F i (y) should not be less than that of F j (y). Therefore SSD can be checked graphically by examining the area under the two cumulative distribution functions.

SSD Graphically 0 y F 1 (y) F 2 (y) Area A Area B

SSD w If area A greater or equal to area B, then a2 dominates a1. Disadvantage of SSD is left tail sensitivity.. Example of this the following two a 1 is 99 with probability 1 a 2 : Y p 2 (y) a 2 does not dominate a 1 although it is far much better option. Graph the options to show that.

SSD Example w Check SSD dominance for the case given in example 2 of FSD. Recall the case w a 1 provides a payoff Y 1 distributed U(-1, 1). w a 2 provides a payoff vector Y 2 distributed U(-2,2). w Check for SSD between a 1 and a 2.

SSD Example = 0 if z <-1 w D 1 (z) = ¼ z 2 + ½ z + ¼ -1  z < 1 = z 1  z = 0 z < -2 D 2 (z) = 1/8 z2 + ½ z + ½ -2  z < 2 = z

Checking SSD Interval F 1 (y) Sign F 2 (y) (- , -2) 0 = 0 [-2, -1) 0 < 1/8 z 2 + ½ z + ½ [-1, 1) ¼ z 2 + ½ z + ¼ < 1/8 z 2 + ½ z + ½ [1, 2) z  1/8 z 2 + ½ z + ½ [2,  ) z = z D 2 (z)  D 1 (z ) for all z. Therefore a 1 dominates a 2 in SSD. The above can be done graphically also.

Some Results w If FSD holds then SSD holds. Proof is obvious. w If SSD holds FSD may not hold. The previous example shows that FSD does not but SSD holds.

Third Stochastic Dominance (TSD) w Assumptions: All the assumptions of SSD Decision maker decreasing risk averse How to check decreasing risk averse w Define the following: w Where D(z) as defined for SSD

TSD Definition w a j dominates a i in TSD iff. E[f j (y)] ≥ E[f i (y)] TD i (t) ≥ TD j (t) for all t w The second condition can be written as double integral or triple integral. Show that in class.

FSD, SSD and TSD Example w Check FSD, SSD and TSD between a 1 and a 2.  p(  ) a 1 a 2 11 2 2  3 1/ /4 44

FSD, SSD and TSD Example w a 1 : Y P 1 (y) Probability mass function F 1 (y) Distribution function a 2 : Y P 2 (y) Probability mass function F 2 (y) Distribution function

FSD, SSD and TSD Example w It is clear that FSD does not hold w Let us check SSD w Let DD(z) = D 2 (z) – D 1 (z) in computing D(z) replace integral by summation because of the discrete nature of the distribution. w DD(10) = 0.25 w DD(11) = -0.25, DD(12) = DD(13) =0 w Since DD(11) is negative, we conclude SSD inconclusive

FSD, SSD and TSD Example w Alternative way for checking SSD w Interval D 1 (z) sign D 2 (z) (- , 10) 0 = 0 [10, 11) 0 < 0.25z – 2.5 [11, 12) 0.75 z – z – 2.5 [12, 13) 0.75 z – 8.25 z [13,  ) z – 11.5 = z – 11.5 SSD inconclusive

TSD w To show TSD hold we need to show that for t = 10, 11, 12, 13 It can be seen that the sum for the above formula at t = 10 is 0.25, at t=11 it is 0 and it stays 0 for all other values. Also E(a 1 ) = E(a 2 ) = 11.5 Therefore TSD holds and a 1 dominates a 2

Nth Stochastic Dominance (NSD) w NSD can similarly be defined by integrating F i (y) and F j (y) n-1 times and examining the difference between the two integrals. If it has the same sign, then dominance holds, otherwise it does not. w IF FSD holds then NSD holds.

Applied Example in Securities w Porter and Carey (1974) applied stochastic dominance to screen 16 randomly selected companies. The companies are: w Company NumberCompany Name 1 Federal Paper Board 2 American Machine and Foundry 3 Smith Kline/French Lab 4 Rayonier 5 American Tobacco 6 Crowell-Collier 7 American Seating 8 Emerson Electric 9 Riegel Textile 10 Cero 11 American Distilling 12 American Investment 13 Johnson and Johnson 14 United Aircraft 15 Dresser Industries 16 Schering

Case Analysis w The rate of retain (ROR) for 54 periods (quarter) from the last quarter of 1963 to the first quarter of The distribution of the rate of return was developed. w ROR t = (P t – P t-1 ) + D t )/P t-1 w P t = price at period t w D t = Dividend in period t

Case Analysis Using FSD w The results of FSD is Company 2dominates 3 Company 5 dominates 3 Company 8 dominates 3, 5, 7

Case Analysis Using SSD w Results of SSD Dominance test Company 2 dominated 3 Company 4dominated10 Company 5dominated1,2,3,7. Company 7dominated1 Company 8 dominated1,2,3,4,5,6,7,9,10,11,13,14 15 Company 9 dominated10 Company 11dominated3 Company 12dominated3, 11 Company 13dominated1, 10 Company 14dominated10 Company 15dominated6, 9, 10, 14 Company 16dominated1, 4, 6, 9, 10, 13, 14, 15 Companies 1, 3, 6, 10dominated none

Case Final Analysis w FSD only screened 3 companies namely 3,5 and 7. w SSD reduced the efficient set to two companies namely companies 8 and 12. w From figure 4.3 in text firm 8 has initially a steeper cumulative distribution function and therefore can not dominate firm 12 under FSD, SSD or TSD. w From figure 4.3 it is clear firm 8 is the better choice.

Comments on Screening w Screening by dominance usually reduces the number of options to a manageable size. w In some cases screening by dominance could leave two options that may be ranked first and last when maximizing expected utility. w Assign example on page 77 for discussion by students