CSCI 1951-G Optimization Methods in Finance Part 07: Portfolio Optimization
|
|
- Lynn Kelly
- 5 years ago
- Views:
Transcription
1 CSCI 1951-G Optimization Methods in Finance Part 07: Portfolio Optimization March 9 16, / 19
2 The portfolio optimization problem How to best allocate our money to n risky assets S 1,..., S n with random returns? µ i : expected return of asset i in a time interval; Stocks Bonds Money Market µ i Σ: variance-covariance n n matrix of returns, with: σ ii : variance of the return of asset i; σ ij : covariance of the returns of assets i and j. Covariance Stocks Bonds MM Stocks Bonds MM / 19
3 The portfolio optimization problem Portfolio x = (x 1,..., x n ), where x i : proportion of money invested in asset i. Expected return: E[x] = µ 1 x µ n x n = µ T x Variance: Var[x] = i,j σ ijx i x j = x T Σx Var[x] 0, so Σ...is positive semidefinite (we assume positive definite) Feasible portfolios: set X = {x : Ax = b, Cx d} One constraint is n x i = 1 i=1 3 / 19
4 The portfolio optimization problem Efficient portfolio w.r.t. R > 0: the portfolio with minimum variance among all those with expected return at least R. (variants possible, e.g., ) Markowitz mean-variance optimization: find the efficient portfolio: min x T Σx s.t. µ T x R Ax = b Cx d This optimization problem is... convex. We assumed Σ 0, so the optimal solution is... unique. 4 / 19
5 The portfolio optimization problem Stocks Bonds MM µ i Covariance Stocks Bonds MM Stocks Bonds MM min x 2 S x S x B x S x M x 2 B x B x M x 2 M s.t x S x B x M R x S + x B + x M = 1 x S 0, x B 0, x M 0 5 / 19
6 The portfolio optimization problem Rate of Return R Variance Stocks Bonds MM Table 8.1: Efficient Portfolios Expected Return (%) Percent invested in different asset classes Stocks Bonds 7 10 MM Standard Deviation (%) Expected return of efficient portfolios (%) Figure 8.1: Efficient Frontier and the Composition of Efficient Portfolios 6 / 19
7 The efficient frontier R min, R max : minimum and maximum expected returns for efficient portfolios. σ(r) : [R min, R max ] R, σ(r) = ( x T RΣx R ) 1/2 where x R is the efficient portfolio w.r.t. R [R min, R max ]. The efficient frontier is the graph E = {(R, σ(r)) : R [R min, R max ]} 7 / 19
8 Maximizing the Sharpe ratio Consider a riskless asset with deterministic return r f R min (why does it make sense?) Consider convex combinations between a risky portfolio x with the riskless asset x θ = [(1 θ)x θ] T As θ varies, (for fixed x) the combinations form a line on the stdev/mean plot: Mean CAL r f V Figure 8.4: Capital Allocation Line For different choices of x, the slope of the line changes. 8 / 19
9 Maximizing the Sharpe ratio Which Capital Allocation Line (CAL) is the best? Mean CAL r f V Figure 8.4: Capital Allocation Line The CAL with the largest slope: the corresponding portfolio will have the lowest stdev for any given value of R r f. 9 / 19
10 Maximizing the Sharpe ratio To which portfolio x does the optimal CAL corresponds to? The feasible x that maximizes the slope: h(x) = µt x r f (x T Σx) 1/2 The quantity h(x) is known as the Sharpe ratio or the reward-to-volatility ratio. 10 / 19
11 Maximizing the Sharpe ratio To find the optimal risky portfolio we solve max µt x r f (x T Σx) 1/2 s.t. Ax = b Cx d The feasible region is polyhedral, but the objective function may be non-concave. Let s build an equivalent convex quadratic program. 11 / 19
12 Maximizing the Sharpe ratio X = {x : Ax = b, Cx d} (includes full alloc. const., assumes ˆx X s.t. µ Tˆx > r f ) X + = {(x, k) : x R n, k R ++, x k X } {(0, 0} The optimal risky portfolio is x = y /k where (y, k ) is the optimal solution of: min y T Σy s.t. (y, k) X + (µ r f 1) T y = 1 This is a quadratic convex program (why?) 12 / 19
13 Returns-based style analysis You are a portfolio manager (!) and would like to understand how your portfolio manager friend Sally, the style of her portfolio i.e., the mix of stocks in it; Sally is secretive on the mix, but publish the returns of her portfolio over time; You also have access to the returns of index funds tracking different sectors of the market; Definition (Return-Based Style Analysis (RBSA)) A technique using constrained optimization to determine the style of a portfolio using the return time series of the portfolio and of a number of other asset classes (factors). 13 / 19
14 RBSA Mathematical Model Fundamentally a linear model for regression. Data R t, t = 1,..., T : the return of Sally s portfolio over T fixed time intervals (e.g., R t is the monthly return, and T = 12 months); F it, i = 1,..., n, t = 1,..., T : the returns of factor i over T fixed time intervals (same intervals as R t ); Model R t = w 1t F 1t + w 2t F 2t + + w nt F nt + ε t = F T t w t + ε t w it : sensitivity of R t to factor i; ε t : non-factor return. 14 / 19
15 Interpretation R t = w 1t F 1t + w 2t F 2t + + w nt F nt + ε t = F T t w t + ε t Assume the F it are returns of passive investments (e.g., index funds); Then: F T t w t is the return of a benchmark portfolio of passive investments; ε t is the difference between the passive benchmark and the active strategy followed by Sally. If the passive investments considered together are representative of the market, then ε t measures the additional (or negative) return due to Sally s ability as a portfolio manager. 15 / 19
16 Optimization problem R t = w 1t F 1t + w 2t F 2t + + w nt F nt + ε t = F T t w t + ε t Additional assumptions/constraints: w it = w i, i.e., the weights do not change over time. w i > 0, n i=1 w t = 1. Constraints of our optimization problem: min??? n s.t. w i = 1 i=1 w i 0, i = 1,..., n What about the objective function? Any idea? 16 / 19
17 Optimization problem (cont.) If ε t measures Sally s ability as a portfolio manager, we can assume that it is approximately constant over time. I.e., we want the plots of the returns of Sally s portfolios and of the benchmark portfolios to be curves with approximately constant distance. I.e., we want ε t to have the smallest possible variance over time. Formulation min Var(e t) = Var(R t F T w R n t w) n s.t. w i = 1 i=1 w i 0, i = 1,..., n The objective function is convex. 17 / 19
18 Objective function Let We have R = R 1. R T, and F = F T 1. F T T, and e = 1. 1 ( Var(R t Ft T w) = 1 T T (R t Ft T w) 2 t=1 (R t Ft T w T T i=1 = 1 ( e T ) 2 T R F (R F w) w 2 T = R 2 2R T F w + w T F T F w T (et R) 2 2e T R w T F T ee T F w T 2 ) 2 18 / 19
19 Objective function Var(R t Ft T w) = R 2 2R T F w + w T F T F w T (et R) 2 2e T R w T F T ee T F w T 2 Reorganizing the terms as function of w: ( R Var(R t Ft T 2 w) = et R) 2 ) ( R T ) F T T 2 2 et R T T 2 et F w ( 1 + w T T F T F 1 ) T F T ee T F w We have 1 T F T F 1 T F T ee T F = 1 T F T ) (I eet F T The matrix M = I ee T /T is symmetric and positive semidefinite (eigenvalues: 0 and 1), and so is F T MF. Hence the objective function is convex. 19 / 19
Techniques for Calculating the Efficient Frontier
Techniques for Calculating the Efficient Frontier Weerachart Kilenthong RIPED, UTCC c Kilenthong 2017 Tee (Riped) Introduction 1 / 43 Two Fund Theorem The Two-Fund Theorem states that we can reach any
More informationLecture IV Portfolio management: Efficient portfolios. Introduction to Finance Mathematics Fall Financial mathematics
Lecture IV Portfolio management: Efficient portfolios. Introduction to Finance Mathematics Fall 2014 Reduce the risk, one asset Let us warm up by doing an exercise. We consider an investment with σ 1 =
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 informationFinancial Mathematics III Theory summary
Financial Mathematics III Theory summary Table of Contents Lecture 1... 7 1. State the objective of modern portfolio theory... 7 2. Define the return of an asset... 7 3. How is expected return defined?...
More informationu (x) < 0. and if you believe in diminishing return of the wealth, then you would require
Chapter 8 Markowitz Portfolio Theory 8.7 Investor Utility Functions People are always asked the question: would more money make you happier? The answer is usually yes. The next question is how much more
More informationFinal Exam Suggested Solutions
University of Washington Fall 003 Department of Economics Eric Zivot Economics 483 Final Exam Suggested Solutions This is a closed book and closed note exam. However, you are allowed one page of handwritten
More informationMean-Variance Analysis
Mean-Variance Analysis Mean-variance analysis 1/ 51 Introduction How does one optimally choose among multiple risky assets? Due to diversi cation, which depends on assets return covariances, the attractiveness
More informationChapter 7: Portfolio Theory
Chapter 7: Portfolio Theory 1. Introduction 2. Portfolio Basics 3. The Feasible Set 4. Portfolio Selection Rules 5. The Efficient Frontier 6. Indifference Curves 7. The Two-Asset Portfolio 8. Unrestriceted
More informationEconomics 424/Applied Mathematics 540. Final Exam Solutions
University of Washington Summer 01 Department of Economics Eric Zivot Economics 44/Applied Mathematics 540 Final Exam Solutions I. Matrix Algebra and Portfolio Math (30 points, 5 points each) Let R i denote
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 informationFinancial Economics: Risk Aversion and Investment Decisions, Modern Portfolio Theory
Financial Economics: Risk Aversion and Investment Decisions, Modern Portfolio Theory Shuoxun Hellen Zhang WISE & SOE XIAMEN UNIVERSITY April, 2015 1 / 95 Outline Modern portfolio theory The backward induction,
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 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 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 informationPORTFOLIO THEORY. Master in Finance INVESTMENTS. Szabolcs Sebestyén
PORTFOLIO THEORY Szabolcs Sebestyén szabolcs.sebestyen@iscte.pt Master in Finance INVESTMENTS Sebestyén (ISCTE-IUL) Portfolio Theory Investments 1 / 60 Outline 1 Modern Portfolio Theory Introduction Mean-Variance
More informationEfficient Portfolio and Introduction to Capital Market Line Benninga Chapter 9
Efficient Portfolio and Introduction to Capital Market Line Benninga Chapter 9 Optimal Investment with Risky Assets There are N risky assets, named 1, 2,, N, but no risk-free asset. With fixed total dollar
More informationOPTIMAL RISKY PORTFOLIOS- ASSET ALLOCATIONS. BKM Ch 7
OPTIMAL RISKY PORTFOLIOS- ASSET ALLOCATIONS BKM Ch 7 ASSET ALLOCATION Idea from bank account to diversified portfolio Discussion principles are the same for any number of stocks A. bonds and stocks B.
More informationWorst-Case Value-at-Risk of Derivative Portfolios
Worst-Case Value-at-Risk of Derivative Portfolios Steve Zymler Berç Rustem Daniel Kuhn Department of Computing Imperial College London Thalesians Seminar Series, November 2009 Risk Management is a Hot
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 informationQR43, Introduction to Investments Class Notes, Fall 2003 IV. Portfolio Choice
QR43, Introduction to Investments Class Notes, Fall 2003 IV. Portfolio Choice A. Mean-Variance Analysis 1. Thevarianceofaportfolio. Consider the choice between two risky assets with returns R 1 and R 2.
More informationRisk and Return. CA Final Paper 2 Strategic Financial Management Chapter 7. Dr. Amit Bagga Phd.,FCA,AICWA,Mcom.
Risk and Return CA Final Paper 2 Strategic Financial Management Chapter 7 Dr. Amit Bagga Phd.,FCA,AICWA,Mcom. Learning Objectives Discuss the objectives of portfolio Management -Risk and Return Phases
More informationProblem Set: Contract Theory
Problem Set: Contract Theory Problem 1 A risk-neutral principal P hires an agent A, who chooses an effort a 0, which results in gross profit x = a + ε for P, where ε is uniformly distributed on [0, 1].
More informationSDMR Finance (2) Olivier Brandouy. University of Paris 1, Panthéon-Sorbonne, IAE (Sorbonne Graduate Business School)
SDMR Finance (2) Olivier Brandouy University of Paris 1, Panthéon-Sorbonne, IAE (Sorbonne Graduate Business School) Outline 1 Formal Approach to QAM : concepts and notations 2 3 Portfolio risk and return
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 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 informationChapter 5 Portfolio. O. Afonso, P. B. Vasconcelos. Computational Economics: a concise introduction
Chapter 5 Portfolio O. Afonso, P. B. Vasconcelos Computational Economics: a concise introduction O. Afonso, P. B. Vasconcelos Computational Economics 1 / 22 Overview 1 Introduction 2 Economic model 3 Numerical
More informationGeneral Notation. Return and Risk: The Capital Asset Pricing Model
Return and Risk: The Capital Asset Pricing Model (Text reference: Chapter 10) Topics general notation single security statistics covariance and correlation return and risk for a portfolio diversification
More informationSolutions to questions in Chapter 8 except those in PS4. The minimum-variance portfolio is found by applying the formula:
Solutions to questions in Chapter 8 except those in PS4 1. The parameters of the opportunity set are: E(r S ) = 20%, E(r B ) = 12%, σ S = 30%, σ B = 15%, ρ =.10 From the standard deviations and the correlation
More informationThe Markowitz framework
IGIDR, Bombay 4 May, 2011 Goals What is a portfolio? Asset classes that define an Indian portfolio, and their markets. Inputs to portfolio optimisation: measuring returns and risk of a portfolio Optimisation
More informationCHAPTER 6: PORTFOLIO SELECTION
CHAPTER 6: PORTFOLIO SELECTION 6-1 21. The parameters of the opportunity set are: E(r S ) = 20%, E(r B ) = 12%, σ S = 30%, σ B = 15%, ρ =.10 From the standard deviations and the correlation coefficient
More informationSolutions to Problem Set 1
Solutions to Problem Set Theory of Banking - Academic Year 06-7 Maria Bachelet maria.jua.bachelet@gmail.com February 4, 07 Exercise. An individual consumer has an income stream (Y 0, Y ) and can borrow
More informationPortfolio Risk Management and Linear Factor Models
Chapter 9 Portfolio Risk Management and Linear Factor Models 9.1 Portfolio Risk Measures There are many quantities introduced over the years to measure the level of risk that a portfolio carries, and each
More informationRisk and Return and Portfolio Theory
Risk and Return and Portfolio Theory Intro: Last week we learned how to calculate cash flows, now we want to learn how to discount these cash flows. This will take the next several weeks. We know discount
More informationVersion A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
More informationQuantitative Risk Management
Quantitative Risk Management Asset Allocation and Risk Management Martin B. Haugh Department of Industrial Engineering and Operations Research Columbia University Outline Review of Mean-Variance Analysis
More informationFIN 6160 Investment Theory. Lecture 7-10
FIN 6160 Investment Theory Lecture 7-10 Optimal Asset Allocation Minimum Variance Portfolio is the portfolio with lowest possible variance. To find the optimal asset allocation for the efficient frontier
More informationSession 8: The Markowitz problem p. 1
Session 8: The Markowitz problem Susan Thomas http://www.igidr.ac.in/ susant susant@mayin.org IGIDR Bombay Session 8: The Markowitz problem p. 1 Portfolio optimisation Session 8: The Markowitz problem
More informationChapter 8. Markowitz Portfolio Theory. 8.1 Expected Returns and Covariance
Chapter 8 Markowitz Portfolio Theory 8.1 Expected Returns and Covariance The main question in portfolio theory is the following: Given an initial capital V (0), and opportunities (buy or sell) in N securities
More informationLecture 10: Performance measures
Lecture 10: Performance measures Prof. Dr. Svetlozar Rachev Institute for Statistics and Mathematical Economics University of Karlsruhe Portfolio and Asset Liability Management Summer Semester 2008 Prof.
More informationMarket Timing Does Work: Evidence from the NYSE 1
Market Timing Does Work: Evidence from the NYSE 1 Devraj Basu Alexander Stremme Warwick Business School, University of Warwick November 2005 address for correspondence: Alexander Stremme Warwick Business
More informationAversion to Risk and Optimal Portfolio Selection in the Mean- Variance Framework
Aversion to Risk and Optimal Portfolio Selection in the Mean- Variance Framework Prof. Massimo Guidolin 20135 Theory of Finance, Part I (Sept. October) Fall 2017 Outline and objectives Four alternative
More informationWorst-Case Value-at-Risk of Non-Linear Portfolios
Worst-Case Value-at-Risk of Non-Linear Portfolios Steve Zymler Daniel Kuhn Berç Rustem Department of Computing Imperial College London Portfolio Optimization Consider a market consisting of m assets. Optimal
More informationMarkowitz portfolio theory
Markowitz portfolio theory Farhad Amu, Marcus Millegård February 9, 2009 1 Introduction Optimizing a portfolio is a major area in nance. The objective is to maximize the yield and simultaneously minimize
More informationCapital Allocation Between The Risky And The Risk- Free Asset
Capital Allocation Between The Risky And The Risk- Free Asset Chapter 7 Investment Decisions capital allocation decision = choice of proportion to be invested in risk-free versus risky assets asset allocation
More informationMATH362 Fundamentals of Mathematical Finance. Topic 1 Mean variance portfolio theory. 1.1 Mean and variance of portfolio return
MATH362 Fundamentals of Mathematical Finance Topic 1 Mean variance portfolio theory 1.1 Mean and variance of portfolio return 1.2 Markowitz mean-variance formulation 1.3 Two-fund Theorem 1.4 Inclusion
More informationSome useful optimization problems in portfolio theory
Some useful optimization problems in portfolio theory Igor Melicherčík Department of Economic and Financial Modeling, Faculty of Mathematics, Physics and Informatics, Mlynská dolina, 842 48 Bratislava
More informationECON FINANCIAL ECONOMICS
ECON 337901 FINANCIAL ECONOMICS Peter Ireland Boston College April 26, 2018 These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike 4.0 International
More informationMATH4512 Fundamentals of Mathematical Finance. Topic Two Mean variance portfolio theory. 2.1 Mean and variance of portfolio return
MATH4512 Fundamentals of Mathematical Finance Topic Two Mean variance portfolio theory 2.1 Mean and variance of portfolio return 2.2 Markowitz mean-variance formulation 2.3 Two-fund Theorem 2.4 Inclusion
More informationLecture 10-12: CAPM.
Lecture 10-12: CAPM. I. Reading II. Market Portfolio. III. CAPM World: Assumptions. IV. Portfolio Choice in a CAPM World. V. Minimum Variance Mathematics. VI. Individual Assets in a CAPM World. VII. Intuition
More informationProblem Set: Contract Theory
Problem Set: Contract Theory Problem 1 A risk-neutral principal P hires an agent A, who chooses an effort a 0, which results in gross profit x = a + ε for P, where ε is uniformly distributed on [0, 1].
More informationChapter 8: CAPM. 1. Single Index Model. 2. Adding a Riskless Asset. 3. The Capital Market Line 4. CAPM. 5. The One-Fund Theorem
Chapter 8: CAPM 1. Single Index Model 2. Adding a Riskless Asset 3. The Capital Market Line 4. CAPM 5. The One-Fund Theorem 6. The Characteristic Line 7. The Pricing Model Single Index Model 1 1. Covariance
More informationminimize f(x) subject to f(x) 0 h(x) = 0, 14.1 Quadratic Programming and Portfolio Optimization
Lecture 14 So far we have only dealt with constrained optimization problems where the objective and the constraints are linear. We now turn attention to general problems of the form minimize f(x) subject
More informationAversion to Risk and Optimal Portfolio Selection in the Mean- Variance Framework
Aversion to Risk and Optimal Portfolio Selection in the Mean- Variance Framework Prof. Massimo Guidolin 20135 Theory of Finance, Part I (Sept. October) Fall 2018 Outline and objectives Four alternative
More informationEconomics 483. Midterm Exam. 1. Consider the following monthly data for Microsoft stock over the period December 1995 through December 1996:
University of Washington Summer Department of Economics Eric Zivot Economics 3 Midterm Exam This is a closed book and closed note exam. However, you are allowed one page of handwritten notes. Answer all
More informationLECTURE NOTES 3 ARIEL M. VIALE
LECTURE NOTES 3 ARIEL M VIALE I Markowitz-Tobin Mean-Variance Portfolio Analysis Assumption Mean-Variance preferences Markowitz 95 Quadratic utility function E [ w b w ] { = E [ w] b V ar w + E [ w] }
More informationElton, Gruber, Brown, and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 6
Elton, Gruber, rown, and Goetzmann Modern Portfolio Theory and Investment nalysis, 7th Edition Solutions to Text Problems: Chapter 6 Chapter 6: Problem The simultaneous equations necessary to solve this
More information2.1 Mean-variance Analysis: Single-period Model
Chapter Portfolio Selection The theory of option pricing is a theory of deterministic returns: we hedge our option with the underlying to eliminate risk, and our resulting risk-free portfolio then earns
More informationFINC3017: Investment and Portfolio Management
FINC3017: Investment and Portfolio Management Investment Funds Topic 1: Introduction Unit Trusts: investor s funds are pooled, usually into specific types of assets. o Investors are assigned tradeable
More informationEE365: Risk Averse Control
EE365: Risk Averse Control Risk averse optimization Exponential risk aversion Risk averse control 1 Outline Risk averse optimization Exponential risk aversion Risk averse control Risk averse optimization
More informationPortfolio models - Podgorica
Outline Holding period return Suppose you invest in a stock-index fund over the next period (e.g. 1 year). The current price is 100$ per share. At the end of the period you receive a dividend of 5$; the
More informationCapital Asset Pricing Model
Capital Asset Pricing Model 1 Introduction In this handout we develop a model that can be used to determine how an investor can choose an optimal asset portfolio in this sense: the investor will earn the
More informationLecture 22. Survey Sampling: an Overview
Math 408 - Mathematical Statistics Lecture 22. Survey Sampling: an Overview March 25, 2013 Konstantin Zuev (USC) Math 408, Lecture 22 March 25, 2013 1 / 16 Survey Sampling: What and Why In surveys sampling
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 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 informationMean Variance Analysis and CAPM
Mean Variance Analysis and CAPM Yan Zeng Version 1.0.2, last revised on 2012-05-30. Abstract A summary of mean variance analysis in portfolio management and capital asset pricing model. 1. Mean-Variance
More informationMean Variance Portfolio Theory
Chapter 1 Mean Variance Portfolio Theory This book is about portfolio construction and risk analysis in the real-world context where optimization is done with constraints and penalties specified by the
More informationOptimal Portfolio Selection
Optimal Portfolio Selection We have geometrically described characteristics of the optimal portfolio. Now we turn our attention to a methodology for exactly identifying the optimal portfolio given a set
More informationLecture 3: Return vs Risk: Mean-Variance Analysis
Lecture 3: Return vs Risk: Mean-Variance Analysis 3.1 Basics We will discuss an important trade-off between return (or reward) as measured by expected return or mean of the return and risk as measured
More informationWeek 1 Quantitative Analysis of Financial Markets Basic Statistics A
Week 1 Quantitative Analysis of Financial Markets Basic Statistics A Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October
More informationQuantitative Portfolio Theory & Performance Analysis
550.447 Quantitative ortfolio Theory & erformance Analysis Week February 18, 2013 Basic Elements of Modern ortfolio Theory Assignment For Week of February 18 th (This Week) Read: A&L, Chapter 3 (Basic
More informationCalculating EAR and continuous compounding: Find the EAR in each of the cases below.
Problem Set 1: Time Value of Money and Equity Markets. I-III can be started after Lecture 1. IV-VI can be started after Lecture 2. VII can be started after Lecture 3. VIII and IX can be started after Lecture
More informationIn terms of covariance the Markowitz portfolio optimisation problem is:
Markowitz portfolio optimisation Solver To use Solver to solve the quadratic program associated with tracing out the efficient frontier (unconstrained efficient frontier UEF) in Markowitz portfolio optimisation
More informationPortfolios that Contain Risky Assets 10: Limited Portfolios with Risk-Free Assets
Portfolios that Contain Risky Assets 10: Limited Portfolios with Risk-Free Assets C. David Levermore University of Maryland, College Park, MD Math 420: Mathematical Modeling March 21, 2018 version c 2018
More informationThe Optimization Process: An example of portfolio optimization
ISyE 6669: Deterministic Optimization The Optimization Process: An example of portfolio optimization Shabbir Ahmed Fall 2002 1 Introduction Optimization can be roughly defined as a quantitative approach
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 informationMean-Variance Portfolio Choice in Excel
Mean-Variance Portfolio Choice in Excel Prof. Manuela Pedio 20550 Quantitative Methods for Finance August 2018 Let s suppose you can only invest in two assets: a (US) stock index (here represented by the
More informationImproving Returns-Based Style Analysis
Improving Returns-Based Style Analysis Autumn, 2007 Daniel Mostovoy Northfield Information Services Daniel@northinfo.com Main Points For Today Over the past 15 years, Returns-Based Style Analysis become
More informationThe mean-variance portfolio choice framework and its generalizations
The mean-variance portfolio choice framework and its generalizations Prof. Massimo Guidolin 20135 Theory of Finance, Part I (Sept. October) Fall 2014 Outline and objectives The backward, three-step solution
More informationAdvanced Financial Modeling. Unit 2
Advanced Financial Modeling Unit 2 Financial Modeling for Risk Management A Portfolio with 2 assets A portfolio with 3 assets Risk Modeling in a multi asset portfolio Monte Carlo Simulation Two Asset Portfolio
More informationRandom Variables and Probability Distributions
Chapter 3 Random Variables and Probability Distributions Chapter Three Random Variables and Probability Distributions 3. Introduction An event is defined as the possible outcome of an experiment. In engineering
More informationPortfolios that Contain Risky Assets Portfolio Models 9. Long Portfolios with a Safe Investment
Portfolios that Contain Risky Assets Portfolio Models 9. Long Portfolios with a Safe Investment C. David Levermore University of Maryland, College Park Math 420: Mathematical Modeling March 21, 2016 version
More informationPortfolio theory and risk management Homework set 2
Portfolio theory and risk management Homework set Filip Lindskog General information The homework set gives at most 3 points which are added to your result on the exam. You may work individually or in
More informationECMC49F Midterm. Instructor: Travis NG Date: Oct 26, 2005 Duration: 1 hour 50 mins Total Marks: 100. [1] [25 marks] Decision-making under certainty
ECMC49F Midterm Instructor: Travis NG Date: Oct 26, 2005 Duration: 1 hour 50 mins Total Marks: 100 [1] [25 marks] Decision-making under certainty (a) [5 marks] Graphically demonstrate the Fisher Separation
More informationRisk Reduction Potential
Risk Reduction Potential Research Paper 006 February, 015 015 Northstar Risk Corp. All rights reserved. info@northstarrisk.com Risk Reduction Potential In this paper we introduce the concept of risk reduction
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 informationFreeman School of Business Fall 2003
FINC 748: Investments Ramana Sonti Freeman School of Business Fall 2003 Lecture Note 3B: Optimal risky portfolios To be read with BKM Chapter 8 Statistical Review Portfolio mathematics Mean standard deviation
More informationOptimization in Finance
Research Reports on Mathematical and Computing Sciences Series B : Operations Research Department of Mathematical and Computing Sciences Tokyo Institute of Technology 2-12-1 Oh-Okayama, Meguro-ku, Tokyo
More informationCHAPTER 6. Risk Aversion and Capital Allocation to Risky Assets INVESTMENTS BODIE, KANE, MARCUS
CHAPTER 6 Risk Aversion and Capital Allocation to Risky Assets INVESTMENTS BODIE, KANE, MARCUS McGraw-Hill/Irwin Copyright 011 by The McGraw-Hill Companies, Inc. All rights reserved. 6- Allocation to Risky
More informationMean-Variance Portfolio Theory
Mean-Variance Portfolio Theory Lakehead University Winter 2005 Outline Measures of Location Risk of a Single Asset Risk and Return of Financial Securities Risk of a Portfolio The Capital Asset Pricing
More informationIndexing and Price Informativeness
Indexing and Price Informativeness Hong Liu Washington University in St. Louis Yajun Wang University of Maryland IFS SWUFE August 3, 2017 Liu and Wang Indexing and Price Informativeness 1/25 Motivation
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 5 Continuous Random Variables and Probability Distributions Ch. 5-1 Probability Distributions Probability Distributions Ch. 4 Discrete Continuous Ch. 5 Probability
More informationOptimization Methods in Finance
Optimization Methods in Finance Gerard Cornuejols Reha Tütüncü Carnegie Mellon University, Pittsburgh, PA 15213 USA January 2006 2 Foreword Optimization models play an increasingly important role in financial
More informationAPPENDIX TO LECTURE NOTES ON ASSET PRICING AND PORTFOLIO MANAGEMENT. Professor B. Espen Eckbo
APPENDIX TO LECTURE NOTES ON ASSET PRICING AND PORTFOLIO MANAGEMENT 2011 Professor B. Espen Eckbo 1. Portfolio analysis in Excel spreadsheet 2. Formula sheet 3. List of Additional Academic Articles 2011
More informationKey investment insights
Basic Portfolio Theory B. Espen Eckbo 2011 Key investment insights Diversification: Always think in terms of stock portfolios rather than individual stocks But which portfolio? One that is highly diversified
More informationModeling Portfolios that Contain Risky Assets Risk and Reward III: Basic Markowitz Portfolio Theory
Modeling Portfolios that Contain Risky Assets Risk and Reward III: Basic Markowitz Portfolio Theory C. David Levermore University of Maryland, College Park Math 420: Mathematical Modeling January 30, 2013
More informationChapter 2 Portfolio Management and the Capital Asset Pricing Model
Chapter 2 Portfolio Management and the Capital Asset Pricing Model In this chapter, we explore the issue of risk management in a portfolio of assets. The main issue is how to balance a portfolio, that
More information1.1 Interest rates Time value of money
Lecture 1 Pre- Derivatives Basics Stocks and bonds are referred to as underlying basic assets in financial markets. Nowadays, more and more derivatives are constructed and traded whose payoffs depend on
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 informationFoundations of Finance
Lecture 5: CAPM. I. Reading II. Market Portfolio. III. CAPM World: Assumptions. IV. Portfolio Choice in a CAPM World. V. Individual Assets in a CAPM World. VI. Intuition for the SML (E[R p ] depending
More informationSession 9: The expected utility framework p. 1
Session 9: The expected utility framework Susan Thomas http://www.igidr.ac.in/ susant susant@mayin.org IGIDR Bombay Session 9: The expected utility framework p. 1 Questions How do humans make decisions
More information